https://en.wikipedia.org/wiki/Menger_sponge
(level four was the best they could do the fourth is always transcendent)
MegaMenger is a project aiming to build the largest fractal model, pioneered by Matt Parker of Queen Mary University of London and Laura Taalman of James Madison University. Each small cube is made from 6 interlocking folded business cards, giving a total of 960 000 for a levelfour sponge. The outer surfaces are then covered with paper or cardboard panels printed with a Sierpinski carpet design to be more aesthetically pleasing.[7] In 2014, twenty levelthree Menger sponges were constructed, which combined would form a distributed levelfour Menger sponge.[8]
https://en.wikipedia.org/wiki/Menger_sponge
The construction of a Menger sponge can be described as follows: (four rules)
Begin with a cube (first image).
Divide every face of the cube into 9 squares, like a Rubik's Cube. This will subdivide the cube into 27 smaller cubes.
Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of the larger cube, leaving 20 smaller cubes (second image). This is a level1 Menger sponge (resembling a Void Cube).
Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate ad infinitum.
The second iteration gives a level2 sponge (third image), the third iteration gives a level3 sponge (fourth image), and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.
It is made of crosses
érusalem,_itération_3.png
In mathematics, a Jerusalem Cube is a fractal object described by Eric Baird in 2011. It is created by recursively drilling Greek crossshaped holes into a cube.[1] The name comes from a face of the cube resembling a Jerusalem cross pattern.
It forms crosses
Notice how the demonstrations stop at four recursions. It is because after four is when it becomes too chaotic.
The Mosely snowflake (after Jeannine Mosely) is a Sierpiński–Menger type of fractal obtained in two variants either by the operation opposite to creating the SierpińskiMenger snowflake or Cantor dust i.e. not by leaving but by removing eight of the smaller 1/3scaled corner cubes and the central one from each cube left from the previous recursion (lighter) or by removing only corner cubes (heavier). In one dimension this operation (i.e. the recursive removal of two side line segments) is trivial and converges only to single point. It resembles the original water snowflake of snow.
The fourth is always different. The four square theorem was difficult to discover after the three square theorem.
AdrienMarie Legendre completed the theorem in 1797–8 with his threesquare theorem, by proving that a positive integer can be expressed as the sum of three squares if and only if it is not of the form
4
k
(
8
m
+
7
)
4^{k}(8m+7) for integers
k
k and
m
m. Later, in 1834, Carl Gustav Jakob Jacobi discovered a simple formula for the number of representations of an integer as the sum of four squares with his own foursquare theorem.
Lagrange's foursquare theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares.
p
=
a
0
2
+
a
1
2
+
a
2
2
+
a
3
2
p=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\
where the four numbers
a
0
,
a
1
,
a
2
,
a
3
a_{0},a_{1},a_{2},a_{3} are integers. For illustration, 3, 31 and 310 can be represented as the sum of four squares as follows:
3 =
1
2
+
1
2
+
1
2
+
0
2
31 =
5
2
+
2
2
+
1
2
+
1
2
310 =
17
2
+
4
2
+
2
2
+
1
2
.{\begin{aligned}3&=1^{2}+1^{2}+1^{2}+0^{2}\\[3pt]31&=5^{2}+2^{2}+1^{2}+1^{2}\\[3pt]310&=17^{2}+4^{2}+2^{2}+1^{2}.\end{aligned}}
This theorem was proven by Joseph Louis Lagrange in 1770.
https://en.wikipedia.org/w…/Lagrange%27s_foursquare_theorem
Lagrange's four square theorem formula is also linked to Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles. This is also linked to Apollonian gaskets, which were more recently related to the Ramanujan–Petersson conjecture.[2]
https://en.wikipedia.org/wiki/Descartes%27_theorem
In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643.
http://mathonweb.com/help_ebook/html/cast.htm
The ASTC or CAST or unit circle method for finding two angles having a given sin, cos or tan is called the CAST diagram/quadrant.
In this method you draw a set of axes and label the quadrants 1, 2, 3 and 4 with the letters A, S, T and C respectively as shown in the diagrams below. The letters mean:
A: all three functions, sin, cos and tan are positive in this quadrant
S: only the sin function is positive in this quadrant
T: only the tan function is positive in this quadrant
C: only the cos function is positive in this quadrant
This is known as the CAST or quadrant rule of trigonometry. This is the foundation of trigonometry. No coincidence it's foundation is the quadrant, the Form of Being.
Notice how the CAST unit circle resembles the quadrant model pattern, with the fourth square A representing an amalgamation of the previous three. That is the quadrant model pattern.
Using the Cartesian Plane to define the CAST diagram in order to define which trigonometric ratios will give us a positive value for the angle theta depending on the quadrant of the CAST diagram the angle is in.
In a CAST diagram
When we include negative values, the x and y axes divide the space up into 4 pieces:
Quadrants I, II, III and IV
(They are numbered in a counterclockwise direction)
In Quadrant I both x and y are positive,
in Quadrant II x is negative (y is still positive),
in Quadrant III both x and y are negative, and
in Quadrant IV x is positive again, and y is negative.
https://en.wikipedia.org/wiki/Jones_diagram
A Jones diagram is a type of Cartesian graph developed by Loyd A. Jones in the 1940s, where each axis represents a different variable. In a Jones diagram opposite directions of an axis represent different quantities, unlike in a Cartesian graph where they represent positive or negative signs of the same quantity. The Jones diagram therefore represents four variables. Each quadrant shares the vertical axis with its horizontal neighbor, and the horizontal axis with the vertical neighbor. For example, the top left quadrant shares its vertical axis with the top right quadrant, and the horizontal axis with the bottom left quadrant. The overall system response is in quadrant I; the variables that contribute to it are in quadrants II through IV.
A common application of Jones diagrams is in photography, specifically in displaying sensitivity to light with what are also called "tone reproduction diagrams". These diagrams are used in the design of photographic systems (film, paper, etc.) to determine the relationship between the light a viewer would see at the time a photo was taken to the light that a viewer would see looking at the finished photograph.
The Jones diagram concept can be used for variables that depend successively on each other. Jones's original diagram used eleven quadrants[how?] to show all the elements of his photographic system.
https://en.wikipedia.org/wiki/Klein_fourgroup
In mathematics, the Klein fourgroup (or just Klein group or Vierergruppe (English: fourgroup), often symbolized by the letter V or as K4) is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe by Felix Klein in 1884
https://en.wikipedia.org/wiki/Klein_fourgroup
Algebra[edit]
According to Galois theory, the existence of the Klein fourgroup (and in particular, the permutation representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals, as established by Lodovico Ferrari: the map S4 → S3 corresponds to the resolvent cubic, in terms of Lagrange resolvents.
In the construction of finite rings, eight of the eleven rings with four elements have the Klein fourgroup as their additive substructure.
If R× denotes the multiplicative group of nonzero reals and R+ the multiplicative group of positive reals, R× × R× is the group of units of the ring R × R, and R+ × R+ is a subgroup of R× × R× (in fact it is the component of the identity of R× × R×). The quotient group (R× × R×) / (R+ × R+) is isomorphic to the Klein fourgroup. In a similar fashion, the group of units of the splitcomplex number ring, when divided by its identity component, also results in the Klein fourgroup.
In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. There are only four allHarshad numbers: 1, 2, 4, and 6 (The number 12 is a Harshad number in all bases except octal).
In music composition the fourgroup is the basic group of permutations in the twelvetone technique. In that instance the Cayley table is written;[2]
S I: R: RI:
I: S RI R
R: RI S I
RI: R I S
The smallest noncyclic group has four elements; it is the Klein fourgroup. Four is also the order of the smallest nontrivial groups that are not simple.
https://en.wikipedia.org/wiki/Quartic_function
The fourth is different.The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals. It took a very long time to discover the quartic function equation. Five cannot be done.
In algebra, a quartic function, is a function of the form
f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,
where a is nonzero, which is defined by a polynomial of degree four, called quartic polynomial.
Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form
f(x)=ax^{4}+cx^{2}+e.
A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form
ax^{4}+bx^{3}+cx^{2}+dx+e=0,
where a ≠ 0.
The derivative of a quartic function is a cubic function.
Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.
The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals.
Notice how four degrees is the highest degree. And it took a ton of effort to discover the quartic equation. It has been proven that five degrees is impossible. Any higher than five it is assume it has been impossible but it has not been proven. Four is always different. Five is ultra transcendent.
The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois prior to dying in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result
http://portpcfix.com/wikifusion/index.php/Quartic_function
The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois prior to dying in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, of which this theorem was one result.[6]
The proof for the binomial equation (quadratic equations) is a quadrant model with four squares
https://en.wikipedia.org/wiki/Binomial_theorem
https://en.wikipedia.org/…/File:Binomial_expansion_visualis…
The geometric proof for the binomial theorem consists of four squares (a quadrant model). The geometric proof for caculus also consists of four squares
http://en.wikipedia.org/wiki/File:BinomialTheorem.png
Geometric explanation
BinomialTheorem.png
For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side a + b can be cut into a cube of side a, a cube of side b, three a×a×b rectangular boxes, and three a×b×b rectangular boxes.
In calculus, this picture also gives a geometric proof of the derivative (xn)' = nxn − 1:[6] if one sets a = x and b = Δx, interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an ndimensional hypercube, (x + Δx)n, where the coefficient of the linear term (in Δx) is nxn − 1, the area of the n faces, each of dimension (n − 1):
(x+\Delta x)^n = x^n + nx^{n1}\Delta x + \tbinom{n}{2}x^{n2}(\Delta x)^2 + \cdots.
Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms – (Δx)2 and higher – become negligible, and yields the formula (xn)' = nxn − 1, interpreted as
"the infinitesimal change in volume of an ncube as side length varies is the area of n of its (n − 1)dimensional faces".
If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral \textstyle{\int x^{n1}\,dx = \tfrac{1}{n} x^n} – see proof of Cavalieri's quadrature formula for details.[6]
In the 11th century, the Islamic mathematician Ibn alHaytham (known as Alhazen in Europe) computed the integrals of cubics and quartics (degree three and four) via mathematical induction, in his Book of Optics.[6]
The highest is four four is different
four square quadrants
The derivative
(
x
n
)
′
=
n
x
n
−
1
(x^n)'=nx^{n1} can be geometrized as the infinitesimal change in volume of the ncube, which is the area of n faces, each of dimension n − 1.
Integrating this picture – stacking the faces – geometrizes the fundamental theorem of calculus, yielding a decomposition of the ncube into n pyramids, which is a geometric proof of Cavalieri's quadrature formula.
https://en.wikipedia.org/wiki/Euclidean_geometry
Euclid solved all of his geometry with four postulates. He had a fifth postulate but mathematicians have proven that it was false, and even showed that Euclid did not use it, or it was not necessary. Some suggest that Euclid knew that only four postulates were needed, but added the QUESTIONABLE FIFTH, so that later mathematicians would use that to find different geometries. The fourth is different, and the fifth is quesitonableultra transcendent
18th century[edit]
Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763 at least 28 different proofs had been published, but all were found incorrect.[31]
Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include doubling the cube and squaring the circle. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two,[32] while doubling a cube requires the solution of a thirdorder equation.
Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).
https://en.wikipedia.org/wiki/Euclidean_geometry
Birkhoff's axioms: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed experimentally with scale and protractor. This system relies heavily on the properties of the real numbers.[50][51][52] The notions of angle and distance become primitive concepts.[53]
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms.[1] These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a modelbased introduction to Euclidean geometry.
Birkhoff's axiom system was utilized in the secondaryschool textbook by Birkhoff and Beatley.[2] These axioms were also modified by the School Mathematics Study Group to provide a new standard for teaching high school geometry, known as SMSG axioms. A few other textbooks in the foundations of geometry use variants of Birkhoff's axioms.[3][page needed]
Postulates[edit]
Postulate I: Postulate of line measure. A set of points {A, B, ...} on any line can be put into a 1:1 correspondence with the real numbers {a, b, ...} so that b − a = d(A, B) for all points A and B.
Postulate II: Pointline postulate. There is one and only one line, ℓ, that contains any two given distinct points P and Q.
Postulate III: Postulate of angle measure. A set of rays {ℓ, m, n, ...} through any point O can be put into 1:1 correspondence with the real numbers a (mod 2π) so that if A and B are points (not equal to O) of ℓ and m, respectively, the difference am − aℓ (mod 2π) of the numbers associated with the lines ℓ and m is ∠ AOB. Furthermore, if the point B on m varies continuously in a line r not containing the vertex O, the number am varies continuously also.
Postulate IV: Postulate of similarity. Given two triangles ABC and A'B'C' and some constant k > 0, d(A', B' ) = kd(A, B), d(A', C' ) = kd(A, C) and ∠ B'A'C' = ±∠ BAC, then d(B', C' ) = kd(B, C), ∠ C'B'A' = ±∠ CBA, and ∠ A'C'B' = ±∠ ACB.
The fourth is different, fifth is questionable.
For two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not selfevident. If the order the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it.[10] Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by his own axiom.
Proclus (410485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give a postulate which is equivalent to the fifth postulate.
Ibn alHaytham (Alhazen) (9651039), an Arab mathematician, made an attempt at proving the parallel postulate using a proof by contradiction,[11] in the course of which he introduced the concept of motion and transformation into geometry.[12] He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn alHaytham–Lambert quadrilateral",[13] and his attempted proof contains elements similar to those found in Lambert quadrilaterals and Playfair's axiom.[14]
The Persian mathematician, astronomer, philosopher, and poet Omar Khayyám (1050–1123), attempted to prove the fifth postulate from another explicitly given postulate (based on the fourth of the five principles due to the Philosopher (Aristotle), namely, "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."[15] He derived some of the earlier results belonging to elliptical geometry and hyperbolic geometry, though his postulate excluded the latter possibility.[16] The Saccheri quadrilateral was also first considered by Omar Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.[13] Unlike many commentators on Euclid before and after him (including Giovanni Girolamo Saccheri), Khayyám was not trying to prove the parallel postulate as such but to derive it from his equivalent postulate. He recognized that three possibilities arose from omitting Euclid's fifth postulate; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's fifth postulate, otherwise, they must be either acute or obtuse. He showed that the acute and obtuse cases led to contradictions using his postulate, but his postulate is now known to be equivalent to the fifth postulate.
Nasir alDin alTusi (1201–1274), in his Alrisala alshafiya'an alshakk fi'lkhutut almutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250), wrote detailed critiques of the parallel postulate and on Khayyám's attempted proof a century earlier. Nasir alDin attempted to derive a proof by contradiction of the parallel postulate.[17] He also considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them.[16]
Euclidean, elliptical and hyperbolic geometry. The Parallel Postulate is satisfied only for models of Euclidean geometry.
Nasir alDin's son, Sadr alDin (sometimes known as "PseudoTusi"), wrote a book on the subject in 1298, based on his father's later thoughts, which presented one of the earliest arguments for a nonEuclidean hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."[17][18] His work was published in Rome in 1594 and was studied by European geometers. This work marked the starting point for Saccheri's work on the subject[17] which opened with a criticism of Sadr alDin's work and the work of Wallis.[19]
Giordano Vitale (16331711), in his book Euclide restituo (1680, 1686), used the KhayyamSaccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Girolamo Saccheri (16671733) pursued the same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case (proceeding, like Euclid, from the implicit assumption that lines can be extended indefinitely and have infinite length), but failing to refute the acute case (although he managed to wrongly persuade himself that he had).
In 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyám, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the nonEuclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further.[20]
Where Khayyám and Saccheri had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the logically consistent geometries which result. In 1829, Nikolai Ivanovich Lobachevsky published an account of acute geometry in an obscure Russian journal (later republished in 1840 in German). In 1831, János Bolyai included, in a book by his father, an appendix describing acute geometry, which, doubtlessly, he had developed independently of Lobachevsky. Carl Friedrich Gauss had also studied the problem, but he did not publish any of his results. Upon hearing of Bolyai's results in a letter from Bolyai's father, Farkas Bolyai, Gauss stated:
"If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirtyfive years."[21]
The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and elliptic geometry (the obtuse case). The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868.
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in twodimensional geometry:
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
This postulate does not specifically talk about parallel lines,[1] it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23[2] just before the five postulates.[3]
Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. A geometry where the parallel postulate does not hold is known as a nonEuclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or, in other places known as neutral geometry).
The unit circle for trigonometry is fundamental to trigonometry. It is a circle with a quadrant inside of it. The four angle of the unit circle are 0, pi over 2, pi, and 3pi over 2 in radian angles.
In quadrant 1 cosine and sin are positive. In quadrant 2 cosine is negative and sign is positive. In quadrant 3 of the cartesian coordinate system cosine is negative and sin is negative. In quadrant 4 of the unit circle cosine is positive and sin is negative.
Cosine is the x coordinate and sine is the y coordinate.
The unit circle is the basis for trigonometry and it reflects the quadrant model image.
graphs of sine, cosine and tangent are made from information in the unit circle.
https://en.wikipedia.org/wiki/File:Unit_circle.svg
The unit circle has four quadrants
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere.
https://en.wikipedia.org/wiki/Unit_circle
http://www.basicmathematics.com/latticemethodformultipl…
Although the lattice method for multiplication is no longer being used right now in school, it is easy understand
I will illustrate with two good examples. Study them carefully and follow the steps exactly as shown
Example #1:
Multiply 42 and 35
Arrange 42 and 35 around a 2 × 2 grid as shown below:
Draw the diagonals of the small squares as shown below:
Multiply 3 by 4 to get 12 and put 12 in intersection of the first row and the first column as show below.
Notice that 3 is located in the first row and 4 in the first column. That is why the answer goes in the intersection.
By the same token, multiply 5 and 2 and put the answer in the intersection of second row and the second column
And so forth...
Then, going from right to left, add the numbers down the diagonals as indicated with the arrows.
The first diagonal has only 0. Bring zero down.
The second diagonal has 6, 1, 0. Add these numbers to get 7 and bring it down.
And so forth...
After the grid is completed, what you see in red is the answer that is 1470
Example #2:
Multiply 658 and 47
Arrange 657 and 47 around a 3 × 2 grid as shown below:
Draw the diagonals of the small squares, find products, and put the answers in intersecting rows and columns as already demonstrated:
Then, going from right to left, add the numbers down the diagonals as shown before.
The first diagonal has only 6. Bring 6 down.
The second diagonal has 2, 5, and 5. Add these numbers to get 12. Bring 2 down and carry the 1 over to the next diagonal.
The third diagonal has 3, 0, 3, and 2. Add these numbers to get 8 and add 1 (your carry) to 8 to get 9.
and so forth...
After the grid is completed, what you see in red is the answer to the multiplication that is 30926
I understand that this may be your first encounter with the lattice method for multiplication. It may seem that it is tough. Just practice with other examples and you will be fine.
Any questions about the lattice method for multiplication? Just contact me
The lattice method employs quadrants and many math teachers think it is the ideal way to solve mathematical problems
In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials—hence the method may be referred to as the FOIL method. The word FOIL is an acronym for the four terms of the product:
First ("first" terms of each binomial are multiplied together)
Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)
Last ("last" terms of each binomial are multiplied)
The general form is:
(a+b)(c+d)=\underbrace {ac} _{\mathrm {first} }+\underbrace {ad} _{\mathrm {outside} }+\underbrace {bc} _{\mathrm {inside} }+\underbrace {bd} _{\mathrm {last} }
Note that a is both a "first" term and an "outer" term; b is both a "last" and "inner" term, and so forth. The order of the four terms in the sum is not important, and need not match the order of the letters in the word FOIL.
The Foil method is the basis for algebra and multiplying binomials. It has four components fitting the quadrant model image. Even if you represent the foil model pictorially, you get four squares/ the quadrant model image.
http://www.basicmathematics.com/latticemethodformultipl…
Although the lattice method for multiplication is no longer being used right now in school, it is easy understand
I will illustrate with two good examples. Study them carefully and follow the steps exactly as shown
Example #1:
Multiply 42 and 35
Arrange 42 and 35 around a 2 × 2 grid as shown below:
Draw the diagonals of the small squares as shown below:
Multiply 3 by 4 to get 12 and put 12 in intersection of the first row and the first column as show below.
Notice that 3 is located in the first row and 4 in the first column. That is why the answer goes in the intersection.
By the same token, multiply 5 and 2 and put the answer in the intersection of second row and the second column
And so forth...
Then, going from right to left, add the numbers down the diagonals as indicated with the arrows.
The first diagonal has only 0. Bring zero down.
The second diagonal has 6, 1, 0. Add these numbers to get 7 and bring it down.
And so forth...
After the grid is completed, what you see in red is the answer that is 1470
Example #2:
Multiply 658 and 47
Arrange 657 and 47 around a 3 × 2 grid as shown below:
Draw the diagonals of the small squares, find products, and put the answers in intersecting rows and columns as already demonstrated:
Then, going from right to left, add the numbers down the diagonals as shown before.
The first diagonal has only 6. Bring 6 down.
The second diagonal has 2, 5, and 5. Add these numbers to get 12. Bring 2 down and carry the 1 over to the next diagonal.
The third diagonal has 3, 0, 3, and 2. Add these numbers to get 8 and add 1 (your carry) to 8 to get 9.
and so forth...
After the grid is completed, what you see in red is the answer to the multiplication that is 30926
I understand that this may be your first encounter with the lattice method for multiplication. It may seem that it is tough. Just practice with other examples and you will be fine.
Any questions about the lattice method for multiplication? Just contact me
The lattice method employs quadrants and many math teachers think it is the ideal way to solve mathematical problems
https://en.wikipedia.org/wiki/Quadratic_function
Most students memorize the quadratic equation and it is the only equation really memorized in school. Quad means four it means to square/make a quadrant.
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highestdegree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:
f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,
with at least one of the coefficients a, b, c, d, e, or f of the seconddegree terms being nonzero.
A quadratic polynomial with two real roots (crossings of the x axis) and hence no complex roots. Some other quadratic polynomials have their minimum above the x axis, in which case there are no real roots and two complex roots.
A univariate (singlevariable) quadratic function has the form[1]
f(x)=ax^{2}+bx+c,\quad a\neq 0
in the single variable x. The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the yaxis, as shown at right.
If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.
The bivariate case in terms of variables x and y has the form
f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f\,\!
with at least one of a, b, c not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a hyperbola).
In general there can be an arbitrarily large number of variables, in which case the resulting surface is called a quadric, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.
Swastika curve
From Wikipedia, the free encyclopedia
The swastika curve.
The swastika curve is the name given by Cundy and Rollett[1] to the quartic plane curve with the Cartesian equation
y
4
−
x
4
=
x
y
,
y^{4}x^{4}=xy,\,
or, equivalently, the polar equation
r
2
=
−
tan
(
2
θ
)
/
2.
r^{2}=\tan(2\theta )/2.\,
The curve looks similar to the righthanded swastika. It can be inverted with respect to a unit circle to resemble a lefthanded swastika. The Cartesian equation then becomes
x
4
−
y
4
=
x
y
.
x^{4}y^{4}=xy.\,
Jump up ^
The quadratrix is mentioned in the works of Proklos (412–485), Pappos (3rd and 4th centuries) and Iamblichus (c. 240–325). Proklos names Hippias as the inventor of a curve called quadratrix and describes somewhere else how Hippias has applied the curve on the trisection problem. Pappos only mentions how a curve named quadratrix was used by Dinostratos, Nicomedes and others to square the circle. He neither mentions Hippias nor attributes the invention of the quadratrix to a particular person. Iamblichus just writes in a single line, that a curve called quadratrix was used bei Nicomedes to square the circle.[10][11][12]
Though based on Proklos' name for the curve it is conceivable that Hippias himself used it for squaring the circle or some other curvilinear figure, most math historians assume that Hippias invented the curve but used it only for the trisection of angles. Its use for squaring the circle only occurred decades later and was due to mathematicians like Dinostratos and Nicomedes. This interpretation of the historical sources goes back to the German mathematician and historian Moritz Cantor.
The quadratrix of Dinostratus (also called the quadratrix of Hippias) was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it affected a mechanical solution of squaring the circle. Pappus, in his Collections, treats its history, and gives two methods by which it can be generated.
Let a helix be drawn on a right circular cylinder; a screw surface is then obtained by drawing lines from every point of this spiral perpendicular to its axis. The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix.
A right cylinder having for its base an Archimedean spiral is intersected by a right circular cone which has the generating line of the cylinder passing through the initial point of the spiral for its axis. From every point of the curve of intersection, perpendiculars are drawn to the axis. Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix.
Quadratrix of Dinostratus (in red)
Another construction is as follows. DAB is a quadrant in which the line DA and the arc DB are divided into the same number of equal parts. Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to AB and through the corresponding points on the radius DA. The locus of these intersections is the quadratrix.
Quadratrix of Dinostratus with a central portion flanked by infinite branches
Letting A be the origin of the Cartesian coordinate system, D be the point (a,0), a units from the origin along the x axis, and B be the point (0,a), a units from the origin along the y axis, the curve itself can be expressed by the equation[1]
Because the cotangent function is invariant under negation of its argument, and has a simple pole at each multiple of π, the quadratrix has reflection symmetry across the y axis, and similarly has a pole for each value of x of the form x = 2na, for integer values of n, except at x = 0 where the pole in the cotangent is canceled by the factor of x in the formula for the quadratrix. These poles partition the curve into a central portion flanked by infinite branches. The point where the curve crosses the y axis has y = 2a/π; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a rational multiple of 1/π, leading to a solution of the classical problem of squaring the circle. Since this is impossible with compass and straightedge, the quadratrix in turn cannot be constructed with compass and straightedge. An accurate construction of the quadratrix would also allow the solution of two other classical problems known to be impossible with compass and straightedge, doubling the cube and trisecting an angle.
In mathematics, quadrature is a historical term which means determining area. Quadrature problems served as one of the main sources of problems in the development of calculus, and introduce important topics in mathematical analysis.
Mathematicians of ancient Greece, according to the Pythagorean doctrine, understood determination of area of a figure as the process of geometrically constructing a square having the same area (squaring), thus the name quadrature for this process. The Greek geometers were not always successful (see quadrature of the circle), but they did carry out quadratures of some figures whose sides were not simply line segments, such as the lunes of Hippocrates and the quadrature of the parabola. By Greek tradition, these constructions had to be performed using only a compass and straightedge.
For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side (the geometric mean of a and b). For this purpose it is possible to use the following: if one draws the circle with diameter made from joining line segments of lengths a and b, then the height (BH in the diagram) of the line segment drawn perpendicular to the diameter, from the point of their connection to the point where it crosses the circle, equals the geometric mean of a and b. A similar geometrical construction solves the problems of quadrature of a parallelogram and of a triangle.
The area of a segment of a parabola is 4/3 that of the area of a certain inscribed triangle.
Problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible. Nevertheless, for some figures (for example a lune of Hippocrates) a quadrature can be performed. The quadratures of the surface of a sphere and a parabola segment discovered by Archimedes became the highest achievement of analysis in antiquity.
The area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere.
The area of a segment of a parabola determined by a straight line cutting it is 4/3 the area of a triangle inscribed in this segment.
For the proof of these results, Archimedes used the method of exhaustion[1]:113 of Eudoxus.
In medieval Europe, quadrature meant the calculation of area by any method. Most often the method of indivisibles was used; it was less rigorous than the geometric constructions of the Greeks, but it was simpler and more powerful. With its help, Galileo Galilei and Gilles de Roberval found the area of a cycloid arch, Grégoire de SaintVincent investigated the area under a hyperbola (Opus Geometricum, 1647),[1]:491 and Alphonse Antonio de Sarasa, de SaintVincent's pupil and commentator, noted the relation of this area to logarithms.[1]:492[2]
John Wallis algebrised this method; he wrote in his Arithmetica Infinitorum (1656) some series which are equivalent to what is now called the definite integral, and he calculated their values. Isaac Barrow and James Gregory made further progress: quadratures for some algebraic curves and spirals. Christiaan Huygens successfully performed a quadrature of the surface area of some solids of revolution.
The quadrature of the hyperbola by SaintVincent and de Sarasa provided a new function, the natural logarithm, of critical importance. With the invention of integral calculus came a universal method for area calculation. In response, the term quadrature has become traditional (some[who?] would say archaic), and instead the modern phrase finding the area is more commonly used for what is technically the computation of a univariate definite integral
The vedic square is made of quadrants
In ancient Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table. The entry in each cell is the digital root of the product of the column and row headings i.e. the remainder when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9).
In Euclidean space, quadrics have dimension D = 2, and are known as quadric surfaces. By making a suitable Euclidean change of variables, any quadric in Euclidean space can be put into a certain normal form by choosing as the coordinate directions the principal axes of the quadric. In threedimensional Euclidean space there are 16 such normal forms.[2] Of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all.[3]
16 is the number of squares in the quadrant model
Four normed division algebras RHCO
Abstract. The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.
https://ncatlab.org/nlab/show/normed+division+algebra
i read a guys article where he talked about how the fourth norm division algebra is trabscendent and enconpasses the previous three
A normed division algebra is a notnecessarily associative algebra, over some ground field, that is both a division algebra as well as a multiplicatively normed algebra.
It turns out that over the real numbers there are precisely only four normed division algebras up to isomorphism: the algebras of
real numbers,
complex numbers,
quaternions,
octonions.
In this sense real normed division algebras may be thought of as a natural generalization of the more familiar real and complex numbers.
Moreover, if one regards the real numbers as a staralgebra, then each step in this sequence is given by applying the CayleyDickson construction. Applied to the octonions it yields the sedenions, which however are no longer a division algebra.
This classification turns out to closely connect to various other systems of exceptional structures in mathematics and physics:
The Hopf invariant one theorem says that the only continuous functions between spheres of the form S2n−1→Sn
S
2
n
−
1
→
S
n
whose Hopf invariant is equal to 1 are the Hopf constructions on the four real normed division algebras, namely
the real Hopf fibration;
the complex Hopf fibration;
the quaternionic Hopf fibration;
the octonionic Hopf fibration.
Patterns related to Majorana spinors in spin geometry are intimately related to the four normed division algebras, and, induced by this, so is the classification of supersymmetry in the form of super Poincaré Lie algebras and super Minkowski spacetimes (which are built from these real spin representations). For more on this see at supersymmetry and division algebras.
(Moreover, apparently these two items are not unrelated, see here.)
2. Definition
A normed division algebra is
a division algebra;
that is also a Banach algebra.
While the norm in a Banach algebra is in general only submultiplicative (∥xy∥≤∥x∥∥y∥
‖
x
y
‖
≤
‖
x
‖
‖
y
‖
), the norm in a normed division algebra must be multiplicative (∥xy∥=∥x∥∥y∥
‖
x
y
‖
=
‖
x
‖
‖
y
‖
). Accordingly, this norm is considered to be an absolute value and often written −

−

instead of ∥−∥
‖
−
‖
. There is also a converse: if the norm on a Banach algebra is multiplicative (including ∥1∥=1
‖
1
‖
=
1
), then it must be a division algebra. While the term ‘normed division algebra’ does not seem to include the completeness condition of a Banach algebra, in fact the only examples have finite dimension and are therefore complete.
Accordingly, a normed division algebras is in particular a division composition algebra.
3. Properties
Classification
Over the complex numbers, the only normed division algebra is the algebra of complex numbers themselves.
The Hurwitz theorem says that over the real numbers there are, up to isomorphism, exactly four finitedimensonal normed division algebras :
R
ℝ
, the algebra of real numbers,
C
ℂ
, the algebra of complex numbers,
H
ℍ
, the algebra of quaternions,
O
𝕆
, the algebra of octonions.
In fact these are also exactly the real alternative division algebras (Zorn 30).
Each of these is produced from the previous one by the Cayley–Dickson construction; when applied to O
𝕆
, this construction produces the algebra of sedenions, which do not form a division algebra.
The Cayley–Dickson construction applies to an algebra with involution; by the abstract nonsense of that construction, we can see that the four normed division algebras above have these properties:
R
ℝ
is associative, commutative, and with trivial involution,
C
ℂ
is associative and commutative but has nontrivial involution,
H
ℍ
is associative but noncommutative and with nontrivial involution,
O
𝕆
is neither associative, commutative, nor with trivial involution.
However, these algebras do all have some useful algebraic properties; in particular, they are all alternative (a weak version of associativity). They are also all composition algebras.
A normed field is a commutative normed division algebra; it follows from the preceding that the only normed fields over R
ℝ
are R
ℝ
and C
ℂ
(e.g. Tornheim 52).
It is in fact true that all unital normed division algebras over R
ℝ
are already finite dimensional, by (UrbanikWright 1960) (the authors give a reference on a nonunital infinitedimensional normed division algebra). Hence the Hurwitz theorem together with UrbanikWright 1960 says that the above four exhaust all real normed division algebras.
For purely inseparable characteristic 2 field extensions one can apparently get infinitedimensional examples; see this MathOverflow answer for reference.
Automorphisms
The automorphism groups of the real normed division algebras, as normed algebras, are
Aut(R)=1
Aut
(
ℝ
)
=
1
, the trivial group
Aut(C)=Z/2
Aut
(
ℂ
)
=
ℤ
/
2
the group of order 2, acting by complex conjugation;
Aut(H)=SO(3)
Aut
(
ℍ
)
=
SO
(
3
)
, the special orthogonal group acting via its canonical representaiton on the 3dimensional space of imaginary octonions;
Aut(O)=G2
Aut
(
𝕆
)
=
G
2
, the exceptional Lie group G2.
Relation to Hspace structures on sphere (Hopf invariant one)
The Hopf invariant one theorem says that the spheres carrying Hspace structure are precisely the unit spheres in one of the normed division algebras
(Adams 60)
Magic square
The Freudenthal magic square is a special square array of Lie algebras/Lie groups labeled by pairs of real normed division algebras and including all the exceptional Lie groups except G2.
4. Related concepts
Hopf invariant one
Cayley–Dickson construction
Lorentzian spacetime dimension spin group normed division algebra brane scan entry
3=2+1
3
=
2
+
1
Spin(2,1)≃SL(2,R)
Spin
(
2
,
1
)
≃
SL
(
2
,
ℝ
)
R
ℝ
the real numbers
4=3+1
4
=
3
+
1
Spin(3,1)≃SL(2,C)
Spin
(
3
,
1
)
≃
SL
(
2
,
ℂ
)
C
ℂ
the complex numbers
6=5+1
6
=
5
+
1
Spin(5,1)≃SL(2,H)
Spin
(
5
,
1
)
≃
SL
(
2
,
ℍ
)
H
ℍ
the quaternions little string
10=9+1
10
=
9
+
1
Spin(9,1)≃"SL(2,O)"
Spin
(
9
,
1
)
≃
"
SL
(
2
,
𝕆
)
"
O
𝕆
the octonions heterotic/type II string
see division algebra and supersymmetry
normed division algebra A
𝔸
Riemannian A
𝔸
manifolds Special Riemannian A
𝔸
manifolds
real numbers R
ℝ
Riemannian manifold oriented Riemannian manifold
complex numbers C
ℂ
Kähler manifold CalabiYau manifold
quaternions H
ℍ
quaternionKähler manifold hyperkähler manifold
octonions O
𝕆
Spin(7)manifold G2manifold
(Leung 02)
5. References
The classification of real divsion composition algebras is originally due (Hurwitz theorem) to
Adolf Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898) 309–316
The alternative classification as real alternative division algebras is due to
Max Zorn, Theorie der alternativen Ringe, Abhandlungen des Mathematischen Seminars der Universität Hamburg 8 (1930), 123147
General discussion includes includes
Leonard Tornheim, Normed fields over the real and complex fields, Michigan Math. J. Volume 1, Issue 1 (1952), 6168. (Euclid)
Silvio Aurora, On normed rings with monotone multiplication, Pacific J. Math. Volume 33, Number 1 (1970), 1520 (JSTOR)
The result about removing the assumption of finitedimensionality from unital normed division algebras appears in:
Kazimierz Urbanik and Fred B. Wright, _ Absolutevalued algebras_, Proc. Amer. Math. Soc. 11 (1960), 861866, doi:10.1090/S00029939196001202646
Exposition with emphasis on the octonions is in
John Baez, Normed Division Algebras
John Baez, This Week’s Finds — Week 59
Discussion of Riemannian geometry and special holonomy modeled on the different normed division algebras is in
Naichung Conan Leung, Riemannian Geometry Over Different Normed Division Algebras, J. Differential Geom. Volume 61, Number 2 (2002), 289333. (euclid)
first quartile (designated Q1) also called the lower quartile or the 25th percentile (splits off the lowest 25% of data from the highest 75%)
second quartile (designated Q2) also called the median or the 50th percentile (cuts data set in half)
third quartile (designated Q3) also called the upper quartile or the 75th percentile (splits off the highest 25% of data from the lowest 75%)
interquartile range (designated IQR) is the difference between the upper and lower quartiles. (IQR = Q3  Q1)
In the case of quartiles, the Interquartile Range (IQR) may be used to characterize the data when there may be extremities that skew the data; the interquartile range is a relatively robust statistic (also sometimes called "resistance") compared to the range and standard deviation. There is also a mathematical method to check for outliers and determining "fences", upper and lower limits from which to check for outliers.
After determining the first and third quartiles and the interquartile range as outlined above, then fences are calculated using the following formula:
{\text{Lower fence}}=Q_{1}1.5(\mathrm {IQR} )\,
{\text{Upper fence}}=Q_{3}+1.5(\mathrm {IQR} ),\,
where Q1 and Q3 are the first and third quartiles, respectively. The Lower fence is the "lower limit" and the Upper fence is the "upper limit" of data, and any data lying outside these defined bounds can be considered an outlier. Anything below the Lower fence or above the Upper fence can be considered such a case. The fences provide a guideline by which to define an outlier, which may be defined in other ways. The fences define a "range" outside of which an outlier exists; a way to picture this is a boundary of a fence, outside of which are "outsiders" as opposed to outliers.
Quartiles are a foundation of statistics and one of the forst things you learn in statistics. They reflect the quadrant model image
In descriptive statistics, the quartiles of a ranked set of data values are the three points that divide the data set into four equal groups, each group comprising a quarter of the data. A quartile is a type of quantile. The first quartile (Q1) is defined as the middle number between the smallest number and the median of the data set. The second quartile (Q2) is the median of the data. The third quartile (Q3) is the middle value between the median and the highest value of the data set.
In applications of statistics such as epidemiology, sociology and finance, the quartiles of a ranked set of data values are the four subsets whose boundaries are the three quartile points. Thus an individual item might be described as being "in th
https://briangrimmerblog.wordpress.com/…/themetamodelbe…/
This guy introduced his "Meta Model" where he noticed the fourth transcends the previous three in 2014 like 9 years after I discovered the quadrant model and years after I made it public on a large scale as the "theory of everything"
The three dimensional Star Tetrahedron (Merkaba) represents the notion of reflectivity between the quadrants while the four dimensional Hypercube (Tesseract) represents the combined properties of reflectivity and reflexivity.
Time and again over the many years of meta modelling we have found ourselves pondering the nature of the tesseract. For example if we were to reflex a dual startetrahedron then we can begin to imagine the 16 vertices and 32 vectors of the tesseract progressing in four dimensions.
cross multiplication
In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can crossmultiply to simplify the equation or determine the value of a variable.
Given an equation like:
{\frac {a}{b}}={\frac {c}{d}}
(where b and d are not zero), one can crossmultiply to get:
ad=bc\qquad \mathrm {or} \qquad a={\frac {bc}{d}}.
In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles.
The axes of a twodimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two halfaxes.
These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are (+,+)), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counterclockwise starting from the upper right ("northeast") quadrant.
Merk Diezle shared a link.
Aug 29, 2016 11:42pm
Buckminster Fuller  Thinking Out Loud (1996)
youtube.com
At 14 minutes it describes how Fuller called himself "4D". I also described how Snowden called himself "Four" in one of my books. It also discusses at the beginning how Fuller had poor vision and when asked to make a structure out of sticks and olives he made a TETRAHEDRON while all the other kids made squares and the teachers were amazed by this and wondered what the figure was that he made. The tetrahedron he would describe as the most stable of all structures and the most important especially in the isotropic vector matrix and Haramein discusses the vector equilibrium and how its related to the Kaballistic tree of life
A Jones diagram is a type of Cartesian graph developed by Loyd A. Jones in the 1940s, where each axis represents a different variable. In a Jones diagram opposite directions of an axis represent different quantities, unlike in a Cartesian graph where they represent positive or negative signs of the same quantity. The Jones diagram therefore represents four variables. Each quadrant shares the vertical axis with its horizontal neighbor, and the horizontal axis with the vertical neighbor. For example, the top left quadrant shares its vertical axis with the top right quadrant, and the horizontal axis with the bottom left quadrant. The overall system response is in quadrant I; the variables that contribute to it are in quadrants II through IV.
A common application of Jones diagrams is in photography, specifically in displaying sensitivity to light with what are also called "tone reproduction diagrams". These diagrams are used in the design of photographic systems (film, paper, etc.) to determine the relationship between the light a viewer would see at the time a photo was taken to the light that a viewer would see looking at the finished photograph.
The Jones diagram concept can be used for variables that depend successively on each other. Jones's original diagram used eleven quadrants[how?] to show all the elements of his photographic system.
https://en.wikipedia.org/wiki/Quaternion_group
IMAGE IS A QUADRANT MODEL
https://en.wikipedia.org/…/File:Quaternion_group;_Cayley_ta…
As Richard Dean showed in 1981, the QUATernion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field, over Q, of the polynomial
x^8  72 x^6 + 180 x^4  144 x^2 + 36.
The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.
Quaternion is a quadrant model16 squares
I'm a paragraph. Click here to add your own text and edit me. It's easy.
I'm a paragraph. Click here to add your own text and edit me. It's easy.
In mathematics, the Klein fourgroup (or just Klein group or Vierergruppe (English: fourgroup), often symbolized by the letter V or as K4) is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2. It was named Vierergruppe by Felix Klein in 1884
In music composition the fourgroup is the basic group of permutations in the twelvetone technique. In that instance the Cayley table is written;[2]
In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. There are only four allHarshad numbers: 1, 2, 4, and 6 (The number 12 is a Harshad number in all bases except octal).
Data Levels of Measurement
A variable has one of four different levels of measurement: Nominal, Ordinal, Interval, or Ratio. (Interval and Ratio levels of measurement are sometimes called Continuous or Scale). It is important for the researcher to understand the different levels of measurement, as these levels of measurement, together with how the research question is phrased, dictate what statistical analysis is appropriate. In fact, the Free download below conveniently ties a variable's levels to different statistical analyses.
Guide showing which Statistical Tests Correspond to a Variable's Level of Measurement
In descending order of precision, the four different levels of measurement are:

NominalLatin for name only (Republican, Democrat, Green, Libertarian)

OrdinalThink ordered levels or ranks (small8oz, medium12oz, large32oz)

IntervalEqual intervals among levels (1 dollar to 2 dollars is the same interval as 88 dollars to 89 dollars)

RatioLet the "o" in ratio remind you of a zero in the scale (Day 0, day 1, day 2, day 3, ...)
The first level of measurement is nominal level of measurement. In this level of measurement, the numbers in the variable are used only to classify the data. In this level of measurement, words, letters, and alphanumeric symbols can be used. Suppose there are data about people belonging to three different gender categories. In this case, the person belonging to the female gender could be classified as F, the person belonging to the male gender could be classified as M, and transgendered classified as T. This type of assigning classification is nominal level of measurement.
The second level of measurement is the ordinal level of measurement. This level of measurement depicts some ordered relationship among the variable's observations. Suppose a student scores the highest grade of 100 in the class. In this case, he would be assigned the first rank. Then, another classmate scores the second highest grade of an 92; she would be assigned the second rank. A third student scores a 81 and he would be assigned the third rank, and so on. The ordinal level of measurement indicates an ordering of the measurements.
The third level of measurement is the interval level of measurement. The interval level of measurement not only classifies and orders the measurements, but it also specifies that the distances between each interval on the scale are equivalent along the scale from low interval to high interval. For example, an interval level of measurement could be the measurement of anxiety in a student between the score of 10 and 11, this interval is the same as that of a student who scores between 40 and 41. A popular example of this level of measurement is temperature in centigrade, where, for example, the distance between 940C and 960C is the same as the distance between 1000C and 1020C.
The fourth level of measurement is the ratio level of measurement. In this level of measurement, the observations, in addition to having equal intervals, can have a value of zero as well. The zero in the scale makes this type of measurement unlike the other types of measurement, although the properties are similar to that of the interval level of measurement. In the ratio level of measurement, the divisions between the points on the scale have an equivalent distance between them.
The researcher should note that among these levels of measurement, the nominal level is simply used to classify data, whereas the levels of measurement described by the interval level and the ratio level are much more exact.
Related pages:
Level of measurement or scale of measure is a classification that describes the nature of information within the numbers assigned to variables.[1] Psychologist Stanley Smith Stevens developed the best known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio.[1][2] This framework of distinguishing levels of measurement originated in psychology and is widely criticized by scholars in other disciplines.
Stevens's typology[edit]
Overview[edit]
Stevens proposed his typology in a 1946 Science article titled "On the theory of scales of measurement".[2] In that article, Stevens claimed that all measurement in science was conducted using four different types of scales that he called "nominal," "ordinal," "interval," and "ratio," unifying both "qualitative" (which are described by his "nominal" type) and "quantitative" (to a different degree, all the rest of his scales). The concept of scale types later received the mathematical rigour that it lacked at its inception with the work of mathematical psychologists Theodore Alper (1985, 1987), Louis Narens (1981a, b), and R. Duncan Luce (1986, 1987, 2001). As Luce (1997, p. 395) wrote:
S. S. Stevens (1946, 1951, 1975) claimed that what counted was having an interval or ratio scale. Subsequent research has given meaning to this assertion, but given his attempts to invoke scale type ideas it is doubtful if he understood it himself ... no measurement theorist I know accepts Stevens's broad definition of measurement ... in our view, the only sensible meaning for 'rule' is empirically testable laws about the attribute.
Comparison[edit]
Incremental
Progress
Measure Property Mathematical
Operators
Advanced
Operations
Central
Tendency
Nominal Classification, Membership =, != Grouping Mode
Ordinal Comparison, Level >, < Sorting Median
Interval Difference, Affinity +,  Yardstick Mean,
Deviation
Ratio Magnitude, Amount *, / Ratio Geometric Mean,
Coeff. of Variation
Nominal level[edit]
The nominal type differentiates between items or subjects based only on their names or (meta)categories and other qualitative classifications they belong to; thus dichotomous data involves the construction of classifications as well as the classification of items. Discovery of an exception to a classification can be viewed as progress. Numbers may be used to represent the variables but the numbers do not have numerical value or relationship: for example, a Globally unique identifier.
Examples of these classifications include gender, nationality, ethnicity, language, genre, style, biological species, and form.[6][7] In a university one could also use hall of affiliation as an example. Other concrete examples are
in grammar, the parts of speech: noun, verb, preposition, article, pronoun, etc.
in politics, power projection: hard power, soft power, etc.
in biology, the taxonomic ranks below domains: Archaea, Bacteria, and Eukarya
in software engineering, type of faults: specification faults, design faults, and code faults
Nominal scales were often called qualitative scales, and measurements made on qualitative scales were called qualitative data. However, the rise of qualitative research has made this usage confusing. The numbers in nominal measurement are assigned as labels and have no specific numerical value or meaning. No form of mathematical computation (+, x etc.) may be performed on nominal measures. The nominal level is the lowest measurement level used from a statistical point of view.
Mathematical operations[edit]
Equality and other operations that can be defined in terms of equality, such as inequality and set membership, are the only nontrivial operations that generically apply to objects of the nominal type.
Central tendency[edit]
The mode, i.e. the most common item, is allowed as the measure of central tendency for the nominal type. On the other hand, the median, i.e. the middleranked item, makes no sense for the nominal type of data since ranking is meaningless for the nominal type.[8]
Ordinal scale[edit]
Further information: Ordinal data
The ordinal type allows for rank order (1st, 2nd, 3rd, etc.) by which data can be sorted, but still does not allow for relative degree of difference between them. Examples include, on one hand, dichotomous data with dichotomous (or dichotomized) values such as 'sick' vs. 'healthy' when measuring health, 'guilty' vs. 'notguilty' when making judgments in courts, 'wrong/false' vs. 'right/true' when measuring truth value, and, on the other hand, nondichotomous data consisting of a spectrum of values, such as 'completely agree', 'mostly agree', 'mostly disagree', 'completely disagree' when measuring opinion.
Central tendency
The median, i.e. middleranked, item is allowed as the measure of central tendency; however, the mean (or average) as the measure of central tendency is not allowed. The mode is allowed.
In 1946, Stevens observed that psychological measurement, such as measurement of opinions, usually operates on ordinal scales; thus means and standard deviations have no validity, but they can be used to get ideas for how to improve operationalization of variables used in questionnaires. Most psychological data collected by psychometric instruments and tests, measuring cognitive and other abilities, are ordinal, although some theoreticians have argued they can be treated as interval or ratio scales. However, there is little prima facie evidence to suggest that such attributes are anything more than ordinal (Cliff, 1996; Cliff & Keats, 2003; Michell, 2008).[9] In particular,[10] IQ scores reflect an ordinal scale, in which all scores are meaningful for comparison only.[11][12][13] There is no absolute zero, and a 10point difference may carry different meanings at different points of the scale.[14][15]
Interval scale[edit]
The interval type allows for the degree of difference between items, but not the ratio between them. Examples include temperature with the Celsius scale, which has two defined points (the freezing and boiling point of water at specific conditions) and then separated into 100 intervals, date when measured from an arbitrary epoch (such as AD), percentage such as a percentage return on a stock,[16] location in Cartesian coordinates, and direction measured in degrees from true or magnetic north. Ratios are not meaningful since 20 °C cannot be said to be "twice as hot" as 10 °C, nor can multiplication/division be carried out between any two dates directly. However, ratios of differences can be expressed; for example, one difference can be twice another. Interval type variables are sometimes also called "scaled variables", but the formal mathematical term is an affine space (in this case an affine line).
Central tendency and statistical dispersion[edit]
The mode, median, and arithmetic mean are allowed to measure central tendency of interval variables, while measures of statistical dispersion include range and standard deviation. Since one can only divide by differences, one cannot define measures that require some ratios, such as the coefficient of variation. More subtly, while one can define moments about the origin, only central moments are meaningful, since the choice of origin is arbitrary. One can define standardized moments, since ratios of differences are meaningful, but one cannot define the coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of) a central moment.
Ratio scale[edit]
The ratio type takes its name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind (Michell, 1997, 1999). A ratio scale possesses a meaningful (unique and nonarbitrary) zero value. Most measurement in the physical sciences and engineering is done on ratio scales. Examples include mass, length, duration, plane angle, energy and electric charge. In contrast to interval scales, ratios are now meaningful because having a nonarbitrary zero point makes it meaningful to say, for example, that one object has "twice the length" of another (= is "twice as long"). Very informally, many ratio scales can be described as specifying "how much" of something (i.e. an amount or magnitude) or "how many" (a count). The Kelvin temperature scale is a ratio scale because it has a unique, nonarbitrary zero point called absolute zero.
Central tendency and statistical dispersion
The geometric mean and the harmonic mean are allowed to measure the central tendency, in addition to the mode, median, and arithmetic mean. The studentized range and the coefficient of variation are allowed to measure statistical dispersion. All statistical measures are allowed because all necessary mathematical operations are defined for the ratio scale.
Elements of Dynamic was the debut of the term crossratio for a fourargument function frequently used in geometry.
THE FIRST THREE PERFECT NUMBERS WERE KNOWN IN ANTIQUITY. (they are close to each other and similar) IT TOOK A WHILE LATER TO DISCOVER THE FOURTH PERFECT NUMBER BUT THE ANCIENT GREEKS KNEW THE FIRST FOUR. THE FOURTH IS ALWAYS DIFFERENT. IT TOOK UNTIL THE 1500S FOR THE FIFTH TO BE DISCOVERED. THE FIFTH IS ALWAYS ULTRA TRANSCENDENT.
THIS STAME PATTERN HOLDS TRUE FOR MANY MANY TYPES OF NUMBERS. LITERALLY I WOULD WATCH LECTURES AND I FORGET WHAT THEY WERE CALLED BUT THERE WAS ONE THAT WAS IMPORTANT FOR PROGRAMMING AND THE FIRST FOUR WERE FOUND BY THE ANCIENTS AND IT TOOK UNTIL COMPUTERS TO FIND THE FIFTH AND THE FOURTH WAS DIFFERENT.
Perfect numbers
A perfect number is a number that's the sum of all of its divisors (excluding itself). For example, 6 is a perfect number because its divisors (apart from 6 itself) are 1, 2 and 3, and
6 = 1 + 2 + 3.
The next perfect number is 28, which has divisors 1, 2, 4, 7 and 14, and:
28 = 1 + 2 + 4 + 7 + 14.
Leonhard Euler
Euclid, depicted with a compass in Raphael's painting The school of Athens.
The next three perfect numbers are 496, 8128 and 33,550,336.
The gaps between perfect numbers are as wide as their discovery has been painstaking. The first four perfect numbers seem to have been known to the Greeks, the fifth and sixth weren't written down explicitly until the 15th century and the seventh followed in the 16th century. Today we know of 48 perfect number, the largest of which has over 34 million digits. All of these 48 are even. This raises two questions:
THIS ISNT THE ONE I SAW THE LECTURE ABOUT HOPEFULLY I CAN FIND THAT. I WATCHED THE LECTURE BEFORE I RECORDED ALL THE STUFF ON FACEBOOK. BUT NOTICE THE PATTERN WITH THE FERMAT NUMBERS. THE FIRST THREE ARE SIMILAR. THE FOURTH IS LARGE A LOT DIFFERENT. THE FIFTH IS WAY WAY WAY DIFFERENT SUPER DIFFERENT EXTREMELY LARGE. THE FOURTH IS ALWAYS TRANSCENDENT. FIFTH ULTRA TRANSCENDENT.
In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form
F
n
=
2
2
n
+
1
,
{\displaystyle F_{n}=2^{2^{n}}+1,}
where n is a nonnegative integer. The first few Fermat numbers are:
3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … (sequence A000215 in the OEIS).
THE FIRST TWO ARE THE DUALITY 3, 5 very similar. THIRD IS MORE SOLID AND THERE IS SOME SPACE BETWEEN THE NUMBERS IT IS 17. THE FOURTH IS WAY DIFFERENT 257. THE FIFTH IS OVER 600000 WAY WAY WAY DIFFERENT
THESE ARE THE ONLY KNOWN FERMAT PRIMES AGAIN THE FOURTH IS DIFFERENT AND THE FIFTH ULTRA TRANSCENDENT. IT GOES TO F4
Factorizations of the first twelve Fermat numbers are:
F0 = 21 + 1 = 3 is prime
F1 = 22 + 1 = 5 is prime
F2 = 24 + 1 = 17 is prime
F3 = 28 + 1 = 257 is prime
F4 = 216 + 1 = 65,537 is the largest known Fermat prime
THE FOURTH IS DIFFERENT THE FIFTH IS ULTRA
TRANSCENDENT. THE FIRST FOUR WERE KNOWN IN ANTIQUITY THE FOURTH IT TOOK A WHILE TO FIGURE OUT BUT IT WAS KNOWN IN ANTIQUITY. IT TOOK UNTIL the 1400s to find the fifth. THE FIFTH IS ALWAYS ULTRA TRANSCENDNET FOURTH IS TRANSCENDENT. I LEARNED ABOUT THESE PRIMES ON A TEACHING COMPANY COURSE
The first four Mersenne primes M2 = 3, M3 = 7, M5 = 31 and M7 = 127 were known in antiquity. The fifth, M13 = 8191, was discovered anonymously before 1461; the next two (M17 and M19) were found by Pietro Cataldi in 1588. After nearly two centuries, M31 was verified to be prime by Leonhard Euler in 1772. The next (in historical, not numerical order) was M127, found by Édouard Lucas in 1876, then M61 by Ivan Mikheevich Pervushin in 1883. Two more (M89 and M107) were found early in the 20th century, by R. E. Powers in 1911 and 1914, respectively.
The best method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime p > 2, Mp = 2p − 1 is prime if and only if Mp divides Sp − 2, where S0 = 4 and Sk = (Sk − 1)2 − 2 for k > 0.
During the era of manual calculation, all the exponents up to and including 257 were tested with the Lucas–Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and 229.[11] Unfortunately for those investigators, the interval they were testing contains the largest known gap between Mersenne primes, in relative terms: the next prime exponent would turn out to be more than four times larger than the previous record of 127.
THERE ARE ONLY FOUR DOUBLE MERSENNE PRIMES
The first four terms of the sequence of double Mersenne numbers are[1] (sequence A077586 in the OEIS):
M
M
2
=
M
3
=
7
M_{M_2} = M_3 = 7
M
M
3
=
M
7
=
127
M_{M_3} = M_7 = 127
M
M
5
=
M
31
=
2147483647
M_{M_5} = M_{31} = 2147483647
M
M
7
=
M
127
=
170141183460469231731687303715884105727
M_{M_7} = M_{127} = 170141183460469231731687303715884105727
M_{M_{61}}, or 22305843009213693951 − 1. Being approximately 1.695×10694127911065419641, this number is far too large for any currently known primality test. It has no prime factor below 4×1033.[2] There are probably no other double Mersenne primes than the four known.[1][3]
In popular culture[edit]
In the Futurama movie The Beast with a Billion Backs, the double Mersenne number
M
M
7
M_{M_7} is briefly seen in "an elementary proof of the Goldbach conjecture". In the movie, this number is known as a "martian prime".
AGAIN THE FOURTH IS DIFFERENT. THE FOURTH ULTRA TRANSCENDENT
The first few Wagstaff primes are:
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, … (sequence A000979 in the OEIS)
NOTICE AGAINT HE FOURTH IS DIFFERENT/TRANSCENDENT THE FIFTH ULTRA TRANSCENDENT EUCLID NUMBERS
The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871,
A Euclid number of the second kind (also called Kummer number) is an integer of the form En = pn# − 1, where pn# is the nth primorial, the first few such numbers are:
1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869
FOURTH POWER 16 IS THE SQUARES OF THE QUADRANT MODEL
In mathematics, a quartan prime is a prime number of the form x4 + y4, where x > 0, y > 0. The odd quartan primes are of the form 16n + 1.
For example, 17 is the smallest odd quartan prime: 17 = 14 + 24.
Due to the rules behind even and odd numbers, only one of the two integers can be odd. If both are odd or even, the resulting integer will be even, and not prime (excluding the trivial solution of 14 + 14 = 2).
The first few quartan primes are
2, 17, 97, 257, 337, 641, 881, … (sequence A002645 in the OEIS).
SORT OF THE SAME PATTERN FOURTH DIFFERENT HERE
Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a prime number, its index n cannot be of the form 3x + 2 for x > 0. The first few Carol numbers that are also prime are 7, 47, 223, 3967, 16127
SAME PATTERN FOURTH NUMBER IS DIFFERENT FIFTH ULTRA TRANSCENDENT
A Kynea number is an integer of the form
4
n
+
2
n
+
1
−
1
4^n + 2^{n + 1}  1.
An equivalent formula is
(
2
n
+
1
)
2
−
2
(2^n + 1)^2  2.
This indicates that a Kynea number is the nth power of 4 plus the (n + 1)th Mersenne number. Kynea numbers were studied by Cletus Emmanuel who named them after a baby girl.[1]
The sequence of Kynea numbers starts with:
7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, ... (sequence A093069 in the OEIS).
FOURTH DEFINITELY TRANSCENDENT FIFTH ULTRA
The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins
2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, ... (sequence A051254 in the OEIS).
Illustration of how a phase portrait would be constructed for the motion of a simple pendulum.
SQUAREING THE CIRCLE WAS ALSO CALLED QUADRATURE OF THE CIRCLE QUAD IS FOUR
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. It may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. It had been known for some decades before then that the construction would be impossible if π were transcendental, but π was not proven transcendental until 1882. Approximate squaring to any given nonperfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.
The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.[1]
The term quadrature of the circle is sometimes used to mean the same thing as squaring the circle, but it may also refer to approximate or numerical methods for finding the area of a circle.
ARCHIMEDES PROOF IS A QUADRANT MODEL FOR QUADRATURE
Sum of the series[edit]
Archimedes' proof that 1/4 + 1/16 + 1/64 + ... = 1/3
To complete the proof, Archimedes shows that
1
+
1
4
+
1
16
+
1
64
+
⋯
=
4
3
.
1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,{\frac {1}{64}}\,+\,\cdots \;=\;{\frac {4}{3}}.
The formula above is a geometric series—each successive term is one fourth of the previous term. In modern mathematics, that formula is a special case of the sum formula for a geometric series.
Archimedes evaluates the sum using an entirely geometric method,[2] illustrated in the adjacent picture. This picture shows a unit square which has been dissected into an infinity of smaller squares. Each successive purple square has one fourth the area of the previous square, with the total purple area being the sum
1
4
+
1
16
+
1
64
+
⋯
.
{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,{\frac {1}{64}}\,+\,\cdots .
However, the purple squares are congruent to either set of yellow squares, and so cover 1/3 of the area of the unit square. It follows that the series above sums to 4/3.
QUADRATURE QUAD IS FOUR
In James Joyce's novel Ulysses, Leopold Bloom dreams of becoming wealthy by squaring the circle, unaware that the quadrature of the circle had been proved impossible 22 years earlier and that the British government had never offered a reward for its solution.
QUINCUNX IS CROSS
The Galton board, also known as a quincunx or bean machine, is a device for statistical experiments named after English scientist Sir Francis Galton. It consists of an upright board with evenly spaced nails (or pegs) driven into its upper half, where the nails are arranged in staggered order, and a lower half divided into a number of evenlyspaced rectangular slots. The front of the device is covered with a glass cover to allow viewing of both nails and slots. In the middle of the upper edge, there is a funnel into which balls can be poured, where the diameter of the balls must be much smaller than the distance between the nails. The funnel is located precisely above the central nail of the second row so that each ball, if perfectly centered, would fall vertically and directly onto the uppermost point of this nail's surface (Kozlov and Mitrofanova 2002). The figure above shows a variant of the board in which only the nails that can potentially be hit by a ball dropped from the funnel are included, leading to a triangular array instead of a rectangular one.
MADE OF QUADRANTS LOOKS LIKE CROSS
In combinatorial mathematics, an Aztec diamond of order n consists of all squares of a square lattice whose centers (x,y) satisfy x + y ≤ n. Here n is a fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both x and y are halfintegers.[1]
The Aztec diamond theorem states that the number of domino tilings of the Aztec diamond of order n is 2n(n+1)/2.[2] The arctic circle theorem says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle.[3]
THERE ARE 16 BOOLEAN OPERATIONS0 16 SQUARES QMR
More generally one may complement any of the eight subsets of the three ports of either an AND or OR gate. The resulting sixteen possibilities give rise to only eight Boolean operations, namely those with an odd number of 1's in their truth table. There are eight such because the "oddbitout" can be either 0 or 1 and can go in any of four positions in the truth table. There being sixteen binary Boolean operations, this must leave eight operations with an even number of 1's in their truth tables. Two of these are the constants 0 and 1 (as binary operations that ignore both their inputs); four are the operations that depend nontrivially on exactly one of their two inputs, namely x, y, ¬x, and ¬y; and the remaining two are x⊕y (XOR) and its complement x≡y.
16 SQUARES QMR
Euclid's Elements, Book IX[edit]
The geometric progression 1, 2, 4, 8, 16, 32, … (or, in the binary numeral system, 1, 10, 100, 1000, 10000, 100000, … ) is important in number theory. Book IX, Proposition 36 of Elements proves that if the sum of the first n terms of this progression is a prime number (means, a Mersenne prime mentioned above), then this sum times the nth term is a perfect number. For example, the sum of the first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is a prime number. The sum 31 multiplied by 16 (the 5th term in the series) equals 496, which is a perfect number.
Book IX, Proposition 35, proves that in a geometric series if the first term is subtracted from the second and last term in the sequence, then as the excess of the second is to the first—so is the excess of the last to all those before it. (This is a restatement of our formula for geometric series from above.) Applying this to the geometric progression 31, 62, 124, 248, 496 (which results from 1, 2, 4, 8, 16 by multiplying all terms by 31), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31, 62, 124, 248. Therefore, the numbers 1, 2, 4, 8, 16, 31, 62, 124 and 248 add up to 496 and further these are all the numbers that divide 496. For suppose that p divides 496 and it is not amongst these numbers. Assume p q is equal to 16 × 31, or 31 is to q as p is to 16. Now p cannot divide 16 or it would be amongst the numbers 1, 2, 4, 8 or 16. Therefore, 31 cannot divide q. And since 31 does not divide q and q measures 496, the fundamental theorem of arithmetic implies that q must divide 16 and be amongst the numbers 1, 2, 4, 8 or 16. Let q be 4, then p must be 124, which is impossible since by hypothesis p is not amongst the numbers 1, 2, 4, 8, 16, 31, 62, 124 or 248.
PRIME NUMBERS EXTREMELY IMPORTANT IN NUMBER THEORY
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:
Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n2 + 1? (sequence A002496 in the OEIS).
As of 2017, all four problems are unresolved.
FAMOUS GOLDBACH CONJECTURE ONE OF LAUNDAUS FOUR PROBLEMS IS THAT "ALL EVEN NUMBERS GREATER THAN FOUR ARE GOLDBACH NUMBERS" FOUR
A Goldbach number is a positive integer that can be expressed as the sum of two odd primes.[4] Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.
NOTICE HOW THE FOURTH PRIME QUADRUPLET IS DIFFERENT THE FIFTH IS ULTRA TRANSCENDENT
A prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form {p, p+2, p+6, p+8}.[1] This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
Contents [hide]
1 Prime quadruplets
2 Prime quintuplets
3 Prime sextuplets
4 References
Prime quadruplets[edit]
The first eight prime quadruplets are:
{5, 7, 11, 13}, {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, {1481, 1483, 1487, 1489}, {1871, 1873, 1877, 1879}, {2081, 2083, 2087, 2089} (sequence A007530 in the OEIS)
All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} for some integer n. (This structure is necessary to ensure that none of the four primes is divisible by 2, 3 or 5). A prime quadruplet of this form is also called a prime decade.
A prime quadruplet contains two pairs of twin primes or can be described as having two overlapping prime triplets.
It is not known if there are infinitely many prime quadruplets. A proof that there are infinitely many would imply the twin prime conjecture, but it is consistent with current knowledge that there may be infinitely many pairs of twin primes and only finitely many prime quadruplets. The number of prime quadruplets with n digits in base 10 for n = 2, 3, 4, ... is 1, 3, 7, 26, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 (sequence A120120 in the OEIS).
As of March 2016 the largest known prime quadruplet has 5003 digits.[2] It starts with p = 4122429552750669 * 216567  1, found by Peter Kaiser.
The constant representing the sum of the reciprocals of all prime quadruplets, Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets:
B
4
=
(
1
5
+
1
7
+
1
11
+
1
13
)
+
(
1
11
+
1
13
+
1
17
+
1
19
)
+
(
1
101
+
1
103
+
1
107
+
1
109
)
+
⋯B_{4}=\left({\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}\right)+\left({\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+{\frac {1}{19}}\right)+\left({\frac {1}{101}}+{\frac {1}{103}}+{\frac {1}{107}}+{\frac {1}{109}}\right)+\cdots
with value:
B4 = 0.87058 83800 ± 0.00000 00005.
This constant should not be confused with the Brun's constant for cousin primes, prime pairs of the form (p, p + 4), which is also written as B4.
The prime quadruplet {11, 13, 17, 19} is alleged to appear on the Ishango bone although this is disputed.
Excluding the first prime quadruplet, the shortest possible distance between two quadruplets {p, p+2, p+6, p+8} and {q, q+2, q+6, q+8} is q  p = 30. The first occurrences of this are for p = 1006301, 2594951, 3919211, 9600551, 10531061, ... (OEIS A059925).
DIFFER BY FOUR
In mathematics, cousin primes are prime numbers that differ by four.[
Sexy prime quadruplets[edit]
Sexy prime quadruplets (p, p + 6, p + 12, p + 18) can only begin with primes ending in a 1 in their decimal representation (except for the quadruplet with p = 5). The sexy prime quadruplets below 1000 are (OEIS A023271, OEIS A046122, OEIS A046123, OEIS A046124):
(5,11,17,23), (11,17,23,29), (41,47,53,59), (61,67,73,79), (251,257,263,269), (601,607,613,619), (641,647,653,659).
In November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits:
p = 411784973 · 2347# + 3301.[3]
In September 2010 Ken Davis announced a 1004digit quadruplet with p = 23333 + 1582534968299.[4]
FOUR BANDS QUADRANTS
The dark blue band separates pairs of Gaussian prime numbers whose minimax path length is 2 or more.
These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least FOUR primes between the squares of consecutive primes greater than 2.
FOUR PRIMES
In number theory, Brocard's conjecture is a conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, for n > 1, where pn is the nth prime number.[1] It is widely believed that this conjecture is true. However, it remains unproven as of 2016.
16 BOOLEAN FUNCTIONS 16 SQUARES QMR
16 SQUARES QMR
Spatial symmetries reduce the 16 twovariable functions to four distinguishable forms
Here are the sixteen Boolean cubes for two variables.
There are four varieties and their reflections. Reflections of the zero and oneface varieties are also rotations. For comparison here are the sixteen Venn diagrams for two variables.
Again the structural emphases change. bounding does not align with reflections, parens nesting does not align with faces or spaces. Each of these different organizational approaches highlights different aspects of the same abstract Boolean twovariable structure. None correspond to the group theoretic organization of twovariable Boolean functions as a complemented, distributed lattice in 4space.
You Can Make Any Number Out of Four 4s Because Math Is Amazing
Up to infinity!
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Here's a fun math puzzle to brighten your day. Say you've got four 4s—4 4 4 4—and you're allowed to place any normal math symbols around them. How many different numbers can you make?
According to the fantastic YouTube channel Numberphile, you can make all of them. Really. You just have to have some fun and get creative.
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When you first start out, the problem seems pretty simple. So, for example, 4  4 + 4  4 = 0. To make 1, you can do 4 / 4 + 4  4. In fact, you can make all the numbers up to about 20 using only the basic arithmetic operations of addition, subtraction, multiplication, and division.
But soon that's not enough. To start reaching bigger numbers, the video explains, you must pull in more sophisticated operations like square roots, exponents, factorials (4!, or 4 x 3 x 2 x 1), and concatenation (basically, turning 4 and 4 into 44).
This way will get you up to about 100, but going really big requires one more tool: the log. Here, things start to get more complicated, but the back half of the video explains how incorporating the log into the four 4s problem allows you to work out ways to make every number up to infinity.
The solution, involving multiple logs and square roots, broke open this puzzle in the 1930s. But it's still a fun exercise if you ever find yourself in possession of a lot of free time and a big piece of paper.
THE FIRST THREE MESSAGES SOLVED THE FOURTH NOT THE FOURTH ALWAYS DIFFERENT
Kryptos is a sculpture by the American artist Jim Sanborn located on the grounds of the Central Intelligence Agency (CIA) in Langley, Virginia. Since its dedication on November 3, 1990, there has been much speculation about the meaning of the four encrypted messages it bears. Of the four messages, the first three have been solved, while the fourth message remains as one of the most famous unsolved codes in the world. The sculpture continues to be of interest to cryptanalysts, both amateur and professional, who are attempting to decipher the fourth section. The artist has so far given two clues to this section.
AGAIN ACCORDING TO MATHEMATICIANS THE FIFTH POSTULATE OF EUCLID WAS WRONG. ALL GEOMETRY WAS FOUNDED UPON EUCLIDS POSTULATES AND THEY WERE THE FOUNDATION OF ONE OF THE MOST POPULAR BOOKS OF ALL TIME BEHIND THE BIBLE BUT THE FIFTH WAS WRONG. SOME MATHEMATICIANS THINK HE PURPOSEFULLY MADE THE FIFTH WRONG SO THAT NEW GEOMETRIES COULD BE FOUND FROM IT THE FIRST THREE ARE SIMILAR CALLED COMPASS AND STRAIGHT EDGE POSTULATES THE FOURTH IS DIFFERENT CALLED A POSTULATE FROM EXPERIENCE. THE FIFTH IS QUESTIONABLE AND ULTRA TRANSCENDENT (TWO LINES GOING FOREVER) AND IN FACT WRONG
Euclid's Fifth Postulate
Besides 23 definitions and several implicit assumptions, Euclid derived much of the planar geometry from five postulates.
A straight line may be drawn between any two points.
A piece of straight line may be extended indefinitely.
A circle may be drawn with any given radius and an arbitrary center.
All right angles are equal.
If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.
The fifth postulate refers to the diagram on the right. If the sum of two angles A and B formed by a line L and another two lines L1 and L2 sum up to less than two right angles then lines L1 and L2 meet on the side of angles A and B if continued indefinitely.
Postulates 1 and 3 set up the "ruler and compass" framework that was a standard for geometric constructions until the middle of the 19th century. They may be said to be based on man's practical experience. The second postulate gives an expression to a commonly held belief that straight lines may not terminate and that the space is unbounded. By the Definition 10, an angle is right if it equals its adjacent angle. Thus the fourth postulate asserts homogeneity of the plane: in whatever directions and through whatever point two perpendicular lines are drawn, the angle they form is one and the same and is called right. We may think of the fourth postulate as having been justified by the everyday experience acquired by man in the finite, inhabited portion of the universe which is our world and extrapolated (much as the Postulate 2) to that part of the world whose existence (and infinite expense) we sense and believe in.
Elaborateness of the fifth postulate stands in a stark contrast to the simplicity of the first four. Euclid himself, probably, had mixed feelings about it as he did not make use of it until Proposition I.29. The postulate looks more like another proposition than a basic truth. Here's, for example, Proposition I.27 which, combined with Proposition I.13, claims that, if (in the diagram above) angles A and B sum up to two right angles, then the lines L1 and L2 are parallel.)
If a straight line crossing two straight lines makes the alternate angles equal to one another, the straight lines will be parallel to one another.
(By Definition 23, two straight line in the same plane are parallel if they do not meet even when produced indefinitely in both directions.)
Proposition I.17 is actually a converse of the fifth postulate:
In any triangle two angles taken together in any manner are less than two right angles.
The postulate (also known as the Parallel Postulate) attracted immediate attention. The commentator Proclus (c. 410485) tells us that the postulate was attacked from the very start. He wrote, "This postulate ought even to be struck out of Postulates altogether; for it is a theorem..." Now we know that it is impossible to derive the Parallel Postulate from the first four. The numerous (and failed) attempts to do that gave rise to a slew of statements equivalent to the postulate itself. Several are cited by S.Brodie. Following are a few more:
The exists a pair of similar noncongruent triangles.
There exists a pair of straight lines everywhere equidistant from one another.
For any three noncollinear points, there exists a circle passing through them.
If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
If a straight line intersects one of two parallels it will intersect the other.
Straight lines parallel to a third line are parallel to each other.
Two straight lines that intersect one another cannot be parallel to a third line.
There is no upper limit to the area of a triangle.
The last one seems especially intuitive. The reverse holds in nonEuclidean geometries of Lobachevsky and Riemann. Coxeter mentions the fact that Lewis Carroll could not accept this assertion and considered it as a proof of the contradictory nature of nonEuclidean geometries.
In one of his books R. Smullyan tells of an experiment he ran in a remedial geometry class. He drew the famous Pythagoras' diagram and asked whether the two small squares are bigger or smaller than the square on the hypotenuse. Half the class thought the sum was bigger, another half thought it was smaller. By all accounts, the Pythagorean Theorem is far from obvious. It is amazing that the Parallel Postulate, being equivalent to such intuitive statements as 1 and 8 above, is also equivalent to the Pythagorean Theorem.
NonEuclidean Geometries, Introduction
The Fifth Postulate
The Fifth Postulate is Equivalent to the Pythagorean Theorem
The Fifth Postulate, Attempts to Prove.
Similarity and the Parallel Postulate
NonEuclidean Geometries, Drama of the Discovery.
NonEuclidean Geometries, As Good As Might Be.
The ManyFaced Geometry
The Exterior Angle Theorem  an appreciation
Angles in Triangle Add to 180°
PROCLUS AN ANCIENT MATHEMATICIAN IMMEDIATELY UNDERSTOOD THAT EUCLIDS FIFTH POSTULATE WAS QUESTIONABLE AND NOT NEEDED (PEOPLE ALSO RECOGNIZED THE FOURTH WAS DIFFERENT) AND PROCLUS SAID THAT EUCLIDS FIFTH SHOULD BE TAKEN OUT I LISTENED TO A TEACHING COMPANY COURSE ON THIS AND THE PROFESSOR SUGGESTED EUCLID INTENTIONALLY MADE THE FIFTH FALSE SO THAT NEW GEOMETRIES WOULD BE MADE FROM IT
I DISCUSSED EUCLID HAD FIVE POSTULATES AND FIVE AXIOMS THAT WERE THE BASIS OF HIS BOOK EUCLIDS ELEMENTS THE MOST POPULAR BOOK BEHIND THE BIBLE BUT IN BOTHE CASES THE FOURTH WAS DIFFERENT AND THE FIFTH WAS QUESTIONABLE THE FIFTH POSTULATE WAS PROVEN WRONG ALSO THE FIFTH AXIOM THE WHOLE IS GREATER THAN THE SUM OF ITS PARTS WAS PROVEN WRONG BY QUANTUM MECHANICS AND FRACTALS FOURTH IS ALWAYS DIFFERENT FIFTH ALWAYS QUESTIONABLE
AGAIN THE FIRST THREE SQUARES OF THE QUADRANT MODEL ARE ALWAYS SIMILAR THE FOURTH IS ALWAYS DIFFERENT THE FIFTH ALWAYS QUESTIONABLE. THE ANCIENT PHILOSOPHER PROCLUS DID NOT JUST HAVE A PROBLEM WITH THE FIFTH POSTULATE WHICH HE THOUGHT WAS FALSE (MORE MODERN MATHEMATICIANS PROVED IT WAS FALSE). HE ALSO HAD A PROBLEM WITH THE FOURTH POSTULATE THINKING IT WAS NOT A POSTULATE IT WAS A THEOREM HE NOTED THAT "THE FIRST THREE ARE SIMILAR THE FOURTH IS DIFFERENT". (THAT IS THE QUADRANT PATTERN). (ACTUALLY THE FOURTH POSTULATE IS ABOUT RIGHT ANGLES WHICH IS INTERESTINGLY THE POSTULATE OF THE QUADRANT)
What’s the Deal with Euclid’s Fourth Postulate?
In February, I wrote about Euclid’s parallel postulate, the black sheep of the big, happy family of definitions, postulates, and axioms that make up the foundations of Euclidean geometry.
By Evelyn Lamb on April 21, 2014
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An illustration from Oliver Byrne's 1847 edition of Euclid's Elements. Euclid's fourth postulate states that all the right angles in this diagram are congruent. Image: Public domain, via Wikimedia Commons.
In February, I wrote about Euclid's parallel postulate, the black sheep of the big, happy family of definitions, postulates, and axioms that make up the foundations of Euclidean geometry. I included the text of the five postulates, from Thomas Heath's translation of Euclid's Elements:
"Let the following be postulated:
1) To draw a straight line from any point to any point.
2) To produce a finite straight line continuously in a straight line.
3) To describe a circle with any centre and distance.
4) That all right angles are equal to one another.
5) That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
The first three postulates have a similar feel to them: we're defining a few things we can do when constructing figures to use in proofs. Those postulates say that if we want to, we can connect two points by a line, draw lines that continue indefinitely, and draw circles wherever we want and of whatever size we want. Fair enough.
But why the heck do we need a postulate that says that all right angles are equal to one another? You probably remember learning in a middle or high school geometry class that right angles are 90 degree angles, and two angles are congruent if they have the same degree measure. We don't need a whole postulate that says this. It's just part of the way we define angles. Why not a postulate that says that all 45 degree angles are equal to one another? Or all 12 degree angles? The fourth postulate seems a bit bizarre. But Euclid knew what he was doing, so there must be a reason for this postulate.
To understand what it would have meant to Euclid, we need to go back and look at Euclid's treatment of angles. In the beginning of the book, he includes a few definitions relating to angles. Definition 8 states, "A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line." Definition 10 says, "When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands." Definitions 11 and 12 are for obtuse and acute angles, which are defined as being greater than or less than a right angle, respectively. Intuitively, we can all imagine what greater and less mean for angles: angle A is greater than angle B if it's "more open" than angle B. It's less if it's "more closed." We know it when we see it.
But Euclid never tells us exactly how to compare two angles. He never discusses degrees, radians, or how to measure an angle using a protractor. Contemporary Greek astronomers and mathematicians used degrees, and Euclid was probably aware of them, but he doesn't use them in the Elements. Without a way to measure angles, what might Euclid have meant by angles being equal?
The axioms might shed some light. Again, from Heath's translation:
"1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part."
On its face, Axiom 4 seems to say that a thing is equal to itself, but it looks like Euclid also used it justify the use of a technique called superposition to prove that things are congruent. Basically, superposition says that if two objects (angles, line segments, polygons, etc.) can be lined up so that all their corresponding parts are exactly on top of each other, then the objects are congruent.
For example, in Book 1, Proposition 4, Euclid uses superposition to prove that sides and angles are congruent. Proposition 4 is the theorem that sideangleside is a way to prove that two triangles are congruent. In Oliver Byrne's translation, which I think is a bit more poetic on this point than Heath's, the proof starts, "Let the two triangles be conceived, to be so placed, that the vertex of the one of the equal angles shall fall upon that of the other…" In other words, Euclid seems to describe physically placing one triangle on top of the other one. When he does this, he shows that all their parts line up and concludes that they are congruent.
Now it makes a little more sense that Euclid would want a postulate that states that right angles are congruent. We need to know that creating a pair of right angles on one piece of paper is the same as creating them on another piece of paper. We need to be able to put the pieces of paper on top of each other and have the angles line up exactly. In effect, the fourth postulate establishes the right angle as a unit of measurement for all angles. Although Euclid never uses degrees or radians, he sometimes describes angles as being the size of some number of right angles. In this light, Euclid's fourth postulate doesn't seem quite so bizarre.
But if you are a bit put off by the fourth postulate, you are not alone. Proclus, a 5th century CE Greek mathematician who wrote an influential commentary on the Elements, thought that the fourth postulate should be a theorem and provided a "proof" of it in his commentary. But his proof relies on assuming that angles "look" the same wherever we are in space, a property that Heath referred to in his 1908 commentary as the homogeneity of space. Basically, Heath states that Proclus's proof replaces the fourth postulate with a different, unstated, postulate.
Heath writes,
"While this Postulate asserts the essential truth that a right angle is a determinate magnitude so that it really serves as an invariable standard by which other (acute and obtuse) angles may be measured, much more than this is implied, as will easily be seen from the following consideration. If the statement is to be proved, it can only be proved by the method of applying one pair of right angles to another and so arguing their equality. But this method would not be valid unless on the assumption of the invariability of figures, which would therefore have to be asserted as an antecedent postulate. Euclid preferred to assert as a postulate, directly, the fact that all right angles are equal; and hence his postulate must be taken as equivalent to the principle of invariability of figures or its equivalent, the homogeneity of space."
Even if we do want accept the postulate without proof, Proclus would prefer that we call it an axiom, rather than a postulate. He thought the postulates should be about construction—something we do—while the axioms should be selfevident notions that we observe. (The axioms are sometimes called "common notions.") But Heath sees a good reason that the fourth postulate should be placed where it is.
"As to the raison d'être and the place of Post. 4 one thing is quite certain. It was essential from Euclid's point of view that it should come before Post. 5, since the condition in the latter than a certain pair of angles are together less than two right angles would be useless unless it were first made clear that right angles are angles of determinate and invariable magnitude."
As a side note, I found Heath's interpretation of the difference between axioms, which he calls common notions, and postulates interesting:
"As regards the postulates we may imagine [Euclid] saying: 'Besides the common notions there are a few other things which I must assume without proof, but which differ from the common notions in that they are not selfevident. The learner may or may not be disposed to agree to them; but he must accept them at the outset on the superior authority of his teacher, and must be left to convince himself of their truth in the course of the investigation which follows.'"
In 1899, the German mathematician David Hilbert published a book that sought to put Euclidean geometry on more solid axiomatic footing, as the standards and style of mathematical proof had changed quite a bit in the two millennia since Euclid's life. Hilbert uses a different set of definitions and axioms, and in his formulation, the equality of right angles is a theorem, not an assumption. But with Euclid's original set of postulates and axioms, the fourth postulate is necessary. In effect it establishes the right angle as the universal ruler for angles. It's not what we're used to now, but it works just as well as degrees or radians.
To explore Euclid's Elements further, check out David E. Joyce's page. You can read the commentaries of Proclus and Heath on Google Books, and if you just can't get enough axiomatic geometry, Hilbert's Foundations of Geometry (pdf) is on Project Gutenberg. I'd like to thank Colin McKinney of Wabash College for his help with some of the details of this post. All errors are mine.
Forty is a composite number, an octagonal number,[1] and as the sum of the first four pentagonal numbers, it is a pentagonal pyramidal number.[2]
MANDELBROT SET IS THE MOST FAMOUS FRACTAL IMAGE AND THE IMAGE LEAD TO THE STUDY OF FRACTALS AND CHAOS THEORY IT HAS 16 RAYS THE GUY ON THE SACRED SWASTIKA WEBSITE DID NOT KNOW ABOUT THE QUADRANT MODEL HE DID NOT KNOW THAT 16 SQUARES IS THE FORM OF EXISTENCE BUT HE DID NOTICE THAT THEMANDELBROT SET HAS 16 RAYS AND SO DOES THE COMPASS AT THE VATICAN
The Vatican SS, a.k.a. the Sweet Sixteen – note how both St. Peter’s compass rose indicates 16 winds and the Mandelbrot (below) has 16 rays converging/emanating out from the center, indicated by the red dots
THE FOURFOLD AUM IS THE SWASTIKA I MENTIONED THIS BEFORE AND POSTED ARTICLES ON IT
Proof that OM is equivalent to Swastika – author Rim sim
According to Yajurveda Swastika is the symbolic representation of om in Hinduism
Swastika – Wikipedia, the free encyclopedia
Jacobi's foursquare theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares.
FAMOUS RHIND PAPYRUS FOUR SECTIONS FOUR MATHEMATICAL PROBLEMS
This table summarizes the content of the Rhind Papyrus by means of a concise modern paraphrase. It is based upon the twovolume exposition of the papyrus which was published by Arnold Buffum Chace in 1927, and in 1929.[6] In general, the papyrus consists of four sections: a title page, the 2/n table, a tiny "19/10 table", and 91 problems, or "numbers". The latter are numbered from 1 through 87 and include four mathematical items which have been designated by moderns as problems 7B, 59B, 61B, and 82B. Numbers 8587, meanwhile, are not mathematical items forming part of the body of the document, but instead are respectively: a small phrase ending the document, a piece of "scrappaper" used to hold the document together (having already contained unrelated writing), and a historical note which is thought to describe a time period shortly after the completion of the body of the papyrus. These three latter items are written on disparate areas of the papyrus' verso (back side) , far away from the mathematical content. Chace therefore differentiates them by styling them as numbers as opposed to problems, like the other 88 numbered items.
RHIND PAPYRUS HAS FOUR SECTIONS PROBLEM 47 HAS A LOT OF QUADRUPLES
Problem 47 is a table with fractional equalities which represent the ten situations where the physical volume quantity of "100 quadruple heqats" is divided by each of the multiples of ten, from ten through one hundred. The quotients are expressed in terms of Horus eye fractions, sometimes also using a much smaller unit of volume known as a "quadruple ro". The quadruple heqat and the quadruple ro are units of volume derived from the simpler heqat and ro, such that these four units of volume satisfy the following relationships: 1 quadruple heqat = 4 heqat = 1280 ro = 320 quadruple ro. Thus,
100/10 quadruple heqat = 10 quadruple heqat
100/20 quadruple heqat = 5 quadruple heqat
100/30 quadruple heqat = (3 + 1/4 + 1/16 + 1/64) quadruple heqat + (1 + 2/3) quadruple ro
100/40 quadruple heqat = (2 + 1/2) quadruple heqat
100/50 quadruple heqat = 2 quadruple heqat
100/60 quadruple heqat = (1 + 1/2 + 1/8 + 1/32) quadruple heqat + (3 + 1/3) quadruple ro
100/70 quadruple heqat = (1 + 1/4 + 1/8 + 1/32 + 1/64) quadruple heqat + (2 + 1/14 + 1/21 + 1/42) quadruple ro
100/80 quadruple heqat = (1 + 1/4) quadruple heqat
100/90 quadruple heqat = (1 + 1/16 + 1/32 + 1/64) quadruple heqat + (1/2 + 1/18) quadruple ro
100/100 quadruple heqat = 1 quadruple heqat [1]
THERE ARE FOUR DIVISION ALGEBRAS
Four division algebras R, C, H, O
FOUR FAMILIES OF SEMISIMPLE LIE ALGEBRAS
Examples of semisimple Lie algebras, with notation coming from classification by Dynkin diagrams, are:
A
n
:
A_{n}:
s
l
n
+
1
{\mathfrak {sl}}_{{n+1}}, the special linear Lie algebra.
B
n
:
B_{n}:
s
o
2
n
+
1
{\mathfrak {so}}_{{2n+1}}, the odddimensional special orthogonal Lie algebra.
C
n
:
C_{n}:
s
p
2
n
{\mathfrak {sp}}_{{2n}}, the symplectic Lie algebra.
D
n
:
D_{n}:
s
o
2
n
{\mathfrak {so}}_{{2n}}, the evendimensional special orthogonal Lie algebra.
These Lie algebras are numbered so that n is the rank. Except certain exceptions in low dimensions, many of these are simple Lie algebras, which are a fortiori semisimple. These four families, together with five exceptions (E6, E7, E8, F4, and G2), are in fact the only simple Lie algebras over the complex numbers.
Classification[edit]
See also: Root system
The simple Lie algebras are classified by the connected Dynkin diagrams.
Every semisimple Lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple Lie algebras (by definition), and the finitedimensional simple Lie algebras fall in four families – An, Bn, Cn, and Dn – with five exceptions E6, E7, E8, F4, and G2. Simple Lie algebras are classified by the connected Dynkin diagrams, shown on the right, while semisimple Lie algebras correspond to not necessarily connected Dynkin diagrams, where each component of the diagram corresponds to a summand of the decomposition of the semisimple Lie algebra into simple Lie algebras.
The classification proceeds by considering a Cartan subalgebra (maximal abelian Lie algebra; corresponds to a maximal torus in a Lie group) and the adjoint action of the Lie algebra on this subalgebra. The root system of the action then both determines the original Lie algebra and must have a very constrained form, which can be classified by the Dynkin diagrams.
The classification is widely considered one of the most elegant results in mathematics – a brief list of axioms yields, via a relatively short proof, a complete but nontrivial classification with surprising structure. This should be compared to the classification of finite simple groups, which is significantly more complicated.
The enumeration of the four families is nonredundant and consists only of simple algebras if
n
≥
1
n\geq 1 for An,
n
≥
2
n\geq 2 for Bn,
n
≥
3
n\geq 3 for Cn, and
n
≥
4
n\geq 4 for Dn. If one starts numbering lower, the enumeration is redundant, and one has exceptional isomorphisms between simple Lie algebras, which are reflected in isomorphisms of Dynkin diagrams; the En can also be extended down, but below E6 are isomorphic to other, nonexceptional algebras.
THERE ARE FOUR CLASSICAL LIE ALGEBRAS THE ULTRA TRANSCENDENT FIFTH STRING THEORY THAT IS BEYOND THE TRANSCENDENT FOURTH IS NAMED AFTER E8 WHICH IS THE LARGEST OF THE EXCEPTIONAL LIE ALGEBRAS WHICH ARE NOT THE FOUR CLASSICAL LIE ALGEBRAS
Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" An, Bn, Cn and Dn, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E8 is the largest of these.
QUAD IS FOUR
The Treatise on Quadrature of Fermat (c. 1659), be sides containing the first known proof of the computation of the area under a higher parabola, x+m/n dx, or under a higher hy perbola, x−m/n dx— with the appropriate limits of integration in each case—, has a second part which was not understood by Fer mat’s contemporaries. This second part of the Treatise is obscure and difficult to rea... See More
THE FAMOUS BABYLONIAN CLAY TABLE HAS A QUADRANT ON IT
Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...[1]
FOUR TYPES FRICTION
FOUR COLOR THEOREM THE FOURTH IS DIFFERENT FIVE COLOR THEOREM WAS PROVEN FOUR COLOR THEOREM NEEDED A COMPUTER TO PROVE
The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps. While trying to color a map of the counties of England, Francis Guthrie postulated the four color conjecture, noting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. Guthrie’s brother passed on the question to his mathematics teacher Augustus de Morgan at University College, who mentioned it in a letter to William Hamilton in 1852. Arthur Cayley raised the problem at a meeting of the London Mathematical Society in 1879. The same year, Alfred Kempe published a paper that claimed to establish the result, and for a decade the four color problem was considered solved. For his accomplishment Kempe was elected a Fellow of the Royal Society and later President of the London Mathematical Society.[1]
In 1890, Heawood pointed out that Kempe’s argument was wrong. However, in that paper he proved the five color theorem, saying that every planar map can be colored with no more than five colors, using ideas of Kempe. In the following century, a vast amount of work and theories were developed to reduce the number of colors to four, until the four color theorem was finally proved in 1976 by Kenneth Appel and Wolfgang Haken. The proof went back to the ideas of Heawood and Kempe and largely disregarded the intervening developments.[2] The proof of the four color theorem is also noteworthy for being the first major computeraided proof.
16 TIMES 9 IS 144
The Number 144: Properties and Meanings
Prime Factors of 144=2x2x2x2x3x3.
144 can be Partitioned 73 times with each term no larger than 2.
144 can be Partitioned 1801 times with each term no larger than 3.
144 is a Fibonacci Number.
144 is a Square Number(12x12).
144 is a 49gonal Number.
144 is an Abundant Number.
The Rosicrucians think that there exists in the Universe 144 kinds of different atom, even if the science has not still discovered them.
Bruce Cathie speculated that all the mathematical evidence so far indicates that the maximum number of elements (matter) to be found in the universe will be 144.
The Graal comprised 144 facets.
There are 144 Kriyas or breathing techniques in Kriya Yoga.
144 is a dozen dozens, or one gross.
144 is the measurement, in cubits, of the wall of New Jerusalem shown by the seventh angel (Holy Bible, Revelations 21:17).
The Year 144 AD
"The Holy Grail was removed from the planet 144 years after the birth of Christ". (Channeled Message?)
In the year 144 AD the church in Rome declared Marcion's views heretical.
In the year 144 AD a Chinese Buddhist saint, Fa Hsien, was caught in a storm and landed in JavaDwipa, or Java Island, where he stayed for five months
In the year 144 AD the Patriarch of Constantinople changed from Patriarch Polycarpus II to Patriarch Athendodorus.
In the year 144 AD the construction of the Antonine Wall was completed.
all quadrant numbers
https://en.wikipedia.org/wi…/Decimal64_floatingpoint_format
Floating point precisions
IEEE 754
16bit: Half (binary16)
32bit: Single (binary32), decimal32
64bit: Double (binary64), decimal64
128bit: Quadruple (binary128), decimal128
256bit: Octuple (binary256)
Extended precision formats
192 IS 16 TIMES 12  256 16 times 4 FOUR BY FOUR COLUMNS QUADRANT MODELS IT WORKS ON 4096 IS 16 TIMES 256
AES operates on a 4 × 4 columnmajor order matrix of bytes, termed the state, although some versions of Rijndael have a larger block size and have additional columns in the state. Most AES calculations are done in a particular finite field.
For instance, if there are 16 bytes,
The ShiftRows step operates on the rows of the state; it cyclically shifts the bytes in each row by a certain offset. For AES, the first row is left unchanged. Each byte of the second row is shifted one to the left. Similarly, the third and fourth rows are shifted by offsets of two and three respectively. For blocks of sizes 128 bits and 192 bits, the shifting pattern is the same. Row
n
n is shifted left circular by
n
−
1
n1 bytes. In this way, each column of the output state of the ShiftRows step is composed of bytes from each column of the input state. (Rijndael variants with a larger block size have slightly different offsets). For a 256bit block, the first row is unchanged and the shifting for the second, third and fourth row is 1 byte, 3 bytes and 4 bytes respectively—this change only applies for the Rijndael cipher when used with a 256bit block, as AES does not use 256bit blocks. The importance of this step is to avoid the columns being linearly independent, in which case, AES degenerates into four independent block ciphers.
On systems with 32bit or larger words, it is possible to speed up execution of this cipher by combining the SubBytes and ShiftRows steps with the MixColumns step by transforming them into a sequence of table lookups. This requires four 256entry 32bit tables, and utilizes a total of four kilobytes (4096 bytes) of memory—one kilobyte for each table. A round can then be done with 16 table lookups and 12 32bit exclusiveor operations, followed by four 32bit exclusiveor operations in the AddRoundKey step.[12]
On systems with 32bit or larger words, it is possible to speed up execution of this cipher by combining the SubBytes and ShiftRows steps with the MixColumns step by transforming them into a sequence of table lookups. This requires four 256entry 32bit tables, and utilizes a total of four kilobytes (4096 bytes) of memory—one kilobyte for each table. A round can then be done with 16 table lookups and 12 32bit exclusiveor operations, followed by four 32bit exclusiveor operations in the AddRoundKey step.[12]
If the resulting fourkilobyte table size is too large for a given target platform, the table lookup operation can be performed with a single 256entry 32bit (i.e. 1 kilobyte) table by the use of circular rotates.
Using a byteoriented approach, it is possible to combine the SubBytes, ShiftRows, and MixColumns steps into a single round operation.[13]
QUADRANT NUMBERS
The register width of a processor determines the range of values that can be represented. Typical binary register widths for unsigned integers include:
8 bits: maximum representable value 28 − 1 = 255
16 bits: maximum representable value 216 − 1 = 65,535
32 bits: maximum representable value 232 − 1 = 4,294,967,295 (the most common width for personal computers as of 2005),
64 bits: maximum representable value 264 − 1 = 18,446,744,073,709,551,615 (the most common width for personal computer CPUs, as of 2017),
128 bits: maximum representable value 2128 − 1 = 340,282,366,920,938,463,463,374,607,431,768,211,455
128 BIT ERA IS 16 TIMES 8
In the history of video games, the sixthgeneration era (sometimes referred to as the 128bit era; see "Bits and system power" below) refers to the computer and video games, video game consoles, and video game handhelds available at the turn of the 21st century which was from 1998 to 2008. Platforms of the sixth generation include the Sega Dreamcast, Sony PlayStation 2, Nintendo GameCube, and Microsoft Xbox. This era began on November 27, 1998 with the Japanese release of the Dreamcast, and it was joined by the PlayStation 2 in March 2000 and the GameCube and Xbox in 2001. The Dreamcast was discontinued in 2001, the GameCube in 2007, Xbox in 2009 and PlayStation 2 in 2013. Though the seventh generation of consoles started in November 2005 with the launch of the Xbox 360, the sixth generation did not end until January 2013, when Sony announced that the PlayStation 2 had been discontinued worldwide.[1]
QUADRANT NUMBERS 1024 IS 16 TIMES 64
–Noll–Vo_hash_function
The current versions are FNV1 and FNV1a, which supply a means of creating nonzero FNV offset basis. FNV currently comes in 32, 64, 128, 256, 512, and 1024bit flavors. For pure FNV implementations, this is determined solely by the availability of FNV primes for the desired bit length; however, the FNV webpage discusses methods of adapting one of the above versions to a smaller length that may or may not be a power of two.[2][3]
224 IS 16 TIMES 14 16 TIMES 76 IS 1216 16 TIMES 24 IS 384 16 TIMES 12 IS 192 16 TIMES 10 IS 160 and 16 TIMES 32 IS 512 ALL QUADRANT NUMBERS PLUS THE USUALS
Unkeyed cryptographic hash functions[edit]
Main article: Cryptographic hash function
See also: Comparison of cryptographic hash functions
Name Length Type
BLAKE256 256 bits HAIFA structure[6]
BLAKE512 512 bits HAIFA structure[6]
BLAKE2s Up to 256 bits HAIFA structure[6]
BLAKE2b Up to 512 bits HAIFA structure[6]
ECOH 224 to 512 bits hash
FSB 160 to 512 bits hash
GOST 256 bits hash
Grøstl Up to 512 bits hash
HAS160 160 bits hash
HAVAL 128 to 256 bits hash
JH 224 to 512 bits hash
MD2 128 bits hash
MD4 128 bits hash
MD5 128 bits Merkle–Damgård construction
MD6 Up to 512 bits Merkle tree NLFSR (it is also a keyed hash function)
RadioGatún Up to 1216 bits hash
RIPEMD 128 bits hash
RIPEMD128 128 bits hash
RIPEMD160 160 bits hash
RIPEMD320 320 bits hash
SHA1 160 bits Merkle–Damgård construction
SHA224 224 bits Merkle–Damgård construction
SHA256 256 bits Merkle–Damgård construction
SHA384 384 bits Merkle–Damgård construction
SHA512 512 bits Merkle–Damgård construction
SHA3 (originally known as Keccak) arbitrary Sponge function
Skein arbitrary Unique Block Iteration
Snefru 128 or 256 bits hash
Spectral Hash 512 bits Wide pipe Merkle–Damgård construction
Streebog 256 or 512 bits Merkle–Damgård construction
SWIFFT 512 bits hash
Tiger 192 bits Merkle–Damgård construction
Whirlpool 512 bits hash
FOURIER IS EXTREMELY IMPORTANT IN TONS OF CLASSES I WENT TO AT UCSD IT WAS FOURIER THIS AND FOURIER THAT I POSTED A LONG TIME AGO THE DIAGRAMS SHOWING THE FOURIER FOUR PARTS AND ALL OF THAT IT IS A QUADRANT MODEL HOPEFULLY I CAN GO BACK INTO MY TIMELINE AND FIND IT BUT HIS BOOK IS CALLED "THEORY FO THE FOUR MOVEMENTS"
Fourier, Charles. Théorie des quatre mouvements et des destinées générales (Theory of the four movements and the general destinies), appeared anonymously in Lyon in 1808.
Jones, Gareth Stedman, and Ian Patterson, eds. Fourier: The Theory of the Four Movements. Cambridge Texts in the History of Political Thought. Cambridge: Cambridge UP, 1996.
THE FOUR TYPES FOURIER  I POSTED OTHER STUFF ON THIS YEARS AGO FROM WIKIPEDIA POSTING THESE QUADRANT DIAGRAMS
In particular, Table 4.1 on page 14, pictured above, shows the relationships between the continuoustime Fourier series (CTFS), discretetime Fourier series (DTFS), continuoustime Fourier transform (CTFT), and discretetime Fourier transform (DTFT). Note the similarities and differences among the four operations:
“Series”: periodic in time, discrete in frequency
“Transform”: aperiodic in time, continuous in frequency
“Continuous Time”: continuous in time, aperiodic in frequency
“Discrete Time”: discrete in time, periodic in frequency
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Merk Diezle
THE FOUR TYPES FOURIER  I POSTED OTHER STUFF ON THIS YEARS AGO FROM WIKIPEDIA POSTING THESE QUADRANT DIAGRAMS
https://stevetjoa.com/633/
In particular, Table 4.1 on page 14, pictured above, shows the relationships between the continuoustime Fourier series (CTFS), discretetime Fourier series (DTFS), continuoustime Fourier transform (CTFT), and discretetime Fourier transform (DTFT). Note the similarities and differences among the four operations:
“Series”: periodic in t...
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Merk Diezle
FOUR POINT AND 8 POINT 16 AND 64
https://en.wikipedia.org/wiki/DFT_matrix
Fourpoint[edit]
The fourpoint DFT matrix is as follows:
W
=
[
ω
0
ω
0
ω
0
ω
0
ω
0
ω
1
ω
2
ω
3
ω
0
ω
2
ω
4
ω
6
ω
0
ω
3
ω
6
ω
9
]
=
[
1 1 1 1 1 −i −1 i 1 −1 1 −1 1 i −1 −i
]
{\displaystyle W={\begin{bmatrix}\omega ^{0}&\omega ^{0}&\omega ^{0}&\omega ^{0}\\\omega ^{0}&\omega ^{1}&\omega ^{2}&\omega ^{3}\\\omega ^{0}&\omega ^{2}&\omega ^{4}&\omega ^{6}\\\omega ^{0}&\omega ^{3}&\omega ^{6}&\omega ^{9}\\\end{bmatrix}}={\begin{bmatrix}1&1&1&1\\1&i&1&i\\1&1&1&1\\1&i&1&i\end{bmatrix}}}
where
ω
=
e
−
π
i
2
=
−
i
{\displaystyle \omega =e^{{\frac {\pi i}{2}}}=i}.
Eightpoint[edit]
The first nontrivial integer power of two case is for eight points:
THERE ARE FOUR COMMON TYPES THERE ARE FOUR MAIN TYPES
There are eight standard DCT variants, of which four are common.
Illustration of the implicit even/odd extensions of DCT input data, for N=11 data points (red dots), for the four most common types of DCT (types IIV).
However, because DCTs operate on finite, discrete sequences, two issues arise that do not apply for the continuous cosine transform. First, one has to specify whether the function is even or odd at both the left and right boundaries of the domain (i.e. the minn and maxn boundaries in the definitions below, respectively). Second, one has to specify around what point the function is even or odd. In particular, consider a sequence abcd of four equally spaced data points, and say that we specify an even left boundary. There are two sensible possibilities: either the data are even about the sample a, in which case the even extension is dcbabcd, or the data are even about the point halfway between a and the previous point, in which case the even extension is dcbaabcd (a is repeated).
In other words, DCT types IIV are equivalent to realeven DFTs of even order (regardless of whether N is even or odd), since the corresponding DFT is of length 2(N−1) (for DCTI) or 4N (for DCTII/III) or 8N (for DCTIV). The four additional types of discrete cosine transform[16] correspond essentially to realeven DFTs of logically odd order, which have factors of N ± ½ in the denominators of the cosine arguments.
The figure to the adjacent shows the four stages that are involved in calculating 3D DCTII using VR DIF algorithm. The first stage is the 3D reordering using the index mapping illustrated by the above equations. The second stage is the butterfly calculation. Each butterfly calculates eight points together as shown in the figure just below., where
THERE ARE 16 POSSIBLITIES ( 16 SQUARES QMR) EIGHT OF THEM ARE THE STANDARD DCT VARIANTS THERE ARE FOUR MAIN ONES/FOUR COMMON ONES
These choices lead to all the standard variations of DCTs and also discrete sine transforms (DSTs). Each boundary can be either even or odd (2 choices per boundary) and can be symmetric about a data point or the point halfway between two data points (2 choices per boundary), for a total of 2 × 2 × 2 × 2 = 16 possibilities. Half of these possibilities, those where the left boundary is even, correspond to the 8 types of DCT; the other half are the 8 types of DST.
THE FOUR COLOR THEOREM
Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored using only four colors so that no areas of the same color touched. Kenneth Appel and Wolfgang Haken proved this in 1976.[9]
The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).[9]
I'm a paragraph. Click here to add your own text and edit me. It's easy.I WATCHED TEACHING COMPANY COURSES AND STUFF ON MATHEMATICS AND OTHER LECTURES AND IT WAS ALL THE QUADRANT MODEL ALL THE BIG THEOREMS EVERYTHING IT WAS BASED ON FOUR BEING DIFFERENT FIFTH QUESTIONABLE EVEN SQUARING THE CIRCLE THING I SAW BUT I FORGET HOW TO EXPLAIN IT ID HAVE TO WATCH THE DVDS AGAIN HOPEFULLY BUT EVEN THERE WAS THIS THING WHERE THEY WERE TRYING TO FIND THESE SPECIAL NUMBERS IT WAS HARD TO FIND THE FOURTH THEY NEEDED COMPUTERS TO FIND THE FIFTH I FORGET WHAT IT WAS CALLED THOUGH NOW
FOUR TRIANGLES
Newton's theorem (quadrilateral)
From Wikipedia, the free encyclopedia
P lies on the Newton line EF
In Euclidean geometry Newton's theorem states, that in any tangential quadrilateral other than a rhombus the center of its incircle lies on its Newton line.
Let ABCD be a tangential quadrilateral with at most one pair of parallel sides. Furthermore, let E and F the midpoints of its diagonals AC and BD and P be the center of its incircle. Given such a configuration the point P is located on the Newton line, that is line EF connecting the midpoints of the diagonals.
A tangential quadrilateral with two pairs of parallel sides is a rhombus. In this case both midpoints and the center of the incircle coincide and by definition no Newton line exists.
Newton's theorem can easily be derived from Anne's theorem considering that in tangential quadrilaterals the combined lengths of opposite sides are equal (Pitot theorem: a + c = b + d). Now according to Anne's theorem showing that the combined areas of opposite triangles PAD and PBC and the combined areas of triangles PAB and PCD are equal is sufficient to ensure that P lies on EF. Let r be the radius of the incircle, then r is also the altitude of all four triangles.
A(△PAB)+A(△PCD) =
1
2
ra+
1
2
rc=
1
2
r(a+c) =
1
2
r(b+d)=
1
2
rc+
1
2
rd = A(△PBC)+A(△PAD){\displaystyle {\begin{aligned}&A(\triangle PAB)+A(\triangle PCD)\\=&{\tfrac {1}{2}}ra+{\tfrac {1}{2}}rc={\tfrac {1}{2}}r(a+c)\\=&{\tfrac {1}{2}}r(b+d)={\tfrac {1}{2}}rc+{\tfrac {1}{2}}rd\\=&A(\triangle PBC)+A(\triangle PAD)\end{aligned}}}
FOUR BASIC MATH OPERATIONS
By the predynastic Naqada period in Egypt, people had fully developed a numeral system.[26] The importance of mathematics to an educated Egyptian is suggested by a New Kingdom fictional letter in which the writer proposes a scholarly competition between himself and another scribe regarding everyday calculation tasks such as accounting of land, labor and grain.[27] Texts such as the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus show that the ancient Egyptians could perform the four basic mathematical operations—addition, subtraction, multiplication, and division—use fractions, compute the volumes of boxes and pyramids, and calculate the surface areas of rectangles, triangles, circles and even spheres.[citation needed] They understood basic concepts of algebra and geometry, and could solve simple sets of simultaneous equations.[28]
THERE ARE 256 LOGICALLY DISTINCT SYLLOGISMS 256 IS FOUR TO THE FOURTH POWER THERE ARE FOUR TYPES OF SYLLOGISM
There are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes the form:
Major premise: All M are P.
Minor premise: All S are M.
Conclusion: All S are P.
(Note: M – Middle, S – subject, P – predicate. See below for more detailed explanation.)
The premises and conclusion of a syllogism can be any of FOUR types, which are labeled by letters[12] as follows. The meaning of the letters is given by the table:
Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, though this makes no difference logically). Each premise and the conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogism BARBARA below is AAA1, or "AAA in the first figure".
The vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms. Even some of these are sometimes considered to commit the existential fallacy, meaning they are invalid if they mention an empty category. These controversial patterns are marked in italics. All but four of the patterns in italics (felapton, darapti, fesapo and bamalip) are weakened moods, i.e. it is possible to draw a stronger conclusion from the premises.
ALL OF THE NUMBERS PRESENTED ARE DIVISIBLE BY 16 256 IS 16 TIMES 16
SHA2 includes significant changes from its predecessor, SHA1. The SHA2 family consists of six hash functions with digests (hash values) that are 224, 256, 384 or 512 bits: SHA224, SHA256, SHA384, SHA512, SHA512/224, SHA512/256.
SHA256 and SHA512 are novel hash functions computed with 32bit and 64bit words, respectively. They use different shift amounts and additive constants, but their structures are otherwise virtually identical, differing only in the number of rounds. SHA224 and SHA384 are simply truncated versions of the first two, computed with different initial values. SHA512/224 and SHA512/256 are also truncated versions of SHA512, but the initial values are generated using the method described in Federal Information Processing Standards (FIPS) PUB 1804. SHA2 was published in 2001 by the National Institute of Standards and Technology (NIST) a U.S. federal standard (FIPS). The SHA2 family of algorithms are patented in US patent 6829355.[5] The United States has released the patent under a royaltyfree license.[6]
In 2005, an algorithm emerged for finding SHA1 collisions in about 2,000 times fewer steps than was previously thought possible.[7] In 2017, an example of a SHA1 collision was published.[8] The security margin left by SHA1 is weaker than intended, and its use is therefore no longer recommended for applications that depend on collision resistance, such as digital signatures. Although SHA2 bears some similarity to the SHA1 algorithm, these attacks have not been successfully extended to SHA2.
Currently, the best public attacks break preimage resistance for 52 rounds of SHA256 or 57 rounds of SHA512, and collision resistance for 46 rounds of SHA256, as shown in the Cryptanalysis and validation section below.[1][2]
256 FOUR TO THE FOURTH POWER HIGHEST RATE DEPLOYED IS 256
The highest rate commonly deployed is the OC768 or STM256 circuit, which operates at rate of just under 38.5 Gbit/s.[13] Where fiber exhaustion is a concern, multiple SONET signals can be transported over multiple wavelengths on a single fiber pair by means of wavelengthdivision multiplexing, including dense wavelengthdivision multiplexing (DWDM) and coarse wavelengthdivision multiplexing (CWDM). DWDM circuits are the basis for all modern submarine communications cable systems and other longhaul circuits.
BOTH DIVISIBLE BY 16
OC768 / STM256[edit]
OC768 is a network line with transmission speeds of up to 39,813.12 Mbit/s (payload: 38,486.016 Mbit/s (38.486016 Gbit/s); overhead: 1,327.104 Mbit/s (1.327104 Gbit/s)).
On October 23, 2008, AT&T announced the completion of upgrades to OC768 on 80,000 fiberoptic wavelength miles of their IP/MPLS backbone network.[3] OC768 SONET interfaces have been available with shortreach optical interfaces from Cisco since 2006. Infinera made a field trial demonstration data transmission on a live production network involving the service transmission of a 40 Gbit/s OC768/STM256 service over a 1,969 km terrestrial network spanning Europe and the U.S. In November 2008, an OC768 connection was successfully brought up on the TAT14/SeaGirt transatlantic cable,[4] the longest hop being 7,500 km.
OC1920 / STM640[edit]
OC1920 is a network line with transmission speeds of up to 99,532.8 Mbit/s.
256 FOUR TO THE FOURTH POWER
SHA256 is a member of the SHA2 cryptographic hash functions designed by the NSA. SHA stands for Secure Hash Algorithm. Cryptographic hash functions are mathematical operations run on digital data; by comparing the computed "hash" (the output from execution of the algorithm) to a known and expected hash value, a person can determine the data's integrity. A oneway hash can be generated from any piece of data, but the data cannot be generated from the hash.
SHA256 is used in several different parts of the Bitcoin network:
Mining uses SHA256 as the Proof of work algorithm.
SHA256 is used in the creation of bitcoin addresses to improve security and privacy.
4cube 24 virtual puzzle, one cubie is highlighted to show how the stickers are distributed across the cube. Note that there are four stickers on each of the cubies of the 24 puzzle but only three are highlighted, the missing one is on the hidden cell.
1999, four further probably prime repunits have been found, but it is unlikely that any of them will be proven prime in the foreseeable future because of their huge size./wiki/Landau%27s_problems
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:

Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?

Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?

Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?

Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n2 + 1? (sequence A002496 in the OEIS).
As of 2017, all four problems are unresolved.
The fourth number is different, the fifth is ultra transcendent.
In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form
Fn=22n+1,
where n is a nonnegative integer. The first few Fermat numbers are:
the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:

Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?

Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?

Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?

Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n2 + 1? (sequence A002496 in the OEIS).
As of 2017, all four problems are unresolved.
The fourth number is different, the fifth is ultra transcendent.
In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form
Fn=22n+1,
where n is a nonnegative integer. The first few Fermat numbers are:

Two conic sections generally intersect in four points, some of which may coincide. To properly account for all intersection points, it may be necessary to allow complex coordinates and include the points on the infinite line in the projective plane. For example:

Two circles never intersect in more than two points in the plane, while Bézout's theorem predicts four. The discrepancy comes from the fact that every circle passes through the same two complex points on the line at infinity. Writing the circle

ézout%27s_theorem

A general conic is defined by five points: given five points in general position, there is a unique conic passing through them. If three of these points lie on a line, then the conic is reducible, and may or may not be unique. If no four points are collinear, then five points define a unique conic (degenerate if three points are collinear, but the other two points determine the unique other line). If four points are collinear, however, then there is not a unique conic passing through them – one line passing through the four points, and the remaining line passes through the other point, but the angle is undefined, leaving 1 parameter free. If all five points are collinear, then the remaining line is free, which leaves 2 parameters free.
Given four points in general linear position (no three collinear; in particular, no two coincident), there are exactly three pairs of lines (degenerate conics) passing through them, which will in general be intersecting, unless the points form a trapezoid (one pair is parallel) or a parallelogram (two pairs are parallel).
In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization.
The rotation is described by four Euler parameters due to Leonhard Euler. The Rodrigues formula (named after Olinde Rodrigues), a method of calculating the position of a rotated point, is used in some software applications, such as flight simulators and computer games.
–Rodrigues_formula
In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization.
The rotation is described by four Euler parameters due to Leonhard Euler. The Rodrigues formula (named after Olinde Rodrigues), a method of calculating the position of a rotated point, is used in some software applications, such as flight simulators and computer games.
FOUR NORMED DIVISIONS
Abstract. The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.
1. Introduction
There are exactly four normed division algebras: the real numbers (R), complex numbers (C), quaternions (H), and octonions (O). The real numbers are the de
Of the four normed trialities, the one that gives the octonions has an interesting property that the rest lack. To see this property, one must pay careful attention to the difference between a normed triality and a normed division algebra. To construct a normed division K algebra from the normed triality t : V1 × V2 × V3 → R, we must arbitrarily choose unit vectors in two of the three spaces, so the symmetry group of K is smaller than that of t. More precisely, let us define an automorphism of the normed triality t : V1 × V2 × V3 → R to be a triple of norm–preserving maps fi: Vi →Vi such that
t(f1(v1), f2(v2), f3(v3)) = t(v1, v2, v3)
To set the stage, we first recall the most mysterious of the four infinite series of Jordan algebras listed at the beginning of Section 3: the spin factors. We described these quite concretely, but a more abstract approach brings out their kinship to Clifford algebras. Given an ndimensional real inner product space V , let the spin factor J(V ) be the Jordan algebra freely generated by V modulo relations
FOUR MAIN CHARACTERS
For a small number of the early episodes, Lightman would team up with Torres to work on a case, while Foster and Loker would team up on a separate case. Occasionally, their work would intertwine, or Foster and/or Lightman would provide assistance on each other's cases. As the first season progressed, the cases became more involved, and all four of the main characters would work together on one case for each episode.
FOUR CONSTRUCTION PHASES Olmec bar four dot motifCertainly the most famous of the La Venta monumental artifacts are the four colossal heads. Seventeen colossal heads have been unearthed in the Olmec area, four of them at La Venta, officially named Monuments 1 through 4.
Complex A is a mound and plaza group located just to the north of the Great Pyramid (Complex C). The centerline of Complex A originally oriented to Polaris (true north) which indicates the Olmec had some knowledge of astronomy.[9] Surrounded by a series of basalt columns, which likely restricted access to the elite, it was erected in a period of four construction phases that span over four centuries (1000 – 600 BCE). Beneath the mounds and plazas were found a vast array of offerings and other buried objects, more than 50 separate caches by one count, including buried jade, polished mirrors made of ironores, and five large "Massive Offerings" of serpentine blocks. It is estimated that Massive Offering 3 contains 50 tons of carefully finished serpentine blocks, covered by 4,000 tons of clay fill.[10]
Also unearthed in Complex A were three rectangular mosaics (also known as "Pavements") each roughly 4.5 by 6 metres (15 by 20 feet) and each consisting of up to 485 blocks of serpentine. These blocks were arranged horizontally to form what has been variously interpreted as an ornate Olmec barandfourdots motif, the Olmec Dragon,[11] a very abstract jaguar mask,[12] a cosmogram,[13] or a symbolic map of La Venta and environs.[14] Not intended for display, soon after completion these pavements were covered over with colored clay and then many feet of earth.
Certainly the most famous of the La Venta monumental artifacts are the four colossal heads. Seventeen colossal heads have been unearthed in the Olmec area, four of them at La Venta, officially named Monuments 1 through 4.
PEIRCE SAID FOURTHNESS REDUNDANT UNNECESSARY FOURTH DIFFERNET
The philosophy of Charles S. Peirce is chockfull of triples, but especially present are his three universal categories of experience. Threes aren’t really my specialty, but while reading a chapter of Richard Bernstein’s book on the “pragmatic turn”, I was reminded of Peirce’s relational ontology: Firstness, Secondness, and Thirdness. Wondering how these could be extended to a Fourthness, I immediately found a fair amount of work on the subject.
Of course, Peirce argued that such a Fourthness was redundant, unnecessary to the structure of his systematic philosophy. He used various reasons for his conclusions, including mathematical, logical, and semiological. There is also a wealth of subsequent work by later researchers on defending this claim, but what is interesting is that others have investigated extending his three into a four.
So, what might be Fourthness? Some of the aspects of fourfolds collected here have commonalities with some of the attributes of Firstness, Secondness, and Thirdness. For Firstness: feeling, quality, possibility; For Secondness: will, fact, existence; For Thirdness: knowledge, law, representation. I really don’t have anything to add at the present time and I have merely gathered these notions together for my future consideration.
PIERCE AND THE TRANSCNEDNET FOURTH FOURTHNESS
Peirce's Sign: Why Triadic?
It should now be very clear that triads and trichotomies form the warp and woof of Peirce's approach to semiotics. The debate over this triadicity has taken two divergent and incompatible avenues: proposals for a category of Fourthness which question the sufficiency of Peirce's semiotic, and proposals for a reduction to dyadicity which would render the semiotic triad unnecessary.
"One, Two, Three... But Where Is the Fourth?"
At least three proposals have been made for a Peircean category of Fourthness: those of Donald Mertz, of Herbert Schneider and Carl Hausman, and of Carl Vaught.
A more detailed proposal for Fourthness comes from Herbert Schneider. Schneider concedes three categories to be adequate for dealing with cognitive processes, but argues for "importance" as a category of Fourthness. He notes that, for Peirce, any purpose or good has meaning only in relation to a completely general summum bonum. "No Kantian idealist could have stated this conception of moral science more formally."23
Schneider observes this scheme does not accommodate norms which might apply "even in the absence of a summum bonum," itches that call to be scratched for their own sake. Such norms he proposes as a phenomenological aspect of Fourthness: logical import is Thirdness, vital importance Fourthness. Satisfaction may comprise either the Thirdness of achievement or the Fourthness of satiety or contentment. The moral selfcontrol of Thirdness in pursuit of an abstract summum bonum is only an abstract "intellectual framework" until it is taken up into the "concrete universal" of the moral selfcriticism of Fourthness. Fourthness supplies what depth psychology, but not the Kantian "moral law within," acknowledges.24
In logical terms, Fourthness would constitute a temporal sequence, though one which is an absolutely discontinuous string of points superimposed on the triadic continuum. Triadic semiosis is "prospective and cumulative"; tetradic semiosis adds a fourth factor which is noncumulative, but "retrospective" along the hierarchy of categories, giving "meaningful individuality" to instances of Firstness. Since Firstness and Secondness "look 'forward'" to Thirdness while Fourthness "looks back" to Firstness, Thirdness in a sense "governs" Fourthness while Fourthness provides the steam to "drive" Thirdness.25
Carl Hausman provisionally adopts Schneider's scheme, applied to both ethics and aesthetics with "importance" or "value" as a possible category of Fourthness. Hausman notes that Schneider's suggestion rejects "Peirce's own principle that a highest good makes specific goods intelligible," but that, quite regardless of this, Schneider puts a finger both on the problematic status of value in Peirce's semiotic and on an apparent "special connection" between value and Firstness.26
Hausman notes that the categories can be related by what Peirce calls discrimination or distinction, prescission, and dissociation (1.353). We can prescind Firstness and Secondness from value, but neither value nor Thirdness from each other. Hausman speculates that value and Thirdness could be separated by "discrimination," "so interdependent that they are copresent as mutual grounds for one another," though this is difficult to determine since Peirce is none too clear as to what he meant by the term.30
In a parallel manner, on the logical level, Peirce insisted that analogy is reducible to a combination of univocity and equivocity, and is not a mediating third between them (3.421, 3.483485, 1.34). Thus, a fourterm analogical relation would be reducible to a combination of simpler terms. But Vaught argues that just such irreducible analogical tetrads occur in Peircean semiosis.
For semiosis gives rise to a sequence of signs and interpretants, each at a slightly different moment in time, and as Peirce's later distinction between immediate and dynamical object implies, the object itself is not static but constitutes a corresponding temporal sequence of objects in interaction with the sign/interpretant sequence. Within the vagueness inherent in Peircean semiosis then, says Vaught, lies precisely a similarity, irreducible to identity and difference, which embraces interpretant at t1, interpretant at t2, dynamical object at t1, and dynamical object at t2 in a tetradic analogical relationship which due to the vagueness is not precisely reducible to any combination of triads. As an example of this, consider a legisign which grows and develops over time: its form at any two instants can, under this proposed Fourthness, be seen as related by an irreducible analogical similarity. Likewise, the similarity between right and left can be seen only through a judgment of analogy.35
Vaught's argument is both closely reasoned and richly textured, very much in the spirit of Peirce's own approach. If similarity is not to be understood (as Peirce saw it) as reducible to some combination of univocity and equivocity, then Vaught's argument is probably correct. But I think counterarguments can be mounted on both logical and phenomenological fronts.
Logically, if the sequence of interpretants in Vaught's argument on analogy are considered, not as a sequence of discrete frames in a movie film (as Vaught takes them), but rather as a genuinely continuous flow of interpretants (and likewise the flow of dynamical objects continuous), then the need for the fourterm relationship vanishes and Thirdness suffices. The situation becomes logically similar to, and no more problematic than, an account of a continuous function in differential calculus.
On the phenomenological front, I note that mathematicians define the "orientation" or handedness of a space by dyadic and triadic arguments alone. The proof is rather technical, but it enables one to speak of left or righthandedness in space (of three dimensions, or even more) without resort to any tetradic combinations and without any prior invocation of right or left.36
ADDITION OF THE TRANSCENDENT FOURTHNESS PEIRCE
QUADRANT
Armand four quadrant diagrams are necessary for complex mathematics with imaginary numbers and phasors.
In Mathematics as both the real and imaginary parts of a complex number in the rectangular form can be either a positive number or a negative number, then both the real and imaginary axis must also extend in both the positive and negative directions. This then produces a complex plane with four quadrants called an Argand Diagram as shown below.
On the Argand diagram, the horizontal axis represents all positive real numbers to the right of the vertical imaginary axis and all negative real numbers to the left of the vertical imaginary axis. All positive imaginary numbers are represented above the horizontal axis while all the negative imaginary numbers are below the horizontal real axis. This then produces a two dimensional complex plane with four distinct quadrants labelled, QI, QII, QIII, and QIV.
The Argand diagram above can also be used to represent a rotating phasor as a point in the complex plane whose radius is given by the magnitude of the phasor will draw a full circle around it for every 2π/ω seconds.
Then we can extend this idea further to show the definition of a complex number in both the polar and rectangular form for rotations of 90o.
definition of complex numbers
Complex Numbers can also have “zero” real or imaginary parts such as: Z = 6 + j0 or Z = 0 + j4. In this case the points are plotted directly onto the real or imaginary axis. Also, the angle of a complex number can be calculated using simple trigonometry to calculate the angles of rightangled triangles, or measured anticlockwise around the Argand diagram starting from the positive real axis.
Then angles between 0 and 90o will be in the first quadrant ( I ), angles ( θ ) between 90 and 180o in the second quadrant ( II ). The third quadrant ( III ) includes angles between 180 and 270o while the fourth and final quadrant ( IV ) which completes the full circle, includes the angles between 270 and 360o and so on. In all the four quadrants the relevant angles can be found from:
tan1(imaginary component ÷ real component)
Addition and Subtraction of Complex Numbers
The addition or subtraction of complex numbers can be done either mathematically or graphically in rectangular form. For addition, the real parts are firstly added together to form the real part of the sum, and then the imaginary parts to form the imaginary part of the sum and this process is as follows using two complex numbers A and B as examples.
Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and v is a column vector describing the position of a point in space, the product Rv yields another column vector describing the position of that point after that rotation. If v is a row vector, the same transformation can be obtained using vRT, where RT is the transpose of R.
THE CROSS PRODUCT AND CROSS MULTIPLICAITON IN MATH
THEY ARE QUADRANTS 16 BY 16
Kakuro or Kakkuro (Japanese: カックロ) is a kind of logic puzzle that is often referred to as a mathematical transliteration of the crossword
The canonical Kakuro puzzle is played in a grid of filled and barred cells, "black" and "white" respectively. Puzzles are usually 16×16 in size, although these dimensions can vary widely.
USES QUARTILES
In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed midrange or 25% midsummary; it is an Lestimator.
The midhinge is complemented by the Hspread, or interquartile range, which is the difference of the third and first quartiles and which is a measure of statistical dispersion, in sense that if one knows the midhinge and the interquartile range, one can find the first and third quartiles.
The use of the term "hinge" for the lower or upper quartiles derives from John Tukey's work on exploratory data analysis,[1] and "midhinge" is a fairly modern term dating from around that time. The midhinge is slightly simpler to calculate than the trimean, which originated in the same context and equals the average of the median and the midhinge.
USING QUARTILES
Tukey's EDA was related to two other developments in statistical theory: robust statistics and nonparametric statistics, both of which tried to reduce the sensitivity of statistical inferences to errors in formulating statistical models. Tukey promoted the use of five number summary of numerical data—the two extremes (maximum and minimum), the median, and the quartiles—because these median and quartiles, being functions of the empirical distribution are defined for all distributions, unlike the mean and standard deviation; moreover, the quartiles and median are more robust to skewed or heavytailed distributions than traditional summaries (the mean and standard deviation). The packages S, SPLUS, and R included routines using resampling statistics, such as Quenouille and Tukey's jackknife and Efron's bootstrap, which are nonparametric and robust (for many problems).
Pirates base their decisions on four factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins each receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.[2] And finally, the pirates do not trust each other, and will neither make nor honor any promises between pirates apart from a proposed distribution plan that gives a whole number of gold coins to each pirate.
IT HAS OTHER NUMBERS BUT ALSO HAS QUATTRO
https://en.wikipedia.org/wiki/Wilkinson_Sword
Wilkinson Sword Quattro: a fourbladed razor for men, introduced in 2003. The Quattro Midnight and Quattro Titanium are models with redesigned handles and different color schemes from the original Quattro.
Quattro Power: A motorized version of the Quattro; it is supposed to reduce friction. The Quattro Titanium Power is a Quattro Power with a different color scheme and Quattro Titanium cartridges. The Quattro Power is powered by a single AAA battery.
Quattro Titanium: includes a titanium coating on the blades that is claimed to reduce irritation.
Quattro for Women: A modified version of the Quattro with a feminine color scheme.
EAST ARABIAN FIGURES FOUR STROKES ONLY COUNTED ONE THROUGH FOUR
Quadrant
ROMAN GREEK AND SOUTH ARABIAN NOTATION OF NUMBERS ONE THROUGH FOUR THE SAME ONLY TO FOUR
Quadrant
Talented and Gifted Animals
One stone,
two houses
three ruins
four gravediggers,
one garden,
some flowers
One raccoon.
Jacques Prevert, Inventaire
FOUR PRESSES LEVER
Quadrant
SEES UP TO FOUR
Dehaene included, believe that our minds perceive up to three or four objects all at once, without having to mentally “spotlight” them one by one. Getting the computer model to subitize the way humans and animals did was possible, he found, only if he built in “number neurons” tuned to fire with maximum intensity in response to a specific number of objects. His model had, for example, a special four neuron that got particularly excited when the computer was presented with four objects.
AZTECS BASED ON FOURS FOUR TIMES FIVE OWN SYMBOL ONE THROUGH FOUR HAD NAMES 400 HAD OWN SYMBOL AND 400 TIMES 20 ALL BUILT AROUND FOUR
The symbol for 20 was a little flag or banner (see Pic 2), and the Aztec word for 20 was cempoalli meaning ‘one count’.
The first four Aztec numbers had simple names in their language, Náhuatl:
1 = ce
2 = ome
3 = yei
4 = nahui
Pic 3: The Mexica symbol for one  a finger, circle or dot
Pic 3: The Mexica symbol for one  a finger, circle or dot (Click on image to enlarge)
Up to 20 you could show numbers just by the right number of dots (or sometimes fingers). It was common among other ancient Mexican peoples to use a bar for 5, but for some funny reason the Aztecs insisted on being ‘dotty’..............!
Pic 4: The Mexica symbol for 400  a feather (or hair)
Pic 4: The Mexica symbol for 400  a feather (or hair) (Click on image to enlarge)
From 20 up to 400 you could join flags together (100 would be 5 flags alongside each other) and add dots to them if need be.
400  which is 20 x 20  had its own symbol, a feather (see Pic 4). Some people say it was more like a hair or even a fir tree: the idea is the same, “as numerous as hairs or the ‘barbs’ [branches] of a feather...” The Náhuatl word for 400 was tzontli or hair.
Pic 5: The Mexica symbol for 8,000  an incense bag
Pic 5: The Mexica symbol for 8,000  an incense bag (Click on image to enlarge)
Finally, 20 x 400 = 8,000, and the symbol for this was the incense bag or pouch (see Pic 5). The Náhuatl word for this was xiquipilli. Why a bag? Perhaps to show the almost uncountable contents of a sack of cacao beans. So drawing 8,000 of something was a bit like saying ‘a sackload’ of whatever it was... In fact the Aztecs/Mexica always measured their tribute by count and volume rather than by weight.
HAWAIIANS COUNTED IN BASE FOUR AND HAD SPECIAL WORDS FOR 40, 400, 4000 FOUR WAS VERY SPECIAL TO THEM
Another thing to keep in mind is that Hawaiian speakers didn’t really think in base ten, so the words for the powers of ten we’re so familiar with (e.g, hundred, thousand, million, billion, etc.) were all borrowed in from English (haneli, kaukani, miliona, biliona, etc.). If anything, precolonial Hawaiians counted in some other base (4? 20?), having special terms for 4 (kāuna), 20 (iwakālua), 40 (kaʻau, ʻiako for “counting tapas, canoes or feathers”), 400 (lau), 4,000 (mano), 40,000 (kini) and 400,000 (lehu). The larger numbers were not generally used for specific amounts (e.g., 44,421 would not be rendered — to my knowledge — as something like hoʻokahi lehu hoʻokahi kini hoʻokahi lau iwakāluakumamākahi), which is precisely how Mandarin and Japanese handle large numbers, using their large unique numbers 10,000 (wàn 万 in Mandarin) and 100,000,000 (yì 亿 in Mandarin). (This means that in Mandarin “one million” would be actually be rendered “one hundred ten thousands” (yībăiwàn 一百万)). Instead, Hawaiians used these big number words to capture the feel of an amount. For example, lau ā lau would mean “hundreds,” while lehu ā lehu would be more like “millions.” You will, however, also encounter generic amounts like ‘elua kini “two 40,000s” in speech and literature, but again the math is not precise.
FOURNESS IMPORTANT IN HAWAII AND DESCRIBING HOW HAWAIIANS HAVE UNIQUE NUMBERS FOR NUMBERS WITH FOUR
As for counting in fours, someone on Facebook reminded me that Hawaiian has unique numbers for 4 and 40 (as well as 400, 4,000 and 400,000), so fourness is important to them – and to other languages as well, so not unique to Hawaiian, but “odd” to the Western sensibility. Further, and to your other point, the Missionaries did much to regularize Hawaiian into something they recognized. I’ll be writing about the one that bugs me the most in the future: delimiting “ua” to simply the past tense marker when it more adequately captures the completive aspect almost exactly like the Mandarin lè 了.
AZTEC NUMBER CHANGE AFTER FOUR AT 400 AND AT 400 TIMES 20
In terms of expressing numbers, the Aztecs used a vigesimal (base 20) number system. A single dot (•) represented the number 1. Occasionally, a finger was also used to indicate the number. Following in this sequence, the numbers 2, 3, and 4 were represented by two, three, and four dots, respectively. 5, though, introduced a new symbol; it was represented by five dots (•••••) or a full bar, as seen in the accompanying figure. From 6 through 9, dots alone or a combination of dots and a bar represented numbers. 10 was represented by a rhombus, two bars, or ten dots. The numbers between 10 and 20 were expressed as a combination of a rhombus, bars, and dots. 20 required a new symbol and resembled a flag, a shell, or a vase with grass growing out of it. After 20, higher numbers were expressed as combinations of the symbols already mentioned. The numbers for 100, 200, and 300 were expressed not only with the mentioned symbols but also by a feather with barbs, each representing twenty units. 100 had five barbs on the feather, 200 had ten, and 300 had fifteen. The next change in symbol occurred at the number 400, which was sometimes represented by twenty barbs on the feather by a bundle of stems tied together. The number 8000 was written as a symbol that represented a bag. The idea that the Aztec system was vigesimal stems from the fact that new symbols were produced for every power of 20; this reasoning, though, is not completely satisfactory because the symbols for higher powers (204, 205, etc.) are unknown.
80000 400 TIMES 20 HAS CROSS
Quadrant
MAYANS USE DOTS UP TO FOUR
The Mayan system is interesting as they developed it without any contact with the other systems on this website. It is similar to the Babylonians but the Mayans chose different numbers as their bases. They used dots to represent numbers under five, so four is four dots. Five is represented by a line. So six is a line and a dot, and seven is a line and two dots, and thirteen is two lines and three dots. This is a unary system, but using five as a base rather than the more common ten.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
This works up to nineteen, but rather than twenty being four lines, they started a new count above the first one. Zero is represented by a shell. So twenty is a single dot above a shell. This stacking of numbers rather than having them in a line is a little disconcerting at first! You have to talk about rows rather than columns. This second system, for counting the twenties and powers of twenty, is positional, and even has a zero, to show that you have no digit in this row.
AZTECS FOUR DOTS
The Mayans developed a number system that was many centuries ahead of anyone else in terms of their ability to express large numbers. They invented the number zero, and using a symbol for zero and a system of dots for the numbers 14 and bars representing a value of 5, it was possible for them to express astronomically large numbers. The Maya number system is similar to ours but instead of the decimal system we have today, the Maya used the vigesimal system for their calculations  a system based on base 20 mathematics rather than base 10. This means that instead of the 1, 10, 100, 1,000 and 10,000 of our mathematical system, the Maya used 1, 20, 400, 8000 and 160,000.
The Mayan place values are vertical, whereas ours are horizontal. For instance, we would write the number 29 horizontally  the number two, then the number nine to the right of it. The Mayans, however, would write 29 vertically  their symbol for nine (a bar representing five units with four dots over it) would be on the bottom, and the symbol for 20 (a dot on the line above) would be directly over it.
BASE FOUR HAWAII
This 15 minute video covers the traditional Hawaiian base4 number system used to group objects, much in the same way that English uses "a dozen" or "four score". These are the beautiful numbers 4, 40, 400, 4000, 40,000, and 400,000. What a pattern!
HAWAIIAN EMPHASIS ON NUMBER FOUR PHILOLOGISTS THINK IT MIGHT BE BECAUSE OF THE CROSS AND BASKET WEAVING
The references report older counting words than those Usted in tourist books. The evidence hinges on the word forfour
and multiples of itby ten.Kauna is the ear
lier term for four; ka'au antedates kanaka
for forty. In groups, therefore, the number words are as follows:
ah? kahi
(four ones) 'umi kauna
(ten fours) 'umi ka'au
Brief Guide for Pronouncing Hawaiian Words
The Hawaiian alphabet has only twelve letters: seven consonants (A, k, /,
m, n,p, w) and five vowels (a, e, /,o, u). The consonants ^ire pronounced as in English, except for w.When w follows e or /, it is given a sound. Otherwise w is
generally pronounced as in English. The vowels are pronounced thus:
(tenforties) 'umilau m:ano
(400) e (4000)
(40000) i (40000) o
'umimano 'umi kini 'umi lehu
!kini :lehu :nalowale
:kauna :ka'au :lau
(4)
(40) a
.. uh (in unaccented syllables)
ah (in accented syllables) .. eh
ah (when marked e or?) .. ee
.. oh .. 00.
(4 000 000) For instance, the following are found:
u
12= ekolu kauna 20 = elima kauna 760 = ekahi laume
iwa ka'au
(three fours) (fivefours)
(one four hundred and
nine forties)
If a bar is over a vowel, the vowel sound is held just a bit longer. If an inverted comma, as in ali'i, precedes a vowel, the breath is stopped momentarily (glottal
stop). Syllables consist of (1) a con sonant and vowel?in that order or (2) a
single vowel. There are no diphthongs. The accent ordinarily falls on thenext to last syllable, unless the final vowel has a
bar. In this case, it gets the accent. For
instance, kanah? (40) is pronounced
k?hnuhh?H, and kanakolu (30) is pro nounced k?hnuhk?Hloo. Spoken
properly, the Hawaiian language seems like gentle waves; the syllables rise and
fall.
the missionaries, computations were per
formed mentally or by counting on fingers. The number words simply indicated the re sults of computations. Hawaiians who were
particularly skilled in computing were much in demand by the ali% local chief.
Their job was to keep an account of tapas, mats, fish, and other property that the chief would distribute to dependents. Thus, be
fore the arrival of missionaries, there was little, ifany, interest in an academic appre
ciation of arithmetic.
The firstmissionaries arrived fromNew
Numbers too large to count were often de noted by kinikini or lehulehu. In fact, the modern mathematical term "infinity" is
represented by nalowale, which also means
"out of sight."
Why the emphasis on "four"? The an
swer is difficult to come by. Alexander
(1968 reprint) remarks that the significance of four arose from the custom of counting
fish, coconuts, taro and such by taking a couple in each hand or by tying them in bundles of four. The Hawaiian philologist
Rubellite Kawena Johnson thinks that the
importance of the number four may have
come from basket weaving and astronomy.
To begin a basket, two pieces of pe'a (i.e.,
"cross") were placed at right angles to one another; hence, there are four times two
strands (or 8). This figure is found likewise
in the constellation H?naiakam?lama, Southern Cross, called Peka (i.e., cross, "8") inTaumotuan.
HAWAIIAN SUPREME BEING SOUNDS LIKE TETRAGRAMMATON
Quadrant
HAWAIIANS VEDICS AND CUSHITES HAVE BASE FOUR
Quadrant
CHALDEANS AND HAWAIIANS BASE FOUR NUMBERING
And FOUR is even more widely used as a way of dividing almost everything. The ancient Greeks used FOUR to in the form of the basic elements of experience (Fire, Water, Air, and Earth). The more ancient Chaldeans used a base of four for their mathematics, dividing the day into four parts, and eventually turning it into a base twelve system that is still in use today for dividing time all over the world, and was used well into modern times by the English for their monetary and measuring systems. A base twelve system is still used in the USA for measuring length. And a surprising number of psychological systems are based on divisions of four types. The Hawaiians, also, have a traditional base four numbering system.
FOUR MOST PRACTICAL APPLICAITONS
So where do we go from here? Into practicality, that's where. Of all the numbers mentioned, FOUR seems to have the most practical applications. Perhaps that's because it's easy to remember things in lists of four, or perhaps it's a brainstructure thing that makes it easy to remember. In any case, here is a New Year's gift for you based on the number FOUR.
I mentioned that one of the most ancient of concepts is the symbolic representation of life processes and states by Fire, Air, Water, and Earth. To superficial minds, it means only that the ancients were acquainted with the four states of matter: Plasma, Gas, Liquid, and Solid. But the meaning goes far deeper than that, and meditation on each of the four symbols and their interrelationships could reveal to you many of the secrets of life and living. They can also refer, respectively, to the spiritual, physical, emotional, and intellectual aspects of man. For the time being, I ask you to consider them as a useful memory device for a daily Total Health practice. In these terms:
Fire = practice positive emotions.
Earth = practice positive posture.
Air = practice deeper breathing.
Water = practice blessing the world around you.
16 NODES DEGREE FOUR MAP 16 NODESKummer's
16
6
16_{6} configuration[edit]
There are several crucial points which relate the geometric, algebraic, and combinatorial aspects of the configuration of the nodes of the kummer quartic:
Any symmetric odd theta divisor on
J
a
c
(
C
)
Jac(C) is given by the set points
{
q
−
w

q
∈
C
}
\{qwq\in C\}, where w is a Weierstrass point on
C
C. This theta divisor contains six 2torsion points:
w
′
−
w
w'w such that
w
′
w' is a Weierstrass point.
Two odd theta divisors given by Weierstrass points
w
,
w
′
w,w' intersect at
0
{\displaystyle 0} and at
w
−
w
′
ww'.
The translation of the Jacobian by a two torsion point is an isomorphism of the Jacobian as an algebraic surface, which maps the set of 2torsion points to itself.
In the complete linear system

2
Θ
C

2\Theta _{C} on
J
a
c
(
C
)
Jac(C), any odd theta divisor is mapped to a conic, which is the intersection of the Kummer quartic with a plane. Moreover, this complete linear system is invariant under shifts by 2torsion points.
Hence we have a configuration of
16
16 conics in
P
3
{\mathbb {P}}^{3}; where each contains 6 nodes, and such that the intersection of each two is along 2 nodes. This configuration is called the
16
6
16_{6} configuration or the Kummer configuration.
The Weil Pairing[edit]
The 2torsion points on an Abelian variety admit a symplectic bilinear form called the Weil pairing. In the case of Jacobians of curves of genus two, every nontrivial 2torsion point is uniquely expressed as a difference between two of the six Weierstrass points of the curve. The Weil pairing is given in this case by
⟨
p
1
−
p
2
,
p
3
−
p
4
⟩
=
#
{
p
1
,
p
2
}
∩
{
p
3
,
p
4
}
\langle p_{1}p_{2},p_{3}p_{4}\rangle =\#\{p_{1},p_{2}\}\cap \{p_{3},p_{4}\}. One can recover a lot of the group theoretic invariants of the group
S
p
4
(
2
)
Sp_4(2) via the geometry of the
16
6
16_{6} configuration.
Group theory, algebra and geometry[edit]
Below is a list of group theoretic invariants and their geometric incarnation in the 166 configuration.
Polar lines
Apolar complexes
Klein's 6015 configuration
Fundamental quadrics
Fundamental tetrahedra
Rosenhain tetrads
Gopel tetrads
K\subset {\mathbb {P}}^{3} be a quartic surface with an ordinary double point p, near which K looks like a quadratic cone. Any projective line through p then meets K with multiplicity two at p, and will therefore meet the quartic K in just two other points. Identifying the lines in
P
3
{\mathbb {P}}^{3} through the point p with
P
2
\mathbb {P} ^{2}, we get a double cover from the blow up of K at p to
P
2
\mathbb {P} ^{2}; this double cover is given by sending q ≠ p ↦
p
q
¯\scriptstyle \overline {pq}, and any line in the tangent cone of p in K to itself. The ramification locus of the double cover is a plane curve C of degree 6, and all the nodes of K which are not p map to nodes of C.
By the genus degree formula, the maximal number possible number of nodes on a sextic curve is obtained when the curve is a union of
6
6 lines, in which case we have 15 nodes. Hence the maximal number of nodes on a quartic is 16, and in this case they are all simple nodes (to show that
p
p is simple project from another node). A quartic which obtains these 16 nodes is called a Kummer Quartic, and we will concentrate on them below.
Finally, since we know that every Kummer quartic is a Kummer variety of a Jacobian of a hyperelliptic curve, we show how to reconstruct Kummer quartic surface directly from the Jacobian of a genus 2 curve: The Jacobian of
C
C maps to the complete linear system

O
J
a
c
(
C
)
(
2
Θ
C
)

≅
P
2
2
−
1
O_{{Jac(C)}}(2\Theta _{C})\cong {\mathbb {P}}^{{2^{2}1}} (see the article on Abelian varieties). This maps factors through the Kummer variety as a degree 4 map which has 16 nodes at the images of the 2torsion points on
DEGREE OF FOUR 16 POINTS 16 SQUARES QMR CLASSIC IN MATHEMATICS KUMMERS QUARTIC SURFACES
https://en.wikipedia.org/wiki/Kummer_surface
In algebraic geometry, a Kummer quartic surface, first studied by Kummer (1864), is an irreducible nodal surface of degree 4 in
P
3
{\mathbb {P}}^{3} with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution x ↦ −x. The Kummer involution has 16 fixed points: the 16 2torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a (possibly nonalgebraic) torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces.
KUMER CONFIGURATIONS ARE CLASSIC IN MATHEMATICS 16 POINTS 16 PLANES
In geometry, the Kummer configuration, named for Ernst Kummer, is a geometric configuration of 16 points and 16 planes such that each point lies on 6 of the planes and each plane contains 6 of the points. The 16 nodes and 16 tropes of a Kummer surface form a Kummer configuration.
KUMER SURFACE 16 TROPES 16 SQUARES QMR QUARTIC SURFACE QUARTIC IS FOUR
In geometry, trope is an archaic term for a singular (meaning special) tangent space of a variety, often a quartic surface. The term may have been introduced by Cayley (1869, p. 202), who defined it as "the reciprocal term to node". It is not easy to give a precise definition, because the term is used mainly in older books and papers on algebraic geometry, whose definitions are vague and different, and use archaic terminology. The term trope is used in the theory of quartic surfaces in projective space, where it is sometimes defined as a tangent space meeting the quartic surface in a conic; for example Kummer's surface has 16 tropes.
HIGHEST NUMBER 16 FOUR COPIES CIRCLE
A quartic surface is one defined by a polynomial equation of degree 4. A nodal surface is one whose only singularities are ordinary double points: that is, points where it looks like the origin of the cone in 3dimensional space defined by
x
2
+
y
2
=
z
2
.
x
2
+
y
2
=
z
2
.
The Kummer surfaces are the quartic nodal surfaces with the largest possible number of ordinary double points, namely 16. In the above picture, Abdelaziz Nait Merzouk has drawn the real points of a Kummer surface.
There are many different Kummer surfaces, but they all have the same topology and they can all be constructed in the same way. Start with a smooth complex curve of genus 2, say
C
C
:
Genus Two Curve  Oleg Alexandrov
Genus Two Curve – Oleg Alexandrov
Then, form its Jacobian variety
J
(
C
)
J
(
C
)
. This is the space of isomorphism classes of holomorphic line bundles over
C
C
with vanishing first Chern class. As a topological space, this Jacobian is always product of 4 copies of a circle, but it is equipped with the structure of a complex surface in a way that depends on
C
C
. It is an abelian group, thanks to our ability to tensor line bundles, so it has an automorphism
x
↦
−
x
x
↦
−
x
called the Kummer involution. The quotient of the Jacobian by this involution is a Kummer surface. The ordinary double points come from the points in
J
(
C
)
J
(
C
)
with
x
=
−
x
x
=
−
x
; there are
16
=
2
4
16
=
2
4
of these in a product of 4 copies of the circle group.
Here is a ‘cut’ view of the same Kummer surface. Again, only the real points are shown:
FOUR SPECIAL TETRADS
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4x4 square of the MOG. These
correspond to the 35 sets of four parallel 4point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
16 SQUARES QMR
Observatory scene from "Magic in the Moonlight"
"The sixteen nodes… can be parametrized
by the sixteen points in affine fourspace
over the tiny field F2 with two elements."
FOUR ROWS SIX COLUMNS
The Miracle Octad Generator is an array of coordinates, arranged in four rows and six columns, capable of describing any point in 24dimensional space. It is remarkable in the fact that it reflects all of the symmetries of the Mathieu group M24, despite its simplicity. More specifically, it preserves the maximal subgroups of M24, namely the monad group, duad group, triad group, octad group, octern group, sextet group, trio group and duum group. This makes it invaluable, as it can be used to study all of these symmetries without having to visualise 24dimensional space.
CONWAY FOUR BY THREE ARRAY TETRACODE
John Horton Conway developed a 4 × 3 array known as the MiniMOG. The MiniMOG provides the same function for the Mathieu group M12 and ternary Golay code as the Miracle Octad Generator does for M24 and binary Golay code, respectively. Instead of using a quaternary hexacode, the MiniMOG uses a ternary tetracode.
64
In the mid1960s with Michael Guy, son of Richard Guy, Conway established that there are sixtyfour convex uniform polychora excluding two infinite sets of prismatic forms. They discovered the grand antiprism in the process, the only nonWythoffian uniform polychoron. Conway has also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation.
TRANSCENDENT FOURTH FUZZY GENERATOR
All numbers are positive, negative, or zero, and we say that a game is positive if Left will win, negative if Right will win, or zero if the second player will win. Games that are not numbers have a fourth possibility: they may be fuzzy, meaning that the first player will win. * is a fuzzy game.[4]
HORTON QUADRATIC FORM TETRACODE
Quadrant
BARTH AFFINE FOUR SPACE
The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3space." The affine 4space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."
"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
nonisotropic planes in this affine space over the finite field."
The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)."
FOUR BY FOUR IN THE MOG CARMICHAEL
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4space over the 2element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
QUATENARY  TWO PLUS TWO ARE FOUR BUT 2 PLUS THREE IS FIVE MINKOWSKI FOUR DIMENSIONAL MODEL KUMERIAN QUARTICS
NUMBERS UP TO FOUR AS ADJECTIVES "BOUNDARY ASPECT OF THE NUMBER FOUR" LANGUAGES HAVE A TRIAL AND QUATERNAL CASE ABOVE DUAL BUT NEVER MORE
Quadrant
IN CHINA FIVE REPRESENTS THE CENTERED FOUR QUINCUNX CROSS
Quadrant
CHINA THE FIVE AS THE CENTER OF FOUR QUINCUNX CROSS PLATO THE ELEMENTS AND JUNG AND THE QUATERNIO
Quadrant