FOUR EIGENVALUES

Eigenvalues and eigenvectors

The eigenvalues of the DFT matrix are simple and well-known, whereas the eigenvectors are complicated, not unique, and are the subject of ongoing research.

Consider the unitary form

U

\mathbf {U} defined above for the DFT of length N, where

.

\mathbf {U} _{m,n}={\frac {1}{\sqrt {N}}}\omega _{N}^{(m-1)(n-1)}={\frac {1}{\sqrt {N}}}e^{-{\frac {2\pi i}{N}}(m-1)(n-1)}.

This matrix satisfies the matrix polynomial equation:

.

\mathbf {U} ^{4}=\mathbf {I} .

This can be seen from the inverse properties above: operating

U

\mathbf {U} twice gives the original data in reverse order, so operating

U

\mathbf {U} four times gives back the original data and is thus the identity matrix. This means that the eigenvalues

λ\lambda satisfy the equation:

λ

4

=

1.

\lambda ^{4}=1.

Therefore, the eigenvalues of

U

\mathbf {U} are the fourth roots of unity:

λ\lambda is +1, −1, +i, or −i.

Since there are only four distinct eigenvalues for this

N

×

N

N\times N matrix, they have some multiplicity. The multiplicity gives the number of linearly independent eigenvectors corresponding to each eigenvalue. (Note that there are N independent eigenvectors; a unitary matrix is never defective.)

The problem of their multiplicity was solved by McClellan and Parks (1972), although it was later shown to have been equivalent to a problem solved by Gauss (Dickinson and Steiglitz, 1982). The multiplicity depends on the value of N modulo 4, and is given by the following table:

Multiplicities of the eigenvalues λ of the unitary DFT matrix U as a function of the transform size N (in terms of an integer m).

size N λ = +1 λ = −1 λ = -i λ = +i

4m m + 1 m m m − 1

4m + 1 m + 1 m m m

4m + 2 m + 1 m + 1 m m

4m + 3 m + 1 m + 1 m + 1 m

THE FOUR DIFFERENT FORMS
http://fourier.eng.hmc.edu/…/le…/Image_Processing/node6.html
Fourier transform can be generalized to higher dimensions. For example, many signals $f(x,y)$ are functions of 2D space defined over an x-y plane. Two-dimensional Fourier transform also has four different forms depending on whether the 2D signal is periodic and discrete.

Aperiodic, continuous signal, continuous, aperiodic spectrum

\begin{displaymath}F(u,v)=\int \int_{-\infty}^{\infty} f(x,y) e^{-j2\pi(ux+vy)} dx dy \end{displaymath}

\begin{displaymath}f(x,y)=\int \int_{-\infty}^{\infty} F(u,v) e^{j2\pi(ux+vy)} du dv \end{displaymath}

where $u$ and $v$ are spatial frequencies in $x$ and $y$ directions, respectively, and $F(u,v)$ is the 2D spectrum of $f(x,y)$.
Aperiodic, discrete signal, continuous, periodic spectrum

\begin{displaymath}F(u,v)=\sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty}
f[m,n]e^{-j2\pi(umx_o+vny_o)} \end{displaymath}

\begin{displaymath}f[m,n]=\frac{1}{UV}\int_0^U \int_0^V F(u,v)
e^{j2\pi(umx_o+vny_o)} du dv \end{displaymath}

where $x_o$ and $y_o$ are the spatial intervals between consecutive signal samples in the $x$ and $y$ directions, respectively, and $U=1/x_o$ and $V=1/y_o$ are sampling rates in the two directions, and they are also the periods of the spectrum $F(u,v)$.
Periodic, continuous signal, discrete, aperiodic spectrum

\begin{displaymath}F[k,l]=\frac{1}{X Y}\int_0^{X} \int_0^{Y} f_{XY}(x,y)
e^{j2\pi(kxu_o+lyv_o)} dx dy \end{displaymath}

\begin{displaymath}f_{XY}(x,y)=\sum_{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty} F[k,l]
e^{-j2\pi(xku_o+ylv_o)} \end{displaymath}

where $X$ and $Y$ are periods of the signal in $x$ and $y$ directions, respectively, and $u_o=1/X$ and $v_o=1/Y$ are the intervals between consecutive samples in the spectrum $F[k,l]$.
Periodic, discrete signal, discrete and periodic spectrum

\begin{displaymath}F[k,l]=\frac{1}{\sqrt{MN}}\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}f[m,n]
e^{-j2\pi(\frac{mk}{M}+\frac{nl}{N})} \end{displaymath}

\begin{displaymath}f[m,n]=\frac{1}{\sqrt{MN}}\sum_{l=0}^{N-1}\sum_{k=0}^{M-1}F[k,l]
e^{j2\pi(\frac{mk}{M}+\frac{nl}{N})} \end{displaymath}

\begin{displaymath}\;\;\;\;(0 \leq m, k \leq M-1,\;\;\;0 \leq n,l \leq N-1) \end{displaymath}

where $M=X/x_0=U/u_0$ and $N=Y/y_0=V/v_0$ are the numbers of samples in $x$ and $y$ directions in both spatial and spatial frequency domains, respectively, and $F[k,l]$ is the 2D discrete spectrum of $f[m,n]$. Both $f[m,n]$ and $F[k,l]$ can be considered as elements of two $M$ by $N$ matrices ${\bf x}$ and ${\bf F}$, respectively.

THE FOUR CATEGORIES BASED ON TWO DICHOTOMIES- RIGHT NOW YOU ARE SAYING "WELL ITS PROBABLY A LOT MORE COMPLEX THAN THAT"- NO I STUDIED EVERY CLASS AT UCSD AND BOUGHT LECTURES AND STUDIED ALL TYPES OF STUFF FOR ALMOST TEN YEARS ALL THAT WAS TAUGHT WAS THE QUADRANT MODEL THE QUADRANT MODEL WAS THE MAIN THING AND PRETTY MUCH ONLY THING TAUGHT

A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these two features generates the four categories, described below and illustrated in the following figure.

FOUR CONSTANT MATRICES

We have four constant matrices: A, B, C, and D. We will explain these matrices below:

Matrix A

Matrix A is the system matrix, and relates how the current state affects the state change x' . If the state change is not dependent on the current state, A will be the zero matrix. The exponential of the state matrix, eAt is called the state transition matrix, and is an important function that we will describe below.

Matrix B

Matrix B is the control matrix, and determines how the system input affects the state change. If the state change is not dependent on the system input, then B will be the zero matrix.

Matrix C

Matrix C is the output matrix, and determines the relationship between the system state and the system output.

Matrix D

Matrix D is the feed-forward matrix, and allows for the system input to affect the system output directly. A basic feedback system like those we have previously considered do not have a feed-forward element, and therefore for most of the systems we have already considered, the D matrix is the zero matrix.

BOARD DIVIDED INTO FOUR ZONES- SQUARES OF THE FIRST FOUR NUMBERS

The board is commonly viewed to be divided into four zones. The first zone is the division of the board into the White v. Black Half into groups of four Ranks. Traditionally, white’s half of the board consists of all of the squares from Ranks 1 to 4; Black’s Half includes all the squares from Ranks 5 to 8. A similar, but vertical division creates a second zone delineating the Kings Side versus the Queen’s Side of the board. The Queen’s Side includes all squares from Files a through d, whereas the King’s Side includes all squares from Files E to F. The third zone is actually a combination of the Center and the Expanded Center of the Board. The Center includes the four most center squares on the chessboard; whereas the Expanded Center includes the 16 squares in the board center. The fourth and final zone consists of the Flanks. There are 6 Flanking squares in total, 3 squares on the Queen’s Side (Files a to c) and 3 squares on the Kingside (Files f to h).

Other accounts of the origin of the game[V] hold that it began in India as early as the time of the great Indian grammarian Pānini (circa 500 BC) where it was a game known as “Chaturanga” (Sanskrit meaning “four members”) which required four participants, and which possibly included the use of dice

A somewhat lesser-known allusion concerning both the Chess Board and the Mosaic Pavement also exists in the Masonic allegory of the 3, 5, and 7 steps leading to the middle chamber. The numbers 3, 5, and 7 represent the difference of squares of the first four integers as follows:

12 = 1,

22 = 4,

32 = 9,

42 = 16

FOUR QUADRANTS- FOUR COCENTRIC RINGS- THE GOLDEN SPIRAL

The Chess Board may be discovered to have a common center, which can be located at the point of intersection of diagonals constructed from the four corners of the board. This is true also of the Masonic Pavement[XII]. In a manner identical to that used in the layout of the Masonic Pavement, the Chess Board is constructed from its center outwards in four concentric rings with each quadrant of the board encompassing four groups of nested squares (See Figure 4) which are identical to the nested squares associated with the 3, 5, 7 sequence shown in Figure 3. Among other things, the arrangement of the nested 3, 5, 7 Difference in Squares incorporates an approximation of the Golden Spiral[XIII].

One of the key features of the Board Layout when envisioned as nested squares derived from the 3, 5, and 7 sequence is that when all four quadrants are considered there is a central common group of four contiguous squares (shown in black outline in Figure 4). The only two Chess pieces which are capable of simultaneously controlling all four of this group of squares are the King and Queen. Four (4) is a cubic number (and 4 X 4 X 4 = 64). In this regards, the four squares, which in a Masonic Mosaic Pavement would be occupied by the Altar, represent the Holy of Holies in Solomon’s Temple. The traditional symbolism of the Chess Pieces (the King and Queen) are that the King represents the Sun, and the Queen represents the Moon. This symbolism identifies the four squares as the center of the universe from the Astrological standpoint, and places two of the three Lesser Lights of Freemasonry on the altar.

FOUR STEPPED ZIGGURAT CHESS BOARD

Thus, if we consider each nested square to have a height equal to its length and width them stacking the nested cubes results in a four-stepped Ziggurat as shown in Figure 5. This Ziggurat has been interpreted[XIV] as representing the “world mountain” which the ancients believed was the central axis of the revolving cosmos.

There are a significant number of other esoteric characteristics incorporated into the Chess Board Layout[XV]. The natural symmetric division of the Chess Board into four quadrants taken in the context of astrological symbolism represent the four opposing seasonal cycles. Extension of this symbolism into the art and science of Alchemy would find a correlation between the Chess Board Layout and the so-called “Square of Opposition” of the elements (Figure 6) proposed by Aristotle.

GEOMETRIC PROOF CALCULUS- THE FOURTH SQUARE THE BLACK ONE IS DIFFERENT
https://en.wikipedia.org/wiki/Product_rule
https://en.wikipedia.org/…/File:Illustration-for-leibniz-pr…

A rigorous proof of the product rule can be given using the definition of the derivative as a limit, and the basic properties of limits.

Let h(x) = f(x) g(x), and suppose that f and g are each differentiable at x0. (Note that x0 will remain fixed throughout the proof). We want to prove that h is differentiable at x0 and that its derivative h′(x0) is given by f′(x0) g(x0) + f(x0) g′(x0).

Let Δh = h(x0 + Δx) − h(x0); note that although x0 is fixed, Δh depends on the value of Δx, which is thought of as being "small".

The function h is differentiable at x0 if the limit

lim
x
\lim _{\Delta x\to 0}{\Delta h \over \Delta x}
exists; when it does, h′(x0) is defined to be the value of the limit.

As with Δh, let Δf = f(x0 + Δx) − f(x0) and Δg = g(x0 + Δx) − g(x0) which, like Δh, also depends on Δx. Then f(x0 + Δx) = f(x0) + Δf and g(x0 + Δx) = g(x0) + Δg.

It follows that h(x0 + Δx) = f(x0 + Δx) g(x0 + Δx) = (f(x0) + Δf) (g(x0) + Δg); applying the distributive law, we see that

h(x_{0}+\Delta x)=f(x_{0}+\Delta x)g(x_{0}+\Delta x)=f(x_{0})g(x_{0})+\Delta fg(x_{0})+f(x_{0})\Delta g+\Delta f\Delta g

(
)
While it is not necessary for the proof, it can be helpful to understand this product geometrically as the area of the rectangle in this diagram:

THIS IS AN AWESOME PAINTING OF PYTHAGORAS FULL OF QUADRANT MODEL-- IN THE BACK IS THE FOUR ROWS OF THE TETRACTYS TRIANGLE- TETRA IS FOUR IT IS FOUR LINES OF 10 DOTS- THE PYTHAGOREANS SAW IT AS THE ULTIMATE SYMBOL LIKE THE ISRAELITES SAW THE TETRAGRAMMATON AS THE ULTIMATE--- BEHIND HIM IS ALSO A TREE WITH FOUR BRANCHES AND PYTHAGORAS TAUGHT MUSIC THROUGH THE TETRACHORD PRIMARILY TETRA IS FOUR--- HE ALSO HAS THE PYTHAGOREAN THEOREM BEHIND HIM AND IN IT YOU SEE A DIAGRAM OF THE SIXTEEN SQUARES OF THE 345 TRIANGLE OF THE SIDE FOUR

QUARANT

PAYOFF MATRICES ARE QUADRANT MODELS- ALL THE GAMES ARE QUADRANT MODELS- NASH WON A NOBEL PRIZE FOR DEVELOPING THIS

These matrices only represent games in which moves are simultaneous (or, more generally, information is imperfect). The above matrix does not represent the game in which player 1 moves first, observed by player 2, and then player 2 moves, because it does not specify each of player 2's strategies in this case. In order to represent this sequential game we must specify all of player 2's actions, even in contingencies that can never arise in the course of the game. In this game, player 2 has actions, as before, Left and Right. Unlike before he has four strategies, contingent on player 1's actions. The strategies are:

Left if player 1 plays Top and Left otherwise

Left if player 1 plays Top and Right otherwise

Right if player 1 plays Top and Left otherwise

Right if player 1 plays Top and Right otherwise

On the right is the normal-form representation of this game.

The matrix to the right is a normal-form representation of a game in which players move simultaneously (or at least do not observe the other player's move before making their own) and receive the payoffs as specified for the combinations of actions played. For example, if player 1 plays top and player 2 plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player (in this case player 2).

Other representations

Often, symmetric games (where the payoffs do not depend on which player chooses each action) are represented with only one payoff. This is the payoff for the row player. For example, the payoff matrices on the right and left below represent the same game.

Both players

Stag Hare

Stag 3, 3 0, 2

Hare 2, 0 2, 2

Just row

Stag Hare

Stag 3 0

Hare 2 2

Uses of normal form

Dominated strategies

The Prisoner's Dilemma

Cooperate Defect

Cooperate −1, −1 −5, 0

Defect 0, −5 −2, −2

The payoff matrix facilitates elimination of dominated strategies, and it is usually used to illustrate this concept. For example, in the prisoner's dilemma (to the right), we can see that each prisoner can either "cooperate" or "defect". If exactly one prisoner defects, he gets off easily and the other prisoner is locked up for a long time. However, if they both defect, they will both be locked up for a shorter time. One can determine that Cooperate is strictly dominated by Defect. One must compare the first numbers in each column, in this case 0 > −1 and −2 > −5. This shows that no matter what the column player chooses, the row player does better by choosing Defect. Similarly, one compares the second payoff in each row; again 0 > −1 and −2 > −5. This shows that no matter what row does, column does better by choosing Defect. This demonstrates the unique Nash equilibrium of this game is (Defect, Defect).

STAG HUNT IS A QUADRANT GAME

In game theory, the stag hunt is a game that describes a conflict between safety and social cooperation. Other names for it or its variants include "assurance game", "coordination game", and "trust dilemma". Jean-Jacques Rousseau described a situation in which two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, they must have the cooperation of their partner in order to succeed. An individual can get a hare by themself, but a hare is worth less than a stag. This has been taken to be a useful analogy for social cooperation, such as international agreements on climate change.[1]

The stag hunt differs from the Prisoner's Dilemma in that there are two pure strategy Nash equilibria: when both players cooperate and both players defect. In the Prisoner's Dilemma, in contrast, despite the fact that both players cooperating is Pareto efficient, the only pure Nash equilibrium is when both players choose to defect.

An example of the payoff matrix for the stag hunt is pictured in Figure 2.

Stag Hare

Stag a, a c, b

Hare b, c d, d

Fig. 1: Generic symmetric stag hunt

Stag Hare

Stag 2, 2 0, 1

Hare 1, 0 1, 1

Fig. 2: Stag hunt example

Contents [hide]

Formal definition

Formally, a stag hunt is a game with two pure strategy Nash equilibria—one that is risk dominant and another that is payoff dominant. The payoff matrix in Figure 1 illustrates a generic stag hunt, where

a>b\geq d>c. Often, games with a similar structure but without a risk dominant Nash equilibrium are called assurance game. For instance if a=2, b=1, c=0, and d=1. While (Hare, Hare) remains a Nash equilibrium, it is no longer risk dominant. Nonetheless many would call this game a stag hunt.

THE GAME IS A QUADRANT MODEL

The game has also been used to describe the mutual assured destruction of nuclear warfare, especially the sort of brinkmanship involved in the Cuban Missile Crisis.[3]

The game of chicken, also known as the hawk-dove game or snowdrift[1] game, is a model of conflict for two players in game theory. The principle of the game is that while it is to both players’ benefit if one player yields, the other player's optimal choice depends on what his opponent is doing: if his opponent yields, the player should not, but if the opponent fails to yield, the player should.

A formal version of the game of Chicken has been the subject of serious research in game theory.[6] Two versions of the payoff matrix for this game are presented here (Figures 1 and 2). In Figure 1, the outcomes are represented in words, where each player would prefer to win over tying, prefer to tie over losing, and prefer to lose over crashing. Figure 2 presents arbitrarily set numerical payoffs which theoretically conform to this situation. Here, the benefit of winning is 1, the cost of losing is -1, and the cost of crashing is -10.

Both Chicken and Hawk-Dove are anti-coordination games, in which it is mutually beneficial for the players to play different strategies. In this way, it can be thought of as the opposite of a coordination game, where playing the same strategy Pareto dominates playing different strategies. The underlying concept is that players use a shared resource. In coordination games, sharing the resource creates a benefit for all: the resource is non-rivalrous, and the shared usage creates positive externalities. In anti-coordination games the resource is rivalrous but non-excludable and sharing comes at a cost (or negative externality).

Because the loss of swerving is so trivial compared to the crash that occurs if nobody swerves, the reasonable strategy would seem to be to swerve before a crash is likely. Yet, knowing this, if one believes one's opponent to be reasonable, one may well decide not to swerve at all, in the belief that he will be reasonable and decide to swerve, leaving the other player the winner. This unstable situation can be formalized by saying there is more than one Nash equilibrium, which is a pair of strategies for which neither player gains by changing his own strategy while the other stays the same. (In this case, the pure strategy equilibria are the two situations wherein one player swerves while the other does not.)

Related strategies and games

Brinkmanship

"Chicken" and "Brinkmanship" are often used synonymously in the context of conflict, but in the strict game-theoretic sense, "brinkmanship" refers to a strategic move designed to avert the possibility of the opponent switching to aggressive behavior. The move involves a credible threat of the risk of irrational behavior in the face of aggression. If player 1 unilaterally moves to A, a rational player 2 cannot retaliate since (A, C) is preferable to (A, A). Only if player 1 has grounds to believe that there is sufficient risk that player 2 responds irrationally (usually by giving up control over the response, so that there is sufficient risk that player 2 responds with A) player 1 will retract and agree on the compromise.

War of attrition

Like "Chicken", the "War of attrition" game models escalation of conflict, but they differ in the form in which the conflict can escalate. Chicken models a situation in which the catastrophic outcome differs in kind from the agreeable outcome, e.g., if the conflict is over life and death. War of attrition models a situation in which the outcomes differ only in degrees, such as a boxing match in which the contestants have to decide whether the ultimate prize of victory is worth the ongoing cost of deteriorating health and stamina.

Penis game

A childish activity where people (usually schoolchildren) compete to shout "Penis!" in an increasingly loud voice while trying not to get in trouble with some authority figure (usually a teacher) [16]

Kiss game

Also a childish activity where two people sit across from each other and move closer to each other, eventually going face to face, and the first person who moves/flinches away from the "kiss" gets to be called a "wussy".

HARM THY NEIGBOR GAME QUADRANT MODELS

Examples of differences between Nash Equilibria and ESSes

Cooperate Defect

Cooperate 3, 3 1, 4

Defect 4, 1 2, 2

Prisoner's Dilemma

A B

A 2, 2 1, 2

B 2, 1 2, 2

Harm thy neighbor

In most simple games, the ESSes and Nash equilibria coincide perfectly. For instance, in the Prisoner's Dilemma there is only one Nash equilibrium, and its strategy (Defect) is also an ESS.

Some games may have Nash equilibria that are not ESSes. For example, in Harm thy neighbor both (A, A) and (B, B) are Nash equilibria, since players cannot do better by switching away from either. However, only B is an ESS (and a strong Nash). A is not an ESS, so B can neutrally invade a population of A strategists and predominate, because B scores higher against B than A does against B. This dynamic is captured by Maynard Smith's second condition, since E(A, A) = E(B, A), but it is not the case that E(A,B) > E(B,B).

C D

C 2, 2 1, 2

D 2, 1 0, 0

Harm everyone

Swerve Stay

Swerve 0,0 -1,+1

Stay +1,-1 -20,-20

Chicken

Nash equilibria with equally scoring alternatives can be ESSes. For example, in the game Harm everyone, C is an ESS because it satisfies Maynard Smith's second condition. D strategists may temporarily invade a population of C strategists by scoring equally well against C, but they pay a price when they begin to play against each other; C scores better against D than does D. So here although E(C, C) = E(D, C), it is also the case that E(C,D) > E(D,D). As a result, C is an ESS.

Even if a game has pure strategy Nash equilibria, it might be that none of those pure strategies are ESS. Consider the Game of chicken. There are two pure strategy Nash equilibria in this game (Swerve, Stay) and (Stay, Swerve). However, in the absence of an uncorrelated asymmetry, neither Swerve nor Stay are ESSes. There is a third Nash equilibrium, a mixed strategy which is an ESS for this game (see Hawk-dove game and Best response for explanation).

This last example points to an important difference between Nash equilibria and ESS. Nash equilibria are defined on strategy sets (a specification of a strategy for each player), while ESS are defined in terms of strategies themselves. The equilibria defined by ESS must always be symmetric, and thus have fewer equilibrium points.

ALL KNOTS UP TO 16 CROSSINGS - 16 SQUARES QMR

In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings (Hoste, Thistlethwaite & Weeks 1998). In 2003 Rankin, Flint, and Schermann, tabulated the alternating knots through 22 crossings (Hoste 2005).

FOURTH DIMENSION IS DIFFERENT

Higher dimensions

A knot in three dimensions can be untied when placed in four-dimensional space. This is done by changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle.

In fact, in four dimensions, any non-intersecting closed loop of one-dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain.

Knotting spheres of higher dimension

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two-dimensional sphere embedded in a four-dimensional ball. Such an embedding is unknotted if there is a homeomorphism of the 4-sphere onto itself taking the 2-sphere to a standard "round" 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots.

In addition to the rows, columns, and diagonals, a 5×5 pandiagonal magic square also shows its magic sum in four "quincunx" patterns, which in the above example are:

17+25+13+1+9 = 65 (center plus adjacent row and column squares)

21+7+13+19+5 = 65 (center plus the remaining row and column squares)

4+10+13+16+22 = 65 (center plus diagonally adjacent squares)

20+2+13+24+6 = 65 (center plus the remaining squares on its diagonals)

ORDER 4N- SIXTEEN SQUARE MAGIC SQUARE IN JAIN TEMPLE

Most-perfect magic square from

the Parshvanath Jain temple in Khajuraho

Apart from the trivial case of the first order square, most-perfect magic squares are all of order 4n. In their book, Kathleen Ollerenshaw and David S. Brée give a method of construction and enumeration of all most-perfect magic squares. They also show that there is a one-to-one correspondence between reversible squares and most-perfect magic squares.

SMALLEST ANTIMAGIC SQUARES HAVE ORDER FOUR---- quadrants--- THERE ARE FOUR ANTIMAGIC SQUARES IN TOTAL

An antimagic square of order n is an arrangement of the numbers 1 to n2 in a square, such that the sums of the n rows, the n columns and the two diagonals form a sequence of 2n + 2 consecutive integers. The smallest antimagic squares have order 4.[1]

In each of these two antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38.[2] In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71.[2] In the antimagic square on the right, the rows, columns and diagonals add up to numbers between 59-70.[1]

16 PIECES OF SELF TILING SET- GEOMAGIC SQUARES- EACH OF THE 16 PIECES COMPOSED OF FOUR PIECES- FAMOUS

Figure 6: A geomagic square whose pieces comprise a self-tiling tile set

In addition to being geomagic, there exist squares with auxiliary properties making them even more distinctive. In Figure 6, for example, which is magic on rows and columns only, the 16 pieces form a so-called Self-tiling tile set. Such a set is defined as any set of n distinct shapes, each of which can be tiled by smaller replicas of the complete set of n shapes.[13]

A second example is Figure 4, which is a so-called 'self-interlocking' geomagic square. Here the 16 pieces are no longer contained within separate cells, but define the square cell shapes themselves, so as to mesh together to complete a square-shaped jigsaw.

I'm a

WITHIN CHINA'S MACAU STAMP THERE IS FOUR BY FOUR QUADRANT MODELS- it is a five by five quadrant geomagic square itself

On October 9, 2014 the post office of Macao in the People's Republic of China issued a series of stamps based on magic squares.[14] The stamp below, showing one of the geomagic squares created by Sallows, was chosen to be in this collection.[15]

THE PROBLEM BEGAN WITH FINDING SETS OF N=4--- THE BIG QUESTION THAT WAS ASKED AROUND SELF TILING CENTERED AROUND FINDING N EQUALS FOUR-

Figure 1: A 'perfect' self-tiling tile set of order 4

A self-tiling tile set, or setiset, of order n is a set of n shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of n shapes. That is, the n shapes can be assembled in n different ways so as to create larger copies of themselves, where the increase in scale is the same in each case. Figure 1 shows an example for n = 4 using distinctly shaped decominoes. The concept can be extended to include pieces of higher dimension. The name setisets was coined by Lee Sallows in 2012,[1][2] but the problem of finding such sets for n = 4 was asked decades previously by C. Dudley Langford, and examples for polyaboloes (discovered by Martin Gardner, Wade E. Philpott and others) and polyominoes (discovered by Maurice J. Povah) were previously published by Gardner.[3]

PROBLEM BEGAN AROUND FINDING N=4

From the above definition it follows that a setiset composed of n identical pieces is the same thing as a 'self-replicating tile' or rep-tile, of which setisets are therefore a generalization.[4] Setisets using n distinct shapes, such as Figure 1, are called perfect. Figure 2 shows an example for n = 4 which is imperfect because two of the component shapes are the same.

THE FOUR BY FOUR ALPHAMAGIC SQUARE (MADE OF QUADRANTS)- IS DIFFERENT- TRANSCENDENT

A surprisingly large number of 3 × 3 alphamagic squares exist—in English and in other languages. French allows just one 3 × 3 alphamagic square involving numbers up to 200, but a further 255 squares if the size of the entries is increased to 300. For entries less than 100, none occurs in Danish or in Latin, but there are 6 in Dutch, 13 in Finnish, and an incredible 221 in German. Yet to be determined is whether a 3 × 3 square exists from which a magic square can be derived that, in turn, yields a third magic square—a magic triplet. Also unknown is the number of 4 × 4 and 5 × 5 language-dependent alphamagic squares.

GALOIS THEORY SHOWS WHY IT IS POSSIBLE TO SOLVE DEGREES FOUR AND LOWER (FOUR IS DIFFERENT)- THERE IS NO FORMULA FOUR FIFTH ROOT OR HIGHER FIFTH IS ALWAYS QUESTIONABLE DIFFERENT

The birth and development of Galois theory was caused by the following question, whose answer is known as the Abel–Ruffini theorem:

Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?

Galois theory not only provides a beautiful answer to this question, it also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do. Further, it gives a conceptually clear, and often practical, means of telling when some particular equation of higher degree can be solved in that manner.

SOME QUINTICS CAN BE SOLVED- THE FIFTH IS QUESTIONABLE- BUT FORMULAS ARE ONLY FROM ONE TO FOUR- FIFTH IS ALWAYS QUESTIOANBLE

While Ruffini and Abel established that the general quintic could not be solved, some particular quintics can be solved, such as (x − 1)5=0, and the precise criterion by which a given quintic or higher polynomial could be determined to be solvable or not was given by Évariste Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its Galois group – had a certain structure – in modern terms, whether or not it was a solvable group. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degree.

Listen at 23:20 there is no proof for five- and he describes for four it is complex and different

FOURTH IS ALWAYS DIFFERENT FIFTH QUESTIONABLE- AS THE AZTECS SAW THINGS AZTECS SAW FOUR AS DIFFERENT AND THEY SAW FIVE AS CHAOS- I POSTED THE BOOK THAT DESCRIBED HOW THE AZTECS SAW THE NUMBER FOUR AS DIFFERENT AND TRANSCENDENT AND THEY SAW FIVE AS CHAOS- I STUDIED TONS OF LECTURES ON MATH THE PATTERN WAS EVERYWHERE

OK SO THIS IS WHAT I WANT TO EXPLAIN---- I WATCHED A TON OF LECTURES AND DOCUMENTARIES ON MATH LIKE TEACHING COMPANY AND THE PROOFS IT WAS ALL THE SAME PATTERN THE FOURTH DIFFERENT FIFTH QUESTIONABLE IF AT ALL- THAT WAS FOR ALL OF THE BIG PROOFS AND FAMOUS PROBLEMS- IF I HAD A DVD PLAYER I WOULD GO BACK AND GIVE YOU THOSE EXAMPLES ESPECIALLY IF IT PLAYED FAST MOTION- BUT HERE IS ONE EXAMPLE FROM A LECTURE I SAW ON ERDOS OF THAT THEME  -- FOR INSTANCE I HAD A LECTURE SERIES ON CHAOS THEORY IT DESCRIED AT THE FOURTH THINGS GOT DIFFERENT AT FIVE CHAOS I WATCHED ALL THESE LECTURES ON MATH I WENT TO LECTURES ALL THE TIME AT UCSD IT WAS ALL THE QUADRANT MODEL I JUST FORGET THE STUFF NOW

IF YOU LISTEN TO THIS AT 23:20 HE TALKS ABOUT THE FOUR LINES AND HOW AT FOUR IT BECOMES DIFFERENT AND COMPLEX- THEN HE SAYS "THERE IS OF YET NO ANSWER FOR FIVE POINTS"- FIVE IS ALWAYS QUESTIOANBLE- I AM TELLING YOU THAT I LISTENED TO LECTURES OF TEACHING COMPANY FROM GREATEST PROFESSORS THAT SAME PATTERN WAS ALWAYS THE CASE IN ALL OF THE HUGE THEOREMS AND IT WAS BIZARRE EVEN FOR THIS ONE THEOREM THEY COULD ONLY FIND THE FIRST FOUR NUMBERS IT TOOK UNTIL THE INVENTION OF ADVANCED COMPUTERS FOR THEM TO FIND THE FIFTH AND IT WAS AN EXTREMELY IMPORTANT THEOREM I JUST CANT REMEMBER THEM NOW

TETRAMAGIC CUBES- TETRA IS FOUR- THE TETRAMAGIC IS TRNASCENDENT

As in the case of magic squares, a bimagic cube has the additional property of remaining a magic cube when all of the entries are squared, a trimagic cube remains a magic cube under both the operations of squaring the entries and of cubing the entries.[1] (Only two of these are known, as of 2005.) A tetramagic cube remains a magic cube when the entries are squared, cubed, or raised to the fourth power.

THESE WORD SQUARES ARE QUADRANTS OF LETTERS

A word square is a special type of acrostic. It consists of a set of words written out in a square grid, such that the same words can be read both horizontally and vertically. The number of words, which is equal to the number of letters in each word, is known as the "order" of the square. For example, this is an order 5 square:

H E A R T

E M B E R

A B U S E

R E S I N

T R E N D

A popular puzzle dating well into ancient times, the word square is sometimes compared to the magic square, though apart from the fact that both use square grids there is no real connection between the two.

Contents [hide]

1 Early history

1.1 Sator Square

1.2 Abramelin the Mage

2 Modern English squares

2.1 Order 10 squares

3 Vocabulary

4 Variant forms

4.1 Double word squares

4.2 Diagonal word squares

4.3 Word rectangles

4.4 Other forms

5 References

Early history

Sator Square in Corinium (Cirencester), England

Sator Square

The Sator Square is a famous word square in Latin. Its canonical form reads as follows:

S A T O R

A R E P O

T E N E T

O P E R A

R O T A S

In addition to satisfying the basic properties of word squares, the Sator Square spread widely due to several other attributes: it is palindromic; it can be read as a sentence of obscure meaning; and additional meaning such as reference to the Christian Paternoster prayer can be derived from its letters. However, the word "Arepo" appears nowhere else in Latin literature; most of those who have studied the Sator Square agree that it is to be taken as a proper name, either an adaptation of a non-Latin word or, more likely, a name invented specifically for this sentence.[1] Thus the square consists of a palindrome ("tenet"), a reversal ("sator" and "rotas"), and a word ("opera") which can be reversed into a passably coined name ("Arepo").

Abramelin the Mage

If the "words" in a word square need not be true words, arbitrarily large squares of pronounceable combinations can be constructed. The following 12×12 array of letters appears in a Hebrew manuscript of The Book of the Sacred Magic of Abramelin the Mage of 1458, said to have been "given by God, and bequeathed by Abraham". An English edition appeared in 1898. This is square 7 of Chapter IX of the Third Book, which is full of incomplete and complete "squares".

I S I C H A D A M I O N

S E R R A R E P I N T O

I R A A S I M E L E I S

C R A T I B A R I N S I

H A S I N A S U O T I R

A R I B A T I N T I R A

D E M A S I C O A N O C

A P E R U N O I B E M I

M I L I O T A B U L E L

I N E N T I N E L E L A

O T I S I R O M E L I R

N O S I R A C I L A R I

No source or explanation is given for any of the "words", so this square does not meet the modern standards for legitimate word squares. Modern research indicates that a 12-square would be essentially impossible to construct from indexed words and phrases, even using a large number of languages. However, equally large English-language squares consisting of arbitrary phrases containing dictionary words are relatively easy to construct; they too are not considered true word squares, but they have been published in The Enigma and other puzzle magazines as "Something Different" squares.

Modern English squares

A specimen of the order-six square (or 6-square) was first published in English in 1859; the 7-square in 1877; the 8-square in 1884; and the 9-square in 1897.[2]

Here are examples of English word squares up to order eight:

B I T C A R D H E A R T G A R T E R B R A V A D O L A T E R A L S

I C E A R E A E M B E R A V E R S E R E N A M E D A X O N E M A L

T E N R E A R A B U S E R E C I T E A N A L O G Y T O E P L A T E

D A R T R E S I N T R I B A L V A L U E R S E N P L A N E D

T R E N D E S T A T E A M O E B A S R E L A N D E D

R E E L E D D E G R A D E A M A N D I N E

O D Y S S E Y L A T E E N E R

S L E D D E R S

The following is one of several "perfect" nine-squares (all words in major dictionaries, uncapitalized, and unpunctuated):[3]

A C H A L A S I A

C R E N I D E N S

H E X A N D R I C

A N A B O L I T E

L I N O L E N I N

A D D L E H E A D

S E R I N E T T E

I N I T I A T O R

A S C E N D E R S

Order 10 squares

A 10-square is naturally much harder to find, and a "perfect" 10-square has been hunted since 1897.[2] It has been called the Holy Grail of logology.

Various methods have produced partial results to the 10-square problem:

Tautonyms

Since 1921, 10-squares have been constructed from reduplicated words and phrases like "Alala! Alala!" (a reduplicated Greek interjection). Each such square contains five words appearing twice, which in effect constitutes four identical 5-squares. Darryl Francis and Dmitri Borgmann succeeded in using near-tautonyms (second- and third-order reduplication) to employ seven different entries by pairing "orangutang" with "urangutang" and "ranga-ranga" with "tanga-tanga", as follows:[4]

O R A N G U T A N G

R A N G A R A N G A

A N D O L A N D O L

N G O T A N G O T A

G A L A N G A L A N

U R A N G U T A N G

T A N G A T A N G A

A N D O L A N D O L

N G O T A N G O T A

G A L A N G A L A N

However, "word researchers have always regarded the tautonymic ten-square as an unsatisfactory solution to the problem."[2]

80% solution

In 1976, Frank Rubin produced an incomplete ten-square containing two nonsense phrases at the top and eight dictionary words. If two words could be found containing the patterns "SCENOOTL" and "HYETNNHY", this would become a complete ten-square.

Constructed vocabulary

From the 1970s, Jeff Grant had a long history of producing well-built squares; concentrating on the ten-square from 1982 to 1985, he produced the first three traditional ten-squares by relying on reasonable coinages such as "Sol Springs" (various extant people named Sol Spring) and "ses tunnels" (French for "its tunnels"). His continuing work produced one of the best of this genre, making use of "impolarity" (found on the Internet) and the plural of "Tony Nader" (found in the white pages), as well as words verified in more traditional references:

D I S T A L I S E D

I M P O L A R I T Y

S P I N A C I N E S

T O N Y N A D E R S

A L A N B R O W N E

L A C A R O L I N A

I R I D O L I N E S

S I N E W I N E S S

E T E R N N E S S E

D Y S S E A S S E S

Personal names

By combining common first and last names and verifying the results in white-pages listings, Steve Root of Westboro, Massachusetts, was able to document the existence of all ten names below (total number of people found is listed after each line):

L E O W A D D E L L 1

E M M A N E E L E Y 1

O M A R G A L V A N 5

W A R R E N L I N D 9

A N G E L H A N N A 2

D E A N H O P P E R 10+

D E L L A P O O L E 3

E L V I N P O O L E 3

L E A N N E L L I S 3

L Y N D A R E E S E 5

Geographic names

Around 2000, Rex Gooch of Letchworth, England, analyzed available wordlists and computing requirements and compiled one or two hundred specialized dictionaries and indexes to provide a reasonably strong vocabulary. The largest source was the United States Board on Geographic Names National Imagery and Mapping Agency. In Word Ways in August and November 2002, he published several squares found in this wordlist. The square below has been held by some word square experts as essentially solving the 10-square problem (Daily Mail, The Times), while others anticipate higher-quality 10-squares in the future.[2][5]

D E S C E N D A N T

E C H E N E I D A E

S H O R T C O A T S

C E R B E R U L U S

E N T E R O M E R E

N E C R O L A T E R

D I O U M A B A N A

A D A L E T A B A T

N A T U R E N A M E

T E S S E R A T E D

There are a few "imperfections": "Echeneidae" is capitalized, "Dioumabana" and "Adaletabat" are places (in Turkey and Guinea respectively), and "nature-name" is hyphenated.

Many new large word squares and new species[clarification needed] have arisen recently. However, modern combinatorics has demonstrated why the 10-square has taken so long to find, and why 11-squares are extremely unlikely to be constructible using English words (even including transliterated place names). However, 11-squares are possible if words from a number of languages are allowed (Word Ways, August 2004 and May 2005).

Vocabulary

It is possible to estimate the degree of difficulty of constructing word squares. 5-squares can be constructed with as little as a 250-word vocabulary. Roughly, for each step upwards, one needs four times the number of words. For a 9-square, one needs over 60,000 9-letter words, which is practically all of those in single very large dictionaries.

For large squares, the vocabulary prevents selecting more "desirable" words (i.e. words that are unhyphenated, in common use, without contrived inflections, and uncapitalized), and any resulting word squares use exotic words. The opposite problem occurs with small squares: a computer search will produce millions of examples, most of which use at least one obscure word. In such cases finding a word square with "desirable" (as described above) words is performed by elimination of the more exotic words or by using a smaller dictionary with only common words. Smaller word squares, used for amusement, are expected to have simple solutions, especially if set as a task for children; but vocabulary in most eight-squares tests the knowledge of an educated adult.

Variant forms

Double word squares

Word squares that form different words across and down are known as "double word squares". Examples are:

T O O

U R N

B E E L A C K

I R O N

M E R E

B A K E S C E N T

C A N O E

A R S O N

R O U S E

F L E E T A D M I T S

D E A D E N

S E R E N E

O P I A T E

R E N T E R

B R E E D S

The rows and columns of any double word square can be transposed to form another valid square. For example, the order 4 square above may also be written as:

L I M B

A R E A

C O R K

K N E E

Double word squares are somewhat more difficult to find than ordinary word squares, with the largest known fully legitimate English examples (dictionary words only) being of order 8. Puzzlers.org gives an order 8 example dating from 1953, but this contains six place names. Jeff Grant's example in the February 1992 Word Ways is an improvement, having just two proper nouns ("Aloisias", a plural of the personal name Aloisia, a feminine form of Aloysius, and "Thamnata", a Biblical place-name):

T R A T T L E D

H E M E R I N E

A P O T O M E S

M E T A P O R E

N A I L I N G S

A L O I S I A S

T E N T M A T E

A S S E S S E D

Diagonal word squares

Diagonal word squares are word squares in which the main diagonals are also words. There are four diagonals: top-left to bottom-right, bottom-right to top-left, top-right to bottom-left, and bottom-left to top-right. In a Single Diagonal Square (same words reading across and down), these last two will need to be identical and palindromic because of symmetry. The 8-square is the largest found with all diagonals: 9-squares exist with some diagonals.

This is an example of a diagonal double square of order 4:

B A R N

A R E A

L I A R

L A D Y

Word rectangles

Word rectangles are based on the same idea as double word squares, but the horizontal and vertical words are of a different length. Here are 4×8 and 5×7 examples:

F R A C T U R E

O U T L I N E D

B L O O M I N G

S E P T E T T E G L A S S E S

R E L A P S E

I M I T A T E

S M E A R E D

T A N N E R Y

Again, the rows and columns can be transposed to form another valid rectangle. For example, a 4×8 rectangle can also be written as an 8×4 rectangle.

Other forms

Numerous other shapes have been employed for word-packing under essentially similar rules. The National Puzzlers' League maintains a full list of forms which have been attempted.

FOUR COCENTRIC CIRCLES

Magic circles were invented by the Song dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical. One of his magic circles was constructed from 33 natural numbers from 1 to 33 arranged on four concentric circles, with 9 at the center.

I ATTENDED ALMOST ALL OF THE MATH LECTURES FROM ALMOST EVERY CLASS AT UCSD THERE WAS SO MUCH MATRIX MATHEMATICS AND MATRICES ARE QUADRANTS

16 SQUARES VERY FAMOUS- THE IMAGE OF THE MYSTIC DIAGRAM YANTRA HE CREATS IN A FOUR BY FOUR 16 SQUARE QUADRANT MODEL

Image of Sri Rama Chakra as a magic square given in the Panchangam published by Sringeri Sharada Peetham.

In some almanacs, for example, in the Panchangam published by the Sringeri Sharada Peetham[1] or the Pnachangam published by Srirangam Temple,[2] the diagram takes the form of a magic square of order 4 with certain special properties.

Sri Rama Chakras

Sringeri/Srirangam Panchangams

9 16 5 4

7 2 11 14

12 13 8 1

6 3 10 15

This is a magic square of order 4. The sum of the numbers in every row, every column and each diagonal are all equal to 34.

Sri Rama Chakra as a strongly magic square

Let M be a magic square of order 4 and let it be represented by matrix as follows:

{\displaystyle M={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\a_{41}&a_{42}&a_{43}&a_{44}\\\end{bmatrix}}}

The numbers in each row, and in each column, and the numbers that run diagonally in both directions, all add up to the number 34. M is called a strongly magic square if the following condition is satisfied:[5]

For all m, n such that 1 ≤ m ≤ 4, 1 ≤ n ≤ 4, we have

{\displaystyle a_{m,n}+a_{m,n+1}+a_{m+1,n}+a_{m+1,n+1}=34},

where it is assumed that if a subscript exceeds 4 it is replaced by 1 (wrapping around rows and columns).

For example in a strongly magic square M the following must be true.

{\displaystyle a_{44}+a_{41}+a_{14}+a_{11}=34} (taking m = 4, n = 4)

One can easily verify that the magic square represented by the Sri Rama Chakra is a strongly magic square.

16 SQUARE HOLES

Nyctography is a form of substitution cipher writing created by Charles Lutwidge Dodgson (better known as Lewis Carroll) in 1891.

The device consisted of a gridded card with sixteen square holes, each a quarter inch wide, and system of symbols representing an alphabet of Dodgson's design, which could then be transcribed the following day.

THERE ARE FOUR ELEMENTARY MATHEMATICAL OPERATIONS

Multiplication (often denoted by the cross symbol "×", by a point "·", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

FOUR FOURS

Four fours

Four fours is a mathematical puzzle. The goal of four fours is to find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical symbols and the digit four (no other digit is allowed). Most versions of four fours require that each expression have exactly four fours, but some variations require that each expression have the minimum number of fours.

FOUR INTEGERS IT IS A QUADRANT (24 is six times four the circle is divided into four in the game)

The 24 Game is an arithmetical card game in which the objective is to find a way to manipulate four integers so that the end result is 24. For example, for the card with the numbers 4, 7, 8, 8, a possible solution is

The game has been played in Shanghai since the 1960s,[citation needed] using playing cards. It is similar to the card game Maths24.

I POSTED THE ARTICLE THAT SAYS THAT EUCLIDS FIFTH POSTULATE WAS WRONG AND HIS FOURTH WAS DIFFERENT SO DIFFERENT MANY ANCIENT MATHEMATICIANS SAID IT WAS NOT A POSTULATE THE FOURTH IS ALWAYS DIFFERENT AND TRANSCENDENT- THE FOUR POSTULATES WERE THE BASIS OF ALL OF GEOMETRY- EUCLID NEVER NEEDED TO USE THE FIFTH POSTULATE AND IT WAS WRONG ANYWAYS SO THERE WAS ONLY FOUR THE FOURTH WAS DIFFERENT THE QUADRANT PATTERN

Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates:

1. To draw a straight line from any point to any point.

2. To produce [extend] a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance [radius].

4. That all right angles are equal to one another.

For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction, including Ibn al-Haytham (Alhazen, 11th century),[1] Omar Khayyám (12th century), Nasīr al-Dīn al-Tūsī (13th century), and Giovanni Girolamo Saccheri (18th century).

Giordano is most noted nowadays for a theorem on Saccheri quadrilaterals that he proved in his 1668 book Euclide restituo (named after Borelli's Euclides Restitutus of 1658).

In examining Borelli's proof of the parallel postulate, Giordano noted that it depended upon the assumption that a line everywhere equidistant from a straight line is itself straight. This in turn is due to Clavius, whose proof of the assumption in his 1574 Commentary on Euclid is faulty.[1][2] So using a figure he found in Clavius, now called a Saccheri quadrilateral, Giordano tried to come up with his own proof of the assumption, in the course of which he proved:

If ABCD is a Saccheri quadrilateral (angles A and B right angles, sides AD and BC equal) and HK is any perpendicular from DC to AB, then

(i) the angles at C and D are equal, and

(ii) if in addition HK is equal to AD, then angles C and D are right angles, and DC is equidistant from AB.

The interesting bit is the second part (the first part had already been proved by Omar Khayyám in the 11th century), which can be restated as:

If 3 points of a line CD are equidistant from a line AB then all points are equidistant.

Which is the first real advance in understanding the parallel postulate in 600 years.[3][4]

FOUR IDENTITCAL RIGHT TRIANGLES- THE FAMOUS 345 PYTHAGOREAN TRIANGLE HAD THE FOUR SQUARED WHICH IS 16 WHICH EGYPTIANS SAW AS THEIR MAIN GOD- THEY SAW THE 16 sQUARES AS THEIR MAIN GOD- BECAUSE THE QUADRANT MODEL IS THE FORM OF BEING

Animation showing proof by rearrangement of four identical right triangles

THE PYTHAGOREAN THEOREM INVOLVES FOUR RECTANGLES (two squares two rectangles making a sort of quadrant)- THE PROOF ATTRIBUTED TO PYTHAGORAS WAS SORT OF A QUADRANT--- PYTHAGORAS SAID HE HAD LIVED FOUR LIVES

The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it.[2] In any event, the proof attributed to him is very simple, and is called a proof by rearrangement.

The two large squares shown in the figure each contain four identical triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, Q.E.D.[9]

That Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus.[10] Several other proofs of this theorem are described below, but this is known as the Pythagorean one.

FOUR LEMMATA

For the formal proof, we require four elementary lemmata:

If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (side-angle-side).

The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.

The area of a rectangle is equal to the product of two adjacent sides.

The area of a square is equal to the product of two of its sides (follows from 3).

FOUR COPIES OF RIGHT TRIANGLES

The theorem can be proved algebraically using four copies of a right triangle with sides a, b and c, arranged inside a square with side c as in the top half of the diagram.[22] The triangles are similar with area

{\tfrac {1}{2}}ab, while the small square has side b − a and area (b − a)2. The area of the large square is therefore

.

{\displaystyle (b-a)^{2}+4{\frac {ab}{2}}=(b-a)^{2}+2ab=b^{2}-2ab+a^{2}+2ab=a^{2}+b^{2}.\,}

But this is a square with side c and area c2, so

.

c^{2}=a^{2}+b^{2}.\,

A similar proof uses four copies of the same triangle arranged symmetrically around a square with side c, as shown in the lower part of the diagram.[23] This results in a larger square, with side a + b and area (a + b)2. The four triangles and the square side c must have the same area as the larger square,

(b+a)^{2}=c^{2}+4{\frac {ab}{2}}=c^{2}+2ab,\,

giving

.

{\displaystyle c^{2}=(b+a)^{2}-2ab=b^{2}+2ab+a^{2}-2ab=a^{2}+b^{2}.\,}

Diagram of Garfield's proof

A related proof was published by future U.S. President James A. Garfield (then a U.S. Representative).[24][25] Instead of a square it uses a trapezoid, which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. The area of the trapezoid can be calculated to be half the area of the square, that is

.

{\frac {1}{2}}(b+a)^{2}.

The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of

1

2

{\frac {1}{2}}, which is removed by multiplying by two to give the result.

KNOWLEDGE OF THE PYTHAGOREAN THEOREM FOUR PARTS

The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.

THE FOUR SQUARED SIDE THE 16 SQUARES IS THE QUADRANT MODEL- THE EGYPTIANS WORSHIPPED THOSE 16 SQUARES AS MAIN GOD

Animation demonstrating the simplest Pythagorean triple, 3 squared + 4 squared = 5 squared

WITHIN THE VESICA PISCES IS THE QUADRANT- PLATO SAID THAT IT WAS THE CORE OF ALL SOLIDS

According to his disciple and follower, Plato, the circle and the interaction of two circles, where the center of each circle lies on the circumference of the other (the so-called Vesica Piscis), became the core of all solids. Following this train of thought, Geometry

Vesica Piscis with figures within

LOOK HOW ISIS REPRESENTS THE 16 SQUARES

So what term can we use to replace the term “Pythagorean Theorem”? We can see the civil engineers and architects of the Giza complex built the proof of the theorem into the design of the Pyramid complex at Giza. So the term we use as a replacement for the term “Pythagorean Theorem” should pay homage to the African Pyramid builders, architects, and engineers. Dr. Kaba Kamene (Dr. Booker T. Coleman) suggests that the name “Pythagoras” comes from a Greek amalgamation of the names of African Egyptian deities Ptah and Horus (Heru). Interestingly enough, the deities Ptah and Horus were patron deities of the Ancient African Pyramid builders, architects, and engineers. Ptah was a patron deity of builders and craftsmen, and Horus as Heru-Behutet was a patron deity of blacksmiths and workers in metal, The Great Chiefs of the Hammer.

Additionally, Ptah and Ausar were combined in Egyptian Mythology, and both Ptah as Ausar, and Horus are present as the upright and hypotenuse of the 3-4-5 right triangle ratio represented by Ausar, Aset, and Heru respectively.

Considering the aforementioned phonetic and symbolic relationships of Ptah and Horus to “Pythagoras”, then the proposition is put forth that the Pythagorean Theorem should henceforth and forever be known as the Ptah-Horus Theorem.

FOUR BAGUA

What can be said of the Eight Trigrams of Taoist cosmology, also known as the Bagua, that hasn’t already been said on ten thousand other web sites? Here I show the trigrams and their duals together in an arrangement that places one or two of them at a site of the four elements.

When considering the binary values of the trigrams, this arrangement is reminiscent of my Marriage of Opposites, Part 2. In doing this each link between them represents a common value for a trigram line. For example between Heaven / Earth and Water / Fire the 2nd line is yang for both Heaven and Water, and the 2nd line is yin for both Earth and Fire. Opposite this link is its reverse: the 2nd line is yin for both Thunder and Mountain, and yang for both Wind and Lake. All six links have this quality.

Two smaller diagrams show the common English names as well as the corresponding attributes of the trigrams. Interestingly, if Heaven and Wind are considered Air, Lake is considered Water, Mountain is considered Earth, and Thunder is considered Fire, then you have each element mentioned twice, once above and once below, and a crossed loop of pairs: Air / Earth, Fire / Air, Water / Fire, Earth / Water, and then back to Air / Earth. Nice!

16 SQUARES QMR- AFRICAN DIVINATION- FOUR BIT COMBINATIONS IFA FOUR TABLET SYSTEM SOUTH AFRICA SIKIDY FOUR BY FOUR MATRIX SYSTEM- 16 TETRAGRAMS

The Sixteen Tetragrams

One of the 65,536 possible Sikidy tableaux

The Chinese have mostly been using their binary combinatorics for

philosophical and religious things. This is also true for the African use of

binary systems.

There are actually at least three types of African binary divination

techniques, all of which use four-bit combinations: Ifa (West Africa), the

four-tablet system (South Africa) and Sikidy (Madagascar). The former two

are quite straight-forward (randomise a combination and interpret its

meaning), but Sikidy requires more advanced computation.

The first step in Sikidy is to randomise four columns of four bits (a

four-by-four matrix). The randomisation of one bit usually happens by

grabbing a handful of seeds from a bag and removing two seeds at a time

until only one or two seeds are left. (This gives a totally new meaning to

the concept of "random number seed"). The one or two seeds are then placed

in their proper position on the Sikidy board.

The figure above shows an example of a completed Sikidy board. The

randomised columns (called "Mother-Sikidy") are in the upper right corner.

The values of the columns from right to left, bottom to top are: 1010, 1001,

1011, 0010.

The next thing to do is to form the "Daughter-Sikidy" by rotating and

flipping the matrix. The rightmost column of the Mother-Sikidy (bottom to

top) becomes the top row (left to right) of the Daughter-Sikidy, and so

forth. Our Daughter-Sikidy (placed to the left of the Mother-Sikidy) is

therefore: 0110, 1101, 0000, 0111.

The rest is pure binary arithmetic. The columns below the Mother-Sikidy and

Daughter-Sikidy are formed by eXclusive-ORing each pair of columns: 1010 XOR

1001 = 0011, 1011 XOR 0010 = 1001, etc. This process is then repeated to all

the new lines until there is only one column left (the bottom column, 0110

in the example).

We now have a complete Sikidy tableau and what is left is the

interpretation: each of the sixteen binary values has its own meaning, and

each of the "memory slots" also has a designated meaning. The interpretation

requires a high level of expertise (or perhaps just a wild imagination).

The Sikidy system was also adopted by Arabs (under the name of "ilm

al-raml", "the science of sand"), and from Arabs it even transferred to

Europe in the Middle Ages. In Europe, it was known as "Arabic geomancy", a

small branch of Arabic occultism. All kinds of freaks extended to system to

include relationships with astrology, numerology, tarot and other things.

Binary Base-16/10 Latin name Direction Gender Element

------ ---------- ---------- --------- ------ -------

0000 0 Populus (people) both F Water

0001 1 Laetitia (joy) up M Air

0010 2 Rubeus (red) up M Fire

0011 3 Fortuna Minor (small fortune) up F Earth

0100 4 Albus (white) down F Water

0101 5 Amissio (loss) up M Fire

0110 6 Conjunctio (reunion) both M Air

0111 7 Cauda Draconis (dragon's tail) up F Earth

1000 8 (-8) Tristitia (sadness) down F Earth

1001 9 (-7) Carcer (prison) both F Water

1010 A (10, -6) Acquisitio (gain) down M Air

1011 B (11, -5) Puer (boy) up M Air

1100 C (12, -4) Fortuna Major (big fortune) down M Fire

1101 D (13, -3) Puella (girl) down F Water

1110 E (14, -2) Caput Draconis (dragon's head) down M Fire

1111 F (15, -1) Via (way) both F Water

BOOK OF SQUARE NUMBERS DAVINCI QUAD

In the 1220s Leonardo was invited to appear before the emperor Frederick II at Pisa. For the next several years, he corresponded with Frederick II and his scholar. In 1225, he dedicated his Liber quadratorum ("Book of Square Numbers") to Frederick.

AT 2:30 THE PROFESSOR SAYS THAT THE 16 SQUARE MAGIC SQUARE (THE QUADRANT MODEL) IS RELATED TO THE STRENGTH OF THE ELECTRIC FIELD TO THE QUANTUM- AND HE SAYS IT IS PROBABLY "COINCIDENCE" BUT "WHO KNOWS"

I DESCRIBED I USED TO WATHC LECUTRES LIKE THIS AND I SAW THE QUADRANT EVERYWHERE IT LITERALLY WAS THE FOUNDAITON THE QUADRANT PATTERN WHERE THE FOURTH WAS DIFFERENT I JUST FORGET THEM NOW

IN DAOIST TEMPLES THE DAOIST PRIESTS WALK IN ACCORDANCE TO THE QUADRANTS OF THE MAGIC SQUARE BELIEVING THIS GIVES THEM SUPERNATURAL POWERS AND HE POINTS OUT THAT THE ODD NUMBERS IN THE MAGIC SQUARE MAKE A CROSS- THEY MAKE A QUADRANT- THAT IS THE THREE BY THREE SQUARE WHERE THERE IS A CROSS INSIDE- THEN HE TALKS ABOUT THE JAINA MAGIC SQUARE WHICH IS 16 SQUARES (THE QUADRANT MODEL)- HE SAYS PEOPLE WOULD WEAR MAGIC SQUARES FOR PROTECTION AND GOOD LUCK- the four corners squares add up to it IT IS CALLED "THE MOST PERFECT MAGIC SQUARE POSSIBLE"- durers magic square up to 16- DURA AND AGRIPPAS FOUR BY FOUR SQUARES HE TALKS ABOUT- AGRIPPA THOUGHT THAT MAGIC SQUARE WAS TOO POWERFUL SO HE KEPT IT SECRET THE JUPITER SQUARE WAS THE FOUR BY FOUR- JUPITER WAS CALLED THE KING OF THE GODS EVEN THE ONLY GOD RELATED TO NUMBER FOUR- USED FOR SYGILS

I DESCRIBED I USED TO WATHC LECUTRES LIKE THIS AND I SAW THE QUADRANT EVERYWHERE IT LITERALLY WAS THE FOUNDAITON THE QUADRANT PATTERN WHERE THE FOURTH WAS DIFFERENT I JUST FORGET THEM NOW

IN DAOIST TEMPLES THE DAOIST PRIESTS WALK IN ACCORDANCE TO THE QUADRANTS OF THE MAGIC SQUARE BELIEVING THIS GIVES THEM SUPERNATURAL POWERS AND HE POINTS OUT THAT THE ODD NUMBERS IN THE MAGIC SQUARE MAKE A CROSS- THEY MAKE A QUADRANT- THAT IS THE THREE BY THREE SQUARE WHERE THERE IS A CROSS INSIDE- THEN HE TALKS ABOUT THE JAINA MAGIC SQUARE WHICH IS 16 SQUARES (THE QUADRANT MODEL)- HE SAYS PEOPLE WOULD WEAR MAGIC SQUARES FOR PROTECTION AND GOOD LUCK- the four corners squares add up to it IT IS CALLED "THE MOST PERFECT MAGIC SQUARE POSSIBLE"- durers magic square up to 16- DURA AND AGRIPPAS FOUR BY FOUR SQUARES HE TALKS ABOUT- AGRIPPA THOUGHT THAT MAGIC SQUARE WAS TOO POWERFUL SO HE KEPT IT SECRET THE JUPITER SQUARE WAS THE FOUR BY FOUR- JUPITER WAS CALLED THE KING OF THE GODS EVEN THE ONLY GOD RELATED TO NUMBER FOUR- USED FOR SYGILS

THE SATURN MAGIC SQUARE (LHO SHU SQUARE) HAS A CROSS/QUADRANT INSIDE OF IT OF THE ODD NUMBERS AND A SIGIL IS MADE FROM IT

AT 5:13 THE SIGIL FOR THE MOON MAGIC SQUARE IS FOUR MOONS IN FOUR QUADRANTS

AT 3:48 THE SATOR SQUARE AND THE CROSS

PATERNOSTER AND TENET CROSS IN THE MIDDLE AND THE FOUR QUADRANTS THE AS AND THE OS ARE THE ALPHAS AND OMEGAS IN THE FOUR QUADRANTS

FOUR MINUTES- THE CROSS/QUADRANT AND THE SIGIL OF JUPITER 16 SQUARE JUPITER MAGIC SQUARE

AT 40 minutes TALKING ABOUT TETRACTYS AND TETRAGRAMMATON

SHE DESCRIBES THERE ARE TWO DISTINCT TYPES OF MAGIC SQUARE IN REGARDS TO CONSTRUCTION THOSE DIVISIBLE BY FOUR AND THOSE NOT DIVISIBLE BY FOUR- AT 1:40 SHE ELUCIDATES THE CHI/X- AT 4:40 THE SIGIL QUADRANT IS PRESENTED

TETRACTYS IS ALL SEEING EYE- HE DESCRIBES "PYTHAGORAS WAS OBSESSED WITH FOURTHNESS"

Mercury (MERK) also represented by the number four AND WAS CONSIDERED THE SAME AS ODIN- his square was 8 by 8 64 EIGHT IS TWO FOURS- AT 2:00 LOOK AT THE QUADRANT HE ALSO CARRIES AN ANKH A CROSS- SYMBOL OF MERCURY IS CRESCENT CIRCLE AND CROSS

The four basic forms of symmetry used in architecture to reinforce the concepts of grouping, order and patterning are:

1. Translation: defined as the parallel movement of a plane-figure from one position to another.

(figure available in print form)

2. Rotation: defined as the movement of a plane figure or object around an axis.

(figure available in print form)

3. Reflection: defined as the bending or folding back of an object upon itself.

(figure available in print form)

4. Glide reflection which is a combination of translation and reflection

PAULI SAW THE NUMBER 137 AS MYSTICAL IT IS THE FOURTH STERN PRIME NOTICE HOW THE FIRST THREE ARE SIMILAR THE FOURTH IS DIFFERENT A HUGE JUMP TO THE FOURTH

A Stern prime, named for Moritz Abraham Stern, is a prime number that is not the sum of a smaller prime and twice the square of a non zero integer. Or, to put it algebraically, if for a prime q there is no smaller prime p and nonzero integer b such that q = p + 2b², then q is a Stern prime. The known Stern primes are

2, 3, 17, 137, 227, 977, 1187, 1493 (sequence A042978 in the OEIS).

TETRA IS FOUR
Plato's Lambda and its tetrahedral
generalisation as the arithmetic version of sacred geometries
http://www.smphillips.mysite.com/plato%27s-lambda.html
Tetrahedron is four

PLATOS LAMBDA IS FOUR LEVELS TWO TETRACTYS- THE WORLD SOUL IS TWO TETRACTYS TWO SERIES OF FOUR TERMS

In Timaeus, Plato’s treatise on Pythagorean cosmology, the central character, Timaeus of Locri (possibly a real person), describes how the Demiurge divided the World Soul into harmonic intervals. Having blended the three ingredients of the World Soul — Sameness, Difference and Existence — into a kind of malleable stuff, the Demiurge took a strip of it and divided its length into portions measured by the numbers forming two geometrical series of four terms each: 1, 2, 4, 8 and 1, 3, 9, 27, generated by multiplying 1 by 2 and 3 (Fig. 1). This became known as "Plato’s Lambda" because of its resemblance to Λ, the Greek letter lambda. Then, according to Timaeus:

“he went on to fill up both the double and the triple intervals, cutting off yet more parts from the original mixture and placing them between the terms, so that within each interval there were two means, the one (harmonic)* exceeding the one extreme and being exceeded by the other by the same fraction of the extremes, the other (arithmetic) exceeding the one extreme by the same number whereby it was exceeded by the other.These links gave rise to intervals of 3/2 and 4/3 and 9/8 within the original intervals. And he went on to fill up all the intervals of 4/3 (i.e., fourths) with the interval of 9/8 (the tone), leaving over in each a fraction. This remaining interval of the fraction had its terms in the numerical proportion of 256 to 243 (semitone). By this time, the mixture from which he was cutting off these portions was all used up.”

AZTECS USED NUMBER SOLELY USING 1-4-- AGAIN I DISCUSSED THE FOUR PLUS ... PATTERN- AZTECS SAW FOUR AS SACRED AND FIVE AS CHAOS AND EXCESS- FIVE WAS REPRESENTED AS FOUR PLUS ONE- NUMBERS WERE DONE THROUGH FOURS THROUGH QUADRANTS- THIRTEEN WAS FOUR PLUS FOUR PLUS FOUR PLUS ONE THREE QUADRANTS PLUS ONE- THAT IS THE QUADRANT MODEL

There are four types of attractors. Figure 1 describes these types: fixed point, limit-cycle, limit-torus, and strange attractor

III. Classifying an Attractor

In this section, we look at four distinct attractors and consider the problem of identifying some of their prop- erties. We are mainly interested in how to identify their general classification: nonstrange, strange nonchaotic, or strange chaotic. We use Lyapunov exponents to identify chaotic behavior and we use what we know about fractals to identify strangeness. Plots of the orbits in each model also help us gain an intuitive feel for the type of attracting set.

Properties and classifications of the four examples of attractors discussed in Section III.

Attractor

Lyapunov spectrum

Dimension Phase sens.

Classification

Buckling Rayleigh GOPY Lorenz

λ1 = −0.25 λ1 =0

λ1 =0

λ1 =0.81

λ2 = −0.25

λ2 =−1.06

λ2 =−1.53

λ2 =0.02 λ3 =−14.50

DKY = 0 DKY = 1 DKY = 1 DKY = 2.06

n/a

n/a

ΓN = (2.1)N1.08 n/a

nonstrange [fixed point] nonstrange [limit cycle] strange nonchaotic strange chaotic

FOUR CLASSES

Wolfram, in A New Kind of Science and several papers dating from the mid-1980s, defined four classes into which cellular automata and several other simple computational models can be divided depending on their behavior. While earlier studies in cellular automata tended to try to identify type of patterns for specific rules, Wolfram's classification was the first attempt to classify the rules themselves. In order of complexity the classes are:

Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state. Any randomness in the initial pattern disappears.[36]

Class 2: Nearly all initial patterns evolve quickly into stable or oscillating structures. Some of the randomness in the initial pattern may filter out, but some remains. Local changes to the initial pattern tend to remain local.[36]

Class 3: Nearly all initial patterns evolve in a pseudo-random or chaotic manner. Any stable structures that appear are quickly destroyed by the surrounding noise. Local changes to the initial pattern tend to spread indefinitely.[36]

Class 4: Nearly all initial patterns evolve into structures that interact in complex and interesting ways, with the formation of local structures that are able to survive for long periods of time.[37] Class 2 type stable or oscillating structures may be the eventual outcome, but the number of steps required to reach this state may be very large, even when the initial pattern is relatively simple. Local changes to the initial pattern may spread indefinitely. Wolfram has conjectured that many, if not all class 4 cellular automata are capable of universal computation. This has been proven for Rule 110 and Conway's game of Life.

FOUR CLASSES FOURTH DIFFERENT

There have been several attempts to classify cellular automata in formally rigorous classes, inspired by the Wolfram's classification. For instance, Culik and Yu proposed three well-defined classes (and a fourth one for the automata not matching any of these), which are sometimes called Culik-Yu classes; membership in these proved undecidable.[40][41][42] Wolfram's class 2 can be partitioned into two subgroups of stable (fixed-point) and oscillating (periodic) rules.[43]

The idea that there are 4 classes of dynamical system came originally from nobel-prize winning chemist Ilya Prigogine who identified these 4 classes of for thermodynamical systems - (1) systems in thermodynamic equilibrium, (2) spatially/temporally uniform systems, (3) chaotic systems, and (4) complex far-from-equilibrium systems with dissipative structures (see figure 1 in Nicolis' paper (Prigogine's student)): [44] (Nicolis was Prigogine's student).

THE FOUR TYPES OF CELLS

There are four types of cells: neuron body, axon, dendrite and blank. The growth phase is followed by a signaling- or processing-phase

FOUR STATES

A Wireworld cell can be in one of four different states, usually numbered 0–3 in software, modeled by colors in the examples here:

Empty (Black)

Electron tail (Red)

Conductor (Yellow)

As in all cellular automata, time proceeds in discrete steps called generations (sometimes "gens" or "ticks"). Cells behave as follows:

Empty → Empty

Electron tail → Conductor

Conductor → Electron head if exactly one or two of the neighbouring cells are electron heads, or remains Conductor otherwise.

Wireworld uses what is called the Moore neighborhood, which means that in the rules above, neighbouring means one cell away (range value of one) in any direction, both orthogonal and diagonal.

These simple rules can be used to construct logic gates (see below).

https://en.wikipedia.org/wiki/Langton%27s_ant
Squares on a plane are colored variously either black or white.We arbitrarily identify one square as the "ant". The ant can travel in any of the four cardinal directions at each step it takes. The "ant" moves according to the rules below:

At a white square, turn 90° right, flip the color of the square, move forward one unit
At a black square, turn 90° left, flip the color of the square, move forward one unit

THE FOUR CLASSES OF CELLULAR AUTOMATA

For larger cellular automaton rule space, it is shown that class 4 rules are located between the class 1 and class 3 rules.[63] This observation is the foundation for the phrase edge of chaos, and is reminiscent of the phase transition in thermodynamics.

FOUR TYPES

In his influential paper "University and Complexity in Cellular Automata" [Physica D 10 (1984) 1-35; all page numbers refer to the reprint in Cellular Automata and Complexity, Addison-Wesley 1994, pp. 115-157], Stephen Wolfram proposed a classification of cellular automaton rules into four types, according to the results of evolving the system from a "disordered" initial state:

Evolution leads to a homogeneous state.

Evolution leads to a set of separated simple stable or periodic structures.

Evolution leads to a chaotic pattern.

Evolution leads to complex localized structures, sometimes long-lived.

AT FOUR BIFURCATIONS IN CHAOS THEORY THINGS GET DIFFERENT- I STUDIED A TEACHING COMPANY COURSE ON THIS I WATCHED THE LECTURE THE PROFESSOR EXPLAINED HOW AT FOUR THINGS GET DIFFERENT AT FIVE IT IS CHAOS THAT IS WHAT THE AZTECS SAID THEY SAID FOUR IS DIFFERENT IT IS SACRED AND FIVE IS CHAOS (they saw five as four four and five are connected)

BAYES THEOREM IS A QUADRANT MODEL AND IT IS CONSIDERED THE MOST IMPORTANT THEOREM IN STATISTICS AT UCSD I SAT IN ON A MATH CLASS THEY TAUGHT IT AS A QUADRANT MODEL

I DISCUSSED THE GEOMETRIC PROOFS FOR ALGEBRA AND CALCULUS ARE LITERALLY QUADRANT MODELS THE WAY THAT INTEGRALS ARE DISCOVERED THE GEOMETRIC PROOF IS A QUADRANT MODEL QUADRANT WITH THE FOURTH PART DIFFERENT I HAD TEACHING COMPANY VIDEOS ON IT AND I ALREADY POSTED STUFF ON THAT

FOURTH IS TRANSCENDENT

The fourth derivative of position, equivalent to the first derivative of jerk, is jounce.

Because of involving third derivatives, in mathematics differential equations of the form

{\displaystyle J\left({\overset {...}{x}},{\ddot {x}},{\dot {x}},x\right)=0}

are called jerk equations. It has been shown that a jerk equation, which is equivalent to a system of three first-order ordinary non-linear differential equations, is in a mathematically well defined sense the minimal setting for solutions showing chaotic behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are accordingly called hyperjerk systems.

FIRST OUTER INNER LAST FOUR PARTS

In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials[1]—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.

Binomial expansions

A visual illustration of the identity (a + b)2 = a2 + 2ab + b2

The binomial theorem supplies patterns of coefficients that permit easily recognized factorizations when the polynomial is a power of a binomial expression.

For example, the perfect square trinomials are the quadratic polynomials that can be factored as follows:

,

a^2 + 2ab + b^2 = (a + b)^2,\,\!

and

.

a^2 - 2ab + b^2 = (a - b)^2.\,\!

Some cubic polynomials are four term perfect cubes that can be factored as:

,

a^3 + 3a^2b + 3ab^2 + b^3 = (a+b)^3,

and

.

a^3 - 3a^2b + 3ab^2 - b^3 = (a-b)^3.

In general, the coefficients of the expanded polynomial

(a+b)^n are given by the n-th row of Pascal's triangle. The coefficients of

(a-b)^n have the same absolute value but alternate in sign.

WE ALL LEARNED COMPLETING THE SQUARE IN ALGEBRA LOOK HOW THE FOURTH PART OF TH SQUARE IS DIFFERENT OTHER THREE- IN THE PROOF FOR CALCULUS THERE IS ALSO FOUR SQUARES WITH THE FOURTH PART DIFFERENT

Completing the Square

Say we have a simple expression like x2 + bx. Having x twice in the same expression can make life hard. What can we do?

Well, with a little inspiration from Geometry we can convert it, like this:

Completing the Square Geometry

As you can see x2 + bx can be rearranged nearly into a square ...

... and we can complete the square with (b/2)2

In Algebra it looks like this:

Solving quadratic equations by completing the square

Consider the equation

​​ +6x=−2x, start superscript, 2, end superscript, plus, 6, x, equals, minus, 2. The square root and factoring methods are not applicable here. [Why is that so?]

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

THERE IS A RELATIONSHIP BETWEEN THE THREE SQUARE THEOREM AND THE TRANSCENDENT FOUR SQUARE THEOREM (four is the highest transcendent)- THAT IS THE DYNAMIC BETWEEN THREE AND FOUR

Relationship to the four-square theorem

This theorem can be used to prove Lagrange's four-square theorem, which states that all natural numbers can be written as a sum of four squares. Gauss[9] pointed out that the four squares theorem follows easily from the fact that any positive integer that is 1 or 2 mod 4 is a sum of 3 squares, because any positive integer not divisible by 4 can be reduced to this form by subtracting 0 or 1 from it. However, proving the three-square theorem is considerably more difficult than a direct proof of the four-square theorem that does not use the three-square theorem. Indeed, the four-square theorem was proved earlier, in 1770.

EULER FOUR SQUARES

The point Fermat is trying to make is that primes of the form 8n − 1 cannot be written as a sum of less than four rational squares. A brief summary of the most important letters concerning sums of squares is given in the following table:

date

written to

content

15.07.1636 02.09.1636 16.09.1636 Sept. 1636

May 1640

Dec. 1640 June 1658 Aug. 1659

Mersenne Mersenne Roberval Mersenne

Mersenne

Mersenne Digby Carcavi

A number n is a sum of exactly three integral squares if and only if a2n is

A number is a sum of three integral squares if and only if it is a sum of three rational squares.

If a and b are rational, and if a2 +b2 = 2(a+b)x+x2, then x and x2 are irrational.

F. asks for solutions of x4 +y4 = z4 and x3 +y3 = z3, and states the polygonal number theorem. He claims that every integer 8n − 1 is the sum of four squares, but not of three; both in integers and fractions. Fermat repeats the problems he communicated in Sept. 1636

Fermat states Two-Squares Theorem

Fermat claims proof of the Two Squares Theorem. Fermat claims proof of the Four Squares Theorem.

FOUR SQUARE THEOREM GOLDBACH

1. The Four-Squares Theorem in the Euler-Goldbach Correspondence

In this article we describe Euler’s efforts at proving the Four-Squares Theorem. As we will see, using the lemma which Euler “almost” proved in his letter no. 141 it is an easy exercise to complete the proof. In order to see how natural Euler’s approach is, we will first discuss a proof of the Two-Squares Theorem based on the same principles. The first published proof of the Four-Squares Theorem is due to Lagrange [10]; immediately afterwards, Euler [2] simplified Lagrange’s version.

There are perhaps no better examples in Goldbach’s correspondence with Euler for illuminating his role as a catalyst than the letters discussing various aspects of the Four-Squares Theorem.

The lady in question (Dr. Muriel Bristol) claimed to be able to tell whether the tea or the milk was added first to a cup. Fisher proposed to give her eight cups, four of each variety, in random order. One could then ask what the probability was for her getting the specific number of cups she identified correct, but just by chance.

David Salsburg reports that a colleague of Fisher, H. Fairfield Smith, revealed that in the test, the woman got all eight cups correct.[5][6] The chance of someone who just guesses getting all correct, assuming she guesses that four had the tea put in first and four the milk, would be only 1 in 70 (the combinations of 8 taken 4 at a time).

In a famous example of hypothesis testing, known as the Lady tasting tea,[18] Dr. Muriel Bristol, a female colleague of Fisher claimed to be able to tell whether the tea or the milk was added first to a cup. Fisher proposed to give her eight cups, four of each variety, in random order. One could then ask what the probability was for her getting the number she got correct, but just by chance. The null hypothesis was that the Lady had no such ability. The test statistic was a simple count of the number of successes in selecting the 4 cups. The critical region was the single case of 4 successes of 4 possible based on a conventional probability criterion (< 5%; 1 of 70 ≈ 1.4%). Fisher asserted that no alternative hypothesis was (ever) required. The lady correctly identified every cup,[19] which would be considered a statistically significant result.

Clairvoyant card game

A person (the subject) is tested for clairvoyance. He is shown the reverse of a randomly chosen playing card 25 times and asked which of the four suits it belongs to. The number of hits, or correct answers, is called X.

As we try to find evidence of his clairvoyance, for the time being the null hypothesis is that the person is not clairvoyant.[21] The alternative is, of course: the person is (more or less) clairvoyant.

If the null hypothesis is valid, the only thing the test person can do is guess. For every card, the probability (relative frequency) of any single suit appearing is 1/4. If the alternative is valid, the test subject will predict the suit correctly with probability greater than 1/4. We will call the probability of guessing correctly p. The hypotheses, then, are:

null hypothesis

:

H

0

:

p

=

1

4

\text{:} \qquad H_0: p = \tfrac 14 (just guessing)

and

alternative hypothesis

:

H

1

:

p

>

1

4

\text{:} H_1: p > \tfrac 14 (true clairvoyant).

IT YOU NOTICE THIS CLASSIC DEPICTION OF THE PYTHAGOREAN THEOREM ON THE THRONE OF ISIS AND HORUS IS THE QUADRANT THERE ARE FOUR SQUARES DEPICTED FORMING A CROSS- I described the pythagorean theorem goes intot he fourth dimension but I forgot all that I'd have to hopefully rewatch a lecture and shot it to all of you hopefully

THE THREE FOUR FIVE TRIANGLE IN EGYPT- THE FOUR IS THE 16 SQUARES QUADRANT MODEL

THE TETRAHEDRAL LAMBDA 216 AND 138- THE TETRAHEDRAL LAMBDA IS WHAT PLATO SAID CREATED EXISTENCE THE TWO TETRACTYS TETRA IS FOUR

Equivalence of the Polyhedral Tree of Life & the Tetrahedral Lambda

When the 1-tree (lowest Tree of Life) constructed from tetractyses is superposed on its inner form — the 14 enfolded, regular polygons — the outer corners of the two triangles coincide with Chesed and Geburah and the top and bottom corners of the two hexagons coincide with Chokmah, Binah, Netzach & Hod. Hence, six of the 70 corners of the 14 polygons are shared and 64 corners are unshared. 62 of these corners are outside the root edge. 74 of the 80 yods in the 1-tree are not corners of the polygons, so that (74+62=136)

(74+62) unshared yods & corners of 1-tree & inner Tree of Life Figure 15. The 62 unshared corners & 74 unshared yods.

yods or corners outside the root edge are unshared (Fig. 15). Their counterpart in the Polyhedral Tree of Life (Fig. 16) are the 74 vertices of the 144 Polyhedron and the 62 vertices of the disdyakis triacontahedron. Their two centres correspond to the endpoints of the root edge. These (74+62+2=138) points are the corners of 216 triangles inside the former polyhedron generated by its 216 edges and of 180 triangles inside the latter generated by its 180 edges, the centre of each polyhedron being the corner shared by each set of triangles. The counterpart in the Tetrahedral Lambda of these 138 points is the fact that 138 is the sum of the 10 red integers (all powers of 2, 3 & 4) that lie on the three edges meeting at the apex to which the number 1 is assigned.

Equivalence of Tetrahedral Lambda & Polyhedral Tree of Life

Figure 16. Equivalence of the Tetrahedral Lambda and the Polyhedral Tree of Life.

The number 138 is a parameter characterizing holistic systems. It appears in the pair of dodecagons — the last polygons of the inner Tree of Life — as the 138 yods outside their shared root edge (Fig. 17). Notice that 64 yods line their 22 unshared sides. Located at a vertex, this is the largest number in the Tetrahedral Lambda. It is the number value of Nogah, the Mundane Chakra of Netzach (see Table 1).

138 yods outside root edge of 2 dodecagons

Figure 17. 138 yods make up the two joined dodecagons of the inner Tree of Life outside their shared side.

The number 138 also manifests in the four Platonic solids whose shapes the ancient Greeks thought were those of the particles of the Elements Earth, Water, Air & Fire. When their 38 faces are constructed from tetractyses, 137 yods are needed on average (see here). Therefore, 138 yods on average occupy their faces and centres. The central yod is the counterpart of the integer 1, the Pythagorean Monad, at the apex of the Tetrahedral Lambda. In each case, it signifies the source of the archetypal object.

THE NUMBER 138 AND THE FOUR PLATONIC SOLIDS

The number 138 also manifests in the four Platonic solids whose shapes the ancient Greeks thought were those of the particles of the Elements Earth, Water, Air & Fire. When their 38 faces are constructed from tetractyses, 137 yods are needed on average (see here). Therefore, 138 yods on average occupy their faces and centres. The central yod is the counterpart of the integer 1, the Pythagorean Monad, at the apex of the Tetrahedral Lambda. In each case, it signifies the source of the archetypal object.

MILTONS NATIVITY ODE BUILT AROUND PLATOS TETRAHEDRAL TWO TETRACTYS LAMBDA number 216 imbeded in it which was Platos number based on Platos two tetractys lambda and also 72 which is gematria of tetractys tetragrammaton AND 36 IS THE NUMBER OF THE GREAT TETRACTYS TETRA IS FOUR

PYTHAGORAS TETRACTYS AND TETRACHORD

The three primary concords can all be constructed from the numbers found in the ‘tetractys’ or the numbers 1–4.

With Archytus we have a second ‘musical’ tetractys: 6, 8, 9, 12. This second tetractys is achieved by taking the harmonic and arithmetic means of within the double interval of the first tetractys: 1, 4/3, 3/2, 2. Multiply by the smallest number that will clear the fractions and it yields the sequence 6, 8, 9, 12. When these are combined with the three musical proportions – the arithmetic, geometric and harmonic – further mathematically satisfying facts are revealed. The number 9 forms the arithmetic mean between 6 and 12. 6:9 expresses the ratio of the fifth, while 9:12 expresses the ratio of the fourth. The number 8 forms the harmonic mean between 6 and 12 and 8:6 is a fourth, while 8:12 is a fifth. The ratio of the extreme terms, 6 and 12, is that of the octave. We can think of the proportion 6, 9, 12 as expressing an octave composed of a fifth followed by a fourth. The proportion 6, 8, 12 is an octave constructed of a fourth followed by a fifth. The ratio 8:9 thus expresses the difference

35 Harm. chapter 6 is copied by Iamblichus in his Life of Pythagoras, chapter 26. See also Macrobius (Som. Scip. II 1, 9–14) and Boethius (Inst. Mus. I 10–11).

36 Adrastus, ap. Theon 56.9–57.10; Aelianus ap. Porphyry, Comm. 33.16 ff.

25

when a fourth is ‘subtracted’ from a fifth.37 This is the ratio of the tone revealed in the ‘musical’ tetractys and it forms the basis for the construction of the Pythagorean diatonic scale that we find in Plato’s Timaeus.

Nêtê (6) Paranêtê Tritê Paramesê (8) Mesê (9) Lichanos Parahypatê Hypatê

4:3

4:3

9:8

9:8 256:243 9:8

9:8

9:8 256:243

This Pythagorean tradition stands opposed to a more empirical approach to harmonics that stems from Aristoxenus. Aristoxenus did conceive of acoustic space as a continuum and this fact helps explain the central point of disagreement with the Pythagorean tradition: the division of the tetrachord. The Pythagoreans equate the fourth with the ratio 4:3 and the tone with 9:8. The interval of the fourth is “bigger”

than two tones, since 98 × 98 < 43 . If the remainder or leimma were exactly half of a

tone, then there would have to be a rational square root of 98 . But this fraction is a

super-particular or epimorion. That is, it has the form n + 1 and super-particular ratio n

is such that it is impossible to insert one or more geometric means between the terms

PROCLUS AND THE MUSICAL TETRACTYS

Proclus provides an elegant demonstration of the relations of the terms in the musical tetractys: 6, 8, 9, 12. If we have four terms in continuous geometric proportion like this, then if one of the intermediate terms forms the arithmetic mean between the extremes, the other forms the harmonic mean. Moreover, if there are four terms a, b, c, d such that c is the arithmetic mean between a and d, while b is the harmonic mean, then the proportion is a geometric one (in Tim. II 173.11–174.10). This demonstration reprises material in the final chapter of Nicomachus’ Introduction to Arithmetic, though Proclus’ presentation is much more succinct and clearer. This reciprocal relation between the harmonic and arithmetic means, on the one hand, and the geometric on the other serves to ground the judgement that the geometric

HE FOUND FOUR FOUR FOUR WITHIN THE CROP CIRCLE QUADRANT QUINCUNX

I studied all of Professor Leahy's articles and later bought his book, "Foundation: Matter the Body Itself."

When I later found the 444 triplet connections in relation to the two quintuplet pyramid crop formations of 1999, I recalled that Professor Leahy wrote an article that included similar shapes. I found it again. The title is "Golden Bowl Structure: The Platonic Line, Fibonacci, And Feigenbaum." This is one of the graphics --

I realized that the image was similar to one quarter of the West Kennet Longbarrow male pyramid quintuplet. I created the graphic below to illustrate the point --

MORE QUINCUNX QUADRANT MODEL CROP CIRCLES

The 2005 Crop Circle Formations

In 2005, two crop circle formations appeared that suggested the quintuplet pyramid geometry. The first one appeared at Hundred Acres (East Field), Alton Priors, Wiltshire around July 3, 2005 --

Image credit: Lucy Pringle

I filled in this diagram of the formation with red to show the "male" part of the quintuplet pyramid --

Diagram credit: Bertold Zugelder

I filled in this inner portion of the diagram with blue to show the "female male" part of the quintuplet pyramid --

The red male and blue female are combined in this illustration --

The other formation appeared at Avebury Manor, near Avebury, Wiltshire, around July 27, 2005. I call this a "quintuplet of quintuplets." In this graphic I drew a pattern in red to illustrate the geometry --

This illustration shows a diagram in red of the basic "quintuplet of quintuplets" inside the Hundred Acres formation --

THE CROP CIRCLE IS A QUINCUNX QUADRANT EVEN QUADRANT MODEL AND WITHIN THE QUINCUNX IS A 16 SQUARE QUADRANT MODEL

THE TRANCENDENT FOURTH LAW

Hamilton (1837–38 lectures on Logic, published 1860): a 4th "Law of Reason and Consequent"

As noted above, Hamilton specifies four laws—the three traditional plus the fourth "Law of Reason and Consequent"—as follows:

"XIII. The Fundamental Laws of Thought, or the conditions of the thinkable, as commonly received, are four: -- 1. The Law of Identity; 2. The Law of Contradiction; 3. The Law of Exclusion or of Excluded Middle; and, 4. The Law of Reason and Consequent, or of Sufficient Reason."[11]

TRANSCENDENT FOURTH

Is Belnap's four valued-logic a boolean algebra?

up vote

1

down vote

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1

Belnap's logic contains the the truth values 'true' (t), 'false' (f), 'unknown' (⊥

) and 'paradox' (T). Each of these is represented by pair a of bits:

t →

(1,0)

f →

(0,1)

(0,0)

T →

(1,1)

The operations are defined as follows:

TRANSCENDENT FOUR VALUED LOGIC

Motivation and Demotivation of a Four-Valued Logic

JOHN FOX*

Abstract Belnap offers two arguments for the usefulness of four-valued logic. I argue that one of them, which rests on interpreting valuations as states of our information, when taken seriously collapses into an argument for two-valued logic in which relevance is lost, and that the other, resting on Scott's thesis, is not an argument for its usefulness.

THE TRANSCENDENT FOUR VALUE LOGIC

Abstract

In his well-known paper “How computer should think” Belnap (1977) argues that four-valued semantics is a very suitable setting for computerized reasoning. In this paper we vindicate this thesis by showing that the logical role that the four-valued structure has among Ginsberg's bilattices is similar to the role that the two-valued algebra has among Boolean algebras. Specifically, we provide several theorems that show that the most useful bilattice-valued logics can actually be characterized as four-valued inference relations. In addition, we compare the use of three-valued logics with the use of four-valued logics, and show that at least for the task of handling inconsistent or uncertain information, the comparison is in favor of the latter.

TRANSCENDENT FOUR VALUE LOGIC AND QUATENARY MULTIPLIER AGAIN IT WAS ORIGIANLLY THREE VALUE LOGIC THEN ADDED TRANSCENDENT FOURTH

This paper presents addition and multiplication in Galois field using multi-valued logic. Multi-valued logic (MVL) has matured to the point where four-valued logic is now part of commercially available VLSI IC’s. Modulo-4 addition and multiplication is also presented in this paper. Logic design of each operation is achieved by reducing the terms using Karnaugh diagrams keeping minimum number of gates and depth of net in to consideration. Quaternary multiplier circuit is proposed to achieve required optimization. Simulation result of each operation is shown separately using Hspice

FOURTH SPATIAL DIMENSION

The Fourth Dimension is a non-fiction work written by Rudy Rucker, the Silicon Valley professor of mathematics and computer science, and was published in 1984 by Houghton Mifflin. The book is subtitled as a guided tour of the higher universes. The foreword included is by Martin Gardner, and the 200+ illustrations are by David Povilaitis. Like other books by Rucker, The Fourth Dimension is dedicated to Edwin Abbott Abbott, author of the novella Flatland.

Synopsis

The Fourth Dimension guides you on a mind-expanding journey; the book is designed to alter the reader's perceptions of the universe through the exploration of a fourth dimension (a fourth spatial dimension, rather than the simpler notion of time as a fourth dimension). The information gives the reader a much better understanding of the concept of higher dimensions, whose existence must be presumed in order to complete some of the mathematical equations of quantum mechanics. Abbott's Flatland is put to use by means of analogies, which are used throughout the book. Rucker compares how a square in Flatland would react to a cube in Spaceland to how a cube in Spaceland would react to a hypercube from the fourth dimension.

In addition to the 200 pages of the guided tour of the higher universes, many puzzles (see mental-skill game) are included to help the reader gain the mental tools necessary to envisioning a fourth dimension.

PROCLUS ON THE TETRAD COMMENTARY ON EUCLID SAYS THE TETRAD COMPREHENDS ALL RATIOS IN ITSELF- HE SAYS THE TETRAD IS THE FIRST SOLID NUMBER AND 1234 MAKE THE MOST CONSANANT MUSICAL SCALES

PYTHAGORAS CONVERSED WITH HEBREWS AND THIS AUTHOR CLAIMS THAT PYTHAGORAS SAW THE TETRACTYS AS GOD/ THE TETRACTYS IS THE TETRAGRAMMATON- THE KABALLAH TREE OF LIFE IS THE TETRACTYS

Pythagoras, it seems, did not only call the supreme Deity a monad, but also a tetrad, or tetractys... It is, in the golden verses, said to be the fountain of the eternal nature; and by Hierocles, the maker of all things, the intelligent god, the cause of the heavenly and sensible god, that is, of the animated world or heaven. The later Pythogoreans endeavour to give reasons why God should be called Tetractys, from certain mysteries in the number four; but... much more probable... this name was really nothing else but the tetragrammaton, or that proper name of the supreme God amongst the Hebrews, consisting of four letters; nor is it strange Pythagoras should be so well acquainted with the name Jehovah, since, besides travelling into other parts of the East, he is by Josephus, Porphyry, and others, to have conversed with the Hebrews also.

Anonymous, The Ancient History of the Jews, and of the Minor Nations of Antiquity (1834) p.10

FOUR THINGS KNOWN OF PYTHAGORAS

The following became universally known: first, that he maintains that the soul is immortal; second, that it changes into other kinds of living things; third, that events recur in certain cycles and that nothing is ever absolutely new; and fourth, that all living things should be regarded as akin. Pythagoras seems to have been the first to bring these beliefs into Greece.

Porphyrius in Life of Pythagoras

PROCLUS CLAIMS THAT SOCRATES ASKS THE QUESTION "BUT WHERE IS THE FOURTH" NOT COINCIDENTALLY BUT AS A CODE REVEALING THE NATURE OF THE TETRACTYS WHICH CREATED EXISTENCE WHERE THE FOURTH IS DIFFERENT- HE CLAIMS THE TETRAD IS THE MOST PERFECT AND SAYS "ALL THINGS EXIST TETRADICALLY" PLATO TETRACHORDS ALSO PLATOS ABCD DIVIDED LINE THEOREM ALSO PLATOS X PLATOS CHIASMA CROSS TETRAD "MOTHER OF ALL THINGS" CLAIMS THE GODS EMERGE FROM THE FOUR ORDERS OF THE TETRAD- HE SAYS THE TETRAD PRODUCED THE DIVINITIES

Socrates having come to the Piraeus for the sake of the Bendidian festival33 and solemn procession, discoursed there concerning a polity with Polemarchus, the son of Cephalus, Glauco and Adimantus, and likewise Thrasymachus the sophist. But on the day after this, he narrates the conference in the Piraeus, as it is laid down in the Republic, in the city, to Timaeus, Hermocrates and Critias, and to another fourth anonymous person. Having, however, made this narration, he calls upon the other associates, to feast him in return on the day after this, with the banquet of discourse. The auditors therefore and speakers assembled together [p.8] on this day, which was the third from the conference in the Piraeus. For in the Republic it is said,34 "I went down yesterday to the Piraeus;" but in this dialogue,35 "Of those who were received by me yesterday at a banquet of discourse, but who ought now in their turn to repay me with a similar repast." Not all of them however, were present at this audition, but the fourth was wanting through indisposition. What, therefore, you will say, are these three auditors of a discussion about the whole world? I reply, that it is fit the father of the discussion should be considered as analogous to [the Demiurgus, or] the father of works.

Plato here, together with the grace and beauty of the words, raises and exalts the whole period. Praxiphanes however, the disciple of Theophrastus, blames Plato, first because he makes an enumeration of one, two, three, in a thing which is manifest to sense and known to Socrates. For what occasion had Socrates to numerate, in order that he might know the multitude of those that assembled to this conference? In the second place he blames him, because he makes a change in using the word fourth, and in so doing, does not accord with what had been said before. For the word four, is consequent to one, two, three; but to the fourth, the first, second, and third are consequent. These, therefore, are the objections of Praxiphanes. The philosopher Porphyry however directly replies to him, and in answer to his second objection observes, that this is the Grecian custom, for the purpose of producing beauty in the diction. Homer48 therefore has said many things of this kind:

Full on the brass descending from above,

Through six bull hides the furious weapon drove,

Till in the seventh it fix'd.

And in a similar manner in many other places. Here also the mutation has a cause. For to numerate the persons that were present, was to point them out. For to say one, two, three, is indicative; but he signifies the person that was absent (since it was impossible to point him out) through the fourth. For we use the term the fourth, of one that is absent. But to the former objection Porphyry replies, that if as many had been present as was requisite, it would have been superfluous to numerate them, but one of them being absent, of whose name we are ignorant, the enumeration of those that are present contains a representation of the one that is wanting, as desiring that which remains, and as being in want of a part of the whole number. Plato therefore indicating this, represents Socrates enumerating the persons that were present, and requiring him who was wanting. For if he had known him, and had been able to manifest him by name, he would perhaps have said, I see Critias, and Timaeus, and Hermocrates, but that man I do not see. Since however, he who was absent was a stranger, and unknown to [p.13] him, he only knew through number that he was wanting, and manifests to us that so many were present. All these observations, therefore, are elegant, and such others of the like kind as may be devised by some in subserviency to the theory of the words before us. But it is necessary to remember that the dialogue is Pythagorean, and that it is requisite interpretations should be made in a way adapted to the philosophers of that sect.

Such ethical Pythagoric dogmas therefore, as the following,49 may be derived from the present text: Those men established friendship and a concordant life, as the scope of all their philosophy. Hence Socrates prior to every thing else adduces this, by giving Timaeus the appellation of friend. In the second place, they thought that the compacts which they made with each other, should be stably preserved by them;50 and for the fulfilment of these, Socrates desires the presence of the fourth person. In the third place, they embraced communion in the invention of dogmas, and the writings of one, were common to all of them. This also Socrates establishes, calling on them to become both guests and hosts, those that fill, and those that are filled, those that teach, and those that learn. Others, therefore, have written arts concerning disciplines through which they think they shall improve the manners of those that are instructed by them; but Plato delineates the forms of appropriate manners, through the imitation of the most excellent men, which have much greater efficacy than those which are deposited in mere rules F alone. For imitation disposes the lives of the auditors, conformably to its own peculiarity. Hence, through these things it is evident what that is about which the philosopher is especially abundant, that it is about the hearing of discussions, and what he conceived to be a true feast; that it is not such as the multitude fancy it to be; for this is of an animal and brutal nature; but that which banquets in us the [true] man.51 Hence too, there is much in Plato about the feast of discourse. These therefore and such particulars as these, are ethical.

The philosopher Porphyry says, that what is delineated in these words: that this is the one cause with wise men of relinquishing such like associations, viz. infirmity of body; and that it is requisite to think that every thing of this kind depends on circumstances and is involuntary. Another thing also is delineated, that friends should make fit apologies for friends, when they appear to have done any thing rightly, which is contrary to common opinion. The present [p.16] words therefore, comprehend both these, indicating the manners of Timaeus, and the necessity of one being absent; exhibiting the former as mild and friendly to truth, but the latter, as an impediment to the life of a lover of learning. But the divine Iamblichus speaking loftily on these words, says that those who are exercised in the survey of intelligibles, are unadapted to the discussion of sensibles; as also Socrates himself says in the Republic,64 "that those who are nurtured in pure splendour, have their eyes darkened when they descend into the cavern, through the obscurity which is there; just as it likewise happens to those who ascend from the cavern, through their inability to look directly to the light." Through this cause therefore, the fourth person is wanting, as being adapted to another contemplation, that of intelligibles. It is also necessary that this his infirmity, should be a transcendency of power, according to which he surpasses the present theory. For as the power of the wicked, is rather impotency than power, thus also imbecility with respect to things of a secondary nature, is transcendency of power. According to Iamblichus therefore, the person who is wanting, is absent in consequence of being incommensurate to physical discussions; but he would have been willingly present, if intelligibles were to have been considered. And nearly with respect to every thing [in this dialogue] prior to physiology, one of these, i.e. Porphyry, interprets every thing in a more political manner, referring what is said to virtues, but the other, Iamblichus, in a more physical way. For it is necessary, that everything should accord with the proposed scope: but the dialogue is physical, and not ethical. Such therefore, are the conclusions of the philosophers about these particulars. For I omit to mention those who labour to evince, that this fourth person was Theaetetus, because he was known to those who came out of the Eleatic school, and because we are informed [elsewhere] that he was ill. Hence he is said to have been now absent on account of illness. For thus Aristocles infers, that the absent person was Theaetetus, who a little before the death of Socrates, became known to Socrates,65 and to the Elean stranger.66 But admitting that he had been long before known to the latter, what is there in common between Timaeus and him? The Platonic Ptolemy however, thinks that the absent person was Clitophon: for in the dialogue which bears his name, he is not thought deserving of an answer by Socrates. But Dercyllides is of opinion that it was Plato: for he was absent through illness, [p.17] when Socrates died.67 These, therefore, as I have said, I omit; since it is well observed by those prior to us, that these men neither investigate what is worthy of investigation, nor assert anything that can be depended on. All of them, likewise, attempt a thing which is of a slippery nature, and which is nothing to the purpose, even if we should discover that which is the object of their search. For to say that it was either Theaetetus or Plato, on account of illness, does not accord with the times. For of these, the former is said to have been ill when Socrates was judged, but the latter when Socrates was dead. But to say it was Clitophon is perfectly absurd. For he was not present on the preceding day, when Socrates narrates what Clitophon said the day before, during the conference in the Piraeus;68 except that thus much is rightly signified by Atticus, that the absent person appears to have been one of those strangers [or guests] that were with Timaeus. Hence Socrates asks Timaeus where that fourth person was; and Timaeus apologizes for him, as a friend, and shows that his absence was necessary, and contrary to his will. And thus much for what is aid by the ancient interpreters.

What, however, our preceptor [Syrianus69] has decided on this subject, must be narrated by us, since it is remarkably conformable to the mind of Plato. He says, therefore, that in proportion as the auditions are about things of a more venerable and elevated nature, in such proportion the multitude of hearers is diminished. But the discussion in the Timaeus becomes, as it proceeds, more mystic and arcane. Hence in the former discussion of a polity during the conference in the Piraeus, the hearers were many, and those who had names were six. But in the second conference, which is narrated by Socrates,70 those who receive the narration are four in number. And in the present conference, the fourth person is wanting; but the auditors are three. And by how much the discussion is more pure, and more intellectual, by so much the more is the number of auditors contracted. For everywhere that which is discussed is a monad. But at one time, it is accompanied with contention; on which account also, the auditors have the indefinite, and the definite is extended into multitude, in which the odd is complicated with the even. At another to me, however, the discussion is narrative, yet is not liberated from opposition, and dialectic contests. Hence also, the auditors are four in number; the tetrad through its tetragonic nature, and alliance to the monad, possessing similitude and sameness; but through the nature of the even, possessing difference and multitude. And at another time the discussion [p.18] is exempt from all agonistic doctrines, the theory being unfolded enunciatively, and narratively. Hence, the triad is adapted to the recipients of it, since this number is in every respect connascent with the monad, is the first odd number, and is perfect. For as of the virtues, some of them subsist in souls the parts of which are in a state of hostility to each other, and measure the hostility of these parts; but others separate indeed from this hostility, yet are not perfectly liberated from it; and other are entirely separated from it; thus also of discussions, some indeed are agonistic, others are enunciative, and others are in a certain respect media between both. Some, indeed, being adapted to intellectual tranquillity, and to the intellectual energy of the soul; but others to doxastic energies; and others to the lives that subsist between these. Moreover, of auditors likewise, some are commensurate to more elevated auditions, but others to such as are of a more grovelling nature. And the auditors indeed of grander subjects, are also capable of attending to such as are subordinate; but those who are naturally adapted to subjects of less importance, are unable to understand such as are more venerable. Thus also with respect to the virtues, he who has the greater possesses likewise the less; but he who is adorned with the inferior, is not entirely a partaker also of the more perfect virtues.

Why, therefore, is it any longer wonderful, if an auditor of discussions about a polity, should not be admitted to hear the discussion about the universe? Or rather, is it not necessary that in more profound disquisitions, the auditors should be fewer in number? Is it not likewise Pythagoric, to define different measures of auditions? For of those who came to the homacoion71 [or common auditory of the Pythagoreans] some were partakers of more profound, but others of more superficial dogmas. Does not this also accord with Plato, who assigns infirmity as the cause of the absence of this fourth person? For the imbecility of the soul with respect to more divine conceptions, separates us from more elevated conferences, in which case the involuntary also takes place. For every thing which benefits us in a less degree, is not conformable to our will. But the falling off from more perfect good is involuntary; or rather it is itself not voluntary. But the falling off which not only separates us from greater goods, but leads us to the infinity of vice, is involuntary. Hence also Timaeus says, that this fourth person was absent not willingly from this conference. For he was not absent in such a way as to be perfectly abhorrent from the theory, but as unable to be initiated in greater speculations. It is possible, therefore, for an auditor of disquisitions about the fabrication of the world, to be also an auditor of discussions about a [p.19] polity. But it is among the number of things impossible, that one who is adapted to receive political discourses, should through transcendency of power, omit to be present at auditions about the universe. This fourth person, therefore, was absent through indigence, and not as some say, through transcendency of power. And it must be said, that the imbecility was not the incommensuration of the others to him, but the inferiority of him to the others. For let there be an imbecility both of those that descend from the intelligible, and of those that ascend from the speculation of sensibles, such as Socrates relates in the Republic;72 yet he who becomes an auditor of political discussions, cannot through a transcendency unknown to those that are present, be absent from the theory of physics. It likewise appears to me, that the words "has befallen him," sufficiently represent to us the difference between him and those that were present, with respect to discussions, and not with respect to transcendency. His being anonymous also, seems to signify, not his being exempt from and circumscribed by those that were present, but the indefiniteness and inferiority of his habit. Plato, therefore, is accustomed to do this in many places. Thus in the Phaedo,73 he does not think him deserving of a name, who in that dialogue answered badly. He also mentions indefinitely,74 the father of Critobulus, who was absent from the discussion of the subjects that were then considered; and likewise very many others. An auditor therefore of this kind would in vain have been present at these discussions; since of those that were present, Critias indeed himself says something; but Hermocrates is silently present, differing only from him who is absent in a greater aptitude to hear, but being inferior to all the rest, through his inability to speak.

For both the tetrad and the decad contain all things in themselves; but the former unitedly, and the latter distributedly. The decad likewise, though it contains all such things as the tetrad contains, yet because it contains them in a more divided manner, it is more imperfect than the tetrad. For the tetrad being nearer to the monad is more perfect; and in proportion as quantity is diminished, the magnitude of power is increased

From this triad, however, the tetrad will shortly after be unfolded, because the natures which are bound together are solids. Hence it is rightly said, that a bond imparts beauty, and an harmonious communion and union. But what this bond is, and how it is inherent in the things that are bound, Plato shows through the following words.

Let us, however, now pass on to similar solids, and survey the media in these. In the first place, therefore, let there be two cubes 8 and 27, the former having for its side 2, and the latter 3. Of these cubes, there will be two media, the one being produced from twice two multiplied by three, i.e. 12, and which on this account is (So/uc;) a beam,856 but the other from thrice three multiplied by two, i.e. 18, and which is therefore a tile. These will make a continued analogy with the before-mentioned cubes, according to a sesquialter ratio. And here you may see how each of the media has two sides from the cube placed next to it, but the remaining side from the other cube.857 This however will be useful to us for the purposes of physiology. Again, if the numbers were not cubes, but similar solids, they will likewise have two analogous middles or means. For let there be two similar solids 24 and 192, the sides of the former being 2, 3, 4, but of the latter 4, 6, 8. And from the duad, the triad, and the ogdoad, 48 will be produced, but from the tetrad, the hexad, and again the tetrad, the product will be 96. Here too, each of the media will have two sides from that similar solid of the extremes which is next to it, but one side from the other cube, in the same manner [p.413] as in the media of the before-mentioned cubes. Hence between similar solids, two media are sufficient; just as Plato says, that two media adapt solids to each other, but never one medium. What then, some one may say, is there not one medium alone of the two solid numbers 64 and 729, which medium is 216? For 64 is a cube produced from 4, but 729 from 9. And 729 is the triple and superparticular ogdoan part of 216; and after the same manner 216 of 64. For each contains the other thrice, and three eighths of it besides.558 And this will not only be the case in these, but also in other numbers: for these are the smallest numbers which admit of this. In answer to this however it must be said, that the abovementioned numbers are cubes and at the same time squares; the one, i.e. 64, being the square of 8, but the other, 2,33 i.e. 729, being the square of 27. Hence they have one mean, not so far as they are cubes, but so far as they have the tetragonic peculiarity. For the tetragonic side of 64, i.e. 8, being multiplied by 27, which is the tetragonic side of 729, produces the analogous mean 216, according to the method delivered [by mathematicians] of finding the mean between two squares. He who makes the objection, therefore, using solids not as solids, binds them together by one medium. But if he had surveyed them so far as they are solid numbers and cubes, he would have found that there are also two media between these, the one being 144, from four times 4 multiplied by 9, but the other 324, from nine times 9 multiplied by 4.

That the tetrad itself of the elements, primarily proceeded from all perfect animal, (for it was the intelligible tetrad) and that on this account all things exist tetradically, becomes I think evident through the words before us; and also that generation proceeds to the tetrad from the monad through the duad. For the world is only begotten and one.884 Afterwards we find it is necessary that there should be the visible and tangible in it; in the next place, we find that these being much separated from each other, are in want of a certain third thing; and in the third [p.430] place, that the medium is biformed, and thus we arrive at the tetrad. This therefore, is what the Pythagoric hymn says about number: That it proceeds from the secret recesses of the monad, until it arrives at the divine tetrad. And this generates the decad, which is the mother of all things. Thus also the father of the Golden Verses, celebrates the tetractys itself, as the fountain of perennial nature. For the world being adorned by the tetrad, which proceeds from the monad and triad, is terminated by the decad, as being comprehensive of all things. That the world likewise is one through analogy, subsisting from these elements, and from such like things according to powers, and from so many according to quantity, Plato clearly manifests by saying, that not the sublunary region, but the body of the universe, was generated from the four elements. But the friendship of the world is the end of the analogy, through which also the world is saved by itself. For every thing which is friendly, wishes to be preservative of that to which it is friendly: but every thing foreign turns from, and does not even wish that to exist to which it is abhorrent; so that the nature which is friendly to, is preservative of itself. The world however, is friendly to itself through analogy and sympathy, and therefore it preserves itself. But it is also preserved by the fabrication of things, receiving from it an ineffable guard. Hence also, the theologist denominates the bond derived from the Demiurgus strong, as Night is represented saying to the Demiurgus,

Plato, therefore, in the diatonic genus, makes a division of tetrachords, and proceeds not only as far as to the diapason, but also as far as to a quadruple diapason and diapente, adding likewise a tone. Or according to Severus, Plato did not produce the tetrachords without a tone, but ended in a leimma, and not in a tone. If, however, some one should doubt, how Plato produced the diagram to such an extent, let him attend to the words of Adrastus. For he says that Aristoxenus, extended the magnitude of his multiform diagram, as far as to the diapason and diatessaron, and the symphony of these, in consequence of preferring the information of the ears to the decision of intellect. But the more modern musicians123 extended the diagram as far as to the fifteenth mode, viz. to the thrice diapason and tone, in so doing looking solely to our utility, and thinking that those who contend in singing could not exceed this, no r their auditors judge clearly beyond [p.58] it. Plato, however, looking to nature, constitutes the soul from all these, in order that it might proceed as far as to solid numbers, as it ought to preside over bodies. For the progression as far as to the quadruple diapason and diapente, necessarily follows the seven terms [or bounding numbers]. But this is evident from the greatest term being twenty seven. And thus much in answer to the doubt.

In short, since Plato makes mention of the three middles, which are comprehended in the geometric middle, let the following theorem be added [as a corollary] to what has been said. If the analogy consists in four terms, and one of the intermediate numbers produces an arithmetical middle, the other will produce an harmonic middle, and vice versa. For let there be four terms, a, b, c, d, so that the first a, is to b, as c is to d, and let b, be an arithmetical middle, [so that a, b, d, are in arithmetical proportion,] I say that c is an harmonic middle. For because the product of a by d is equal to the product of b by c, but b is an arithmetic middle and the product of c by a added to the product of c by d is the double of the product of b by c, as in the arithmetic middle; this being the case, it follows that the product of c by a added to the product of c by d, is the double of the product of a by d.128 But this was the property of the harmonic middle, viz. that the product of the middle by the extremes, is the double of the product of the extremes. Again, let c be an harmonic middle, I say that b is an arithmetic middle. For since the product of c by a added to the product of c by d, is the double of the product of b by c, the sum of a added to d is the double c of b.129 But this is an arithmetical middle, when the sum of the extremes is the double of the middle term. Again of these four terms, let b be an arithmetic, but c an harmonic mean, I say that as a is to b, so is c to d. For because the product of c by a, added to the product of c by d, is the double of the product of a by d, on account of the harmonic middle, but the sum of a added to d, is the double of b on account of the arithmetic middle, hence the product of a by d will be equal to the product of b by c. As a therefore is to b, so is c to d.130 But this was the peculiarity of the geometric middle. Hence those two middles are contained in the geometric [p.61] middle, and reciprocate with each other. Since however we have premised thus much, let us proceed to the text of Plato.

Enneadically, in order that it may not alone contain the universe monadically, but also a proceeding to the last of things from the monad. But tetradically, as collecting the quadripartite division into one.

"Having, therefore, cut all this double composition according to length, so as to produce two from one, he adapted middle to middle, each to the other, as it were in the form of the letter X."*

In the first place, it is requisite to show mathematically of what kind the figure of the soul is, and thus, afterwards, introduce the theory of the things; in order that being led in a becoming manner by the phantasy, we may render ourselves adapted to the scientific apprehension of what is said. All the numbers therefore, must be conceived to be described in one rule, as those who are skilled in music are accustomed to do. And let the rule have the numbers according to the whole of its depth, and be divided according to its length. All the ratios therefore, will be in each of the sections. For if the division was made according to breadth, it would be entirely necessary that some of the numbers should be taken here, but others there. Since however, the section is according to length, but all the numbers are in all the length, there will be the same numbers in each of the parts. For it is evident, that it is not the same thing, to divide the length, and to divide according to the length; since the latter signifies, that the section proceeds through the whole length, but the former, that the length is divided.232 Let the rule, therefore, be thus divided according to length,233 and let the two lengths be applies to each other in the points which bisect the lengths, yet not so as to be at right angles: for neither will the circles be at right angles. Let the two lengths likewise be so incurvated, that they may again be conjoined at the extremities. Two circles therefore, will be formed, of which one will be the interior, but the other exterior, and they will be oblique to each other. One of these likewise, is called the circle of the same, but the other, the circle of the different. And the one indeed, subsists according to the equinoctial circle, but the other, according to the zodiac. For the whole circle of the different revolves about the zodiac, but that of the same about the equinoctial. Hence, we conceive that the right lines ought not to be applied to each other at right angles, but like [p.111] the letter X as Plato says, so as to cause the angles to be equal only at the summit, but those on each side, and the successive angles, to be unequal. For the equinoctial circle does not cut the zodiac at right angles. Such therefore, in short, is the mathematical discussion of the figure of the soul.

The figure X however, produced by this application, has an affinity to the universe, and also to the soul. And as Porphyry relates, a character of this kind, viz. X, surrounded by a circle, is with the Egyptians a symbol of the mundane soul. For perhaps it signifies, through the right lines indeed, the biformed progression of the soul, but through the circle its uniform life, and regression according to an intellectual circle. We must not however conceive, that Plato thought a divine essence could be discovered through these things. For the truth of real beings cannot, as some fancy, be known from characters, positions, and vocal emissions. But these are after another manner symbols of divine natures. For as a certain motion, so likewise a certain figure and colour, are symbols of this kind, as the initiators into mysteries say. For different characters and also different signatures are adapted to different Gods; just as the present character is adapted to the soul. For the complication of the right lines indicates the union of a biformed life. For a right line itself, also, is a symbol of a life which flows from on high. In order however, that we may not, omitting the things themselves, be too busily employed about the theory of the character, Plato adds "as it were," indicating that this is assumed as a veil, and for the sake of concealment, thus endeavouring to invest with figure the unfigured nature of the soul.

For he not only delivers the Demiurgus as a nomenclator, who first gives names to the two circulations of the soul; but prior to these unfolds the essential character of it, viz. two separate right lines, and the X produced from them, and also the two circles formed from these lines; which things theurgy likewise unfolded after him,260 giving completion to the character of the soul from chiasmi261 and semicircles.

"But from Saturn and Rhea, Jupiter, Juno, and all such as we know are called the brethren of these descended, and also the other progeny of these."*

This is the third progression of the Gods who are the fabricators of generation, but the fourth order, closing as a tetrad the nomination of the leading Gods. For the tetrad is comprehensive of the divine orders. But as a duad this progression is assimilated to the first kingdom; because that as well as this is dyadic. There are however, present with it, the all-perfect according to progression, and the uncircumscribed according to number. But Plato here, not only adds the words "such as," as in the progression prior to it, but likewise the word "all," that he may indicate the progression of them to every thing. For we use the term to oson, such as, in speaking of things united, but the term to pantas, all, in speaking of things now divided and multiplied. The total (to olikon) likewise pertains to this progression. For the Gods which are denominated in it, and those that proceed every where together with them, are characterised according to this form of fabrication. For all Demiurgi are total. Who therefore are they, and what kind of order do they possess?

But Tethys establishes every thing in its proper motion; intellectual essences in intellectual, middle essences in psychical, and such as are corporeal in physical, motion; Ocean at the same time collectively moving all things. Saturn alone divides intellectually, Rhea vivifies; and Phorcys distributes spermatic productive principles; Jupiter perfects things apparent from such as are unapparent; and Juno evolves according to the all-various mutations of visible natures. And thus through this ennead all the sublunary world derives its completion, and is fitly arranged; divinely indeed from the Gods, but angelically, as we say, from angels, and daemoniacally from daemons; the Gods indeed subsisting about bodies, souls, and intellects; but angels exhibiting their providence about souls and bodies; and daemons being distributed about the fabrication of nature, and the providential care of bodies. But again, the number of the ennead is adapted to generation. For it proceeds from the monad as far as to the extremities without retrogression; which is the peculiarity of generation. For reasons (i.e. productive principles) fall into matter, and are unable to convert themselves to the principles of their [p.331] existence.656 Moreover, the duad being triadic, for three dyadic orders were assumed,657 manifests the complication here of the perfect and the imperfect, and of bound with infinity. For the celestial natures are definite, and as Aristotle says,658 are always in the end. But things in generation proceed from the imperfect to the perfect, and receive the same boundary indefinitely. Besides this, the tetrad arising from the generation of these divinities is adapted to the orders of the fabricators of the sublunary region; in order that they may contain multitude unitedly, and the partible impartibly; and also to the natures that exist in generation. For the sublunary elements are four; the seasons according to which generation is evolved are four; and the centres are four. And in short, there is an abundant dominion of the tetrad in generation.

The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents x2 (Figure 1). To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square (Figure 2). Figure 2 has area x2 + 10 x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 × 5/2 = 25/4. Hence the outside square in Fig 3 has area 4 × 25/4 + 39 = 25 + 39 = 64. The side of the square is therefore 8. But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.

four conic sections fourth different circle a lot of people say three i learned this on teaxhin company lecture on math and tons of other examples most i forgot but the proof for calculus derivatives everything was quadrant model i just forgot how to explain

Conic Sections have been studied for a quite a long time. Kepler first noticed that planets had elliptical orbits. Depending on the energy of an orbiting body, orbit shapes that are any of the four types of conic sections are possible.

A conic section can be formally defined as a set or locus of a point that moves in the plane of a fixed point called the focus and the fixed line is called the directrix.

Conic Sections have been studied for a quite a long time. Kepler first noticed that planets had elliptical orbits. Depending on the energy of an orbiting body, orbit shapes that are any of the four types of conic sections are possible.

A conic section can be formally defined as a set or locus of a point that moves in the plane of a fixed point called the focus and the fixed line is called the directrix.

UP TO THE FOURTH POWER four groups
https://en.wikipedia.org/wiki/Ibn_al-Haytham
This leads to an equation of the fourth degree.[95] This eventually led Alhazen to derive a formula for the sum of fourth powers, where previously only the formulas for the sums of squares and cubes had been stated. His method can be readily generalized to find the formula for the sum of any integral powers, although he did not himself do this (perhaps because he only needed the fourth power to calculate the volume of the paraboloid he was interested in). He used his result on sums of integral powers to perform what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.[

Alhazen wrote a total of twenty-five astronomical works, some concerning technical issues such as Exact Determination of the Meridian, a second group concerning accurate astronomical observation, a third group concerning various astronomical problems and questions such as the location of the Milky Way; Alhazen argued for a distant location, based on the fact that it does not move in relation to the fixed stars.[124] The fourth group consists of ten works on astronomical theory, including the Doubts and Model of the Motions discussed above.[125]

Alhazen discovered the sum formula for the fourth power, using a method that could be generally used to determine the sum for any integral power. He used this to find the volume of a paraboloid. He could find the integral formula for any polynomial without having developed a general formula.[135]

FORMULAS TO TRANSCENDENT FOURTH POWER BEFORE NOT KNOWN ONLY TO THIRD

The problem comprises drawing lines from two points in a circle meeting at a third point on its circumference and making equal angles with the normal at that point. This is equivalent to finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in order to carom off the edge of the table and hit another ball at a second given point. Thus, its main application in optics is to solve the problem, "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer." This leads to an equation of the fourth degree.[2][3][1]

Sums of powers

Ibn al-Haytham eventually derived a formula for the sum of fourth powers, where previously only the formulas for the sums of squares and cubes had been stated. His method can be readily generalized to find the formula for the sum of any integral powers, although he did not himself do this (perhaps because he only needed the fourth power to calculate the volume of the paraboloid he was interested in). He used his result on sums of integral powers to perform what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.[4]

NOTICE HOW THE FOURTH IS DIFFERENT- THE FIRST THREE ARE KIND OF SIMILAR THEN HUGE JUMP TO FOURTH THEN HUMUNGOUS JUMP TO FIFTH

The Wilson numbers are

THE FOUR SQUARE THEOREM AND GOLDBACH QUATERNIONS AND ONE TRIANGLE FOURTH POWER

Remark. The matrices M[r, s, t, u] form a ring isomorphic to the Lipschitz quater- nions. The proof of the Four-Squares Theorem due to Lagrange and Euler was first translated into the language of quaternions by Hurwitz [8].

Acknowledgement

I thank Martin Mattmu ̈ller for the crucial observation that Euler’s postscript might be sufficient for proving the Four-Squares Theorem, as well as for his help in translating from Latin. I also thank Norbert Schappacher for his valuable com- ments.

“Lately, reading Fermat’s works, I came upon another rather elegant theorem stating that any number is the sum of four squares, or that for any number four square numbers can be found whose sum is equal to the given number”.

Goldbach has not read Fermat’s works.

Euler observes that 104 + 1 is divisible by 37, and that 38 + 28 is divisible by 17. Euler cannot prove that any number is the sum of four squares. He has found another result by Fermat, namely that 1 is the only triangular number that is a fourth power (Several years earlier, Goldbach had sent an erroneous proof of this claim to D. Bernoulli).

THEON TEN TETRACTYS

Sacred Tetractys

One particular triangular number that they especially liked was the number ten. It was called a Tetractys, meaning a set of four things, a word attributed to the Greek Mathematician and astronomer Theon (c. 100 CE). The Pythagoreans identified ten such sets.

Ten Sets of Four Things

Numbers 1 2 3 4

Magnitudes point line surface solid

Elements fire air water earth

Figures pyramid octahedron icosahedron cube

Living Things seed growth in length in breadth in thickness

Societies man village city nation

Faculties reason knowledge opinion sensation

Seasons spring summer autumn winter

Ages of a Person infancy youth adulthood old age

Parts of living things body three parts of the soul

ricardos four magi numbers
https://en.m.wikipedia.org/wiki/David_Ricardo
Ricardo attempted to prove theoretically that international trade is always beneficial.[18] Paul Samuelson called the numbers used in Ricardo's example dealing with trade between England and Portugal the "four magic numbers".[19] "In spite of the fact that the Portuguese could produce both cloth and wine with less amount of labor, Ricardo suggested that both countries would benefit from trade with each other.

TALEB POINTS OUT HOW THE FOURTH QUADRANT IS DIFFERENT

THE FOURTH QUADRANT: A MAP OF THE LIMITS OF STATISTICS

By Nassim Nicholas Taleb [9.14.08]

Introduction by:

John Brockman

Statistical and applied probabilistic knowledge is the core of knowledge; statistics is what tells you if something is true, false, or merely anecdotal; it is the "logic of science"; it is the instrument of risk-taking; it is the applied tools of epistemology; you can't be a modern intellectual and not think probabilistically—but... let's not be suckers. The problem is much more complicated than it seems to the casual, mechanistic user who picked it up in graduate school. Statistics can fool you. In fact it is fooling your government right now. It can even bankrupt the system (let's face it: use of probabilistic methods for the estimation of risks did just blow up the banking system).

But, as he points out, there is also good news.

We can identify where the danger zone is located, which I call "the fourth quadrant", and show it on a map with more or less clear boundaries. A map is a useful thing because you know where you are safe and where your knowledge is questionable. So I drew for the Edge readers a tableau showing the boundaries where statistics works well and where it is questionable or unreliable. Now once you identify where the danger zone is, where your knowledge is no longer valid, you can easily make some policy rules: how to conduct yourself in that fourth quadrant; what to avoid.

Fourth Quadrant: Complex decisions in Extremistan: Welcome to the Black Swan domain. Here is where your limits are. Do not base your decisions on statistically based claims. Or, alternatively, try to move your exposure type to make it third-quadrant style ("clipping tails").

FOUR VOLUME WORK

The Luminous Ground, the fourth book of The Nature of Order, contains what is, perhaps, the deepest revelation in the four-volume work. Alexander addresses the cosmological implications of the theory he has presented. The book begins with a critique of current cosmological thinking, and its separation from personal feeling and value. The outline of a theory in which matter itself is more spirit-like, more personal in character, is sketched. Here is a geometrical view of space and matter seamlessly connected to our own private, personal, experience as sentient and knowing creatures. This is not merely an emotional appendix to the scientific theory of the other books. It is at the core of the entire work, and is rooted in the fact that our two sides - our analytical thinking selves, and our vulnerable emotional personalities as human beings - are coterminous, and must be harnessed at one and the same time, if we are ever to really make sense of what is around us, and be able to create a living world.

THE FOUR GATEWAYS ARE FOUR QUADRANTS

Plan of Meenakshi Amman Temple, Madurai, from 7th century onwards. The four gateways (numbered I-IV) are tall gopurams.

The Meenakshi Amman Temple is a large complex with multiple shrines, with the streets of Madurai laid out concentrically around it according to the shastras. The four gateways are tall towers (gopurams) with fractal-like repetitive structure as at Hampi. The enclosures around each shrine are rectangular and surrounded by high stone walls.[61]

Merk Diezle FOUR TALL MINURETS The historian of Islamic art Antonio Fernandez-Puertas suggests that the Alhambra, like the Great Mosque of Cordoba,[70] was designed using the Hispano-Muslim foot or codo of about 0.62 metres (2.0 ft). In the palace's Court of the Lions, the proportions follow a series of surds. A rectangle with sides 1 and √2 has (by Pythagoras's theorem) a diagonal of √3, which describes the right triangle made by the sides of the court; the series continues with √4 (giving a 1:2 ratio), √5 and so on. The decorative patterns are similarly proportioned, √2 generating squares inside circles and eight-pointed stars, √3 generating six-pointed stars. There is no evidence to support earlier claims that the golden ratio was used in the Alhambra.[10][71] The Court of the Lions is bracketed by the Hall of Two Sisters and the Hall of the Abencerrajes; a regular hexagon can be drawn from the centres of these two halls and the four inside corners of the Court of the Lions.[72]
https://en.wikipedia.org/wiki/Mathematics_and_architecture

Selimiye Mosque, 1569–1575
The Selimiye Mosque in Edirne, Turkey, was built by Mimar Sinan to provide a space where the mihrab could be see from anywhere inside the building. The very large central space is accordingly arranged as an octagon, formed by 8 enormous pillars, and capped by a circular dome of 31.25 metres (102.5 ft) diameter and 43 metres (141 ft) high. The octagon is formed into a square with four semidomes, and externally by four exceptionally tall minarets, 83 metres (272 ft) tall. The building's plan is thus a circle inside an octagon inside a square.[73]

ONE OF THE FIRST THINGS LEARNED IN STATISTICS CLASS QUARTILES DIVIDING THE DATA INTO FOUR PARTS

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.

FOUR POSTULATES

The defense of probability is mainly based on Cox's theorem, which starts from four postulates concerning rational reasoning in the presence of uncertainty. It demonstrates that the only mathematical framework that satisfies these postulates is probability theory. The argument is that any approach other than probability necessarily infringes one of these postulates and the value of that infringement.

FOUR COMPONENTS

Doug Walton developed a distinctive philosophical theory of logical argumentation built around a set of practical methods to help a user identify, analyze and evaluate arguments in everyday conversational discourse and in more structured areas such as debate, law and scientific fields.[16] There are four main components: argumentation schemes,[17] dialogue structures, argument mapping tools, and formal argumentation systems. The method uses the notion of commitment in dialogue as the fundamental tool for the analysis and evaluation of argumentation rather than the notion of belief.[18] Commitments are statements that the agent has expressed or formulated, and has pledged to carry out, or has publicly asserted. According to the commitment model, agents interact with each other in a dialogue in which each takes its turn to contribute speech acts. The dialogue framework uses critical questioning as a way of testing plausible explanations and finding weak points in an argument that raise doubt concerning the acceptability of the argument.

FOUR METATHEORETICAL COMPONENTS

To allow for the systematic integration of the pragmatic and dialectical dimensions in the study of argumentation, the pragma-dialectical theory uses four meta-theoretical principles as its point of departure: functionalization, socialization, externalization and dialectification. Functionalization is achieved by treating discourse as a purposive act. Socialization is achieved by extending the speech act perspective to the level of interaction. Externalization is achieved by capturing the propositional and interactional commitments created by the speech acts performed. And dialectification is achieved by regimenting the exchange of speech acts to an ideal model of a critical discussion. (see Van Eemeren & Grootendorst, 2004, pp.52-53).

PLATOS LAMBDA AND TETRACTYS

The Lambdoma

Pythagoreans swear "By him that gave our family the Tectractys, which contains the Fount and Root of everflowing Nature" (Sextus Empiricus, Adv. Math. VII.94-5). To understand this most sacred symbol of the Pythagoreans, we will begin with the Lambdoma:

[lambdoma image]

The sides of the Tetractys have the values 1 2 3 4 9 8 27, which are called "the seven boundaries of all numbers" (Mead 164); arranged in this way they are called the Lambdoma or Platonic Lambda. Their significance is described below.

In addition to the Tetractys that increases by addition, 1 2 3 4, the Pythagoreans say there is another that increases by multiplication, that is, in geometric proportion. Plato (Tim. 31C) says that a continuous geometric proportion is the most perfect bond, and so we find such a proportion along both sides of the Lambdoma: 1 2 4 8 and 1 3 9 27.

The duple progression (1 2 4 8) represents "the evolution of the vehicle" proceeding out of Unity, that is, the differentiation and division that constitute the physical body. The triple progression (1 3 9 27), in the order of involution (27 9 3 1), represents "the development of consciousness" as a return to unity, that is, the unification and integration of the psyche. (Note that the Pythagoreans considered 1 to be neither even nor odd, nor even a number properly speaking, but the source of both the even and odd numbers; 2 was the first even and 3 the first odd.) In general, the duple axis represents the physical, temporal, divisible and perishable; the triple axis represents the incorporeal, eternal, indivisible and imperishable.

(Aristides III.24; Mead 165-8; Theon II.38)

BEYOND FOUR BIFURCATIONS CHAOS

Figure 1 (left) is a schematic diagram of successive bifurcations taken from the above text of Prigogine (51). Where we see just one branch, the long-term behavior of the system tends towards a fixed, homogenous final state. When we see two branches, the system has bifurcated and the long-term behavior of the system is now alternat- ing between two different states. This is called periodic behavior. Since there are two states the period is two. When we see four branches, the system has undergone a second bifurcation and the period has increased to four. More bifurcations beyond period four lead to chaotic, rather than periodic, behavior of the system.