https://en.wikipedia.org/wiki/Iron_Chef

THREE JUDGES EXPANDED TO FOUR THE THREE PLUS ONE

https://en.wikipedia.org/wiki/Iron_Chef

Throughout the cook-off, running commentary is made in a booth near the cooking area by an announcer, Kenji Fukui; a commentator, Yukio Hattori, and one or two of the guest judges, with one floor reporter (sometimes two; normally Shinichiro Ohta) providing details of the action on each side. The commentators and judges discuss the style of cooking, culinary traditions and unusual food preparation. At the end of the hour, after end-of-battle interviews with both competitors, each dish is presented to the camera, with a description of its properties (written by the show's screenwriters based on the chef's explanation) read by the announcer. Then, a panel of three (later expanded to four and, later still, five) judges, of which typically one is a professional critic, tastes the dishes and judges them based on taste, presentation, and originality. Each chef may be awarded up to 20 points by each judge, with ten given for taste and five each for presentation and originality. The chef with the greatest score wins the competition. (In earlier four-judge episodes, the win went to the chef who won three of the four judges, or, failing that, the chef that makes the highest points total.)

In 2010, UK public television network Channel 4 debuted Iron Chef UK, based on Iron Chef. The show airs five days a week, and is hosted by Olly Smith and Nick Nairn.[15] The four Iron Chefs are Tom AikensMartin Blunos, Sanjay Dwivedi and Judy Joo.

https://en.wikipedia.org/wiki/Activity_vector_analysis

Activity vector analysis (AVA) is a psychometric questionnaire designed to measure four personality factors or vectors: aggressiveness, sociability, emotional control and social adaptability.[1] It is used as an employment test.

https://en.wikipedia.org/wiki/Eysenck_Personality_Questionnaire

Eysenck initially conceptualized personality as two biologically-based independent dimensions of temperament, E and N, measured on a continuum, but then extending this to include a third, P.

E - Extraversion/Introversion: Extraversion is characterized by being outgoing, talkative, high on positive affect (feeling good), and in need of external stimulation. According to Eysenck's arousal theory of extraversion, there is an optimal level of cortical arousal, and performance deteriorates as one becomes more or less aroused than this optimal level. Arousal can be measured by skin conductance, brain waves or sweating. At very low and very high levels of arousal, performance is low, but at a better mid-level of arousal, performance is maximized. Extraverts, according to Eysenck's theory, are chronically under-aroused and bored and are therefore in need of external stimulation to bring them UP to an optimal level of performance. About 16 percent of the population tend to fall in this range. Introverts, on the other hand, (also about 16 percent of the population) are chronically over-aroused and jittery and are therefore in need of peace and quiet to bring them DOWN to an optimal level of performance. Most people (about 68 percent of the population) fall in the midrange of the extraversion/introversion continuum, an area referred to as ambiversion.[2]

N - Neuroticism/Stability: Neuroticism or emotionality is characterized by high levels of negative affect such as depression and anxiety. Neuroticism, according to Eysenck's theory, is based on activation thresholds in the sympathetic nervous system or visceral brain. This is the part of the brain that is responsible for the fight-or-flight response in the face of danger. Activation can be measured by heart rate, blood pressure, cold hands, sweating and muscular tension (especially in the forehead). Neurotic people — who have low activation thresholds, and unable to inhibit or control their emotional reactions, experience negative affect (fight-or-flight) in the face of very minor stressors — are easily nervous or upset. Emotionally stable people — who have high activation thresholds and good emotional control, experience negative affect only in the face of very major stressors — are calm and collected under pressure.

The two dimensions or axes, extraversion-introversion and emotional stability-instability, define four quadrants. These are made up of:

• Stable extraverts (sanguine qualities such as outgoing, talkative, responsive, easygoing, lively, carefree, leadership)

• Unstable extraverts (choleric qualities such as touchy, restless, excitable, changeable, impulsive, irresponsible)

• Stable introverts (phlegmatic qualities such as calm, even-tempered, reliable, controlled, peaceful, thoughtful, careful, passive)

• Unstable introverts (melancholic qualities such as quiet, reserved, pessimistic, sober, rigid, anxious, moody)

Further research demonstrated the need for a third category of temperament:[3]

P - Psychoticism/Socialisation: Psychoticism is associated not only with the liability to have a psychotic episode (or break with reality), but also with aggression. Psychotic behavior is rooted in the characteristics of toughmindedness, non-conformity, inconsideration, recklessness, hostility, anger and impulsiveness. The physiological basis suggested by Eysenck for psychoticism is testosterone, with higher levels of psychoticism associated with higher levels of testosterone.

The following table describes the traits that are associated with the three dimensions in Eysenck's model of personality:

TRNA IS CRUCIFORM

https://en.wikipedia.org/wiki/Transfer_RNA

A transfer RNA (abbreviated tRNA and formerly referred to as sRNA, for soluble RNA[1]) is an adaptor moleculecomposed of RNA, typically 76 to 90 nucleotides in length,[2] that serves as the physical link between the mRNA and the amino acid sequence of proteins. It does this by carrying an amino acid to the protein synthetic machinery of a cell (ribosome) as directed by a three-nucleotide sequence (codon) in a messenger RNA (mRNA). As such, tRNAs are a necessary component of translation, the biological synthesis of new proteins in accordance with the genetic code.

IT HAS 16 FORMS THERE ARE 16 SQUARES IN THE QUARANT MODEL FOUR CHAPTERS

Unlike most styles of silat, lian padukan is offensive and teaches students to advance before the opponent makes a move. Once the fighter closes in on the opponent, they attack with a continuous combination of hand, foot, elbow, and knee strikes until the enemy has been subdued. All styles of buah pukul contain four main techniques called jurus which introduce the use of punches, kicks, elbows, and knee strikes. Lian padukan further includes the knife-hand as one of its primary attacks. The most basic strike is the polek, a hand-strike in which one hand attacks while the other pulls down the opponent's blocking arm. This is delivered as a rapid series of "rolling" punches or chops aimed at the head or upper body. This rolling action, called gulung, is the most fundamental technique in all variants of buah pukul. Kicks are mostly targeted below the waist but the roundhouse kick is much the same as in tomoi.

After learning all the jurus, students begin training in preset forms called lian. Lian are divided into four "chapters", each of which contain four routines, making a total of sixteen forms. The lian allow students to learn strategies and applications of moves.

https://en.wikipedia.org/wiki/Delusion

Delusions are categorized into four different groups:

• Bizarre delusion: Delusions are deemed bizarre if they are clearly implausible and not understandable to same-culture peers and do not derive from ordinary life experiences.[15] An example named by the DSM-5 is a belief that someone replaced all of one's internal organs with someone else's without leaving a scar.

• Non-bizarre delusion: A delusion that, though false, is at least possible, e.g., the affected person mistakenly believes that he is under constant police surveillance.

• Mood-congruent delusion: Any delusion with content consistent with either a depressive or manic state, e.g., a depressed person believes that news anchors on television highly disapprove of him, or a person in a manic state might believe she is a powerful deity.

• Mood-neutral delusion: A delusion that does not relate to the sufferer's emotional state; for example, a belief that an extra limb is growing out of the back of one's head is neutral to either depression or mania.[16]

The cooperative principle can be divided into four maxims, called the Gricean maxims, describing specific rational principles observed by people who obey the cooperative principle; these principles enable effective communication.[3] Grice proposed four conversational maxims that arise from the pragmatics of natural language.[3] Applying the Gricean maxims is a way to explain the link between utterances and what is understood from them.

CRUCIFIXION IN A SEPARATE PEACE

https://www.newspapers.com/newspage/33587037/

With the passage of 15 years and the world's movement from war to peace, Gene reflects on a tortuous summer of his adolescence — the dark summer when as a lonely, introverted intellectual, he acted on his jealousy, jounced a tree : branch that sent his best friend crashing to the ground, and plunged himself from innocence. In his fall, he brought about the death of Finny, whose innate goodness contrasts starkly with Gene's sinfulness. In this beautifully written work, Knowles constructs a parable of the dark forces that brood over life. He writes a garden tale wherein Devon School in that wondrous summer of the war years is Eden. Like Eden, Devon holds a symbolic tree that functions both as the Tree of Knowledge and the crucifix of redemption. Knowies. fuses the figurative crucifixion of Finny with the subsequent wearing of his pink shirt to give his protagonist, Gene, valuable knowledge which aids his passage through adolescence! ., ." ...-..- • Knowles uses the tree with its branches, trunk, and roots not only to link heaven, earth, but also to bond the sinful personality of Gene with the sinless personality of Phineas, the only, two boys at Devon to jump from the tree. Holding the good of Phineas and the evil of Gene, this tree becomes the biblical Tree of Knowledge of Good and Evil. As Gene jounces the limb and commits his first major, intentional sin, he realizes the consequences of his act and gains a knowledge of good and evil. The tree contains further symbolic significance. Its trunk is a vertical line which connects the three worlds of heaven, earth and hell. Pinny's sacrifice takes place from a branch, the horizontal line of the tree. The horizontal on the vertical forms a cross. This cross exhibits Finny as a Christ figure. Later in the novel when Gene wears Finny's pink shirt, he recognizes the significance of the sacrifice. Pink is created by blending red and white. Gene realizes that his innocence, represented by the white, has been altered by the sacrificial blood of Finny, represented by the red. By placing Finny's shirt on his body, Gene allows Finny to become a part of him and performs a rite of redemption. In essence, he becomes the high priest of Phineas' peace. Using the tree and the pink shirt as religious images, Knowles enables his readers to observe Gene as he gains knowledge of good and eviland evolves into a spiritual adult, John Knowies combines the tree and the pink shirt with yet a third image, the Devon River, to manifest Gene's spiritual transformation.

http://crossref-it.info/articles/13/cross-crucifixion

Along with the Nativity, the Crucifixion is one of the most frequently depicted Christian scenes in art. It is also referred to in much literature – for example, in T.S. Eliot's poem The Journey of the Magi, where the scholars travelling to find the Christ-Child see a foreshadowing of his death in the form of ‘three trees on the low sky'. In Golding's Lord of the Flies, Simon runs to tell the boys about ‘a dead man on a hill'.

FOUR STAGES OF THE FRENCH REVOLUTION

http://westerncivguides.umwblogs.org/2012/02/14/stages-of-the-french-revolution-3/comment-page-1/

–The Moderate Stage (1789-1792)

-Formation of The National Assembly (1789)

-Tennis Court Oath (1789)

-Fall of The Bastille (1789)

-Women’s March on Versailles (1789)

-Declaration of the Rights of Man (1789)

-Civil Constitution of The Clergy (1790)

-Constitution of 1791

-Invention of The Guillotine (1791)

-Republic (1792)

-Execution of Louis XVI and Marie Antoinette (1793)

-Constitution of 1793

-Robespierre Comes to Power Under the Committee of Public Safety (1793)

-Thermidorian Reaction and Fall of Robespierre (1794)

-The Directory (1795-1799)

-Napoleon Overthrows Directory and Becomes First Consul of The Consulate (1799)

-The Age of Napoleon (1800-1815)

-Napoleonic Code Created

-Concordat (1801)

-Napoleon Becomes Emperor and Revolution Ends (1804)

-Napoleonic Wars (1804-1815)

WILLIAM JAMES FATHER OF PSYCHOLOGY FOUR ASPECTS OF SELF

https://en.wikipedia.org/wiki/File:James_Selves.PNG

https://en.wikipedia.org/wiki/Hypostatic_model_of_personality

In his Principles of Psychology,[13] William James describes four aspects of the self:

material self (the body and the person's closest possessions and relatives, including the family);

social self (the being-for-others);

spiritual self (the person's inner and subjective being, her psychic faculties and dispositions, taken concretely);

the "pure" ego (the bare principle of personal unity).

Selves according to William James

Many schools of psychotherapy see subpersonalities as relatively enduring psychological structures or entities that influence how a person feels, perceives, behaves, and sees him- or herself.[35] According to the hypostatic model, human personality consists of four components or hypostases, which are patterns of behavior related to specific systems in the brain, and are conceptualized by virtually every culture as being characteristic and/or essential to humans:[36][37][38]

In every specific task of daily life, one of the first four dimensions (cognitive, practical, affective, or expressive) is dominant, being at the center of the experience, whereas the other three are subordinated to it. Regulative and adaptive dimensions are constantly acting as a background throughout the behavioral process.[43]

FOUR PART MANTRA TETRA IS FOUR- THIS MANTRA WAS HUNG IN MY YOGA STUDIO ALONG WITH THE 16 WORD HARE KRISHNA MAHA MANTRA- ENDS WITH FOUR WORDS- ALMOST ALL OF THE FAMOUS MANTRAS AND HINDU SONGS ARE QUATRAINS AND HAVE FOUR PARTS

http://www.eaglespace.com/spirit/gayatribywords.php

https://en.wikipedia.org/wiki/Gayatri_Mantra

Whereas in principle the gāyatrī mantra specifies three pādas of eight syllables each, the text of the verse as preserved in the Samhita is one short, seven instead of eight. Metrical restoration would emend the attested tri-syllabic vareṇyaṃ with a tetra-syllabic vareṇiyaṃ.[10]

The Gayatri mantra is,[9] in Devanagari:

ॐ भूर् भुवः स्वः ।

तत्स॑वि॒तुर्वरेण्यं॒

भर्गो॑ दे॒वस्य॑धीमहि ।

धियो॒ यो नः॑ प्रचो॒दया॑त् ॥

In IAST:

om bhūr bhuvaḥ svaḥ

tát savitúr váreṇ(i)yaṃ

bhárgo devásya dhīmahi

dhíyo yó naḥ prachodayāt

## The Four Parts of the Gayatri Mantra

### AumBhurBhuvah Swah (ॐ भूर्भुव: स्व:)

1. AUM (ॐ), the Supreme name of God. A full explanation of this has been given in a related article.

BHUR BHUVAH SWAH. These three words collectively are known as the "Mahavyahriti". They express the nature of God, and demonstrate his inherent qualities.

2. BHUR (भूर्)

Firstly, the word Bhur implies existence. God is self-existent and independent of all. He is eternal and unchanging. Without beginning and without end, God exists as a continuous, permanent, constant entity. Secondly, the word Bhur can also mean the Earth, on which we are born and sustained. God is the provider of all, and it is through His divine will that we our blessed with all that we require to maintain us through our lives. Finally, Bhur signifies Prana, or life (literally, breath). God is That which gives life to all. Whilst He is independent of all, all are dependent on Him. It is God who has given us life, God who maintains us throughout our lives, and God alone who has the ability to take away our life, when He so chooses. The only permanent entity, all others are subject to His own will

3. BHUVAH (भुव:)

Bhuvah describes the absolute Consciousness of God. God is self-Conscious as well as being Conscious of all else, and thus is able to control and govern the Universe. Also, the word Bhuvah relates to God's relationship with the celestial world. It denotes God's greatness - greater than the sky and space, He is boundless and unlimited. Finally, Bhuvah is also indicative of God's role as the remover of all pain and sufferings (Apaana). We see pain and sorrow all around us. However, through supplication to God, we can be freed from that pain and hardship. God Himself is devoid of any pain. Though He is Conscious of all, and is thus aware of pain, it does not affect Him. It is our own ignorance that makes us susceptible to the effects of Maya, or illusion, which causes us to feel pain. Through true devotion to God, we can be freed from the clutches of Maya, and thus be rid of pain and sorrow.

4. SWAH (स्व:)

Swah indicates the all-pervading nature of God. He is omnipresent and pervades the entire multi-formed Universe. Without Form Himself, He is able to manifest Himself through the medium of the physical world, and is thus present in each and every physical entity. In this way, God is able to interact with the Universe created by Him, and thus sustain and control it, ensuring its smooth and proper running and function.

Also, Swah symbolizes God's bliss. All but God experience pain, suffering and sorrow. Devoid of all such things, God alone is able to experience supreme bliss. Happiness as experienced by humans is temporary, a transient state of mental satisfaction, which soon dissolves back into the mire of worldly troubles. Perfect, and without any form of deficiency, God alone experiences true bliss, permanent and unaffected by worldly pains and woes. One who realizes God is able to join in this bliss, and thus God is able to impart true happiness to those who establish oneness with that Supreme Divinity.

Top of Page | Related article: Gayatri Mantra Summary, Origins

### TAT SAVITUR VARENYAM (तत्सवितुर्वरेण्यं)

5. TAT (तत् s.1)

Literally, this word means "that", being used in Sanskrit to denote the third person. It is also mentioned in the Bhagavad Gita by Sri Krishna Himself, where He implies the selfless nature of the word. Being used in the third person, the word has implicit in it an idea of selflessness. Sri Krishna uses it to imply the selfless nature of charity (charity, or a gift, being used as an analogy for worship, in the form of action, implying that action should be preformed without regard to its fruits, but simply out of devotion and sense of duty, or Dharma). Tat then is used here in the Gayatri Mantra to indicate that the worshipper is referring to [that] God, and that the praise being offered to God in the prayer is purely directed towards Him, without thought of gaining any personal benefit from that praise.

6. SA-VI-TUR (सवितुर् s.2-4)

Savita, from which Savitur is derived, is another name of God, this being the reason that the Gayatri Mantra is often known as the Savitri Mantra. The implication of Savita is of God's status as the fountain, the source of all things. It is through His Divine Grace that the Universe exists, and so this word sums up the Mahavyahriti, by describing God's ability to create the Universe and sustain it, as well as, at the right time, bring about its dissolution.

Savita is also indicative of God's gift to mankind. Humans also have, in limited amount, the power, or shakti, of Savita. This shakti acts as an impetus in humans, and brings about the requirement for them to do something. They cannot sit idle, and are constantly searching for something to do. This is what is commonly known as the "creative urge". It is through this shakti that mankind has created art, and it is through this shakti also that scientific advances are made. The gift of Savita also gives creatures the ability of procreation. Hence, Savita can be thought of as meaning Father (or Mother) also.

Finally, it is the power of Savita that enables mankind to distinguish right from wrong, and vice from virtue. Through this ability, we are able to in some part direct our own selves, and thus, Savita imparts to us a certain self-guiding ability. Thus, by using this word in the mantra, we demonstrate that we are making efforts ourselves also, since God will not help us unless we are willing to help ourselves.

7. VA-RE-NY-AM (वरेण्यं s.5-8)

Varenyam signifies our acceptance of God, and can be translated as meaning "Who is worthy". Ever ready to obtain all the material riches of the world, more often than not, they are a disappointment once they have been achieved. God however is the one who, once realized and achieved, has the ability to truly satisfy. We therefore accept Him as the Highest reality, and it is to Him that we dedicate our efforts.

Varenyam can also be interpreted as signifying one who is eligible. We have chosen Him to be our Leader and our Guide. We place our all into His hands, and accept Him regardless of anything else. We place no conditions on this acceptance, as it is all out of sheer devotion.

Top of Page | Related article: Gayatri Mantra Audio in mp3

### BHARGO DEVASYA DHIMAHI (भर्गो देवस्य धीमहि)

This triplet is a further description of the attributes and qualities of God - His functional and instrumental qualities, rather than intrinsic qualities - and through those qualities, His relationship to us.

8. BHAR-GO (भर्गो s.1,2)

Bhargo is taken to signify the Glorious Light that is God's love and power. It indicates His complete purity - being absolutely pure Himself, God also has the ability to purify those that come into contact with Him. Thus, Bhargo is indicative of God's power to purify, and to destroy all sins and afflictions. In the same way as a metal ore placed into a fire will yield the pure metal, by merging with God, by realizing His Divine Form and establishing unity and oneness with Him, we can cleanse ourselves and be made pure by His Grace.

Though the soul, being itself Divine in nature, possesses that Light, it lacks luster, having been made impure by the sins and vices, which are a result of the darkness of Maya. By removing the veil of Maya, and cleansing our soul, God can enable the soul to realize its true, Divine self, and thus purify it.

9. DE-VAS-YA (देवस्य s.3-5)

The word Deva, from which this word is derived, has been translated by different people in many different ways. It is generally thought of as meaning simply "God". However, its meaning is more complex than that.

Deva, which forms the root of the words "Devata" and "Devi", means "quality" or "attribute", and can be thought of as another word for "Guna". Thus, the various forms of God are given this name, as each of those forms is related to a specific quality and function (for example, Brahma has the quality of Creation, Kamadeva has the quality of love, etc.). Also, Deva is thus used to describe anyone who is considered to possess a special quality.

Since Deva is symbolic of the individual qualities of God, the word demonstrates the inherent oneness of those different Forms, and thus the use of this word can be taken as describing the fundamental unity of God. Thus we see that here, we reaffirm that central belief in the Hindu Dharma that "Ekam sat viprah bahudah vadanti" (Truth, or God, is one, but wise men call Him/It by different names).

Thus, Deva is indicative of the various multifaceted entity that is the absolute Personality of God. It describes in one word all the functions, roles and different attributes of God, and symbolizes therefore his absolutely essential nature - without God, nothing can exist.

10. DHI-MA-HI (धीमहि s.6-8)

Meaning to meditate and focus our mind on God. Meditation on God implies that we remove all other thoughts from our mind, since thoughts of the world render our mind impure, and thus we are unable to conceptualize the absolute purity of God. We must be able to concentrate, and direct our mental energies towards the task in hand - which is communion with God.

Top of Page | Related article: Goddess Gayatri

### DHIYO YO NAH PRACHODAYAT (धीयो यो न: प्रचोदयात्)

Prayer is carried out for four main reasons:

• to praise and glorify God;

• to thank God;

• to ask forgiveness from God;

• or to make a request from God.

Having carried out the other three parts (praise of His greatness, thanks for His generosity in Creation and maintaining us through our lives, and forgiveness by demonstrating our awareness of our own impurity, which we have realized is present and must be cleansed through contact with God), this part is now our request from God. Since our soul is the Light of Life within us, and that acts on our body via the medium of the brain, we ask God to make this contact pure and righteous. The soul is of course inherently pure, being itself Divine in nature. The body is under the complete control of the mind. The link is the mind, which is affected not only by the soul, but also the outside world. We ask in these four words that God help us to improve our intellect, and guide it towards what is right.

11. DHI-YO (धीयो s.1,2)

Sanskrit for "intellect", this is the essence of this part of the Gayatri Mantra. Having firmly set God in our hearts, we now must try to emphasize His presence and influence on our mind and intellect.

Material prosperity holds no true meaning for the person who is truly devoted to God. Pain and suffering are of no consequence to him as, touched by God, he is imbued with God's own Divine Bliss, and all worldly sorrows pale to nothingness in comparison. However, still the individual must live in the world. Thus, it is important that the person's intellect remains focussed on serving God, and that it is able, through the medium of the body, to serve God to the best of its ability.

Physical objects can be obtained very easily, if one is intelligent enough to know how to go about it. Intellect however cannot be obtained, but must be there from the very first. It is by use of this intellect, in fact, that one is able to cultivate all other qualities (building of wealth, "success" in life (in material terms), physical fitness, etc.) Thus, intellect is the key to all else in life, and as such, it is the most important possession. We ask God in the Gayatri Mantra to gift us with the highest intellect, and to help us by showing us the way to use that intellect.

12. YO (यो s.3)

Meaning "Who" or "That", Yo signifies yet again that it is not to anyone else that we direct these prayers, but to God alone. Only God is worthy of the highest adoration, only God is perfect and free from all defects. It is That God to Whom we offer these prayers.

13. NAH (न: s.4)

Nah means "Ours", and signifies the selflessness of the request we make of God in this part of the Gayatri Mantra. We offer this prayer, and make the request of God, not simply for ourselves, but for the whole of humanity. We seek the uplift of the whole of society. Hindu philosophy has since the beginning recognized the concept of "Vasudhaiva Kutumbakam" - "The whole world is one big family". Thus, we pray not only for ourselves, but for each and every member of that great family, that we may all benefit from the greatness and generosity of the All-loving God.

14. PRA-CHO-DA-YAT (प्रचोदयात् s.5-8)

Prachodayat, the final word of the Gayatri Mantra, rounds off the whole mantra, and completes the request we make of God in this final part. This word is a request from God, in which we ask Him for Guidance, and Inspiration. We ask that, by showing us His Divine and Glorious Light (cf. BHARGO), He remove the darkness of Maya from our paths, that we are able to see the way, and in this manner, we ask Him to direct our energies in the right way, guiding us through the chaos of this world, to find sanctuary in the tranquility and peace of God Himself, the root of all Happiness, and the source of true Bliss.

FOUR FEET OF GAYATRI

http://veda.wikidot.com/gayatri-mantra

## Four 'foots' of the Gayatri

This verse talks about the unbounded wealth contained within the four 'foots' of the Gayatri Mantra.

• The first foot (aum bhur bhuvah svaha) is said to be equivalent to the wealth contained in the three worlds put together.

• The second foot (tat savitur varenyam) is said to be equivalent to the wealth contained in the three main vedas.

• If one were to receive a gift extending as far as there are living beings, that would equal the third foot (bhargo devasaya dheemahi).

• The fourth foot (dheeyo yo nah prachodyaat) is based on the glory of the sun, whose power and wealth remains unequaled and unrivaled. Hence, there is no amount of wealth that can equal the fourth foot of the Gayatri!

The Gayatri is considered as Vedasara --"the essence of the Vedas." Veda means knowledge, and this prayer fosters and sharpens the knowledge-yielding faculty.
As a matter of fact, the four mahavakyas or 'core-declarations' enshrined in the four Vedas are implied in this Gayathri mantra.

http://www.dnaalchemy.com/Mantras_Gayatri.html

http://scriptures.ru/gayatr_l.htm

The Gayatri Mantra implies the 4 Maha Vakyas or the 4 core declarations enshrined in the 4 Vedas.

25. What are the four Maha Vakyas?

The four Maha Vakyas are:

1. PRAJNANAM BRAHMA

Consciousness is Brahman (Aitareya Upanishad of the Rg Veda)

2. AHAM BRAHMA ASMI

3. TAT TVAM ASI

That Thou Art (Chandogya Upanishad of the Sama Veda)

4. AYAM ATMA BRAHMA

This Self is Brahman (Mandukya Upanishad of the Atharva Veda)

FOUR PART MANTRA

They are also saying the four syllable Narayan in the interludes

A very loose translation might be:
Om = Salutations! Everyone wake up!
Gam = The secret power sound of Ganesh. It is his "seed syllable" or bija mantra.
Ganapataye = Another name of Ganesh, the breaker of obstacles.
Namaha = Yo! Ganesh! You da God!

The pronunciation is also pretty easy:
Om = ohm or aum
Gam = Somewhere between "gahm" and "gum". In some dialects, it is "guhng".
Ganapataye = gah-nah-paht-ah-yeh
Namaha - Nah-mah-ha

http://humanityhealing.net/2012/01/om-gam-ganapataye-namaha/

THE MANTRA SAYS THAT AUM IS REALITY- THE AUM IS THE FOURFOLD SOUND WITH THE FOURTH PART DIFFERENT

https://en.wikipedia.org/wiki/Om_Tat_Sat

Om Tat Sat (Sanskrit: ओम् तत् सत्,  Om Tat Sat (help·info)) is a mantra in Sanskrit found in verse 17.23 of the Bhagavad Gita. It means "Om, that is Truth", "Om, it is Reality", "Om it is good". It is the threefold designation of the Hindu metaphysical concept called Brahman.[1][2]

'Om' is the eternal sound-pranava. ‘Om’ represents the unmanifest and absolute reality. By the word ‘reality’, here it means total existence. We may even use the word God, reality, existence, Parbrahma or the absolute, are all synonymous terms pointing to one being.

"Om Tat Sat" can be literally translated as the ”Supreme Absolute Truth” or ”all that is.”

THE IDOLS OF GANESHA, WHOM THE MANTRA OM GANAPATEYA NAMO NAMAHAH ABOVE IS DEDICATED TO, IS SAID TO HAVE THE FORM OF THE AUM AND REPRESENT THE FOURFOLD AUM

Om tat sat is a Hindu mantra.

FOUR WORD MANTRA CHANTED BEFORE HARE KRISHNA CLASSES

Om Namo Bhagavate Vasudevaya ( listen (help·info)) (in devanagari: ॐ नमो भगवते वासुदेवाय) is a Hindu mantra. ‘Om Namo Bhagavate Vasudevaya’ is a mantra of Vishnu and Krishna both. It has two traditions—Tantric and Puranic. In Tantrik Tradition, the Rishi of the Mantra is Prajapati, in Puranic Tradition the Rishi is Narada. Both, however, say it is the Supreme Vishnu Mantra. Sharada Tilak Tantram, for example, says "Dvadasharno mahamantrah pradhano Vaishnavagame"—the twelve lettered mantra is the chief among vaishnava mantras. Similarly, this is the ultimate mantra in ShrimadBhagavatam, whose 12 Chapters are taken as extensions of the 12 Letters of this Mantra.[1] This twelve syllable mantra[2] is known as a Mukti (liberation) mantra and a spiritual formula for attaining freedom.[3] This can be chanted like Gayatri Mantra.[4] This is the principal mantra of the Vedic scripture "Srimad Bhagavatam".[5] This mantra can also be found in Vishnu Purana.

Muslims have a celebraiton with babies on the fourth day of the fourth month or something maybe it was hindus I forgot but it is somewhere in the quadrant internet stuff.

FOUR FAIRS NECTAR FELL IN FOUR PLACES

https://en.wikipedia.org/wiki/Kumbh_Mela

Kumbh Mela or Kumbha Mela (/ˌkʊm ˈmeɪlə/ or /ˌkʊm məˈlɑː/) is a mass Hindu pilgrimage of faith in which Hindus gather to bathe in a sacred or holy river. Traditionally, four fairs are widely recognized as the Kumbh Melas: the Haridwar Kumbh Mela, the Allahabad Kumbh Mela, the Nashik-Trimbakeshwar Simhastha, and Ujjain Simhastha. These four fairs are held periodically at one of the following places by rotation: HaridwarAllahabad (Prayaga), Nashik district (Nashik and Trimbak), and Ujjain. The main festival site is located on the banks of a river: the Ganges (Ganga) at Haridwar; the confluence (Sangam) of the Ganges and the Yamuna and the invisible Sarasvati at Allahabad; the Godavari at Nashik; and the Shipra at Ujjain. Bathing in these rivers is thought to cleanse a person of all sins.

The exact age of the festival is uncertain. According to medieval Hindu mythology, Lord Vishnu dropped drops of Amrita (the drink of immortality) at four places, while transporting it in a kumbha (pot). These four places are identified as the present-day sites of the Kumbh Mela. The name "Kumbh Mela" literally means "kumbha fair". It is known as "Kumbh" in Hindi (due to schwa deletion); in Sanskrit and some other Indian languages, it is more often known by its original name "Kumbha".

1.  Kumbh Mela pictured from space – probably the largest human gathering in history BBC News, 26 January 2001.

2. Jump up ^

I remember watching a Hare Krishna video talking about 16 rivers and Krishna

Theres also a famous sikh guru who wrote a three line verse but couldn't do the fourth line but finally did the fourth. There was also an Indian Christian Martyr who wrote a 16 line four by four by four song with the fourth stanza different.

FOUR STAGES ENLIGHTENMENT

https://en.wikipedia.org/wiki/Four_stages_of_enlightenment

Stream-enterer (Sotapanna) is free from:

• 1. Identity view

• 2. Attachment to rites and rituals

• 3. Doubt about the teachings

• 4. Sensual desire

• 5. Ill will

Non-returner (Anāgāmi) is free from:

• 4. Sensual desire

• 5. Ill will

An Arahant is free from all of the five lower fetters and the five higher fetters, which are:

• 6. Craving for existence in the material world

• 7. Craving for existence in the ideal world (heaven)

• 8. Conceit

• 9. Restlessness

• 10. Ignorance

The Sutta Pitaka classifies the four levels according to the levels' attainments. In the Sthaviravada and Theravada traditions, which teach that progress in understanding comes all at once, and that 'insight' (abhisamaya) does not come 'gradually' (successively - anapurva),"[4] this classification is further elaborated, with each of the four levels described as a path to be attained suddenly, followed by the realisation of the fruit of the path.

HASSAN YOUSSEF ON THE FOUR QUADRANTS OF THE ISRAELI PRISON

Every section of the prison had twelve tents. Each tent housed twenty beds and footlockers. Maximum section capacity: 240 prisoners. Picture a rectangular picture frame, bordered with razor wire. Section Five was divided into quarters. A wall, topped with razor wire, bisected the section from north to south, and a low fence bisected it from east to west.

Quadrants One and Two (upper right and left) housed three Hamas tents each. Quadrant Three (lower right) had four tents—one each for Hamas, Fatah, the combined DFLP/PFLP, and Islamic Jihad. And Quadrant Four (lower left) had two tents, one for Fatah and one for the DFLP/PFLP.

Quadrant Four also had the kitchen, toilets, showers, an area for the shaweesh and kitchen workers, and basins for wudu. We lined up in rows for prayer in an open area in Quadrant Two. And, of course, there were guard towers at every corner. The main gate to Section Five was in the fence between Quadrants Three and Four.

One more detaiclass="underline" The fence running east and west had gates between Quadrants One and Three and between Two and Four. They were left open during most of the day, except during head counts. Then they were closed so officials could isolate half a section at a time.

I was assigned to the Hamas tent in the upper corner of Quadrant One, third bunk on the right. After the first head count, we were all sitting around talking when a distant voice shouted, “Bareed ya mujahideen! Bareed! [Mail from the freedom fighters! Mail!].”

https://en.wikipedia.org/wiki/Kaon

In particle physics, a kaon /ˈkeɪ.ɒn/, also called a K meson and denoted
K
,[nb 1] is any of a group of four mesonsdistinguished by a quantum number called strangeness. In the quark model they are understood to be bound states of a strange quark (or antiquark) and an up or down antiquark (or quark).

The four kaons are :

1. K−
, negatively charged (containing a strange quark and an up antiquark) has mass 493.677±0.013 MeV and mean lifetime (1.2384±0.0024)×10−8 s.

2. K+
(antiparticle of above) positively charged (containing an up quark and a strange antiquark) must (by CPT invariance) have mass and lifetime equal to that of
K−
. The mass difference is 0.032±0.090 MeV, consistent with zero. The difference in lifetime is (0.11±0.09)×10−8 s.

3. K0
, neutrally charged (containing a down quark and a strange antiquark) has mass 497.648±0.022 MeV. It has mean squared charge radius of −0.076±0.01 fm2.

4. K0
, neutrally charged (antiparticle of above) (containing a strange quark and a down antiquark) has the same mass.

It is clear from the quark model assignments that the kaons form two doublets of isospin; that is, they belong to the fundamental representation of SU(2) called the 2. One doublet of strangeness +1 contains the
K+
and the
K0
. The antiparticles form the other doublet (of strangeness −1).

FOUR DELTA BARYONS EXIST

https://en.wikipedia.org/wiki/Delta_baryon

The Delta baryons (or Δ baryons, also called Delta resonances) are a family of subatomic particle made of three up or down quarks (u or d quarks).

Four Δ baryons exist:
Δ++
(constituent quarks: uuu),
Δ+
(uud),
Δ0
(udd), and
Δ−
(ddd), which respectively carry an electric charge of +2 e, +1 e, 0 e, and −1 e.

The Δ baryons have a mass of about 1232 MeV/c2, a spin of 3/2, and an isospin of 3/2. In many ways, a Δ baryon is an 'excited' nucleon (symbol N). Nucleons are made of the same constituent quarks, but they are in a lower-energy spin configuration (spin 1/2). The
Δ+
(uud) and
Δ0
(udd) particles are the higher-energy equivalent of the proton (
N+
, uud) and neutron (
N0
, udd), respectively. However, the
Δ++
and
Δ−
have no nucleon equivalent.

POSSIBLE TRANSCENDENT FOURTH NEUTRINO TYPE

https://en.wikipedia.org/wiki/Neutrino_oscillation

Data from the LSND experiment appear to be in conflict with the oscillation parameters measured in other experiments. Results from the MiniBooNE appeared in Spring 2007 and contradicted the results from LSND, although they could support the existence of a fourth neutrino type, the sterile neutrino.[1]

https://en.wikipedia.org/wiki/Mishima:_A_Life_in_Four_Chapters

Mishima: A Life in Four Chapters is a 1985 American biographical drama film co-written and directed by Paul Schrader. The film is based on the life and work of Japanese writer Yukio Mishima (portrayed by Ken Ogata), interweaving episodes from his life with dramatizations of segments from his books The Temple of the Golden PavilionKyoko's House, and Runaway Horses. It was executive produced by Francis Ford Coppola and George Lucas.

Kyoko's House contains four equally ranking storylines, featuring four different protagonists, but Schrader picked out only the one which he considered convenient.[6]

FOUR METHODS
https://en.wikipedia.org/wiki/Scientific_method
In 1877,[19] Charles Sanders Peirce (/ˈpɜːrs/ like "purse"; 1839–1914) characterized inquiry in general not as the pursuit of truth per se but as the struggle to move from irritating, inhibitory doubts born of surprises, disagreements, and the like, and to reach a secure belief, belief being that on which one is prepared to act. He framed scientific inquiry as part of a broader spectrum and as spurred, like inquiry generally, by actual doubt, not mere verbal or hyperbolic doubt, which he held to be fruitless.[91] He outlined four methods of settling opinion, ordered from least to most successful:

The method of tenacity (policy of sticking to initial belief) – which brings comforts and decisiveness but leads to trying to ignore contrary information and others' views as if truth were intrinsically private, not public. It goes against the social impulse and easily falters since one may well notice when another's opinion is as good as one's own initial opinion. Its successes can shine but tend to be transitory.[92]
The method of authority – which overcomes disagreements but sometimes brutally. Its successes can be majestic and long-lived, but it cannot operate thoroughly enough to suppress doubts indefinitely, especially when people learn of other societies present and past.
The method of the a priori – which promotes conformity less brutally but fosters opinions as something like tastes, arising in conversation and comparisons of perspectives in terms of "what is agreeable to reason." Thereby it depends on fashion in paradigms and goes in circles over time. It is more intellectual and respectable but, like the first two methods, sustains accidental and capricious beliefs, destining some minds to doubt it.
The scientific method – the method wherein inquiry regards itself as fallible and purposely tests itself and criticizes, corrects, and improves itself.

http://www.strictly-stress-management.com/types_of_stress.h…
4 TYPES OF STRESS TO KNOW & UNDERSTAND
To manage stress well, you need to understand the four types of stress. The "Four Quadrant Stress Grid" below, uses a simple, well-known color-coding system to rate the four main types of stress. Green means good or go, yellow means proceed with caution and red means stop or bad.

Staying in any of the four zones for extended periods or on a repeated basis, will cause the body to create millions of free radicals, leading to extreme free radical damage or oxidative stress, which then leads to serious chronic degenerative disease (CDDs). More than 200 known CDD's are believed to be caused by oxidative stress, of which about half are inflammatory diseases and the other half are auto-immune diseases.

These four types of stress are perfectly complemented by the four main fields of stress (chemical, physical, electromagnetic and emotional). Being continually or repeatedly stressed out in any of the fields is extremely dangerous and should be avoided.

Of course, the red zone and even the yellow zone do the most damage, because the human body requires significant periods of non-stress from all four fields and all four types for the sake of optimal health and wellness. By respecting the limitations of the four types and the four fields of stress, you will be able to achieve and maintain optimal health and wellness.

DEFINITIONS FOR THE 4 MAIN TYPES OF STRESS
Chronic

adjective: being long-lasting and recurrent or characterized by long suffering ("Chronic indigestion")
adjective: having a habit of long standing ("A chronic smoker")
Acute

having or experiencing a rapid onset and short but severe course ("Acute appendicitis")
adjective: extremely sharp or intense ("Acute pain")
Eustress (Good Stress as in Euphoric Stress)

stress that is deemed healthful or giving one the feeling of fulfillment
the optimal state of stress to be the most productive
Distress (Bad Stress or Destructive Stress)

great pain, anxiety, or sorrow; acute physical or mental suffering; affliction; trouble.
a state of extreme necessity or misfortune.

THE TYPE IV CIVILIZATION IS TRANSCENDENT- BEYOND THE REGULAR THREE

https://en.wikipedia.org/wiki/Kardashev_scale

• In his book Parallel Worlds, Michio Kaku has discussed a Type IV civilization that could harness "extragalactic" energy sources such as dark energy.[30]

TRANSCENDENT FOURTH

https://en.wikipedia.org/wiki/Clarke%27s_three_laws

British science fiction writer Arthur C. Clarke formulated three adages that are known as Clarke's three laws, of which the third law is the best known and most widely cited:

When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.

The only way of discovering the limits of the possible is to venture a little way past them into the impossible.

Any sufficiently advanced technology is indistinguishable from magic.

Proposed fourth law

A fourth law has been proposed for the canon, despite Clarke's declared intention of not going one better than Newton. Geoff Holder quotes: "For every expert, there is an equal and opposite expert,"[5] which is part of American economist Thomas Sowell's "For every expert, there is an equal and opposite expert, but for every fact there is not necessarily an equal and opposite fact", from his 1995 book The Vision of the Anointed. [6]

THE PRETZELS IN CHEX MIX ARE QUADRANTS AND THE CHEX HAVE QUADRANT GRIDS

https://en.wikipedia.org/wiki/Chex

Chex is also the basis for a baked snack called "Chex Mix", in which different kinds of Chex are mixed with nuts, pretzels, and baked crackers,[1] and then often baked again with butter and various other spices (Worcestershire sauce in the original mix) to add flavor. Commercial and homemade varieties exist, and the dish is a common holiday snack in the United States. Chex Mix recipes were regularly featured on Chex cereal boxes, and commercially prepared Chex Mix snacks started to be sold in supermarkets[when?].

Twix bars are packaged with two or four bars in a package. Miniature and bite-size Twix are also available

TWIX EXTRA WITH CROSS TWIX X QUADRANT

https://jlbrooks.co.uk/twix-x-32.html

I read Who Moved My Cheese for Kids to my 9-year-old son recently. It’s a fun little book, based on the eponymous bestseller, about four characters who live in a ‘maze’ and look for ‘cheese’ to nourish them and make them happy. You probably know how the story goes already (it was a bestseller) but if not, or you’ve forgotten, here’s a quick synopsis:

Two of the characters are mice named Sniff and Scurry and two are little people – beings the size of mice who look and act a lot like people. Their names are Hem and Haw. The ‘cheese’ is a metaphor for what you want to have in life – whether it’s a good job, a loving relationship, money, possessions, health, or peace of mind. The ‘maze’ is where you look for what you want – the organization you work in, or the family or community you live in.

In the story, the characters are faced with unexpected change. Eventually, one of the little people deals with it successfully, and writes what he has learned from his experience on the maze walls. When you come to see the handwriting on the wall you can discover for yourself how to deal with change, so that you enjoy less stress and more success (however you define it) in your work and life.

There’s a lot of truth in the book and I thought it would be fun to relate the four characters to the four PSIU forces of Organizational Physics. That way, the next time you’re managing a Hem, Haw, Sniff, or Scurry, you’ll have a better sense for how to handle it.

As a refresher, here’s a matrix that shows the traits of the four universal PSIU forces. If this concept is new to you, you can quickly get a sense of it using the world’s fastest personality test (it takes less than 15 seconds to get a good sense of someone’s style).

The characters of Who Moved My Cheese mapped to the four PSIU forces of Organizational Physics.

In a nutshell:

• Sniff is an Innovator style. He’s got the ability to sense and respond to changes happening in the environment much more quickly than the other styles. He gets excited about creating new things and likes you to get excited with him.

• Scurry is a Producer style. He’s got the ability to run, run, run and do the work from early to late. He gets frustrated when there are obstacles in his path and seeks to run around them or punch through them.

• Hem is a Stabilizer style. He’s got the ability to make things systematized and controllable. In the story, it is Hem who gets left behind because change can be seen as a really big threat to someone who excels at control and stability.

• Haw is a Unifier style. He’s got the ability to empathize and connect well with others. In the story, it is Haw who follows Sniff and Scurry but all the while is concerned about where Hem is and how Hem is doing. Ultimately, Haw leaves the writing on the wall for others like Hem to follow.

FOUR FAMILIES OF AUTOMATA

https://cs.stanford.edu/people/eroberts/courses/soco/projects/2004-05/automata-theory/basics.html

There are four major families of automaton :

• Finite-state machine

• Pushdown automata

• Linear-bounded automata

• Turing machine

The families of automata above can be interpreted in a hierarchal form, where the finite-state machine is the simplest automata and the Turing machine is the most complex. The focus of this project is on the finite-state machine and the Turing machine. A Turing machine is a finite-state machine yet the inverse is not true.

https://en.wikipedia.org/wiki/4D_film

4D TRANSCENDING THE 3D- THE THREE FOUR DYNAMIC
https://en.wikipedia.org/wiki/4D_film
4D film or 4-D film is a marketing term for an entertainment presentation system combining a 3D film with physical effects that occur in the theatre in synchronization with the film. Effects simulated in a 4D film may include rain, wind, temperature changes, strobe lights, and vibration. Seats in 4D venues may vibrate or move a few centimeters during the presentations. Other common chair effects include air jets, water sprays, and leg and back ticklers. Hall effects may include smoke, rain, lightning, air bubbles, and smell.

Notable formats for providing different aspects of a "fourth dimension" to films include Human 4D, Sensurround, Smell-O-Vision and 4DX. As of June 2015, about 530 screens worldwide have installed the technology.[2]

TRAPEZIUM WAS A THREE PLUS ONE PATTERN WITH THE FOURTH BEING FOUND LATER

https://en.wikipedia.org/wiki/Trapezium_Cluster

The Trapezium or Orion Trapezium Cluster, also known by its Bayer designation of Theta1 Orionis, is a tight open cluster of stars in the heart of the Orion Nebula, in the constellation of Orion. It was discovered by Galileo Galilei. On February 4, 1617 he sketched three of the stars (A, C, D), but missed the surrounding nebulosity.[2][3][4] The fourth component (B) was identified by several observers in 1673.

The Trapezium is most readily identifiable by the asterism of four relatively bright stars for which it is named. The four are often identified as A, B, C, and D in order of increasing right ascension. The brightest of the four stars is C, or Theta1 Orionis C, with an apparent magnitude of 5.13. Both A and B have been identified as eclipsing binaries.

Trapezium in optical (left) and infrared light (right) from HubbleNASA photo.

http://www.messier-objects.com/trapezium-cluster/

Theta1 Orionis (θ1 Ori) - is the Trapezium and one of the finest multiple stars in the sky. Located at the heart of the Orion Nebula, a small 80mm (3.1-inch) telescope scope easily splits the main components. The brightest four stars are of magnitudes +5.1(C), +6.7(D), +6.7->7.7(A) and +8.0->8.7(B).

I WATCHED A TEACHING COMPNAY COURSE WHERE THEY SHOWED THE FOUR TYPES OF SKULLS

https://en.wikipedia.org/wiki/Skull

There are four types of amniote skull, classified by the number and location of their temporal fenestrae. These are:

• Anapsida – no openings

• Synapsida – one low opening (beneath the postorbital and squamosal bones)

• Euryapsida – one high opening (above the postorbital and squamosal bones); euryapsids actually evolved from a diapsid configuration, losing their lower temporal fenestra.

• Diapsida – two openings

Evolutionarily, they are related as follows:

THE FOUR COMPONENTS OF THE JOVIAN RING SYSTEM

https://en.wikipedia.org/wiki/Rings_of_Jupiter

The Jovian ring system is faint and consists mainly of dust.[1][5] It has four main components: a thick inner torus of particles known as the "halo ring"; a relatively bright, exceptionally thin "main ring"; and two wide, thick and faint outer "gossamer rings", named for the moons of whose material they are composed: Amalthea and Thebe.[6]

Jupiter's ring system was the third to be discovered in the Solar System, after those of Saturn and Uranus. It was first observed in 1979 by the Voyager 1 space probe.[1] It is composed of four main components: a thick inner torus of particles known as the "halo ring"; a relatively bright, exceptionally thin "main ring"; and two wide, thick and faint outer "gossamer rings", named after the moons of whose material they are composed: Amalthea and Thebe

IT IS SPLIT INTO FOUR PARTS

https://en.wikipedia.org/wiki/The_Gods_Must_Be_Crazy_II

The Gods Must Be Crazy II is a 1989 South African comedy film, a sequel to Jamie Uys' 1980 comedy film, The Gods Must Be Crazy, and it is the second film in The Gods Must Be Crazy film series.

The film is split into four stories:

Xixo trying to find his lost children

Two elephant poachers travelling in a truck on which Xixo's children are stuck

A zoologist and a lawyer stranded in a desert

Two soldiers fighting each other

https://en.wikipedia.org/wiki/Four_Pillars_of_Destiny

The Four Pillars of Destiny (四柱命理) is a Chinese conceptual term describing the four components that supposedly create a person's destiny or fate. The four components within the moment of birth are year, month, day, and hour. The four pillars (a translation of the Chinese dynastic phrase Shēng Chén Bā Zì; Korean Saju) are used alongside fortune-telling practices such as Zǐ wēi dòu shù within the realm of Chinese astrology. Chinese astrology believes the alignment of sun and stars affects a person's destiny. Ba Zi uses the alignment of sun's position, in other words the solar calendar, while Zǐ wēi dòu shù uses the alignment of moon and stars positions.

QUARTILES ARE VERY IMPORTANT IN STATISTICS AND ONE OF THE FIRST THINGS TAUGHT IN STATISTICS CALSS IS THE INNER QUARTILE RANGE

https://en.wikipedia.org/wiki/Quartile

In descriptive statistics, the quartiles of a ranked set of data values are the three points that divide the data set into four equal groups, each group comprising a quarter of the data. A quartile is a type of quantile. The first quartile (Q1) is defined as the middle number between the smallest number and the median of the data set. The second quartile (Q2) is the median of the data. The third quartile (Q3) is the middle value between the median and the highest value of the data set.

In applications of statistics such as epidemiologysociology and finance, the quartiles of a ranked set of data values are the four subsets whose boundaries are the three quartile points. Thus an individual item might be described as being "in the upper quartile".

Boxplot (with quartiles and an interquartile range) and a probability density function (pdf) of a normal N(0,1σ2) population

BOXPLOTS ARE ONE OF THE FIRST THINGS AND MOST IMPORTANT THINGS TAUGHT IN STATISTICS IN MIDDLE SCHOOL AS WELL AS COLLEGe

https://en.wikipedia.org/wiki/Box_plot

In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles. Box plots may also have lines extending vertically from the boxes (whiskers) indicating variability outside the upper and lower quartiles, hence the terms box-and-whisker plot and box-and-whisker diagram.

Box and whisker plots are uniform in their use of the box: the bottom and top of the box are always the first and third quartiles, and the band inside the box is always the second quartile (the median). But the ends of the whiskers can represent several possible alternative values, among them:

• the minimum and maximum of all of the data[1] (as in figure 2)

• the lowest datum still within 1.5 IQR of the lower quartile, and the highest datum still within 1.5 IQR of the upper quartile (often called the Tukey boxplot)[2][3] (as in figure 3)

• one standard deviation above and below the mean of the data

• the 9th percentile and the 91st percentile

• the 2nd percentile and the 98th percentile.

Figure 1. Box plot of data from the Michelson–Morley experiment

QUARTILES DIVIDE INTO FOUR EQUAL PARTS

https://en.wikipedia.org/wiki/Interquartile_range

In descriptive statistics, the interquartile range (IQR), also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles,[1][2] IQR = Q3 −  Q1. In other words, the IQR is the first quartile subtracted from the third quartile; these quartiles can be clearly seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed range, and is the most significant basic robust measure of scale.

The IQR is a measure of variability, based on dividing a data set into quartiles. Quartiles divide a rank-ordered data set into four equal parts. The values that separate parts are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

http://conservatoryplanning.com/conservatory-grid-options/

WINDOWS

A traditional grid consists of multiple horizontal and vertical divisions

TETRA IS FOUR- FULLER TETRAHEDRON CITY

Fuller and Sadao designed this crystalline pyramid for Matsutaro Shoriki, a Japanese financier. The form is a tetrahedron (a four-sided triangular solid), with each edge measuring two miles. The buoyant metropolis was designed to accommodate one million citizens in 300,000 apartment units, and it even includes a huge interior harbor. Fuller, who spent his career searching for "ever higher performance with ever less investment of material resources," envisioned Tetrahedron City as an efficient response to two major problems of architecture and urban planning: construction costs and land acquisition. The tetrahedral composition with an aluminum octet truss system makes the structure's enormous size feasible and economical. Additionally, by designing the city to float at sea (in this case in Tokyo Bay), the cost of real estate is diverted. Fuller and Sadao's radical urban proposal was never built.

CROSS UFOS- QUATERNITIES

https://en.wikipedia.org/wiki/1561_celestial_phenomenon_over_Nuremberg

The 1561 celestial phenomenon over Nuremberg was a mass sighting of celestial phenomena or unidentified flying objects (UFO) above NurembergGermany

The broadsheet describes objects of various shapes including crosses, globes, two lunar crescents, a black spear and tubular objects from which several smaller, round objects emerged and darted around the sky at dawn.

Jung expressed a view that the spectacle was likely a natural phenomenon with religious and military interpretations overlaying it. “If the Ufos were living organisms, one would think of a swarm of insects rising with the sun, not to fight one another but to mate and celebrate the marriage flight.” A military interpretation would view the tubes as cannons and the spheres as cannonballs, emphasize the black spearhead at the bottom of the scene, and Glaser’s own testimony that the globes fought vehemently until exhausted. A religious view would emphasize the crosses. Jung thinks the images of four globes coupled by lines suggested crossed marriage quaternities and forms the model for “the primitive cross cousin marriage.” It could also be an individuation symbol. The association of sunrise suggests “the revelation of the light.”[11]

Celestial phenomenon over the German city of Nuremberg on April 14, 1561 as printed in an illustrated news notice in the same month.

## "Pope's Revenge"

When the sun shines on the Fernsehturm's tiled stainless steel dome, the reflection usually appears in the form of a cross. Berliners nicknamed the luminous cross Rache des Papstes, or the "Pope's Revenge". For the same reasons, the structure was also called "St. Walter" (from Walter Ulbricht). U.S. President Ronald Reagan mentioned this in his Tear down this wall speech on 12 June 1987.[6] It is also affectionately known as the Toothpick and Telespargel (TV-asparagus) due to its shape.[5]

FOUR BALL BILLIARDS- REGULAR BILLIARDS POOL HAS 16 BALLS IN TOTAL- 16 SQUARES QMR

https://en.wikipedia.org/wiki/Four-ball_billiards

Four-ball billiards (often abbreviated to simply four-ball, and sometimes spelled 4-ball or fourball) is a carom billiardsgame, played on a pocketless table with four billiard balls, usually two red and two white, one of the latter with a spot to distinguish it (in some sets, one of the white balls is yellow instead of spotted). Each player is assigned one of the white (or yellow) balls as a cue ball. A point is scored when a shooter's cue ball caroms on any two other balls in the same shot (with the opponent's cue ball serving as an object ball, along with the reds, for the shooter). Two points are scored when the shooter caroms on each of the three object balls in a single shot.[1] A carom on only one ball results in no points, and ends the shooter's inning.

GEORGE WASHINGTON 16 SIDE BARN- 16 SQUARES QUADRANT MODEL

http://www.mountvernon.org/digital-encyclopedia/article/16-sided-barn/

The selection of sixteen sides rather than any other even number is more difficult to explain, although it was also probably related to the octagonal shape of the interior framing system. Since the treading floor was to withstand the weight and punishment of trotting horses, durable white oak was utilized. Most of the remainder of the wood for the barn, except the roof and the ground floor interior octagon posts, was made of pine.

Because the barn has an unusual shape, it must have been a difficult building for Thomas Green, Washington's carpenter, and the Mount Vernon slave carpenters to construct. One of the most important decisions Washington made was to design a building comprised of two nested polygons. This refers to the sixteen-sided outside wall, enclosing the interior octagonal framing system. This design was the most straightforward way to support the roof since the same system for laying out the rafters could be continued from the eaves to the peak of the roof.

The transition from the exterior sixteen sides to the interior eight meant that the structure of the roof had to undergo some subtle adjustments as construction continued upward, including the reduction of the number of rafters from eighty to forty-eight. Since the rafters were the only structural members spanning that space, they served the crucial function of binding the exterior wall to the frame.

Interior of the 16-sided barn showing the roof's framing. - MVLA

16 SIDE BARNS ARE THE HIGHEST

https://en.wikipedia.org/wiki/Round_barn

Round barns date to the 18th and early 19th century. George Washington designed and built a sixteen-sided threshing barn at his Dogue Run Farm in Fairfax County, Virginia in 1793.[4] The first truly round barn in North America was constructed in 1826 at Hancock Shaker Village.[5] A few other round barns appeared on the American landscape before the Civil War.

The "Octagonal Era" of round barn design stretched from about 1850 until 1900. Round barns, such as Washington's, were often multi sided in their earliest incarnations. Multi-sided round barns came in a variety of polygonal shapes, including layouts of six, eight, nine, ten, twelve, fourteen and sixteen sides. Polygonal barns constructed before the advent of balloon framing tended to have interior spaces that were more rectangular than circular.[2]

FOUR MASTED SHIPS- TRANSCENDENT TO THE THREE

https://en.wikipedia.org/wiki/Barque

Most ocean-going windjammers were four-masted barques, since the four-masted barque is considered the most efficient rig available because of its ease of handling, small need of manpower, good running capabilities, and good capabilities of rising toward wind. Usually the main mast was the tallest; that of Moshulu extends to 58 m off the deck. The four-masted barque can be handled with a surprisingly small crew—at minimum, ten—and while the usual crew was around thirty, almost half of them could be apprentices.

THE FAMOUS RHIND PAPYRUS HAS FOUR SECTIONS- A LOT OF THE PROBLEMS THEMSELVES ARE BASED AROUND FOUR.

https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

In general, the papyrus consists of four sections: a title page, the 2/n table, a tiny "1-9/10 table", and 91 problems, or "numbers". The latter are numbered from 1 through 87 and include four mathematical items which have been designated by moderns as problems 7B, 59B, 61B, and 82B. Numbers 85-87, meanwhile, are not mathematical items forming part of the body of the document, but instead are respectively: a small phrase ending the document, a piece of "scrap-paper" used to hold the document together (having already contained unrelated writing), and a historical note which is thought to describe a time period shortly after the completion of the body of the papyrus. These three latter items are written on disparate areas of the papyrus' verso(back side) , far away from the mathematical content. Chace therefore differentiates them by styling them as numbers as opposed to problems, like the other 88 numbered items.

FOUR PROTAGONISTS JOURNEY TO THE WEST

Journey to the West is a Chinese novel published in the 16th century during the Ming dynasty and attributed to Wu Cheng'en. It is one of the Four Great Classical Novels of Chinese literature.

The novel has 100 chapters that can be divided into four unequal parts. The first part, which includes chapters 1–7, is a self-contained introduction to the main story. It deals entirely with the earlier exploits of Sun Wukong, a monkey born from a stone nourished by the Five Elements, who learns the art of the Tao, 72 polymorphic transformations, combat, and secrets of immortality, and through guile and force makes a name for himself, Qitian Dasheng (simplified Chinese: 齐天大圣; traditional Chinese: 齊天大聖), or "Great Sage Equal to Heaven". His powers grow to match the forces of all of the Eastern (Taoist) deities, and the prologue culminates in Sun's rebellion against Heaven, during a time when he garnered a post in the celestial bureaucracyHubris proves his downfall when the Buddha manages to trap him under a mountain, sealing it with a talisman for five hundred years.

18th-century Chinese illustration of a scene from Journey to the West

An illustrated edition of the story

The second part (chapters 8–12) introduces the nominal main character, Xuanzang(Tang Sanzang), through his early biography and the background to his great journey. Dismayed that "the land of the South knows only greed, hedonism, promiscuity, and sins", the Buddha instructs the bodhisattva Avalokiteśvara(Guanyin) to search Tang China for someone to take the Buddhist sutras of "transcendence and persuasion for good will" back to the East. Part of the story here also relates to how Xuanzang becomes a monk (as well as revealing his past life as a disciple of the Buddha named "Golden Cicada" (金蟬子) and comes about being sent on this pilgrimage by Emperor Taizong, who previously escaped death with the help of an official in the Underworld).

The third and longest section of the work is chapters 13–99, an episodic adventure story in which Xuanzang sets out to bring back Buddhist scriptures from Leiyin Temple on Vulture Peak in India, but encounters various evils along the way. The section is set in the sparsely populated lands along the Silk Road between China and India, including XinjiangTurkestan, and Afghanistan. The geography described in the book is, however, almost entirely fantasy; once Xuanzang departs Chang'an, the Tang capital, and crosses the frontier (somewhere in Gansu province), he finds himself in a wilderness of deep gorges and tall mountains, inhabited by demons and animal spirits, who regard him as a potential meal (since his flesh was believed to give immortality to whomever ate it), with the occasional hidden monastery or royal city-state amidst the harsh setting.

Episodes consist of 1–4 chapters and usually involve Xuanzang being captured and having his life threatened while his disciples try to find an ingenious (and often violent) way of liberating him. Although some of Xuanzang's predicaments are political and involve ordinary human beings, they more frequently consist of run-ins with various demons, many of whom turn out to be earthly manifestations of heavenly beings (whose sins will be negated by eating the flesh of Xuanzang) or animal-spirits with enough Taoist spiritual merit to assume semi-human forms.

Chapters 13–22 do not follow this structure precisely, as they introduce Xuanzang's disciples, who, inspired or goaded by Guanyin, meet and agree to serve him along the way in order to atone for their sins in their past lives.

• The first is Sun Wukong, or Monkey, whose given name loosely means "awakened to emptiness", trapped by the Buddha for defying Heaven. He appears right away in chapter 13. The most intelligent and violent of the disciples, he is constantly reproved for his violence by Xuanzang. Ultimately, he can only be controlled by a magic gold ring that Guanyin has placed around his head, which causes him unbearable headaches when Xuanzang chants the Ring Tightening Mantra.

• The second, appearing in chapter 19, is Zhu Bajie, literally "Eight Precepts Pig", sometimes translated as Pigsy or just Pig. He was previously the Marshal of the Heavenly Canopy, a commander of Heaven's naval forces, and was banished to the mortal realm for flirting with the moon goddess Chang'e. A reliable fighter, he is characterised by his insatiable appetites for food and women, and is constantly looking for a way out of his duties, which causes significant conflict with Sun Wukong.

• The third, appearing in chapter 22, is the river ogre Sha Wujing, also translated as Friar Sand or Sandy. He was previously the celestial Curtain Lifting General, and was banished to the mortal realm for dropping (and shattering) a crystal goblet of the Queen Mother of the West. He is a quiet but generally dependable and hard-working character, who serves as the straight foil to the comic relief of Sun and Zhu.

• The fourth is Yulong, the third son of the Dragon King of the West Sea, who was sentenced to death for setting fire to his father's great pearl. He was saved by Guanyin from execution to stay and wait for his call of duty. He appears first in chapter 15, but has almost no speaking role, as throughout the story he mainly appears as a horse that Xuanzang rides on.

Chapter 22, where Sha Wujing is introduced, also provides a geographical boundary, as the river that the travelers cross brings them into a new "continent". Chapters 23–86 take place in the wilderness, and consist of 24 episodes of varying length, each characterised by a different magical monster or evil magician. There are impassably wide rivers, flaming mountains, a kingdom with an all-female population, a lair of seductive spider spirits, and many other fantastic scenarios. Throughout the journey, the four brave disciples have to fend off attacks on their master and teacher Xuanzang from various monsters and calamities.

It is strongly suggested that most of these calamities are engineered by fate and/or the Buddha, as, while the monsters who attack are vast in power and many in number, no real harm ever comes to the four travelers. Some of the monsters turn out to be escaped celestial beasts belonging to bodhisattvas or Taoist sages and deities. Towards the end of the book there is a scene where the Buddha literally commands the fulfillment of the last disaster, because Xuanzang is one short of the 81 tribulations he needs to face before attaining Buddhahood.

In chapter 87, Xuanzang finally reaches the borderlands of India, and chapters 87–99 present magical adventures in a somewhat more mundane (though still exotic) setting. At length, after a pilgrimage said to have taken fourteen years (the text actually only provides evidence for nine of those years, but presumably there was room to add additional episodes) they arrive at the half-real, half-legendary destination of Vulture Peak, where, in a scene simultaneously mystical and comic, Xuanzang receives the scriptures from the living Buddha.

Chapter 100, the last of all, quickly describes the return journey to the Tang Empire, and the aftermath in which each traveller receives a reward in the form of posts in the bureaucracy of the heavens. Sun Wukong (Monkey) and Xuanzang (monk) achieve Buddhahood, Sha Wujing (Sandy) becomes an arhat, the dragon horse is made a nāga, and Zhu Bajie (Pig), whose good deeds have always been tempered by his greed, is promoted to an altar cleanser (i.e. eater of excess offerings at altars).

The four protagonists, from left to right: Sun WukongTang Sanzang (on the White Dragon Horse), Zhu Bajie, and Sha Wujing

THE TETRAHEDRON AND SYNERGETICS

https://en.wikipedia.org/wiki/Synergetics_(Fuller)

Synergetics is the empirical study of systems in transformation, with an emphasis on total system behavior unpredicted by the behavior of any isolated components, including humanity's role as both participant and observer.

Since systems are identifiable at every scale from the quantum level to the cosmic, and humanity both articulates the behavior of these systems and is composed of these systems, synergetics is a very broad discipline, and embraces a broad range of scientific and philosophical studies including tetrahedral and close-packed-sphere geometries, thermodynamics, chemistry, psychology, biochemistry, economics, philosophy and theology. Despite a few mainstream endorsements such as articles by Arthur Loeb and the naming of a molecule "buckminsterfullerene", synergetics remains an iconoclastic subject ignored by most traditional curricula and academic departments.

Buckminster Fuller (1895-1983) coined the term and attempted to define its scope in his two volume work Synergetics.[1][2][3] His oeuvre inspired many researchers to tackle branches of synergetics. Three examples: Haken explored self-organizing structures of open systems far from thermodynamic equilibrium, Amy Edmondson explored tetrahedral and icosahedral geometry, Stafford Beer tackled geodesics in the context of social dynamics, and Nystrom proposed a theory of computational cosmography.[4] Many other researchers toil today on aspects of Synergetics, though many deliberately distance themselves from Fuller's broad all-encompassing definition, given its problematic attempt to differentiate and relate all aspects of reality including the ideal and the physically realized, the container and the contained, the one and the many, the observer and the observed, the human microcosm and the universal macrocosm.

Tetrahedral accounting

A chief hallmark of this system of mensuration was its unit of volume: a tetrahedron defined by four closest-packed unit-radius spheres. This tetrahedron anchored a set of concentrically arranged polyhedra proportioned in a canonical manner and inter-connected by a twisting-contracting, inside-outing dynamic named the Jitterbug Transformation.[citation needed]

https://en.wikipedia.org/wiki/Maui

Maui is part of the State of Hawaii and is the largest of Maui County's four islands, which include MolokaʻiLānaʻi, and unpopulated Kahoʻolawe

https://en.wikipedia.org/wiki/Maui_Nui

Maui Nui or Greater Maui, is a modern geologists' name given to a prehistoric Hawaiian Island built from seven shield volcanoes. Nui means "great/large" in the Hawaiian language.

1.2 million years ago, Maui Nui was 14,600 square kilometres (5,600 sq mi),[1] 40% larger than the present-day island of Hawaiʻi. Sea levels were lower than today's due to distant glaciation locking up the Earth's water during ice ages, thus exposing more land. As the volcanoes slowly settled by subsidence due to the weight of the shield volcanoes and erosion, the saddles between them slowly flooded, forming four islands: MauiMolokaʻiLānaʻi and Kahoʻolawe by about 200,000 years ago. Another former volcanic island lying west of Molokaʻi was completely submerged and covered with a cap of coral; it is now known as Penguin Bank.

The sea floor between these four islands is relatively shallow, about 500 metres (1,600 ft) deep, and all of the islands except Kahoʻolawe were joined during the low sea levels of the last glacial maximum, about 20,000 years ago. But at the outer edges of former Maui Nui, as with the edges of all Hawaiian Islands, the floor plummets to the abyssal ocean floor of the Pacific Ocean. The steep slopes can result in massive landslides due to flank collapse, including one which removed most of the northern half of East Molokaʻi.

https://quoteinvestigator.com/2014/02/28/clever-lazy/

The text was reprinted under the title “Selecting Officers” in the “United States Naval Institute Proceedings” in March 1933 and in the “Review of Military Literature: The Command and General Staff School Quarterly” in September 1933. Boldface has been added to excerpts: 1 2

General Freiherr von Hammerstein-Equord, the present chief of the German Army, has a method of selecting officers which strikes us as being highly original and peculiarly un-­Prussian. According to Exchange, a Berlin newspaper has printed the following as his answer to a query as to how he judged his officers: “I divide my officers into four classes as follows: The clever, the industrious, the lazy, and the stupid. Each officer always possesses two of these qualities.

Those who are clever and industrious I appoint to the General Staff. Use can under certain circumstances be made of those who are stupid and lazy. The man who is clever and lazy qualifies for the highest leadership posts. He has the requisite nerves and the mental clarity for difficult decisions. But whoever is stupid and industrious must be got rid of, for he is too dangerous.”

In 1942 Viscount Swinton (Philip Lloyd-Greame) spoke in the House of Lords in London as recorded in the Hansard. He described the four classes of officers and credited an unnamed German General: 3

I do not know whether your Lordships are familiar with the saying of a German General that there are four types of officer but I think that it is relevant to what we are discussing. He said that there are four types of officer: the clever and lazy, the clever and industrious, the stupid and lazy, and the stupid and industrious.

The clever and lazy you make Chief of Staff, because he will not try to do everybody else’s work, and will always have time to think. The clever and industrious you make his deputy. The stupid and lazy you put into a line battalion, and kick him into doing a job of work. The stupid and industrious you must get rid of at once, because he is a national danger.

A compact instance of the saying appeared in the World War II diary of General Henry H. ‘Hap’ Arnold. An entry dated November 16, 1943 presented the lacerating words that General Douglas MacArthur used to comment about a subordinate: 4

“You can use the brilliant but lazy man as a strategist, a brilliant but energetic man as a Chief of Staff, but God help you with a dumb but energetic man”: MacArthur’s estimate of GB [George Brett] one of the most charming damn fools I have had the pleasure of meeting.

Lieutenant General George Brett was reassigned to the less important Caribbean Defense Command during World War II primarily because of MacArthur’s negative opinion.

In 1953 LIFE magazine printed a condensed version of the analysis and ascribed the words to Hammerstein: 5

General Kurt von Hammerstein-Equord, head of the German army from 1930 to 1933, defined the difference this way: “Officers who are clever and industrious I appoint to the General Staff . . . the man who is clever and lazy qualifies for the highest leadership posts [Command]. He has the requisite nerves and the mental clarity for difficult decisions.”

In 1966 the “Dictionary of Military and Naval Quotations” compiled by Robert Debs Heinl included an instance of the quotation: 6

I divide officers into four classes—the clever, the lazy, the stupid, and the industrious. Each officer possesses at least two of these qualities. Those who are clever and industrious are fitted for the high staff appointments. Use can be made of those who are stupid and lazy.

The man who is clever and lazy is fit for the very highest command. He has the temperament and the requisite nerves to deal with all situations. But whoever is stupid and industrious must be removed immediately.

Attributed to General Kurt von
Hammerstein, c. 1933

By 2004 the scheme for sorting officers was being ascribed to General Erich von Manstein in a book called “Living the 80/20 Way: Work Less, Worry Less, Succeed More, Enjoy More”: 7

German military chief General von Manstein said:
“There are only four types of officers.

First, there are the lazy, stupid ones. Leave them alone, they do no harm.

Second, there are the hard-working intelligent ones. They make excellent staff officers, ensuring that every detail is properly considered.

Third, there are the hard-working, stupid ones. These people are a menace, and must be fired at once. They create irrelevant work for everybody.

Finally, there are the intelligent lazy ones. They are suited for the highest office.”

By 2005 the categorization method was being credited to Helmuth von Moltke. There was some ambiguity because there were two German Generals: Helmuth von Moltke the Elder and the Younger. Here is an example from a website post in 2005: 8

Legend has it that Prussian General von Moltke had a very simple, but elegant, conceptual framework which underlied his approach to leadership and management. He classified all individuals on only two dimensions — intelligence and drive — which he considered key independent variables. According to General von Moltke, people are either smart or stupid and they are either active or lazy.

What often surprises observers is the relative value General von Moltke assigned to these four categories of people. Although most people would reflexively assume that the “Smart Actives” would be the most prized — it was actually the “Smart Lazies” that are the most valuable.

A 2011 book about military strategy attributed an instance of the four-fold classification system to Frederick the Great: 9

Frederick the Great’s typology of officers provides a hint of how one might best think about the issue of military competence. The Prussian king suggested that there were four types of officers. First were the brilliant but lazy. He suggested such officers had the attributes to function at the highest levels of command.

Second were the brilliant, but diligent. They made the best staff officers. Third were the less intelligent but lazy. They made good battalion officers. Finally, there were the less intelligent and the diligent. They were the most dangerous to the proper functioning of any military organization, in both peace and war, because of their penchant for confusing process and work for product.

In conclusion, this was a difficult expression to trace because it was complex, and it could be articulated in myriad ways. Currently, the earliest example located by QI appeared in English in 1933 and was credited to Kurt von Hammerstein-Equord.

If the 1933 citation was accurate then the expression appeared in German in a Berlin newspaper in 1932 or 1933. QI has not yet located this instance.

Image Notes: The table graphic was created by QI.

(Special thanks to Dave Hause who noted the existence of this saying attributed to Helmuth von Moltke the Elder during a discussion of the “lazy man” quotation attributed to Bill Gates. Great thanks to Dan Goncharoff who noted the ascription to Erich von Manstein. Also, thanks to the other ADS discussants: John Baker, Fred Shapiro, Jonathan Lighter, and Victor Steinbok.)

Pollens/Microspores of Lycopersicon esculentum at coenocytic tetrad stage of development observed through oil immersion microscope; the chromosomes of what will become four pollen grains can be seen.

https://en.wikipedia.org/wiki/Pollen

In angiosperms, during flower development the anther is composed of a mass of cells that appear undifferentiated, except for a partially differentiated dermis. As the flower develops, four groups of sporogenous cells form within the anther. The fertile sporogenous cells are surrounded by layers of sterile cells that grow into the wall of the pollen sac. Some of the cells grow into nutritive cells that supply nutrition for the microspores that form by meiotic division from the sporogenous cells.

In a process called microsporogenesis, four haploid microspores are produced from each diploid sporogenous cell (microsporocyte, pollen mother cell or meiocyte), after meiotic division. After the formation of the four microspores, which are contained by callose walls, the development of the pollen grain walls begins. The callose wall is broken down by an enzyme called callase and the freed pollen grains grow in size and develop their characteristic shape and form a resistant outer wall called the exine and an inner wall called the intine. The exine is what is preserved in the fossil record. Two basic types of microsporogenesis are recognised, simultaneous and successive. In simultaneous microsporogenesis meiotic steps I and II are completed prior to cytokinesis, whereas in successive microsporogenesis cytokinesis follows. While there may be a continuum with intermediate forms, the type of microsporogenesis has systematic significance. The predominant form amongst the monocotsis successive, but there are important exceptions.[3]

http://www.yourarticlelibrary.com/biology/stamen-male-reproductive-organ-in-flowering-plants/11816/

The microspore mother cells or microsporocytes develop an internal layer of callose (β-1, 3 glucan) which breaks the plasmodesmal connections among themselves. The separated mother cells round off and undergo meiosis to produce tetrads of haploid microspores or pollen grains.

image

The phenomenon is called micro-sporogenesis. The pollen grains of a tetrad grow and separate from one another. Usually the arrangement of microspores in a tetrad is tetrahedral (most common type) or isobilateral. However, decussate, linear and T-shaped tetrads are also found (Fig. 2.5). In Aristolochia elegans, all the five type of tetrads have been recorded.

clip_image006

Mostly, all the 4 nuclei in a tetrad remain functional to form 4 microspores. However, in Сурегасеае, only one functions and therefore, only one microspore instead of 4 is formed by one meiosis. In some cases, all the 4 pollens remain attached forming compound pollen grains e.g., Juncus, Jatropha, Typha.

In Calotropis and related plants all the pollen grains of an anther lobe remain united in a single sac called pollinium. Two pollinia of adjacent anthers are attached to produce a translator. Polyspory is occurrence of more than four spores in a tetrad. As many as 11 microspores are observed in a ‘tetrad’ in Cuscuta.

BUDDHISM QUADRANT MODEL FOUR RIGHT ASSERTIONS

https://en.wikipedia.org/wiki/Four_Right_Exertions

https://en.wikipedia.org/wiki/Four_Right_Exertions

The Four Right Exertions (also known as, Four Proper Exertions, Four Right Efforts, Four Great Efforts, Four Right Endeavors or Four Right Strivings) (Pali: sammappadhāna; Skt.: samyak-pradhāna or samyakprahāṇa) are an integral part of the Buddhist path to Enlightenment. Built on the insightful recognition of the arising and non-arising of various mental qualities over time and of our ability to mindfully intervene in these ephemeral qualities, the Four Right Exertions encourage the relinquishment of harmful mental qualities and the nurturing of beneficial mental qualities.

The Four Right Exertions are associated with the Noble Eightfold Path's factor of "right effort" (sammā-vāyāma) and the Five Spiritual Faculties' faculty of "energy" (viriya); and, are one of the seven sets of Qualities Conducive to Enlightenment

The Four Right Exertions are found in the Vinaya Pitaka, Sutta Pitaka, Abhidhamma Pitaka and Pali commentaries.[1] Additionally, a similar-sounding but different concept, the "four exertions," is referenced in the literature as well. These two concepts are presented below.

Four Right Exertions

The Four Right Exertions (cattārimāni sammappadhānāni) are defined with the following traditional phrase:

"There is the case where a monk generates desire, endeavors, activates persistence, upholds & exerts his intent for:

"[i] the sake of the non-arising [anuppādāya] of evil, unskillful qualities that have not yet arisen.

"[ii] ... the sake of the abandonment [pahānāya] of evil, unskillful qualities that have arisen.

"[iii] ... the sake of the arising [uppādāya] of skillful qualities that have not yet arisen.

"[iv] ... the maintenance [ṭhitiyā], non-confusion, increase, plenitude, development, & culmination of skillful qualities that have arisen."[2]

This elaboration is attributed to the Buddha in response to the following questions:

"What is right effort?" (SN 45.8,[3] in the context of the Noble Eightfold Path)

"What is the faculty of energy?" (SN 48.10,[4] in the context of the Five Spiritual Faculties)

"What are the four right strivings?" (SN 49.1ff.)[5]

This formulation is also part of an extensive exposition by Ven. Sariputta when addressing the question of "What is this Dhamma that has been well-proclaimed by the Lord [Buddha]?" (DN 33).[6] In addition, in a section of the Anguttara Nikaya known as the "Snap of the Fingers Section" (AN 1.16.6, Accharāsaṇghātavaggo), the Buddha is recorded as stating that, if a monk were to enact one of the four right exertions for the snap of the fingers (or, "only for one moment")[7] then "he abides in jhana, has done his duties by the Teacher, and eats the country's alms food without a debt."[8]

A similar two-part elaboration is provided by the Buddha in SN 48.9, again in the context of the Five Spiritual Faculties, when he states:

"And what, bhikkhus, is the faculty of energy? Here, bhikkhus, the noble disciple dwells with energy aroused for the abandoning of unwholesome states and the acquisition of wholesome states; he is strong, firm in exertion, not shirking the responsibility of cultivating wholesome states. This is the faculty of energy."[9]

What constitutes "unskillful" or "unwholesome" (akusala) and "skillful" or "wholesome" (kusala) qualities is taken up in the Abhidhamma Pitaka and the post-canonical Pali commentaries.[10] In general, the unskillful states are the three defilements (kilesa): greed (lobha), hatred (dosa) and delusion (moha).[11] Skillful states are the defilements' opposites: non-greed (alobha), non-hatred (adosa) and non-delusion (amoha).[12][13]

Four Exertions

Throughout the Pali Canon, a distinction is made between the fourfold "exertions" (padhāna) and the four "Right Exertions" (sammappadhāna). While similarly named, canonical discourses consistently define these different terms differently, even in the same or adjacent discourses.[14]

The four exertions (cattārimāni padhānāni) are summarized as:

Restraint (saṃvara padhāna) of the senses.

Cultivation (bhāvanā padhāna) of Enlightenment Factors.

Preservation (anurakkhaṇā padhāna) of concentration, for instance, using charnel-ground contemplations.[15]

WOLFGANG PAULI AND THE QUATERNITY

Pauli knew and embraced Jung’s early concepts; Myers and Briggs extended them to provide a psychological dynamic of four dimensions with eight polar opposites, a thesis that would have delighted Pauli. The mature Pauli loved any four- part dynamic inside a whole entity, in this instance a “quaternian” dynamic inside the whole “mandala” that makes up one’s personality. Myers and Briggs advanced a classification scheme of personality based upon the axes of Extroversion-Introversion, Sensing-Intuition, Thinking-Feeling, and Judging-Perceiving. I will discuss the Myers- Briggs typology further below.24

Examples of mandalas that Pauli saw in his later years, as they appear in his letters to Jung, may help to elucidate his earlier noticing of kernels. There are several examples of Pauli's use of mandalas, which he alternately refers to as "quaternios." In a letter of 1950, he draws four physical concepts on two axes in a four-part mandala, as shown below. 133 This image has a holistic quality and an internal “symmetry-breaking” feature, as he showed later when he modified it in a letter of 1952, in which the “poles” of “energy” and “space-time” are broken up into three-dimensional momentum plus energy and three-dimensional space plus time.134

Examples of mandalas that Pauli saw in his later years, as they appear in his letters to Jung, may help to elucidate his earlier noticing of kernels. There are several examples of Pauli's use of mandalas, which he alternately refers to as "quaternios." In a letter of 1950, he draws four physical concepts on two axes in a four-part mandala, as shown below. 133 This image has a holistic quality and an internal “symmetry-breaking” feature, as he showed later when he modified it in a letter of 1952, in which the “poles” of “energy” and “space-time” are broken up into three-dimensional momentum plus energy and three-dimensional space plus time.134

https://arxiv.org/pdf/1003.3223.pdf

Thus, I now will identify similar examples as kernels in mathematical physics that Pauli may have encountered between 1913 and 1918, as follows: (1) the complex numbers z = x + i y and z = |z| e i θ; (2) Maxwell’s equations in electrodynamics; and (3) the Lorentz group in special relativity. In later chapters covering later periods in Pauli's life, I propose that Pauli saw kernels in other areas: (1) the mathematical quaternians used by Sommerfeld and Felix Klein to describe the motion of a top; (2) the angular momentum of the electron and its four quantum numbers; (3) the momentum and position variables in quantum systems, that is, the Heisenberg uncertainty relationship; (4) 2 x 2 Pauli spin matrices and perhaps the 4 x 4 Dirac matrix used to model the electron; and (5) the energy-general relativity, and charge-electromagnetism relationships that form a group in the theory of beta decay, that is to say, the conserved quantities that Pauli used to justify his neutrino hypothesis.

When forming a complex number as a real number plus an imaginary component, mathematicians discovered that a diagram could be constructed to illustrate it. A circle drawn in a two-dimensional plane with axes of real numbers and orthogonal imaginary numbers could be used to describe a complex number, as a vector from the origin to a point on the circle, as shown below.

https://arxiv.org/pdf/1003.3223.pdf

The connection of imaginary numbers to the base of the natural logarithms e and to π has a long history, and also points to interest among mathematicians in the intellectual attraction that some call mysticism. The relationship eiπ + 1 = 0 was known to Leonhard Euler--to whom we largely owe the symbolism for the three important mathematical symbols e, i, and π.140 In fact, the symbol e was chosen to honor Euler.141 The mathematical connection that relates these symbols with their disparate histories and applications is striking to any student when first encountering them. The chord that is struck in a student, whether it resonates as a mere example of a mathematical triviality or as a Platonic truth about the mathematical basis of reality, depends upon the personality

138 Ibid., p. 444. Boyer notes that Leibnitz was a prominent Christian theologian, and used numerological connotations to illustrate his theological beliefs.
139 E.T. Bell, “Gauss, the Prince of Mathematicians,” in James R. Newman, ed., The World of Mathematics, Vol. I (New York: Simon and Schuster, 1956), p. 309.

140 Boyer, History of Mathematics (ref. 21), p. 309.
141 Richard Aufmann, Vernon Baker, Richard Nation, College Algebra-Fourth Edition (New York: Houghton Mifflin, 2000), p. 383.

of the student and the student's philosophical orientation. Pauli's naural personality type, INFP, was one in which deep connections and meanings were primary, in contrast to the utilitarian aspects of mathematical equations valued by some other personality types.

When forming a complex number as a real number plus an imaginary component, mathematicians discovered that a diagram could be constructed to illustrate it. A circle drawn in a two-dimensional plane with axes of real numbers and orthogonal imaginary numbers could be used to describe a complex number, as a vector from the origin to a point on the circle, as shown below.

Figure 4. Author's drawing of the “kernel” symbol of a complex number.

Trigonometric functions combine to form an expression for e and the complex number Z = X + iY = | Z| ei θ, where θ is the angle formed between the vector Z and the X axis. The circular visualization of the complex-number plane involves four mathematical symbols, e, i, 1, and π, and may have been the first mathematical “kernel” or mandala that Pauli recognized during his early education. The symbols e, 1, and π are of similar real-number character, but i introduces the need for an almost mystical perspective to appreciate it fully. And how does one comprehend the deep relationship between these seemingly disparate symbols and mathematical traditions? Are they a product of the human mind, or does the mathematical mind uncover an aspect of deep reality? The symbol for the imaginary number i, the square root of the real number -1, is related to e, the base of the natural logarithms, and to π . This striking visual connection between e, i, and trigonometric functions that include π and represent rotations is mysterious. The circle has the image of a mandala, the mathematics is that of a four-part constituency, and

the beauty is expressible by no means other than as a visually complete symbollism, visible only to the “mind’s eye.” Pauli was exposed to this mandala-like image or kernel and to its mathematics early in his education.

Pauli was impressed with the visualizability of the relationships among e, i, and π, and he would write about his impressions in his later years in his philosophical speculations. He likely first encountered this visualization, with its sense of wonder, in his teens when he was learning mathematics. In an unpublished article of June 1948, he wrote of “modern examples of ‘background physics’,” 146 by which he meant visually symbolic representations of physical concepts, symbols of a psychologically objective nature and independent of the person viewing them. These were the cognitive archetypes of Jung, and Pauli was exploring their appearance and role in physics. One of the examples he gave in 1948 was that of the visualizable symbol generated by e, i, and indirectly π, in his self-analysis of one of his dreams. He likely first encountered that same mathematical symbolism in his adolescence, and now in 1948, he was interpreting his awe-inspiring adolescent experience. His mystical awe arose from the image of a

circle, which he referred to as a mandala that was produced by the mathematical relationship of the symbols e, i, and π. Pauli had dreamed of an egg-shaped image that first split into two eggs, then into a third, and the third into two again to generate four eggs. In his dream the four eggs then changed into mathematical symbols of trigonometric functions. Pauli here was interpreting the numerological relationship of a kernel to a quaternity, where the number four indicated a process of splitting from one into four components:

I [Pauli] say, "The whole thing gives eiδ, and that is the circle." The formula vanishes and a circle appears.... One becomes two, two becomes three, and out of the third comes the One as the fourth. The last mentioned typically comes about for me through mathematics. The formula

(cos δ/2 + i sin δ/2) /(cos δ/2 - i sin δ/2 ) = e iδ

is mathematically correct, and in the representation of complex numbers through distances eiδ is a number that always lies on the "unit circle" (the circle with radius 1) ....

The imaginary unit i = √-1 is a typical symbol since it is not continued under the ordinary numbers; the introduction of this symbol gives many mathematical theorems a simple and distinct form. In this dream it has the irrational function of uniting the pairs of opposites and thus producing wholeness.

Without going into mathematical detail, I should nevertheless like to stress here that I cannot acknowledge an antithesis between a mathematical and a symbolic description of nature, since for me the mathematical representation is a symbolic description par excellence.147

In his encyclopedia article of 1921 on the theory of relativity, Pauli illustrated how the Lorentz transformation could be visualized as a rotation in an abstract mathematical space, as shown in the figure below.151 The space-time distance between two four-

Figure 6. Pauli’s illustration of a Lorentz transformation.

dimensional points, x, y, z, ict and x', y', z', ict' is invariant, giving Pauli in his “mind’s eye” another opportunity to visualize a kernel, with the transformation between coordinate systems perceived visually as a rotation. Pauli adapted the above figure from Hermann Minkowski’s famous lecture of September 21, 1908,152 which Pauli would have encountered during his Gymnasium education, since he was completely familiar with relativity theory before he began his studies with Arnold Sommerfeld in Munich in 1918.153 The above image of 1921 reminds one of a kernel, in which the space-time distance has four components and is invariant under a rotational transformation. Some components within the circle change, but not the underlying measure or radius of the circle. The fourth component, moreover, involves the imaginary number i, an extension

in mathematical complexity from the other three components. If Pauli had mystical stirrings, they certainly could have occurred here.

Dynamical systems of four parts, such as complex numbers, electromagnetic fields, and Minkowski space- time, would indicate a “quaternian” way of perceiving the world. Mathematical group transformations would be visualized as rotations of a mandala. Even his own personality would be transformed from being outwardly focused to inwardly consumed, from believing that rationality and the senses were supreme to equally valuing feelings and intuitions.

The topics of a philosophical nature that I believe first appeared in Pauli’s adolescence need further identification. I see them surfacing in his teens as he became aware of kernels, quaternities, rotations, and deep reality. The kernels of Pauli’s adolescence later would become the mandalas of Jung. One might think of kernels as being inert physical wholes larger than the sum of their parts, implying a unity of

mathematics and physics, the “all is number” of Pythagoras. Kernels can be inferred most easily through visualization, with their indicator the circle expressing unity. Pauli’s kernels came in four-part divisions. Another indicator of the presence of kernels was visualization of rotation of these four components.

The four mathematical symbols e, i, π, and 1 comprise the components of the kernel that formed the unit circle in the complex-number plane. The electric and magnetic fields of the four Maxwell equations comprised the kernel that could be visualized as a traveling electromagnetic wave. The four space-time dimensions comprised the kernel found in Minkowski’s view of special relativity, and could be visualized in the Lorentz transformations as rotations. Whenever Pauli encountered rings, circles, wholes dissectible into four parts, and spheres, he became intuitively drawn to them.

By the time Pauli met Jung, Pauli was ready for Jung's concept of mandalas. Jung’s mandalas extended the idea of kernels for Pauli by including animate conscious components. Jung uses the Sanskrit word mandala as a symbolic or “magic circle”154 that captures in the "mind's eye" the psychic center of a transformational process. This idea defies precise definition since the person seeing the mandala image in his or her "mind's eye" is supposed to experience a mystic feeling when viewing the symbol, a feeling going beyond a reductionist attempt at verbal definition. Jung’s mandalas also generally have four components. Thus, the mandala is the “One” whole that is larger than the sum of its four parts.

The four components of most kernels that Pauli saw in his physics later developed into his numerological attraction to the number 4. In his youth, however, I think Pauli was struggling with positivist messages that undermined his trust in his intuition. The number 3, likely derived from his Catholic upbringing and attachment to the Trinity, was another numerological sign to which the young Pauli was sensitive. The number 3 represented an opposition to the number 4; either three or four components existed in a physical system, but not both. The three components of Cartesian space were completely rational and perceptible by the senses. The fourth component in Minkowski space-time, ict, involved a nonrational extension that was not perceptible by the senses; it extended the three to form a unity of four components. Thus, to the youthful Pauli, numerology of

the number 4 clashed with his positivist upbringing and rationality. Four components were not subject to proof; instead they had to be accepted on the basis of intuition, symmetry, and aesthetics. He was not ready to accept the numerology of the number 4 until after he met Jung.

THE BOHR SOMMERFIELD ATOM WAS A QUADRANT QUATERNITY- WHENEVER THE AUTHOR OF THIS ARTICLE SAYS "KERNALS" HE IS REFERRING TO KERNALS WHERE PAULI RECOGNIZED THE QUATERNITY IN PHYSICS

https://arxiv.org/pdf/1003.3223.pdf

Einstein called Bohr’s theory of the atom of 1913 the “highest musicality of human thought,” and Sommerfeld extended it in 1916 by introducing the so-called Sommerfeld quantization conditions. The Bohr-Sommerfeld atom, so named because of Sommerfeld’s theoretical extensions of Bohr’s musicality, was much more of a visual than physical model. One of its striking features was its visually aesthetic appeal. It was a visually beautiful model filled with classical electron orbits, as seen in the figure below.165 The young Pauli criticized it sharply for that very reason: There was no firm theoretical basis on which to calculate these classical orbits.

In 1955 Pauli wrote about the numerological meaning of the number 4, essentially calling for a reconsideration of Platonism:

For [Pythagoras] ... wherever number is, there also is soul, the expression of the unity which is God. Whole-number relationships, as they occur in the proportions of the frequencies of the simple musical intervals, are harmony, that is to say they are what brings unity into contrasts. As part of mathematics number also belongs to an abstract supersensuous eternal world which can be apprehended not by the

171 Enz and von Meyenn, Wolfgang Pauli (ref. 6), p. 64.

senses but only in contemplation by the intellect. Thus for the Pythagoreans mathematics and contemplative meditation (the original meaning of “theoria”) are very closely connected; for them mathematical knowledge and wisdom (sophia) are inseparable. Special significance was attached to the tetraktys, fourfoldedness, and there is a traditional oath of the Pythagoreans: “by him who has committed to our soul the tetraktys, original source and the root of eternal Nature”....172

We have here the fundamental explanation of Pauli’s attraction to the number 4 and the four components of his physical kernels. Pauli was attracted to Kepler in part because he was a bridge in the history of ideas between “Trinitarian” and “quaternian” thinking. Pauli's analysis of the Kepler-Fludd polemic in the 1940s is connected to this issue. For Pauli, the Kepler-Fludd polemic mirrored the debates over the philosophical interpretation of quantum mechanics:

Modern quantum physics has come closer to the quaternary point of view, which was so violently opposed to the natural science that was germinating in the 17th century, to the extent that it takes into greater consideration the role of the observer in physics than is the case in classical physics.173

Sommefeld introduced the dimensionless fine-structure constant α, numerically equal to about 1/137, in his extension of Bohr’s theory of the hydrogen atom where it governed the splitting of its spectral lines. 174 It is comprised most importantly of the

172 Ibid., p. 141.
173 Meier, Atoms and Archetypes (ref. 11), p. 208.
174 Micheal Eckert, Willibald Pricha, Helmut Schubert, and Gisela Torkar, ed., Geheimrat Sommerfeld- Theoretischer Physiker: Eine Dokumentation aus seinem Nachlass(München: Deutches Museum, 1984),

four fundamental physical “quanta” of the charge of the electron e, the speed of light c, Planck’s constant h, and the Pythagorean symbol of mysticism π. During the course of his life, the fine-structure constant appealed to Pauli as a symbolic integer number, a theoretical coincidence, a kernel with quaternity components, and as a symbol of mystical foreboding. It had captured his attention already by the time he published his encyclopedia article on relativity theory in 1921. The visualizable kernel formed by the quaternity of e, c, h, and π, however, eluded Pauli. He never indicated that he had gleaned an image of it, nor did he decipher the mystery of the interconnectedness that he associated with the number 137.

QUATERNION ALGEBRA

https://arxiv.org/pdf/1003.3223.pdf

In his Theory of Relativity, Pauli especially acknowledged his debt to Klein. 193 We can assume he also appreciated Sommerfeld’s role, as Sommerfeld wrote the Preface to it. Klein and Sommerfeld worked with quaternian algebra, building on the efforts of William Rowan Hamilton (1805-1865) and Bernhard Riemann. They all recognized the power of quaternian algebra for mathematics and physics, and some of them saw further

190 Felix Klein and Arnold Sommerfeld, Über die Theorie des Kreisels (New York: Johnson Reprint Corporation, 1965).

191 Pauli had to have reviewed Klein and Sommerfeld’s work when he conducted his comprehensive summary of relativity theory, if not before.

192 Enz and von Meyenn, Wolfgang Pauli (ref. 6), p. 129. 193 Pauli, Theory of Relativity (ref. 16), p. vi.

connections. Thus, Hamilton saw mystical connections that likely were not lost on either Sommerfeld or the young Pauli's Shadow. Pauli used quaternian algebra when he introduced spin matrices in 1927, but he might have picked up its mystical connections to Plato’s Tetractys much earlier by the way it was visualized. Hamilton defined quaternians as:

The quotient of two vectors, or the operator which changes one vector into another....

The SYMBOL OF OPERATION q( )q -1 , where q may be called (as before) the operator quaternion, while the symbol (suppose r) of the operand quaternian is conceived to occupy the place marked by the parentheses ... can be regarded as a conical rotation of the axis of the operand round the axis of the operator, through double the angle thereof....194

Klein and Sommerfeld extended the quaternian algebra of the real and imaginary numbers into an algebra that included vectors, matrices, and generalized transformations. Here was a mathematics suitable for describing rotations. Historian of mathematics Simon Altman has commented on Hamilton’s achievements:

We must stress that Hamilton's everlasting monument ... is his construction of objects that, except for commutativity, obey the same algebra as that of the real and complex numbers: and Hamilton was aware of this–although he could not foresee that quaternions were to receive in 1878, at the hands of [Georg Ferdinand] Frobenius, the supreme accolade of being proved to be the only possible objects with this property.195

Klein's and Sommerfeld's treatise, Über die Theorie des Kriesels196, is perhaps the most underappreciated source for the mathematical development of matrix mechanics. We find in it the mathematics of quaternians and matrices, symmetry analysis, noncommutative matrix-multiplication methods, spin matrices, and a mathematical alethic reality used to describe bodies in motion, namely, the rotation of a spinning top. Sommerfeld no doubt put his and Klein's work into the hands of Pauli who, for example,

194 Simon Altmann, Rotations, Quaternians and Double Groups (Oxford: Clarendon Press, 1986), p. 16. 195 Ibid., p. 15.
196 Klein and Sommerfeld, Theorie des Kreisels (ref. 31).

treated a spinning hydrogen nucleus long before electron spin was considered.197 He mastered matrices, vectors, tensors, quaternians, and, in general, Hamiltonian dynamics long before Heisenberg’s discovery of matrix mechanics. Klein and Sommerfeld had used matrix mathematics, infinitesimal rotations, imaginary and complex numbers, and mathematical visualizations to express the deeply embedded rationality behind classical mechanics. These visualizations often were in an abstract alethic space, and Pauli had to become comfortable with them, such as the one that Herbert Goldstein noted in 1950, when he discussed the equations of motion for a rotating rigid body: “Hence the jabberwockian sounding statement: the polhode rolls without slipping on the herpolhode lying in the invariable plane.”198 To illustrate the challenge, Goldstein included the diagram below.199 He relied on Klein and Sommerfeld’s earlier work, where similar

Figure 9. Herbert Goldstein’s diagram of jabberwockian terms in describing the motion of the inertia ellipsoid relative to the invariable plane.

illustrations of the physical parameters derived from abstract mathematics appear, such as the one below for the movement of Pohlbahnen of a rotating rigid body.200

Such mathematical visualizations are in an alethic reality, in the sense that one cannot evaluate or measure their components in their abstract space, since the mathematics involves not only real but also imaginary and complex numbers. Their mystical connotations could not be ignored: Imaginary numbers that rotate a real number about an axis move beyond naïve visualizations to abstract ones, hidden from observation and measurement. Visualizations of rotations by quaternians extended the abstractions further and simultaneously illustrated the value of group theory. A measureable, real quantity might emerge from the mathematical derivation, but that derivation proceeded in

an abstract space. Mach’s direct connections to sensory information no longer held: The end result of a mathematical calculation might be measurable or observable, but not its intermediary steps. Deep reality was hidden and inaccessible at the foundational mathematical level in Klein and Sommerfeld’s analysis of rotating bodies. Their readers were transported into an analytic world of difficult but visualizable mathematics that expressed an alethic level of reality.

The mathematics of quaternians can be visualized in the “mind’s eye” as a type of abstract rotation similar to that produced by the imaginary number i. Quaternians involve a higher degree of abstractness and of rotations, however. Thus, a quaternian Q is defined as Q = iA + j B + k C + D. As Klein and Sommerfeld declared:

By way of the original definition of the word quaternion whereby we base our concept of rotational transformation, a quaternion means nothing else but the operation of a rotational transformation. It is unequivocally determined by the amount of the rotation (T), by the axis of rotation (a,b,c) and the half angle (ω/2) of the amount of rotation.201

The symbols i, j, and k are extensions of the imaginary numbers and are interrelated by: ii = -1, jj = -1 , kk = -1,

ij= k, jk= i, ki= j,

ji=-k, kj=-i, ik=-j
The multiplication of two quaternians is noncommutative, as would be expected in a system that models rotations. In the “mind’s eye” of a mathematician when learning quaternian algebra, the four elements i, j, k, and 1 that generate the visualizable rotations, and their holistic unity, produce a sense of awe regarding the inherent power of that kernel of symbols. Not surprisingly, Klein, Sommerfeld, and Pauli were struck by the power of quaternian mathematics.202

Klein and Sommerfled not only used matrix and group methods to analyze the difficult classical problem of a rotating top; they also connected the alethic-like

imaginary numbers and the mystical, almost Pythagorean Hamiltonian quaternians to relativity theory. In the fourth part of their text published in 1910, they treat the Lorentz group and Einstein’s theory of special relativity using quaternian mathematics, as Pauli knew, and they used group-theoretical methods that would later reappear in quantum mechanics.203 As Minkowski did in 1908, so Klein and Sommerfeld showed in 1910 how the Lorentz transformations could be viewed as rotations in four dimensions, that is, how quaternian mathematics could be used to represent a transformation between two coordinate systems.204

To Hamilton, quaternians held mystical connotations, which may have fascinated Pauli's Shadow during his years with Sommerfeld owing to their mathematical power in manipulating components of his physical kernels. Quaternians form a system of interrelated mathematical entities. Just as the imaginary number i extends the concept of a real number to include operations that are not allowable for the real numbers, so quaternians extend the algebra of imaginary and real numbers to allow visualization of rotations from one vector system to another.205 The mathematics of rotations employed in the Lorentz transformations is the mathematics of matrices, as noted by Goldstein:

Clearly a spatial rotation between two systems at rest relative to each other is included as a subclass of the Lorentz transformation.... [Any] general Lorentz transformation is a product of a space rotation and a pure Lorentz transformation.206

Klein and Sommerfled’s use of complex numbers, of matrices to describe abstract rotations in a mathematical space, of quaternians, and of symmetry principles, prepared the theoretical physicist Pauli to use these powerful mathematical methods in exploring the physical world. They prepared the ground for a kind of mathematical and Pythagorean abstraction that soon found its way into matrix mechanics. To Pauli, quaternians offered a mathematical tool to handle rotations of abstract mathematical entities, and to visualize them. To him, intuitively visualizable rotations were a clear sign of a buried physical kernel.

Frank Stella Harran II 1967 https://en.wikipedia.org/wiki/Frank_Stella