Quadrant Model of Reality: Quadrant Model of Reality Book 2 Philosophy

Philosophy chapter

Ken Wilber's AQAL, pronounced "ah-qwul", is the basic framework of Integral Theory. It suggests that all human knowledge and experience can be placed in a four-quadrant grid, along the axes of "interior-exterior" and "individual-collective". According to Wilber, it is one of the most comprehensive approaches to reality, a metatheory that attempts to explain how academic disciplines and every form of knowledge and experience fit together coherently.[12]

AQAL is based on four fundamental concepts and a rest-category: four quadrants, several levels and lines of development, several states of consciousness, and "types", topics which don't fit into these four concepts. "Levels" are the stages of development, from pre-personal through personal to transpersonal."Lines" are lines of development, the several domains of development, which may process uneven, with several stages of development in place at the various domains. [note 1] "States" are states of consciousness; according to Wilber persons may have a terminal experience of a higher developmental stage. [note 2] "Types" is a rest-category, for phenomena which don't fit in the other four concepts.[14] In order for an account of the Kosmos to be complete, Wilber believes that it must include each of these five categories. For Wilber, only such an account can be accurately called "integral". In the essay, "Excerpt C: The Ways We Are in This Together", Wilber describes AQAL as "one suggested architecture of the Kosmos".[15]

The model is topped with formless awareness, "the simple feeling of being," which is equated with a range of "ultimates" from a variety of eastern traditions. This formless awareness transcends the phenomenal world, which is ultimately only an appearance of some transcendental reality. According to Wilber, the AQAL categories — quadrants, lines, levels, states, and types - describe the relative truth of the two truths doctrine of Buddhism. According to Wilber, none of them are true in an absolute sense: only formless awareness, "the simple feeling of being", exists absolutely.[citation needed]

According to Frank Visser, Wilber's conception of four quadrants, or dimensions of existence is very similar to E. F. Schumacher's conception of four fields of knowledge.[34] Visser finds Wilber's conception of levels, as well as Wilber's critique of science as one-dimensional, to be very similar to that in Huston Smith's Forgotten Truth.Visser also writes that the esoteric aspects of Wilber's theory are based on the philosophy of Sri Aurobindo as well as other theorists including Adi Da.

The Fourth Turning: Imagining the Evolution of an Integral Buddhism,

Appropriate technology is an ideological movement (and its manifestations) originally articulated as intermediate technology by the economist Dr. Ernst Friedrich "Fritz" Schumacher in his influential work, Small is Beautiful.Appropriate technology was meant to address four problems: extreme poverty, starvation, unemployment and urban migration. Schumacher saw the main purpose for economic development programs was the eradication of extreme poverty and he saw a clear connection between mass unemployment and extreme poverty. Schumacher sought to shift development efforts from a bias towards urban areas and on increasing the output per laborer to focusing on rural areas (where a majority of the population still lived) and on increasing employment.[

Small Is Beautiful: A Study of Economics As If People Mattered is a collection of essays by British economist E. F. Schumacher. The phrase "Small Is Beautiful" came from a phrase by his teacher Leopold Kohr.[1] It is often used to champion small, appropriate technologies that are believed to empower people more, in contrast with phrases such as "bigger is better".

First published in 1973, Small Is Beautiful brought Schumacher's critiques of Western economics to a wider audience during the 1973 energy crisis and emergence of globalization. The Times Literary Supplement ranked Small Is Beautiful among the 100 most influential books published since World War II.[2] A further edition with commentaries was published in 1999.[3]

The book is divided into four parts: "The Modern World", "Resources", "The Third World", and "Organization and Ownership".

Four fields of knowledge[edit]

Schumacher identifies four fields of knowledge for the individual:

I → inner

I → other persons (inner)

other persons → I

I → the world

These four fields arise from combining two pairs: Myself and the World; and Outer Appearance and Inner Experience. He notes that humans only have direct access to fields one and four.

Field one is being aware of your feelings and thoughts and most closely correlates to self awareness. He argues this is fundamentally the study of attention. He differentiates between when your attention is captured by the item it focuses upon, which is when a human being functions much like a machine; and when a person consciously directs their attention according to their choosing. This for him is the difference between being lived and living.

Field two is being aware of what other people are thinking and feeling.

Despite these problems we do experience a 'meeting of minds' with other individuals at certain times. People are even able to ignore the words actually said, and say something like "I don't agree with what you are saying; but I do agree with what you mean." Schumacher argues that one of the reasons we can understand other people is through bodily experience, because so many bodily expressions, gestures and postures are part of our common human heritage.

Schumacher observes that the traditional answer to the study of field two has been "You can understand others to the extent you understand yourself."[7] Schumacher points out that this a logical development of the principle of adequateness, how can you understand someone's pain unless you too have experienced pain?

Field three is understanding yourself as an objective phenomenon. Knowledge in field three requires you to be aware what other people think of you. Schumacher suggests that the most fruitful advice in this field can be gained by studying the Fourth Way concept of external considering.

Schumacher observes that relying on just field one knowledge makes you feel that you are the centre of the universe; while focusing on field three knowledge makes you feel that you are far more insignificant. Seeking self-knowledge via both fields provides more balanced and accurate self-knowledge.

Field four is the behaviourist study of the outside world. Science is highly active in this area of knowledge and many people believe it is the only field in which true knowledge can be gained. For Schumacher, applying the scientific approach is highly appropriate in this field.

Schumacher summarises his views about the four fields of knowledge as follows:

Only when all four fields of knowledge are cultivated can you have true unity of knowledge. Instruments and methodologies of study should be only applied to the appropriate field they are designed for.

Clarity of knowledge depends on relating the four fields of knowledge to the four levels of being.

The instructional sciences should confine their remit to field four, because it is only in the field of appearances that mathematical precision can be obtained. The descriptive sciences, however, are not behaving appropriately if they focus solely on appearances, and must delve in meaning and purpose or they will produce sterile results.

Self-knowledge can only be effectively pursued by balanced study of field one (self awareness) and field three (objective self-knowledge).

Study of field two (understanding other individuals) is dependent on first developing a powerful insight into field one (self awareness).

For Schumacher one of science's major mistakes has been rejecting the traditional philosophical and religious view that the universe is a hierarchy of being. Schumacher makes a restatement of the traditional chain of being.

He agrees with the view that there are four kingdoms: Mineral, Plant, Animal, Human. He argues that there are critical differences of kind between each level of being. Between mineral and plant is the phenomenon of life. Schumacher says that although scientists say we should not use the phrase 'life energy', the difference still exists and has not been explained by science[clarification needed]. Schumacher points out that though we can recognize life and destroy it, we can't create it. Schumacher notes that the 'life sciences' are 'extraordinary' because they hardly ever deal with life as such, and instead content themselves with analyzing the "physico-chemical body which is life's carrier." Schumacher goes on to say there is nothing in physics or chemistry to explain the phenomenon of life.

For Schumacher, a similar jump in level of being takes place between plant and animal, which is differentiated by the phenomenon of consciousness. We can recognize consciousness, not least because we can knock an animal unconscious, but also because animals exhibit at minimum primitive thought and intelligence.

The next level, according to Schumacher, is between Animal and Human, which are differentiated by the phenomenon of self-consciousness or self awareness. Self-consciousness is the reflective awareness of one's consciousness and thoughts.

Schumacher realizes that the terms—life, consciousness and self-consciousness—are subject to misinterpretation so he suggests that the differences can best be expressed as an equation which can be written thus:

'Mineral' = m

'Plant' = m + x

'Animal' = m + x + y

'Human' = m + x + y + z

In his theory, these three factors (x, y and z) represent ontological discontinuities. He argues that the differences can be likened to differences in dimension; and from one perspective it could be argued that only humans have 'real' existence insofar as they possess the three dimensions of life, consciousness and self-consciousness. Schumacher uses this perspective to contrast with the materialistic scientism view, which argues that what is 'real' is inanimate matter, denying the realness of life, consciousness and self-consciousness, despite the fact each individual can verify those phenomena from their own experience.

He directs our attention to the fact that science has generally avoided seriously discussing these discontinuities, because they present such difficulties for strictly materialistic science, and they largely remain mysteries.

Next he considers the animal model of humanity which has grown popular in science. Schumacher notes that within the humanities the distinction between consciousness and self consciousness is now seldom drawn. Consequently, people have become increasingly uncertain about whether there is any difference between animals and humans. Schumacher notes that a great deal of research about humans has been conducted by studying animals. Schumacher argues that this is analogous to studying physics in the hope of understanding life. Schumacher goes on to say that much can be learned about humanity by studying minerals, plants and animals because humans have inherited those levels of being: all, that is, 'except that which makes him [sic] human.'

Schumacher goes on to say that nothing is 'more conducive to the brutalisation of the modern world' than calling humans the 'naked ape'. Schumacher argues that once people begin viewing humans as 'animal machines' they soon begin treating them accordingly.[2]

Schumacher argues that what defines humanity are our greatest achievements, not the common run of the mill things. He argues that human beings are open-ended because of self-awareness, which as distinct from life and consciousness has nothing mechanical or automatic about it. For Schumacher "the powers of self awareness are, essentially, a limitless potentiality rather than an actuality. They have to be developed and 'realized' by each human individual if one is to become truly human, that is to say, a person."[3]

A Guide for the Perplexed is a short book by E. F. Schumacher, published in 1977. The title is a reference to Maimonides's The Guide for the Perplexed. Schumacher himself considered A Guide for the Perplexed to be his most important achievement, although he was better known for his 1973 environmental economics bestseller Small Is Beautiful, which made him a leading figure within the ecology movement. His daughter wrote that her father handed her the book on his deathbed, five days before he died and he told her "this is what my life has been leading to".[1] As the Chicago Tribune wrote, "A Guide for the Perplexed is really a statement of the philosophical underpinnings that inform Small is Beautiful".

Schumacher describes his book as being concerned with how humans live in the world. It is also a treatise on the nature and organisation of knowledge and is something of an attack on what Schumacher calls "materialistic scientism". Schumacher argues that the current philosophical 'maps' that dominate western thought and science are both overly narrow and based on some false premises.

However, this book is only in small part a critique. Schumacher spends the greater part of it putting forward and explaining what he considers to be the four great truths of philosophical map making:

The world is a hierarchical structure with at least four 'levels of being'.

The 'Principle of Adequateness' determines human ability to accurately perceive the world.

Human learning relates to four 'fields of knowledge'.

The art of living requires an understanding of two types of problem: 'convergent' and 'divergent'.

In the first chapter, "The Problem of Production", Schumacher argues that the modern economy is unsustainable. Natural resources (like fossil fuels), are treated as expendable income, when in fact they should be treated as capital, since they are not renewable, and thus subject to eventual depletion. He further argues that nature's resistance to pollution is limited as well. He concludes that government effort must be concentrated on sustainable development, because relatively minor improvements, for example, technology transfer to Third World countries, will not solve the underlying problem of an unsustainable economy.

Schumacher's philosophy is one of "enoughness", appreciating both human needs, limitations and appropriate use of technology. It grew out of his study of village-based economics, which he later termed Buddhist economics, which is the subject of the book's fourth chapter.

He faults conventional economic thinking for failing to consider the most appropriate scale for an activity, blasts notions that "growth is good", and that "bigger is better", and questions the appropriateness of using mass production in developing countries, promoting instead "production by the masses". Schumacher was one of the first economists to question the appropriateness of using gross national product to measure human well-being, emphasizing that "the aim ought to be to obtain the maximum amount of well being with the minimum amount of consumption". In the epilogue he emphasizes the need for the "philosophy of materialism" to take second place to ideals such as justice, harmony, beauty, and health.

Wilber believes that the mystical traditions of the world provide access to, and knowledge of, a transcendental reality which is perennial, being the same throughout all times and cultures. This proposition underlies the whole of his conceptual edifice, and is an unquestioned assumption.[note 4] Wilber juxtaposites this generalisation to plain materialism, presenting this as the main paradigma of regular science.[19][quote 1]

Interior Exterior

Individual Standard: Truthfulness

(1st person)

(sincerity, integrity, trustworthiness) Standard: Truth

(3rd person)

(correspondence,

representation, propositional)

Collective Standard: Justness

(2nd person)

(cultural fit, rightness,

mutual understanding) Standard: Functional fit

(3rd person)

(systems theory web,

Structural functionalism,

social systems mesh)

In his later works, Wilber argues that manifest reality is composed of four domains, and that each domain, or "quadrant", has its own truth-standard, or test for validity:[20]

"Interior individual/1st person": the subjective world, the individual subjective sphere;[21]

"Interior collective/2nd person": the intersubjective space, the cultural background;[21]

"Exterior individual/3rd person": the objective state of affairs;[21]

"Exterior collective/3rd person": the functional fit, "how entities fit together in a system".[21]

Pre/trans fallacy

Upper-Left (UL)

"I"

Interior Individual

Intentional

e.g. Freud

Upper-Right (UR)

"It"

Exterior Individual

Behavioral

e.g. Skinner

Lower-Left (LL)

"We"

Interior Collective

Cultural

e.g. Gadamer

Lower-Right (LR)

"Its"

Exterior Collective

Social

e.g. Marx

Various literary works use or refer to the quincunx pattern for its symbolic value:

Quincunx's are crosses/ quadrants

A quincuncial map is a conformal map projection that maps the poles of the sphere to the centre and four corners of a square, thus forming a quincunx.

The points on each face of a unit cell of a face-centred cubic lattice form a quincunx.

The quincunx as a tattoo is known as the five dots tattoo. It has been variously interpreted as a fertility symbol,[8] a reminder of sayings on how to treat women or police,[9] a recognition symbol among the Romani people,[9] a group of close friends,[10] standing alone in the world,[11] or time spent in prison (with the outer four dots representing the prison walls and the inner dot representing the prisoner).[12] Thomas Edison, whose many inventions included an Electric pen which later became the basis of a tattooing machine created by Samuel O'Reilly, had this pattern tattooed on his forearm.[13]

Republican Roman manipular legions adopted a checkered formation called quincunx when deployed for battle.

The first two stages of the Saturn V moon rocket had engines in a quincunx arrangement.[14]

A baseball diamond forms a quincunx with the four bases and the pitcher's mound.

Quincunx patterns occur in many contexts:

The flag of the Solomon Islands features a quincunx of stars.

A quincuncial map

Cosmatesque pavements with the quincunx pattern

In heraldry, groups of five elements (charges) are often arranged in a quincunx pattern, called in saltire in heraldic terminology. The flag of the Solomon Islands features this pattern, with its five stars representing the five main island groups in the Solomon Islands. Another instance of this pattern occurred in the flag of the 19th-century Republic of Yucatán, where it signified the five departments into which the republic was divided.

A quincunx is a standard pattern for planting an orchard.[2]

Quincunxes are used in modern computer graphics as a pattern for multisample anti-aliasing. Quincunx antialiasing samples scenes at the corners and centers of each pixel. These five sample points, in the shape of a quincunx, are combined to produce each displayed pixel. However, samples at the corner points are shared with adjacent pixels, so the number of samples needed is only twice the number of displayed pixels.[3]

In numerical analysis, the quincunx pattern describes the two-dimensional five-point stencil, a sampling pattern used to derive finite difference approximations to derivatives.[4]

In architecture, a quincuncial plan, also defined as a "cross-in-square", is the plan of an edifice composed of nine bays. The central and the four angular ones are covered with domes or groin vaults so that the pattern of these domes forms a quincunx; the other four bays are surmounted by barrel vaults.[5] In Khmer architecture, the towers of a temple, such as Angkor Wat, are sometimes arranged in a quincunx to represent the five peaks of Mount Meru.[6]

A quincunx is one of the quintessential designs of Cosmatesque inlay stonework.[7]

The quincunx was originally a coin issued by the Roman Republic c. 211–200 BC, whose value was five twelfths (quinque and uncia) of an as, the Roman standard bronze coin. On the Roman quincunx coins, the value was sometimes indicated by a pattern of five dots or pellets. However, these dots were not always arranged in a quincunx pattern.

The OED dates the first appearances of the Latin word in English as 1545 and 1574 (“in the sense ‘five-twelfths of a pound or as’”). The first citation for “A pattern used for planting trees” dates from 1606. The OED also cites a 1647 reference to the German astronomer Kepler to the astronomical/astrological meaning.

A quincunx /ˈkwɪn.kʌŋks/ is a geometric pattern consisting of five points arranged in a cross, with four of them forming a square or rectangle and a fifth at its center.[1] It forms the arrangement of five units in the pattern corresponding to the five-spot on six-sided dice, playing cards, and dominoes. It is represented in Unicode as U+2059 ⁙ five dot punctuation or (for the die pattern) U+2684 ⚄ die face-5.

(6n+3)×(6n+3) panmagic squares, n>0[edit]

A (6n+3)×(6n+3) panmagic square with n>0 can be built by the following algorithm.

Create a (2n+1)×3 rectangle with the first 6n+3 natural numbers so that each column has the same sum. You can do this by starting with a 3×3 magic square and set up the rest cells of the rectangle in meander-style. You can also use the pattern shown in the following examples.

Examples:

For 9×9 square

vertical sum = 15

For 15×15 square

vertical sum = 40

For 21×21 square

10 11 12

15 14 13

16 17 18

21 20 19

vertical sum = 77

Put this rectangle in the left upper corner of the (6n+3)×(6n+3) square and two copies of the rectangle beneath it so that the first 3 columns of the square are filled completely.

Example:

Copy the left 3 columns into the next 3 columns, but shift it ring-wise by 1 row.

Example:

Continue copying the current 3 columns into the next 3 columns, shifted ring-wise by 1 row, until the square is filled completely.

Example:

Build a second square and copy the first square into it but mirror it diagonal. So you have to exchange rows and columns.

Example:

Build the final square by multiplying the second square by 6n+3, adding the first square and subtract 6n+3 in each cell of the square.

Example: A + (6n+3)×AT – (6n+3)

4n×4n panmagic squares[edit]

A 4n×4n panmagic square can be built by the following algorithm.

Put the first 2n natural numbers into the first row and the first 2n columns of the square.

Example:

1 2 3 4

Put the next 2n natural numbers beneath the first 2n natural numbers in inverse sequence. Each vertical pair must have the same sum.

Example:

1 2 3 4

8 7 6 5

Copy that 2×2n rectangle 2n-1 times beneath the first rectangle.

Example:

1 2 3 4

8 7 6 5

1 2 3 4

8 7 6 5

1 2 3 4

8 7 6 5

1 2 3 4

8 7 6 5

Copy the left 4n×2n rectangle into the right 4n×2n rectangle but shift it ring-wise by one row.

Example:

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

Build a second 4n×4n square and copy the first square into it but turn it by 90°.

Square A

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

1 2 3 4 8 7 6 5

8 7 6 5 1 2 3 4

Square B

5 4 5 4 5 4 5 4

6 3 6 3 6 3 6 3

7 2 7 2 7 2 7 2

8 1 8 1 8 1 8 1

4 5 4 5 4 5 4 5

3 6 3 6 3 6 3 6

2 7 2 7 2 7 2 7

1 8 1 8 1 8 1 8

Build the final square by multiplying the second square by 4n, adding the first square and subtract 4n in each cell of the square.

Example: A + 4n×B - 4n

33 26 35 28 40 31 38 29

48 23 46 21 41 18 43 20

49 10 51 12 56 15 54 13

64 7 62 5 57 2 59 4

25 34 27 36 32 39 30 37

24 47 22 45 17 42 19 44

9 50 11 52 16 55 14 53

8 63 6 61 1 58 3 60

If you build a 4n×4n pandiagonal magic square with this algorithm then every 2×2 square in the 4n×4n square will have the same sum. Therefore many symmetric patterns of 4n cells have the same sum as any row and any column of the 4n×4n square. Especially each 2n×2 and each 2×2n rectangle will have the same sum as any row and any column of the 4n×4n square. The 4n×4n square is also a Most-perfect magic square.

(6n±1)×(6n±1) panmagic squares[edit]

A (6n±1)×(6n±1) panmagic square can be built by the following algorithm.

Set up the first column of the square with the first 6n±1 natural numbers.

Example:

Copy the first column into the second column but shift it ring-wise by 2 rows.

Example:

1 6

2 7

3 1

4 2

5 3

6 4

7 5

Continue copying the current column into the next column with ring-wise shift by 2 rows until the square is filled completely.

Example:

1 6 4 2 7 5 3

2 7 5 3 1 6 4

3 1 6 4 2 7 5

4 2 7 5 3 1 6

5 3 1 6 4 2 7

6 4 2 7 5 3 1

7 5 3 1 6 4 2

Build a second square and copy the first square into it but mirror it diagonal. So you have to exchange rows and columns.

1 6 4 2 7 5 3

2 7 5 3 1 6 4

3 1 6 4 2 7 5

4 2 7 5 3 1 6

5 3 1 6 4 2 7

6 4 2 7 5 3 1

7 5 3 1 6 4 2

1 2 3 4 5 6 7

6 7 1 2 3 4 5

4 5 6 7 1 2 3

2 3 4 5 6 7 1

7 1 2 3 4 5 6

5 6 7 1 2 3 4

3 4 5 6 7 1 2

Build the final square by multiplying the second square by 6n±1, adding the first square and subtract 6n±1 in each cell of the square.

Example: A + (6n±1)×AT - (6n±1)

1 13 18 23 35 40 45

37 49 5 10 15 27 32

24 29 41 46 2 14 19

11 16 28 33 38 43 6

47 3 8 20 25 30 42

34 39 44 7 12 17 22

21 26 31 36 48 4 9

(4n+2)×(4n+2) panmagic squares with nonconsecutive elements[edit]

No panmagic square exists of order 4n+2 if consecutive integers are used. But certain sequences of nonconsecutive integers do admit order-(4n+2) panmagic squares.

Consider the sum 1+2+3+5+6+7 = 24. This sum can be divided in half by taking the appropriate groups of three addends, or in thirds using groups of two addends:

1+5+6 = 2+3+7 = 12

1+7 = 2+6 = 3+5 = 8

Note that the consecutive integer sum 1+2+3+4+5+6 = 21, an odd sum, lacks the half-partitioning.

With both equal partitions available, the numbers 1, 2, 3, 5, 6, 7 can be arranged into 6x6 pandigonal patterns A and B, respectively given by:

1 5 6 7 3 2

5 6 1 3 2 7

6 1 5 2 7 3

1 5 6 7 3 2

5 6 1 3 2 7

6 1 5 2 7 3

6 5 1 6 5 1

1 6 5 1 6 5

5 1 6 5 1 6

2 3 7 2 3 7

7 2 3 7 2 3

3 7 2 3 7 2

Then 7xA + B - 7 gives the nonconsecutive pandiagonal 6x6 square:

6 33 36 48 19 8

29 41 5 15 13 47

40 1 34 12 43 20

2 31 42 44 17 14

35 37 3 21 9 45

38 7 30 10 49 16

with a maximum element of 49 and a panmagic sum of 150.

For 10th order a similar construction is possible using the equal partitionings of the sum 1+2+3+4+5+9+10+11+12+13 = 70:

1+3+9+10+12 = 2+4+5+11+13 = 35

1+13 = 2+12 = 3+11 = 4+10 = 5+9 = 14

This leads to squares having a maximum element of 169 and a panmagic sum of 850

5×5 panmagic squares[edit]

There are many 5×5 pandiagonal magic squares. Unlike 4×4 panmagic squares, these can be associative. The following is a 5×5 associative panmagic square:

20 8 21 14 2

11 4 17 10 23

7 25 13 1 19

3 16 9 22 15

24 12 5 18 6

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

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

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

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

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

Each of these quincunxes can be translated to other positions in the square by cyclic permutation of the rows and columns (wrapping around), which in a pandiagonal magic square does not affect the equality of the magic sums. This leads to 100 quincunx sums, including broken quincunxes analogous to broken diagonals.

The quincunx sums can be proved by taking linear combinations of the row, column, and diagonal sums. Consider the panmagic square

A B C D E

F G H I J

K L M N O

P Q R S T

U V W X Y

with magic sum Z. To prove the quincunx sum A+E+M+U+Y = Z (corresponding to the 20+2+13+24+6 = 65 example given above), one adds together the following:

3 times each of the diagonal sums A+G+M+S+Y and E+I+M+Q+U

The diagonal sums A+J+N+R+V, B+H+N+T+U, D+H+L+P+Y, and E+F+L+R+X

The row sums A+B+C+D+E and U+V+W+X+Y

From this sum the following are subtracted:

The row sums F+G+H+I+J and P+Q+R+S+T

The column sum C+H+M+R+W

Twice each of the column sums B+G+L+Q+V and D+I+N+S+X.

The net result is 5A+5E+5M+5U+5Y = 5Z, which divided by 5 gives the quincunx sum. Similar linear combinations can be constructed for the other quincunx patterns H+L+M+N+R, C+K+M+O+W, and G+I+M+Q+S.

The smallest possible non trivial pan diagonal magic sure is 4 by 4. The fourth is always different. The fifth is always questionable

4×4 panmagic squares[edit]

Euler diagram of requirements of some types of 4×4 magic squares. Cells of the same colour sum to the magic constant.

* In 4×4 most-perfect magic squares, any 2 cells that are 2 cells diagonally apart (including wraparound) sum to half the magic constant, hence any 2 such pairs also sum to the magic constant.

The smallest non-trivial pandiagonal magic squares consisting of numbers are 4×4 squares.

1 8 13 12

14 11 2 7

4 5 16 9

15 10 3 6

In 4×4 panmagic squares, the magic constant of 34 can be seen in a number of patterns in addition to the rows, columns and diagonals:

Any of the sixteen 2×2 squares, including those that wrap around the edges of the whole square, e.g. 14+11+4+5, 1+12+15+6

The corners of any 3×3 square, e.g. 8+12+5+9

Any pair of horizontally or vertically adjacent numbers, together with the corresponding pair displaced by a (2, 2) vector, e.g. 1+8+16+9

Thus of the 86 possible sums adding to 34, 52 of them form regular patterns, compared with 10 for an ordinary 4×4 magic square.

There are only three distinct 4×4 pandiagonal magic squares, namely A above and the following:

1 12 7 14

8 13 2 11

10 3 16 5

15 6 9 4

1 8 11 14

12 13 2 7

6 3 16 9

15 10 5 4

These three are very closely related. B and C can be seen to differ only because the components of each semi-diagonal are reversed. It is not as easy to see how A relates to the other two but:

i: if the components of each semi-diagonal of A are reversed (A1) and the left-hand column of A1 is moved to the extreme right (A2), the result is a reflection of B

1 14 7 12

8 11 2 13

10 5 16 3

15 4 9 6

14 7 12 1

11 2 13 8

5 16 3 10

4 9 6 15

ii: if the left-hand column of A is moved to the extreme right (A3), the components of each semi-diagonal of A3 are reversed (A4), and the right-hand column of A4 is moved to the extreme left (A5), the result is C

8 13 12 1

11 2 7 14

5 16 9 4

10 3 6 15

8 11 14 1

13 2 7 12

3 16 9 6

10 5 4 15

1 8 11 14

12 13 2 7

6 3 16 9

15 10 5 4

In any 4×4 pandiagonal magic square, the two numbers at the opposite corners of any 3×3 square add up to 17. Consequently, no 4×4 panmagic squares are associative, though they all fulfil the further requirement for a 4×4 most-perfect magic square, that each 2×2 subsquare sums to 34.

3×3 panmagic squares[edit]

It is easily shown that non-trivial pandiagonal numerical magic squares of order 3 do not exist. However, if the magic square concept is generalized to include geometric shapes instead of numbers—the geometric magic squares discovered by Lee Sallows—a 3×3 panmagic square does exist.

A pandiagonal magic square remains pandiagonally magic not only under rotation or reflection, but also if a row or column is moved from one side of the square to the opposite side. As such, an n×n pandiagonal magic square can be regarded as having 8n2 orientations.

A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

Inquisitors required heretical sympathisers – repentant first offenders – to sew a yellow cross onto their clothes.[68]

Pambu Panchangam

The Sri Rama Chakra as given in Pambu Panchangam is shown below.

1 2 3 4 5 6

7 8 9 1 2 3

4 5 6 7 8 9

1 2 3 4 5 6

7 8 9 1 2 3

4 5 6 7 8 9

Seetha Chakras[edit]

Sringeri/Srirangam Panchangams

The Seetha Chakra as given in the Panchangam published by the Sringeri Sharada Peetham[1] or the one published by Srirangam Temple[2] is shown below.

2 9 4

7 5 3

6 1 8

This is a magic square of order 3. The sum of the numbers in every row, every column and each diagonal are all equal to 15.

Pambu Panchangam

The Seetha Chakra as given in Pambu Panchangam is shown below.

Sri Rama Chakra as a strongly magic square[edit]

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

M=\begin{bmatrix}

a_{11} & a_{12} & a_{13} & a_{14} \\

a_{21} & a_{22} &a_{23} & a_{24} \\

a_{31} & a_{32} &a_{33} & a_{34} \\

a_{41} & a_{42} &a_{43} & a_{44} \\

\end{bmatrix}

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

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

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

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

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

a_{23}+a_{24}+a_{33}+a_{34} = 34 (taking m = 2, n = 3)

a_{24}+a_{21}+a_{34}+a_{31} = 34 (taking m = 2, n = 4)

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

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

Sri Rama Chakras[edit]

Sringeri/Srirangam Panchangams

The Sri Rama Chakra as given in the Panchangam published by the Sringeri Sharada Peetham[1] or the one published by Srirangam Temple[2] is shown below.

9 16 5 4

7 2 11 14

12 13 8 1

6 3 10 15

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

It is a four by four showing the image of the quadrant model

Sriramachakra (also called Sri Rama Chakra, Ramachakra, Rama Chakra, or Ramar Chakra) is a mystic diagram or a yantra given in Tamil almanacs as an instrument of astrology for predicting one's future. The geometrical diagram consists of a square divided into smaller squares by equal numbers of lines parallel to the sides of the square. Certain integers in well defined patterns are written in the various smaller squares. In some almanacs, for example, in the Panchangam published by the Sringeri Sharada Peetham[1] or the Pnachangam published by Srirangam Temple,[2] the diagram takes the form of a magic square of order 4 with certain special properties.[3][4] This magic square belongs to a certain class of magic squares called strongly magic squares which has been so named and studied by T V Padmakumar, an amateur mathematician from Thiruvananthapuram, Kerala.[5][6][7] In some almanacs, for example, in the Pambu Panchangam, the diagram consists of an arrangement of 36 small squares in 6 rows and 6 columns in which the digits 1, 2, ..., 9 are written in that order from left to right starting from the top-left corner, repeating the digits in the same direction once the digit 9 is reached.[8]

There is another smaller mystic diagram, called Seetha Chakra given in Tamil almanacs. In some almanacs[1][2] it is given as a magic square of order 3 whereas in some others[8] it is an arrangement of 9 small squares in 3 rows and 3 columns in which the digits 1, 2, .. 9 are written in that order column-wise from left to right.

Theses Chakras are used by the believers to predict future. A believer takes a small flower, prays to God seeking divine directions and drops the flower randomly on a board containing an inscription of one of the Chakras. The number on which the flower falls is believed to give a broad indication of the future of the believer. For example, if the design is Sri Rama Chakra in the form of a magic square and the number on which the flower has fallen is 11 then the person can expect "victory in his/her future endeavors".[9]

All most-perfect magic squares are panmagic squares.

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

For n = 36, there are about 2.7 × 1044 essentially different most-perfect magic squares.

The second property above implies that each pair of the integers with the same background colour in the 4×4 square below have the same sum, and hence any 2 such pairs sum to the magic constant.

The most perfect magic square is 64 squares that is four quadrant models

and each two by two square (each quadrant) has a special property

within the 64 squares

The Hilbert space filling curve can be divided into 8 cell segments. Each of these segments can be labeled from the beginning of the segment to the end of the segment with the numbers 1 - 8. There are 144 examples of the 8x8 most-perfect magic squares where the 8 cell segments sum to the magic constant of 260 as well as the individual positions in each segment summing to 260. The 1st position is highlighted with a large red font in b) and c) below.

Specific examples of most-perfect magic squares that begin with the 2015 date demonstrate how theory and computer science are able to define this group of magic squares. [1] [2] Only a fraction of the 2x2 cell blocks that sum to 130 are accented by the different colored fonts in the 8x8 example.

Most-perfect magic square from

the Parshvanath Jain temple in Khajuraho

A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 with two additional properties:

Each 2×2 subsquare sums to 2s, where s = n2 + 1.

All pairs of integers distant n/2 along a (major) diagonal sum to s.

A magic square is in Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions apply:

the element at position [1,1] (top left corner) is the smallest of the four corner elements; and

the element at position [1,2] (top edge, second from left) is smaller than the element in [2,1].

Frénicle's published book of 1693 described all the 880 essentially different order-4 magic squares.

This standard form was devised since a magic square remains "essentially similar" if it is rotated or transposed, or flipped so that the order of rows is reversed — there exists 8 different magic squares sharing one standard form. For example, the following magic squares are all essentially similar, with only the final square being in Frénicle standard form:

8 1 6 8 3 4 4 9 2 4 3 8 6 7 2 6 1 8 2 9 4 2 7 6

3 5 7 1 5 9 3 5 7 9 5 1 1 5 9 7 5 3 7 5 3 9 5 1

4 9 2 6 7 2 8 1 6 2 7 6 8 3 4 2 9 4 6 1 8 4 3 8

Generalising the concept of essentially different squares[edit]

384

For each group of magic squares one might identify the corresponding group of automorphisms, the group of transformations preserving the special properties of this group of magic squares. This way one can identify the number of different magic square classes.

From the perspective of Galois theory the most-perfect magic squares are not distinguishable. This means that the number of elements in the associated Galois group is 1. Please compare OEIS A051235 Number of essentially different most-perfect pandiagonal magic squares of order 4n. with OEIS A000012 The simplest sequence of positive numbers: the all 1's sequence.

Magic squares are made up of quadrants and they are 4 by 4, 3 by 3, 6 by 6, or whatever number.

Bernard Frénicle de Bessy (c. 1605 – 17 January 1675), was a French mathematician born in Paris, who wrote numerous mathematical papers, mainly in number theory and combinatorics. He is best remembered for Des quarrez ou tables magiques, a treatise on magic squares published posthumously in 1693, in which he described all 880 essentially different normal magic squares of order 4. The Frénicle standard form, a standard representation of magic squares, is named after him. He solved many problems created by Fermat and also discovered the cube property of the number 1729 (Ramanujan number), later referred to as a taxicab number.

The congruum problem was originally posed in 1225, as part of a mathematical tournament held by Frederick II, Holy Roman Emperor, and answered correctly at that time by Fibonacci, who recorded his work on this problem in his Book of Squares.[2]

Fibonacci was already aware that it is impossible for a congruum to itself be a square, but did not give a satisfactory proof of this fact.[3] Geometrically, this means that it is not possible for the pair of legs of a Pythagorean triangle to be the leg and hypotenuse of another Pythagorean triangle. A proof was eventually given by Pierre de Fermat, and the result is now known as Fermat's right triangle theorem. Fermat also conjectured, and Leonhard Euler proved, that there is no sequence of four squares in arithmetic progression

Fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle. Congrua are also closely connected with congruent numbers: every congruum is a congruent number, and every congruent number is a congruum multiplied by the square of a rational number.

The Book of Squares, (Liber Quadratorum)[1] in the original Latin is a book on algebra by Leonardo Fibonacci, published in 1225. Fibonacci's identity, establishing that the set of all sums of two squares is closed under multiplication, appears in it. The book anticipated the works of later mathematicians like Fermat and Euler.[2] The book examines several topics in number theory,[3] among them an inductive method for finding Pythagorean triples based on the sequence of odd integers, the fact that the sum of the first n odd integers is n^2, and the solution to the congruum problem.

Recall that the word for to square is quadratum because it is to make it a quadrant and the proofs underlying calculus and algebra are based on the quadrant four (a square divided into four sections)

The Fourier operator is the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform.

It may be thought of as a limiting case for when the size of the discrete Fourier transform increases without bound while its spatial resolution also increases without bound, so as to become both continuous and not necessarily periodic.

As a teaching tool the Fourier operator is used widely and it has also been used as an art form,[clarification needed] including the book cover of the book Advances in Machine Vision (ISBN 9810209762).

Visualization of the Fourier transform as the result of the Fourier operator[edit]

The Fourier operator defines a continuous two-dimensional function that extends along time and frequency axes, outwards to infinity in all four directions. This is analogous to the DFT matrix but, in this case, is continuous and infinite in extent. The value of the function at any point is such that it has the same magnitude everywhere. Along any fixed value of time, the value of the function varies as a complex exponential in frequency. Likewise along any fixed value of frequency the value of the function varies as a complex exponential in time. A portion of the infinite Fourier operator is shown in the illustration below, which depicts how it acts on a rectangular pulse to generate its Fourier transform (in this case, a sinc function):

It is within a quadrant matrix, with frequency and time as its axes

The smallest cubefree taxicab number with four representations was discovered by Stuart Gascoigne and independently by Duncan Moore in 2003. It is

1801049058342701083

= 922273 + 12165003

= 1366353 + 12161023

= 3419953 + 12076023

= 6002593 + 11658843

1 + 7 + 2 + 9 = 19

19 × 91 = 1729

Masahiko Fujiwara showed that 1729 is one of four positive integers (with the others being 81, 1458, and the trivial case 1) which, when its digits are added together, produces a sum which, when multiplied by its reversal, yields the original number

Euler conjectured a fourth power cannot be written as the sum of 3 smaller fourth powers, but 200 years later this was disproven (Elkies, Frye) with:

958004 + 2175194 + 4145604 = 4224814.

Conjectured incorrectly by Euler to have no nontrivial solutions. Proved by Elkies to have infinitely many nontrivial solutions, with a computer search by Frye determining the smallest nontrivial solution.

Fermat's right triangle theorem is a non-existence proof in number theory, the only complete proof left by Pierre de Fermat.[1] It has several equivalent formulations:

If three square numbers form an arithmetic progression, then the gap between consecutive numbers in the progression (called a congruum) cannot itself be square.

There do not exist two Pythagorean triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle.

A right triangle for which all three side lengths are rational numbers cannot have an area that is the square of a rational number. An area defined in this way is called a congruent number, so no congruent number can be square.

A right triangle and a square with equal areas cannot have all sides commensurate with each other.

The only rational points on the elliptic curve y^2=x(x-1)(x+1) are the three trivial points (0,0), (1,0), and (−1,0).

The Diophantine equation x^4-y^4=z^2 has no integer solution.

An immediate consequence of the last of these formulations is that Fermat's last theorem is true for the exponent n=4.

Two-point[edit]

The two-point DFT is a simple case, in which the first entry is the DC (sum) and the second entry is the AC (difference).

\frac{1}{\sqrt{2}}

\begin{bmatrix}

1 & 1 \\

1 & -1 \end{bmatrix}

The first row performs the sum, and the second row performs the difference.

The factor of 1/\sqrt{2} is to make the transform unitary (see below).

Four-point[edit]

The four-point DFT matrix is as follows:

W = \frac{1}{\sqrt{4}}

\begin{bmatrix}

\omega^0 & \omega^0 &\omega^0 &\omega^0 \\

\omega^0 & \omega^1 &\omega^2 &\omega^3 \\

\omega^0 & \omega^2 &\omega^0 &\omega^2 \\

\omega^0 & \omega^3 &\omega^2 &\omega^1 \\

\end{bmatrix} = \frac{1}{2}

\begin{bmatrix}

1 & 1& 1 & 1\\

1 &-i&-1 & i\\

1 &-1& 1 &-1\\

1 & i&-1 &-i\end{bmatrix}

where \omega = -i.

Eight-point[edit]

The first non-trivial integer power of two case is for 8 points:

\frac{1}{\sqrt{8}}

\begin{bmatrix}

\omega^0 & \omega^0 &\omega^0 &\omega^0 &\omega^0 &\omega^0 &\omega^0 & \omega^0 \\

\omega^0 & \omega^1 &\omega^2 &\omega^3 &\omega^4 &\omega^5 &\omega^6 & \omega^7 \\

\omega^0 & \omega^2 &\omega^4 &\omega^6 &\omega^0 &\omega^2 &\omega^4 & \omega^6 \\

\omega^0 & \omega^3 &\omega^6 &\omega^1 &\omega^4 &\omega^7 &\omega^2 & \omega^5 \\

\omega^0 & \omega^4 &\omega^0 &\omega^4 &\omega^0 &\omega^4 &\omega^0 & \omega^4 \\

\omega^0 & \omega^5 &\omega^2 &\omega^7 &\omega^4 &\omega^1 &\omega^6 & \omega^3 \\

\omega^0 & \omega^6 &\omega^4 &\omega^2 &\omega^0 &\omega^6 &\omega^4 & \omega^2 \\

\omega^0 & \omega^7 &\omega^6 &\omega^5 &\omega^4 &\omega^3 &\omega^2 & \omega^1 \\

\end{bmatrix} = \frac{1}{2\sqrt{2}}

\begin{bmatrix}

1 &1 &1 &1 &1 &1 &1 &1 \\

1 &\omega &-i &-i\omega &-1 &-\omega &i &i\omega \\

1 &-i &-1 &i &1 &-i &-1 &i \\

1 &-i\omega &i &\omega &-1 &i\omega &-i &-\omega \\

1 &-1 &1 &-1 &1 &-1 &1 &-1 \\

1 &-\omega &-i &i\omega &-1 &\omega &i &-i\omega \\

1 &i &-1 &-i &1 &i &-1 &-i \\

1 &i\omega &i &-\omega &-1 &-i\omega &-i &\omega \\

\end{bmatrix}

In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.

Every positive integer can be expressed as the sum of at most 19 fourth powers; every sufficiently large integer can be expressed as the sum of at most 16 fourth powers (see Waring's problem).

16 is the squares of the quadrant model

Fourth degree equations, which contain a fourth degree (but no higher) polynomial are, by the Abel-Ruffini theorem, the highest degree equations solvable using radicals. The fourth is always different. The fifth is always questionable

In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So:

n4 = n × n × n × n

Fourth powers are also formed by multiplying a number by its cube. Furthermore, they are squares of squares.

The sequence of fourth powers of integers (also known as biquadratic numbers or tesseractic numbers) is:

1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, (sequence A000583 in OEIS)

The last two digits of a fourth power of an integer can be easily shown (for instance, by computing the squares of possible last two digits of square numbers) to be restricted to only twelve possibilities:

Every positive integer can be expressed as the sum of at most 19 fourth powers; every sufficiently large integer can be expressed as the sum of at most 16 fourth powers (see Waring's problem).

Euler conjectured a fourth power cannot be written as the sum of 3 smaller fourth powers, but 200 years later this was disproven (Elkies, Frye) with:

958004 + 2175194 + 4145604 = 4224814.

That the equation x4 + y4 = z4 has no solutions in nonzero integers (a special case of Fermat's Last Theorem), was known, see Fermat's right triangle theorem.

Now let me show you a Greek design involving right triangles." He drew their attention to the tiled floor on which they were standing, and then traced a similar pattern on the sand, outlining the important parts.

"Here the two sides of the right angle are equal, and the same relation holds." With his pointer, he indicated one triangle and the related squares, and they all counted together. (See illustration)

"Look! Two triangles plus two triangles equals four triangles. The total area of the squares on the two sides of this right triangle is likewise equal to the area of the square on its hypotenuse."

Again he waited until the newcomers nodded their assent, and then continued:

"In India the priests know other constructions that give similar results; they guard these numbers closely, but we have found some of them. In Babylon, a priestly astrologer whispered to me that there was a secret about this mystery that had never been penetrated."

Now the attention was almost breathless as Pythagoras in-toned solemnly: "That secret is our problem! Would the same relation always be true of ANY right triangle, no matter what the length of its sides, and how could you show this?"

At this dramatic moment, he withdrew behind a curtain, while attendants played on stringed instruments to indicate an intermission. Pandemonium broke out among the assembled initiates. All the newcomers began talking at once, making suggestions, arguing, and shouting. The older mathematicians, who had worked oil this problem themselves, were less noisy but even more excited.

Finally Pythagoras reappeared. Silence instantly fell over the group as he resumed his lecture.

"I will now show you how to construct a wondrous figure which discloses that the answer is always yes! The older 'mathematicians' will realize that by slowly and carefully defining each step of the construction, and using a few simple theorems that you already know, this demonstration can he made into a rigorous proof. Today I will just draw it quickly, so you can all see my great discovery."

lie signaled to attendants to smooth the sand, and began to draw, using his pointer to emphasize his words.

"Watch this beautiful construction! I make a square frame,

102

any size, and in its corner I place a small square, any size. Next I draw straight lines, continuing the sides of the small square to the edge of the frame.

"Do you see what my frame now contains? A small square and a medium square, and two equal rectangles.

"And next-we are almost there-I simply add diagonal lines across the rectangles!

"This is the figure I need. My frame now contains a small square, a medium square, and four equal right triangles. Now I will ask you to look more closely at this figure."

Pythagoras beckoned to the attendants, who poured colored sand from jars onto the parts of the drawing, so the pattern showed plainly.

"Look again!" He used his pointer and spoke with care. "All the triangles, you know, are equal; each is the same triangle in a different position. Now, notice how the triangles touch the squares, especially Triangle 2. You can see that the same square is the square on the short side of the triangle. And the medium square is the square on the long side of the triangle. So my frame is completely filled by four equal right triangles plus the square on the short side and the square on the long side!"

Pythagoras paused while a low murmur of awe rose from the initiates.

"Now watch!" he intoned. And while they all craned their necks to see, and the attendants poured more colored sand, Pythagoras drew his final masterful figure.

"Watch well! I have only to swing and push these four triangles around, like this, so that they fit perfectly into the four corners of the frame, and my frame is now completely filled by the same four equal right triangles plus the square on the hypotenuse!

"Therefore, in any right triangle, the area of the square on one side plus the area of the square on the other side will add up to the area of the square on the hypotenuse!"

A mighty shout-we can imagine-went up from the assembled inner group of the Secret Brotherhood. For this theorem was a true landmark in the development of geometry by the Pythagoreans. Almost all later geometric work involving lengths. and measurement was based upon it. And this style of solving problems, especially equations, by diagramming them, would remain a chief trait of Greek geometry.

But to the initiate who first heard it, the theorem also partook of a mystical revelation. Tradition says that Pythagoras himself celebrated the occasionly a noble sacrifice-an ox, or a hundred oxen-to his "divine father," Apollo. Some ancient writers dispute this, as the Pythagoreans were vegetarians. Whatever the offering, we can easily picture the festivities described in the verse of legend. Doubtless the "mathematicians" chanted, torches waved, and smoke rose from the sacrificial altar,

The day Pythagoras the famous figure found

For which he brought the gods a sacrifice renowned!

QMRInscapes

by Steven H. Cullinane

In finite geometry and combinatorics,

an inscape is a 4x4 array of square figures,

each figure picturing a subset of the overall 4x4 array:

Inscape001A.gif

Inscapes provide a way of picturing

the following equivalent concepts:

The 60 smile emoticon 15×4) Göpel tetrads in PG(3,2),

The generalized quadrangle GQ(2,2),

Tutte's 8-cage, and

the Cremona-Richmond 153 configuration (pdf):

Cremona-Richmond.gif

Diamond Theory shows that this structure

can also be modeled by an inscape:

Inscape.gif

The illustration below shows how the

points and lines of the inscape may

be identified with those of the

Cremona-Richmond configuration.

Inscape2.gif

Related material on inscapes:

Rosenhain and Göpel Tetrads in PG(3,2)

The 2-Subsets of a 6-Set are the Points of a PG(3,2)

A Symplectic Approach to the Miracle Octad Generator

Inscapes, Inscapes II, Inscapes III, Inscapes IV

QMRRosenhain and Göpel Tetrads in PG(3,2)

by Steven H. Cullinane on March 17, 2013

Hudson's 1905 classic Kummer's Quartic Surface

discusses the 80 Rosenhain odd tetrads and the 60 Göpel even tetrads.

These 140 tetrads occur naturally in the Galois projective 3-space PG(3,2).

IMAGE- Rosenhain and Göpel Tetrads in Hudson

The Rosenhain and Göpel tetrads occur naturally

in the diamond theorem model of the 35 lines of PG(3,2):IMAGE- Rosenhain and Göpel tetrads in PG(3,2)

As in the proof of the diamond theorem, each 4x4 array above has a set of three line diagrams. Arrays containing the Rosenhain odd tetrads (turquoise or white) have either 3 or 1 unidirectional line diagrams (all-horizontal or all-vertical lines). Arrays containing the Göpel even tetrads (yellow or gold) have either 2 or 0 unidirectional line diagrams.

The Göpel tetrads appear in my "Inscapes" notes and in "Picturing the Smallest Projective 3-Space" (April 1986). See Inscapes and Notes on Groups and Geometry, 1978-1986.

Göpel tetrads in an inscape, April 1986

More recently, they have appeared in "Birational Automorphisms of Quartic Hessian Surfaces," by Igor Dolgachev and JongHae Keum, in Transactions of the American Mathematical Society, Volume 354, Number 8, pages 3031-3057 (published online on April 3, 2002).

Dolgachev and Keum represent the Göpel tetrads as occuring in a set of 16 points that they identify with the empty set plus the fifteen 2-element subsets of a 6-set (p. 3037). For an earlier version of this representation, see "The 2-subsets of a 6-set are the points of a PG(3,2)" (May 1986). That note also presents the Göpel tetrads as they relate to the Miracle Octad Generator of R. T. Curtis and to its underlying Moore correspondence.

Update: August 3, 2014 —

The Diamond-Theorem Correlation

Update: August 6-7 and 15, 2014 —

Symplectic Structure*

From Gotay and Isenberg, “The Symplectization of Science,”

Gazette des Mathématiciens 54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….

Its mathematical usage is due to Hermann Weyl who,

in an effort to avoid a certain semantic confusion, renamed

the then obscure ‘line complex group’ the ‘symplectic group.’

… the adjective ‘symplectic’ means ’plaited together’ or ‘woven.’

This is wonderfully apt….”

The above symplectic structure** now appears in the figure

illustrating the diamond-theorem correlation in the webpage

Rosenhain and Göpel Tetrads in PG(3,2).

Some related passages from the literature:

http://www.log24.com/.../100915...

* The title phrase is a deliberate abuse of language.

For some background, see a definition of "symplectic structure"

as the relevant symplectic form. See also the phrase in Wikipedia's

"Symplectic vector space" article.

** See Steven H. Cullinane, Inscapes III, 1986

Update: August 15, 2014 —

Related readings:

Gonzalez-Dorrego on symplectic structure in PG(3,2)

Dolgachev and Keum, 2002, on symplectic structure

Six-set geometry, illustrated by Steven H. Cullinane

Symplectric structure in PG(3,2), 1986, by Steven H. Cullinane

Updates: March 26, 2013, August 3, 2014, August 6-7, 2014, August 15, 2014

The sixteen figures in the lower right quadrant of the von-Franz-style 8x8 array above (from the icing) also appear in a discussion of the work of Charles Sanders Peirce, a logician of the late nineteenth and early twentieth century. See "New Light on Peirce's Iconic Notation for the Sixteen Binary Connectives," by Glenn Clark, in Studies in the Logic of Charles Sanders Peirce, Indiana University Press, 1997.

Peter Loly of the University of Manitoba has also investigated cubic and square arrangements of the 64 hexagrams that have the Karnaugh property (adjacent figures differing in only one place). See his paper "A Logical Way of Ordering the Hexagrams of the Yijing and the Trigrams of the Bagua" (pdf), published in The Oracle-- The Journal of Yijing Studies, vol. 2, no. 12, January 2002, pp. 2-13.

The new illustration below may serve to clarify some of my earlier remarks above and to show how Loly's "unfolding" process may be applied to my 1989 arrangement to yield a flat "chessboard" of hexagrams that, like the cube, has the Karnaugh property.

Hexagrams and the Karnaugh property

QMRSuch a logical connective as converse implication "←" is actually the same as material conditional with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in classical logic) certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy is the classical equivalence between ¬P ∨ Q and P → Q. Therefore, a classical-based logical system does not need the conditional operator "→" if "¬" (not) and "∨" (or) are already in use, or may use the "→" only as a syntactic sugar for a compound having one negation and one disjunction.

There are sixteen Boolean functions associating the input truth values P and Q with four-digit binary outputs.[14] These correspond to possible choices of binary logical connectives for classical logic. Different implementations of classical logic can choose different functionally complete subsets of connectives.

One approach is to choose a minimal set, and define other connectives by some logical form, as in the example with the material conditional above. The following are the minimal functionally complete sets of operators in classical logic whose arities do not exceed 2:

The oldest known proof the oldest proof of the pythagorean theorem was by drawing a quadrant

There is evidence that Pythagoras' Theorem was discovered very early by the Chinese and the Indians (refer to Heath's discussion just after I.47), but exactly how early is not known. The earliest tangible record of Pythagoras' Theorem comes from Babylonian tablets dating to around 1000 B.C. In addition to tablets containing exercises which depend on knowing at least a few specific cases, a number of tablets have been found with pictures which are in effect proofs of the Theorem in the special case where the sides of the right triangle are equal.

In this case of course the Pythagoras' Theorem asserts that the ratio of the diagonal of a square to one of its sides is equal to the square root of 2.

This picture leads rather directly to the irrationality of the square root of 2. The argument we give here is variant of one from Heath's comments on Euclid (volume I, p.400).

Let s and d be the side and diagonal of the large square in the figure above. To say that s and d are commensurable, or equivalently that the ratio d/s (the square root of 2) is a rational number, means that there exists some small segment e such that d and s are both multiples of e. Let s' and d' be the side and diagonal of the smaller (red) square in the figure above. We claim that if s and d are multiples of e then so are s' and d'. But an argument about congruent triangles shows that

s' = d - s

d' = s - s'

which imply the claim.

This leads to a contradiction. The two squares are similar. Suppose the smaller one is equal to the larger one scaled by a constant c, which from the figure is clearly smaller than !. We can apply the argument over and over again to see that all lengths cn s and cn d are multiples of e. This cannot happen since cn becomes arbitrarily small.

Reference

Otto Neugebauer, Keilschrift-Texte, Part II (especially BM 15285, Tafel 3). Springer, 1935.

Otto Neugebauer, The exact sciences in antiquity, published by Dover, 1969. Especially p. 35 and plate VI.a.

[Back to the Pythagoras main page]

1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4x4 square is now available online:

Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:

"In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an 'octad generator'; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code."

(Received July 20 1987)

-- Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.

(Published online by Cambridge University Press 19 Dec 2008.)

In the above article, Curtis explains how two-thirds of his 4x6 MOG array may be viewed as the 4x4 model of the four-dimensional affine space over GF(2). (His earlier 1974 paper (below) defining the MOG discussed the 4x4 structure in a purely combinatorial, not geometric, way.)

For further details, see Geometry of the 4x4 Square and Curtis's original 1974 article, which is now also available online:

A new combinatorial approach to M24, by R. T. Curtis. Abstract:

"In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent."

(Received June 15 1974)

-- Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.

(Published online by Cambridge University Press 24 Oct 2008.)

QMRProjective spaces over a finite field, otherwise known as Galois geometries, find wide application in coding theory, algebraic geometry, design theory, graph theory, and group theory as well as being beautiful objects of study in their own right."

— Oxford University Press on

General Galois Geometries, by

J. W. P. Hirschfeld and J. A. Thas

The simplest Galois geometries are the projective spaces of one, two, and three dimensions over the two-element* Galois field... That is to say, the binary projective line, plane, and 3-space.

Perhaps the simplest models for such geometries are those of Diamond Theory.

The pictures at left show these models.

The pictures at right illustrate some graphic designs with underlying structures based on models like those shown at left.

Click on any of the pictures at right for an introduction, Theme and Variations, to the aesthetics of designs based on Galois geometry.

The appearance of the pictures at right explains the origin of the title "Diamond Theory."

The line diagrams at left are related to the two-color patterns at right as follows.

The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn. Taken as a set, these three line diagrams describe the structure of the bottom colored figure. After coordinatizing the figure in a suitable manner, we find that these three line diagrams are invariant under the group of 16 binary translations acting on the colored figure.

A more remarkable invariance— that of symmetry itself— is observed if we arbitrarily and repeatedly permute rows and/or columns and/or 2x2 quadrants of the colored figure above. Each resulting figure has some ordinary or color-interchange symmetry. The cause of this symmetry-invariance in the colored patterns is the symmetry-invariance of the line diagrams under a group of 322,560 binary affine transformations.

— Steven H. Cullinane

QMRFrom a 2002 review by Stacy G. Langton of Sherman Stein's book on mathematics, How the Other Half Thinks:

"The title of Stein's book (perhaps chosen by the publisher?) seems to refer to the popular left brain/right brain dichotomy. As Stein writes (p. ix): 'I hope this book will help bridge that notorious gap that separates the two cultures: the humanities and the sciences, or should I say the right brain (intuitive, holistic) and the left brain (analytical, numerical). As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role.' Stein does well to avoid identifying mathematics with the activity of just one side of the brain. He would have done better, however, not to have endorsed the left brain/right brain ideology. While it does indeed appear to be the case that the two sides of our brain act in rather different ways, the idea that the right brain is 'intuitive, holistic,' while the left brain is 'analytical, numerical,' is a vast oversimplification, and goes far beyond the actual evidence."

Despite the evidence, it is tempting to view the above pictures as illustrating, on the left, the line-diagram side, a cold, analytical approach to diamond theory, and, on the right, the colored-pattern side, a warm, intuitive approach to the same theory— the grid (left) versus the quilt (right), as it were.

Of course, these sides are reversed when the information on this page reaches the brain, so that the diagrams on the left side of the page go to the right side of the brain, and the patterns on the right side of the page go to the left side of the brain. This page layout may or may not help the reader integrate the analytical and the intuitive natures of the pictures. More likely to be helpful in such an integration, playing the role of a corpus callosum, is the combination of line diagrams and colored pattern in the central illustration, "Invariant."

Page created July 16, 2004.

* Update of September 10, 2011:

The two-element Galois field GF(2) is of course the simplest such field.

The three-element Galois field GF(3) also leads to some structures that

are easily visualized (see below), but that lack, since they involve an odd prime,

some of the intriguing combinatorial properties of structures based on GF(2).

Elementary Galois Geometry over GF(3)

Note that projective points are visualized in these figures over GF(3)

in a different way from those over GF(2) in the main article above.

These models of geometry over GF(3) are based on the standard

definition of points in an n-dimensional projective space as derived

from lines in an (n+1)-dimensional vector space. The main article's

models of projective points in spaces over GF(2), on the other hand—

as partitions— is not based on that standard definition, but is

new (as of 1979) and different.

Plato’s diamond that Socrates used to prove the immortality of the soul was a quadrant

Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.

Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of noncontinuous (and asymmetric) transformations. (These might be called noncontinuous groups, as opposed to so-called discontinuous (or discrete) symmetry groups. See Weyl's Symmetry.)

For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4x4 array. (Details)

By embedding the 4x4 array in a 4x6 array, then embedding A in a supergroup that acts in a natural way on the larger array, one can, as R. T. Curtis discovered, construct the Mathieu group M24 -- which is, according to J. H. Conway, the "most remarkable of all finite groups."

The proof that A preserves symmetry involves the following elementary, but useful and apparently new, result: Every 4-coloring (i.e., every map into a 4-set) can be expressed as a sum of three 2-colorings. It is conceivable that this result might have applications other than to diamond theory. (Details)

The proof that A preserves symmetry also yields some insight into orthogonality of Latin squares, at least in the 4x4 case. In this case, orthogonality turns out to be equivalent to skewness of lines in a finite projective 3-space. (Details)

Diamond theory provides simple ways to visualize

"the simplest non-trivial model for harmonic analysis,"1 the Walsh functions,

"the smallest perfect universe,"2 the finite projective space PG(3,2),

the "distinguished geometry," "remarkable geometric individual," and "gem,"3 the finite projective plane PG(2,4),

the "remarkable"4 outer automorphisms of S6, and

the structural equivalence of certain

factorizations3 of graphs,

parallelisms5 of partitions,

spreads6 of lines, and

sets of mutually orthogonal squares (i.e., n-orthogonal7 nxn matrices).

Diamond Theory

Plato tells how Socrates helped Meno's slave boy "remember" the geometry of a diamond. Twenty-four centuries later, this geometry has a new theorem.

The Diamond Theorem:

Inscribe a white diamond in a black square.

Split the resulting figure along its vertical and horizontal midlines into four quadrants so that each quadrant is a square divided by one of its diagonals into a black half and a white half. Call the resulting figure D.

Let G be the group of 24 transformations of D obtained by randomly permuting (without rotating) the four quadrants of D. Let S4 denote the symmetric group acting on four elements. Then

(1) Every G-image of D has some ordinary or color-interchange symmetry (see below),

(2) G is an affine group generated by S4 actions on parts of D, and

(3) Results (1) and (2) generalize, through intermediate stages, to symmetry invariance under a group of approximately 1.3 trillion transformations generated by S4 actions on parts of a 4x4x4 cube.

The 2x2 case

In the 2x2 case, D is a one-diamond figure (top left, below) and G is a group of 24 permutations generated by random permutations of the four 1x1 quadrants. Every G-image of D (as below) has some ordinary or color-interchange symmetry.

Example of the 4x4 case

In the 4x4 case, D is a four-diamond figure (left, below) and G is a group of 322,560 permutations generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2x2 quadrants. Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

D Let e denote transposition of the first two rows, f denote transposition of the last two columns, g denote transposition of the top left and bottom right quadrants, and h denote transposition of the middle two columns. Then Defgh is as at right. Note that Defgh has rotational color-interchange symmetry like that of the famed yin-yang symbol.

Defgh

Remarks on the 4x4 case:

G is isomorphic to the affine group A on the linear 4-space over GF(2). The 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2). Orthogonality of structures corresponds to skewness of lines. We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4). For a movable JavaScript version of these 4x4 patterns, see The Diamond 16 Puzzle.

QMRABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs-- found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4x4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2x2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:

For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)-- in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4x4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero-- i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).

The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2x2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4x4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4x4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis-- used in the construction of the Steiner system S(5,8,24)-- and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For movable JavaScript versions of these 4x4 patterns, see The Diamond 16 Puzzle and the easier Kaleidoscope Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

For some context, see Reflection Groups in Finite Geometry.

The Diamond 16 Puzzle

Change the pair of patterns at right to any of the pairs at left. (The original pair is at top left.)

To change the patterns, click on a row, then click on the row you want to move it to. Do the same for other rows, columns, or quadrants.

You can make 322,560 pairs of patterns. Each pair pictures a different symmetry of the underlying 16-point space.

Note that the 4x4 arrays in the picture at bottom right may serve as the basis for patterns like those in the picture at top left. The 35 structures in the picture at bottom right may be regarded as exemplifying the aesthetics of James J. Gibson in his 1978 essay "The Ecological Approach to the Visual Perception of Pictures" --

"What modern painters are trying to do, if they only knew it, is paint invariants."

-- James J. Gibson, Leonardo,

Vol. 11, pp. 227-235. Pergamon Press Ltd., 1978

Gibson is discussing Euclidean 3-space rather than binary 4-space, but his remarks on invariants are still relevant.

An example of invariant structure:

The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn. Taken as a set, these three line diagrams describe the structure of the bottom colored figure. After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 binary translations acting on the colored figure.

For another sort of invariance of the colored figure, try applying a symmetry of the square to each of the set of four diagonally-divided squares from which the figure's entries are drawn, and observe the induced effect on the figure itself.

A more remarkable invariance -- that of symmetry itself -- is observed if we arbitrarily and repeatedly permute rows and/or columns and/or 2x2 quadrants of the colored figure above. Each resulting figure has some ordinary or color-interchange symmetry. The cause of this symmetry-invariance in the colored patterns is the symmetry-invariance of the line diagrams under a group of 322,560 binary affine transformations.

Related material on two meanings of "design theory":

In the mathematical sense:

Design Theory, by Beth, Jungnickel, and Lenz

In the artistic sense:

Visual Language, by Karl Gerstner

For more details on the above block designs, see

QMRWe present two results -- one old, one new -- on the geometry of Latin squares.

Result A (old):

There exist n-1 mutually orthogonal nxn Latin squares if and only if there exists a finite projective plane with n2 + n + 1 lines (or, equivalently, an affine plane with n2 + n lines).

Result A is well known. See, for instance, chapter 8 of Discrete Mathematics Using Latin Squares, C. F. Laywine and G. L. Mullen, Wiley Interscience, 1998, or Bose's [1938] Theorem (pdf).

Result B (new):

The six 4x4 Latin squares that have orthogonal Latin mates can be embedded in a set of thirty-five 4x4 arrays so that orthogonality in the set of arrays corresponds to skewness in the set of 35 lines of the finite projective space PG(3,2).

Result B is apparently new, and should not be confused with result A. The closest thing to the diagrams of result B in the refereed literature seems to be the use of diagrams on page 774 of Design Theory, Volume 2, by T. Beth, D. Jungnickel, and H. Lenz, Cambridge U. Press, 1999, in proving Tarry's theorem on the nonexistence of two mutually orthogonal 6x6 Latin squares.

The research note below shows how result B works. Note particularly that the 35-line projective space of result B differs from the 21-line projective plane of result A.

Result B is, of course, highly special, being limited to 4x4 squares. This limitation should not prevent its use as an example in popular introductions to discrete mathematics. Indeed, the 4x4 case figured prominently (and exclusively) on the cover and in a two-page article in the August/September 2001 issue of the Mathematical Association of America's "Focus" newsletter.

Naturally, a more general result than B is desirable; hence the problem stated in the 1978 research note below.

Steven H. Cullinane

Orthogonality of Latin squares viewed as skewness of lines. Dec. 1978.

Shown below is a way to embed the six order-4 Latin squares that have orthogonal Latin mates in a set of 35 arrays so that orthogonality in the set of arrays corresponds to skewness in the set of 35 lines of PG(3,2). Each array yields a 3-set of diagrams that show the lines separating complementary 2-subsets of {0,1,2,3}; each diagram is the symmetric difference of the other two. The 3-sets of diagrams correspond to the lines of PG(3,2). Two arrays are orthogonal iff their 3-sets of diagrams are disjoint, i.e. iff the corresponding lines of PG(3,2) are skew.

This is a new way of viewing orthogonality of Latin squares, quite different from their relationship to projective planes.

PROBLEM: To what extent can this result be generalized?

The apparent conflict between the 2003 paper by Jungnickel et al. and my 1978 note can be resolved as follows:

"The [1954-1964] André/Bruck-Bose construction yields a one-to-one correspondence between spreads of projective space and translation planes(special affine planes). If one feeds a partial spread into this construction, a net results. A net is a point-line geometry which is a natural weakening of an affine plane."

-- John Bamberg, Symplectic spreads

And it has long been known that affine planes are, as noted above, closely related to orthogonal Latin squares.

In other words, the 1954-1964 André/Bruck-Bose construction is the missing (missing, that is, according to Jungnickel, Thas, et al.) link between Latin-square orthogonality and projective-space skewness. Such an orthogonality-skewness link is shown rather more directly and clearly in my 1978 note.

(For details of the André/Bruck-Bose construction, see

Johannes André, Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe, Math Z. 60, pp. 156-186, 1954, and

R. H. Bruck and R. C. Bose, The construction of translation planes from projective spaces, J. Algebra 1, pp. 85-102, 1964.

The following may also be helpful:

-- From Flocks, ovals, and generalized quadrangles (ps), (Four lectures in Napoli, June 2000), by Maska Law and Tim Penttila)

For further background, here is material on finite geometry from the paper Symplectic spreads (pdf), 15 Sept. 2003, by Simeon Ball, John Bamberg, Michel Lavrauw, and Tim Penttila:

QMRWalsh functions have an inherent symmetry that is best seen by regarding the non-constant Walsh functions as hyperplanes in finite geometry. Such symmetry is exhibited by the 15 affine hyperplanes of the 4-space over the 2-element field.

For definitions of Walsh functions, see The first sixteen Walsh functions and Mathworld -- Walsh Function.

For an excellent introduction to Walsh functions, by Benjamin Jacoby (pdf format), click here.

QMRFour Kinds of Symmetry

Each of the four kinds of symmetry used here is named for a letter to help you remember what it is.

H symmetry.

The right side is a

mirror image of the

left, and the top

is a mirror of the

bottom. The letters

O and X also have

H symmetry.

M symmetry.

The right side is a mirror image of the left, but the top is not a mirror of the bottom. The letters A and V also have M symmetry.

S symmetry.

There are no lines of

mirror symmetry, but

the figure is the same

if you rotate it 180

degrees. The letters

N and Z also have

S symmetry.

B symmetry.

The top is a mirror image of the bottom, but the right side is not a mirror of the left. The letters C and E also have B symmetry.

Notice that the quilt blocks do not necessarily look like the letters—they only have the same symmetry.

Now it's your turn. Try to match quilt blocks by choosing a symmetry first. Go!

In the context of quantum information theory, the following structure seems to be of interest--

"... the full two-by-two matrix ring with entries in GF(2), M2(GF(2))-- the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."

-- "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

This ring is another way of looking at the 16 elements of the affine 4-space A4(GF(2)) over the 2-element field. (Arrange the four coordinates of each element-- 1's and 0's-- into a square instead of a straight line, and regard the resulting squares as matrices.) (For more on A4(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.) Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring. This system of 35 objects has a subconfiguration isomorphic to the (2,2) generalized quadrangle W2 (which occurs naturally as a subconfiguration of PG(3,2)-- see Inscapes.) The connection of this generalized quadrangle with PG(3,2) is noted in a later paper by Saniga and others ("The Veldkamp Space of Two-Qubits," cited below).

Saniga concludes "Geometry of Two-Qubits" as follows:

"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus

not only offers a principally new geometrically-underlined insight into their intrinsic nature,

but also gives their applications a wholly new perspective

and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."

It would seem, in the light of Saniga's remarks, that my own study of pure mathematics-- for instance, of the following "diamond ring"--

is not without relevance to the physics of quantum theory.

Other material related to the above Saniga paper:

Quantum Geometry, a list of recent work in this area by Metod Saniga

The Veldkamp Space of Two-Qubits, by Metod Saniga, Michel Planat, Petr Pracna, and Hans Havlicek, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA 3 (2007), 075) (pdf, June 18, 2007, 7 pp.), and the following cited papers:

Pauli Operators of N-Qubit Hilbert Spaces and the Saniga–Planat Conjecture, by K. Thas, Chaos Solitons Fractals, to appear (as of June 18, 2007)

The Geometry of Generalized Pauli Operators of N-Qudit Hilbert Space, by K. Thas, Quantum Information and Computation, submitted (as of June 18, 2007)

This site is about the

Geometry of the 4x4x4 Cube

(the mathematical structure,

not the mechanical puzzle)

and related simpler structures.

For applications of this sort

of geometry to physics, see

Quantum Information Theory.

These traditional and supposedly well-known structures are, surprisingly, closely related to small finite geometries.

These finite geometries underlie some remarkable symmetries of graphic designs. For instance, there is a group G of 322,560 natural transformations which, acting on the four-diamond figure D below, always yields a symmetric image. This group turns out to also be the automorphism group of the 16-point four-dimensional finite affine space.

Example of symmetry in a 4x4 design

For an animated illustration, click here.

Such designs are formed by assembling two-colored square tiles or two-colored cubical blocks into larger squares or cubes, when the number of tiles or blocks in the larger arrays is a power of two.

The structure underlying such graphic symmetries is that of finite projective geometry:

Small finite spaces

Similar structural diagrams can be made for the 64 and 63 points of, respectively, finite affine and projective spaces of 6 and 5 dimensions over the 2-element field, and the graphic symmetries that result generalize the results in fewer dimensions.

For a more detailed example of how affine and projective points are related in such models, click on the image below.

In 1998, the Mathematical Sciences Research Institute at Berkeley published a book, The Eightfold Way, inspired by a new sculpture at the Institute. This note describes another sculpture embodying some of the same concepts in a different guise.

The Eightfold Way deals with Klein's quartic, which, like all non-singular quartic curves, has 28 bitangents. The relationship of the 28 bitangents to the 27 lines of a " Solomon's seal" in a cubic surface is sketched at the Mathworld encyclopedia. For more details, see the excerpt below, from Jeremy Gray's paper in The Eightfold Way.

Both the 28 bitangents and the 27 lines may be represented within the 63-point space PG(5,2), as noted by J. W. P. Hirschfeld in Ch.20 of Finite Projective Spaces of Three Dimensions (Clarendon Press, Oxford, 1985).

The space PG(5,2) also contains a representation of the Klein quadric (as opposed to the Klein quartic discussed in The Eightfold Way). This representation, obtained via the Klein correspondence, may be used to construct the Mathieu group M24. See The Klein Correspondence, Penrose Space-Time, and a Finite Model.

Group actions on the 63 points of the finite projective space PG(5,2) are derived from group actions on the 64 points of the finite affine space AG(6,2).... which may, as my Diamond Theory points out, be visualized as a 4x4x4 cube.

The theorem may be verified by manipulating a JavaScript version of the cube.

Those who like to associate mathematical with religious entities may contemplate the above in the light of the 1931 Charles Williams novel Many Dimensions. Instead of Solomon's seal, this book describes Solomon's cube.

From a review: "Imagine 'Raiders of the Lost Ark' set in 20th-century London, and then imagine it written by a man steeped not in Hollywood movies but in Dante and the things of the spirit, and you might begin to get a picture of Charles Williams's novel Many Dimensions."

From The Eightfold Way, a publication of

the Mathematical Sciences Research Institute

(MSRI Publications Vol. 35, 1998):

From the History of a Simple Group

by Jeremy Gray

Excerpt:

Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton, 2008–

"Here, modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated— indeed, anxious— rather than a naïve relationship with the day-to-day world, which is the de facto view of a coherent group of people, such as a professional or discipline-based group that has a high sense of the seriousness and value of what it is trying to achieve. This brisk definition…."

Brisk? Consider Caesar's "The die is cast," Gray in "Solomon's Cube," and Symmetry, Automorphisms, and Visual Group Theory–

Order-8 group generated by reflections in three midplanes of cube

This is the group of "8 rigid motions

generated by reflections in midplanes"

of "Solomon's Cube."

Related material:

"… the action of G168 in its alternative guise as SL(3; Z/2Z) is also now apparent. This version of G168 was presented by Weber in [1896, p. 539],* where he attributed it to Kronecker."

– Jeremy Gray, "From the History of a Simple Group," in The Eightfold Way, MSRI Publications, 1998

Here MSRI, an acronym for Mathematical Sciences Research Institute, is pronounced "Misery." See Stephen King, K.C. Cole, and Heinrich Weber.

QMRAmong the groups above, GL(4,2) has several nice properties that make it of particular interest to students of group theory, combinatorics, or finite geometry.

GL(4, 2) is one of two nonisomorphic simple groups of the same order, the other being the projective unimodular group PSL(3, 4). (See AR, p. 171, or RO, p. 172.)

GL(4, 2) is isomorphic to A8, and this isomorphism plays a role (described below) in the structure of the quintuply transitive Mathieu group M24.

Underlying the isomorphism of GL(4, 2) with A8 is a structure that by itself is of interest in combinatorics and finite geometry. This structure is a correspondence J between the 35 partitions of an 8-set into two 4-sets and the 35 2-dimensional subspaces of V(4, 2) when the latter is regarded as a vector 4-space over F2. Under the correspondence J, two partitions into 4-sets have a common refinement as a partition into four 2-sets if and only if the corresponding 2-dimensional subspaces have a nontrivial intersection. (See HA or CA, p. 60.)

J. H. Conway [CO] has shown that M24 can be generated by combining three sorts of permutations, one sort being an action of GL(4, 2) on the 16 points of V(4, 2), combined with the corresponding action of A8 on a separate set E of 8 points, the second sort being a translation acting on V(4, 2), with all points of E fixed, and the third sort being an interchange of E with either of two halves of V(4, 2). Conway shows that a family of 8-element subsets (octads) is left invariant by the action of M24 on a 24-set, and also shows that the vectors over F2 corresponding to these subsets span a 12-dimensional space C over F2 that is known as the binary Golay code, in which each vector has 0, 8, 12, 16, or 24 1's. In a related paper, R. T. Curtis [CU] displays the correspondence J, described above, in his "miracle octad generator," which beautifully pictures the locations of octads within a 4x6 array.

GL(4, 2) is the collineation group of a finite projective geometry, PG(3, 2), that as the smallest projective space is a rich source of examples for geometers.

We now describe how to generate eight of the groups on our list above, of orders 6 through approximately 1.3 trillion, by letting the symmetric groups S3 (for linear groups) or S4 (for affine groups) permute natural subdivisions of square or cubical arrays.

We first recall that a permutation group acting on spatially located objects has isomorphic "alias" and "alibi" interpretations, depending on whether the cycle (ab...) means that object a is replaced by object b, etc. ("alias"), or whether it means that the object in location a is replaced by the object in location b, etc. ("alibi"). To avoid confusion we will use the alibi approach throughout, and will apply labels (numbers or (0,1)-vectors) only to locations within square or cubical arrays, and not to the objects permuted, which are "points" represented by unlabeled square or cubical unit cells. (The reader may, of course, label these cells as he pleases.)

Our groups will act on the following arrays of unit cells: a 2x2 square, a 2x2x2 cube, a 4x4 square, and a 4x4x4 cube. In each case, the same mathod of labeling locations is used:

First, think of the locations in a square array, or in the top layer of a cubical array, as labeled by the integers 0, 1, 2, ... in normal reading order (left to right, top to bottom); for a cubical array, continue this process on lower layers, with the numbers increasing as the layers get lower.

Now change the numbers to binary notation.

Finally, regard the string of 0's and 1's labeling a given location as a vector over F2, rather than a binary number.

By a standard result of linear algebra [AR, ch. IV] the group of all unimodular linear transformations on an n-space over a field, i.e. the group of all nxn matrices with determinant 1, is generated by transvections, i.e. by the matrices Bij(t) obtained from the identity matrix by substituting the scalar t for one of the 0's. Since we are working over F2, every nonsingular linear transformation is unimodular, so the transvections Bij are the only generators for GL(n, 2) we need examine.

A quick check shows that for the 2x2 array as labeled above, the Bij for GL(2, 2) generate the action of S3 on the three unit cells not located at the origin (0, 0).

Given this S3action, clearly AGL(2, 2) acts as S4, since any nonzero translation moves the cell at the origin into the S3 orbit.

For the 2x2x2 array, the transvections of GL(3, 2) are easily seen to generate copies of the S3 action on the top, left, and back of the cube, with this action being extended through adjacent layers. In other words, GL(3, 2) is generated by S3 acting in each of 3 sets of 4 parallel 1x1x2 sections of the cube, with the fourth section in each set being fixed under S3's action on that set, since the fourth section contains the cell located at the origin. By the same argument as above, AGL(3, 2) is generated by S4 acting on each of the 3 sets of 4 parallel 1x1x2 sections.

For the 4x4 array, 6 of the Bij generate all 12 transvections, and also generate the action of S3 on the 3 rows, the 3 columns, and the 3 2x2 quadrants of cells that do not contain the cell located at the origin. Again, it is clear that AGL(4, 2) is generated by the action of S4 on rows, on columns, and on quadrants.

For GL(6, 2) acting on the 4x4x4 cube, consider the three 4x4 submatrices of the 6x6 identity matrix that correspond to the 3 sets of parallel 1x4x4 layers of the cube such that 4 of the 6 coordinates are identical throughout the four layers in a set. Together, these three submatrices cover all possible locations for a 1 located outside the diagonal in a 6x6 transvection matrix. It follows that the actions of GL(6 2) on the 4x4x4 cube are generated by S3 permuting any 3 parallel 1x4x4 layers not containing a cell at the origin and by S3 permuting any 3 parallel 2x2x4 sections not containing a cell at the origin. Naturally, AGL(6, 2) is generated by arbitrary permutations of parallel layers and of parallel 2x2x4 sections.

We conclude by mentioning an application to symmetries of ornamental patterns. Consider a cubic cell that is colored white around each of two opposite corners, in an area bounded by diagonals of the faces surrounding these corners, and black elsewhere, so that a band of black running around the cube separates the two white "caps," which are images of one another under a reflection in the center of the cube. If the reader arranges such cubes into a diamond pattern within any of the arrays we have discussed, he is likely to find that the resulting pattern has either an ordinary or a color-interchange symmetry under some rigid motion (rotation, reflection, or inversion) of the entire array. That such symmetry always occurs, given the right sort of starting pattern, is due to the Euclidean symmetries of partitions by affine hyperplanes.

For a summary of these results, see Affine groups on small binary spaces.

For a look at AGL(4, 2) in action, see The Diamond 16 Puzzle.

QMRIn mathematics, the lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent each line in S lie on a hyperbolic quadric, Q known as the Klein quadric.

If the underlying vector space of S is the 4-dimensional vector space V, then T has as the underlying vector space the 6-dimensional exterior square Λ2V of V. The line coordinates obtained this way are known as Plücker coordinates.

These Plücker coordinates satisfy the quadratic relation

p_{12} p_{34}+p_{13}p_{42}+p_{14} p_{23} = 0

defining Q, where

p_{ij} = u_i v_j - u_j v_i

are the coordinates of the line spanned by the two vectors u and v.

The 3-space, S, can be reconstructed again from the quadric, Q: the planes contained in Q fall into two equivalence classes, where planes in the same class meet in a point, and planes in different classes meet in a line or in the empty set. Let these classes be C and C'. The geometry of S is retrieved as follows:

The points of S are the planes in C.

The lines of S are the points of Q.

The planes of S are the planes in C’.

The fact that the geometries of S and Q are isomorphic can be explained by the isomorphism of the Dynkin diagrams A3 and D3.

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in (D + 1)-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates x1, x2, ..., xD+1, the general quadric is defined by the algebraic equation[1]

\sum _{i,j=1}^{D+1}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{D+1}P_{i}x_{i}+R=0

which may be compactly written in vector and matrix notation as:

xQx^{\mathrm {T} }+Px^{\mathrm {T} }+R=0\,

where x = (x1, x2, ..., xD+1) is a row vector, xT is the transpose of x (a column vector), Q is a (D + 1) × (D + 1) matrix and P is a (D + 1)-dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be over real numbers or complex numbers, but a quadric may be defined over any ring. In general, the locus of zeros of a set of polynomials is known as an algebraic set, and is studied in the branch of algebraic geometry.

A quadric is thus an example of an algebraic set. For the projective theory see Quadric (projective geometry).

By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

-- "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ("octads") of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24.

Robert A. Wilson on R. T. Curtis:

"One of his most important contributions is surely the invention of the MOG (Miracle Octad Generator) for facilitating calculation in the Mathieu group M24. Before then, if you wanted to generate octads, you had to look up the list of all 759 of them in a paper of Todd. Afterwards, all the octads were drawn on a postcard (not quite a postage-stamp!) and calculation became immeasurably easier. The MOG was an essential ingredient in the constructions of J4 and the Monster, and remains an indispensable tool for working in many of the sporadic groups."

-- "248 and All That," talk at the R. T. Curtis 60th birthday conference, Sept. 21, 2007

The MOG:

The image “MOGCurtisPlain.gif” cannot be displayed, because it contains errors.

Shown above is a rearranged version of the

Miracle Octad Generator (MOG) of R. T. Curtis

("A new combinatorial approach to M24,"

Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)

The MOG is a pairing of the 35 partitions of an 8-set into two 4-sets with the 35 partitions of AG(4,2) (the affine 4-space over GF(2)) into 4 affine planes. The pairing preserves certain incidence properties. It is used in studying the Steiner system S(5, 8, 24), the large Mathieu group, the extended binary Golay code, the Leech lattice, and subgroups of the Monster.

Note: The above "pairing" of two 35-member sets is Curtis's original approach to the MOG. Many sources now offer an alternate definition-- the MOG as a rectangular array of 4 rows and 6 columns-- based on Conway's hexacode. See "Competing Definitions of the Miracle Octad Generator."

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