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Friday, May 27, 2005

Friday May 27, 2005

Filed under: General,Geometry — m759 @ 12:25 pm

Drama of the Diagonal,
Part Deux

Wednesday’s entry The Turning discussed a work by Roger Cooke. Cooke presents a

“fanciful story (based on Plato’s dialogue Meno).”

The History of Mathematics is the title of the Cooke book.

Associated Press thought for today:

“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
— Henry Kissinger (whose birthday is today)

For Henry Kissinger on his birthday:
a link to Geometry for Jews.

This link suggests a search for material
on the art of Sol LeWitt, which leads to
an article by Barry Cipra,
The “Sol LeWitt” Puzzle:
A Problem in 16 Squares (ps),
a discussion of a 4×4 array
of square linear designs.
Cipra says that

“If you like, there are three symmetry groups lurking within the LeWitt puzzle: the rotation/reflection group of order 8, a toroidal group of order 16, and an ‘existential’* group of order 16. The first group is the most obvious. The third, once you see it, is also obvious.”

* Jean-Paul Sartre,
Being and Nothingness,
Philosophical Library, 1956
[reference by Cipra]

For another famous group lurking near, if not within, a 4×4 array, click on Kissinger’s birthday link above.

Kissinger’s remark (above) on analogy suggests the following analogy to the previous entry’s (Drama of the Diagonal) figure:

The image “http://www.log24.com/log/pix05/021126-diagonH2.jpg” cannot be displayed, because it contains errors.

Logos Alogos II:
Horizon

This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–

From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:

1.	on Page 51:
	“He makes himself known to himself from the other side of the world and he looks from the horizon toward himself to recover his inner being. Man is ‘a being of distances.'”
2.	on Page 154:
	“… impossible, for the for-itself attained by the realization of the Possible will make itself be as for-itself–that is, with another horizon of possibilities. Hence the constant disappointment which accompanies repletion, the famous: ‘Is it only this?’….”
3.	on Page 155:
	“… end of the desires. But the possible repletion appears as a non-positional correlate of the non-thetic self-consciousness on the horizon of the glass-in-the-midst-of-the-world.”
4.	on Page 158:
	“… it is in time that my possibilities appear on the horizon of the world which they make mine. If, then, human reality is itself apprehended as temporal….”
5.	on Page 180:
	“… else time is an illusion and chronology disguises a strictly logical order of deducibility. If the future is pre-outlined on the horizon of the world, this can be only by a being which is its own future; that is, which is to come….”
6.	on Page 186:
	“… It appears on the horizon to announce to me what I am from the standpoint of what I shall be.”
7.	on Page 332:
	“… the boat or the yacht to be overtaken, and the entire world (spectators, performance, etc.) which is profiled on the horizon. It is on the common ground of this co-existence that the abrupt revelation of my ‘being-unto-death’….”
8.	on Page 359:
	“… eyes as objects which manifest the look. The Other can not even be the object aimed at emptily at the horizon of my being for the Other.”
9.	on Page 392:
	“… defending and against which he was leaning as against a wail, suddenly opens fan-wise and becomes the foreground, the welcoming horizon toward which he is fleeing for refuge.”
10.	on Page 502:
	“… desires her in so far as this sleep appears on the ground of consciousness. Consciousness therefore remains always at the horizon of the desired body; it makes the meaning and the unity of the body.”

11.	on Page 506:
	“… itself body in order to appropriate the Other’s body apprehended as an organic totality in situation with consciousness on the horizon— what then is the meaning of desire?”
12.	on Page 661:
	“I was already outlining an interpretation of his reply; I transported myself already to the four corners of the horizon, ready to return from there to Pierre in order to understand him.”
13.	on Page 754:
	“Thus to the extent that I appear to myself as creating objects by the sole relation of appropriation, these objects are myself. The pen and the pipe, the clothing, the desk, the house– are myself. The totality of my possessions reflects the totality of my being. I am what I have. It is I myself which I touch in this cup, in this trinket. This mountain which I climb is myself to the extent that I conquer it; and when I am at its summit, which I have ‘achieved’ at the cost of this same effort, when I attain this magnificent view of the valley and the surrounding peaks, then I am the view; the panorama is myself dilated to the horizon, for it exists only through me, only for me.”

Illustration of the
last horizon remark:

From CIPRA – Slovenia,
the Institute for the
Protection of the Alps

For more on the horizon, being, and nothingness, see

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Sunday, November 21, 2004

Sunday November 21, 2004

Filed under: General — Tags: on041121 — m759 @ 3:00 pm

Trinity and Counterpoint

Today's Roman Catholic meditation is from Gerry Adams, leader of Sinn Fein, the political arm of the Irish Republican Army:

"I certainly regret what happened and I make no bones about that," Adams said on the 30th anniversary of pub bombings that killed 21 on Nov. 21, 1974, in Birmingham, England.

Those who care what Roman Catholics think of the Trinity may read the remarks of St. Bonaventure at math16.com.

That site also offers a less holy but more intelligible trinity based on the irrefutable fact that 3 x 8 = 24 and on a remarkable counterpoint between group actions on a 4×2 array and group actions on a 4×4 array.

For a Protestant view of this trinity, see a website at the University of Birmingham in England.

That site's home page links to Birmingham's City Evangelical Church.

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Tuesday, November 16, 2004

Tuesday November 16, 2004

Filed under: General,Geometry — m759 @ 12:12 pm

Geometry, continued

Added a long footnote on symplectic properties of the 4×4 array to “Geometry of the 4×4 Square.”

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Wednesday, November 12, 2003

Wednesday November 12, 2003

Filed under: General,Geometry — Tags: For Cameron Duncan — m759 @ 9:58 am

The Silver Table

“And suddenly all was changed. I saw a great assembly of gigantic forms all motionless, all in deepest silence, standing forever about a little silver table and looking upon it. And on the table there were little figures like chessmen who went to and fro doing this and that. And I knew that each chessman was the idolum or puppet representative of some one of the great presences that stood by. And the acts and motions of each chessman were a moving portrait, a mimicry or pantomine, which delineated the inmost nature of his giant master. And these chessmen are men and women as they appear to themselves and to one another in this world. And the silver table is Time. And those who stand and watch are the immortal souls of those same men and women. Then vertigo and terror seized me and, clutching at my Teacher, I said, ‘Is that the truth?….’ ”

— C.S. Lewis, The Great Divorce, final chapter

Follow-up to the previous four entries:

St. Art Carney, whom we may imagine to be a passenger on the heavenly bus in The Great Divorce, died on Sunday, Nov. 9, 2003.

The entry for that date (Weyl’s birthday) asks for the order of the automorphism group of a 4×4 array. For a generalization to an 8×8 array — i.e., a chessboard — see

Geometry of the I Ching.

Audrey Meadows, said to have been the youngest daughter of her family, was born in Wuchang, China.

Tui: The Youngest Daughter

“Tui means to ‘give joy.’ Tui leads the common folk and with joy they forget their toil and even their fear of death. She is sometimes also called a sorceress because of her association with the gathering yin energy of approaching winter. She is a symbol of the West and autumn, the place and time of death.”

— Paraphrase of Book III, Commentaries of Wilhelm/Baynes.

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Tuesday, November 11, 2003

Tuesday November 11, 2003

Filed under: General,Geometry — Tags: For Cameron Duncan — m759 @ 11:11 am

11:11

“Why do we remember the past
but not the future?”

— Stephen Hawking,
A Brief History of Time,
Ch. 9, “The Arrow of Time”

For another look at
the arrow of time, see

Time Fold.

Imaginary Time: The Concept

The flow of imaginary time is at right angles to that of ordinary time. “Imaginary time is a relatively simple concept that is rather difficult to visualize or conceptualize. In essence, it is another direction of time moving at right angles to ordinary time. In the image at right, the light gray lines represent ordinary time flowing from left to right – past to future. The dark gray lines depict imaginary time, moving at right angles to ordinary time.”

Is Time Quantized?

Yes.

Maybe.

We don’t really know.

Let us suppose, for the sake of argument, that time is in fact quantized and two-dimensional. Then the following picture,

from Time Fold, of “four quartets” time, of use in the study of poetry and myth, might, in fact, be of use also in theoretical physics.

In this event, last Sunday’s entry, on the symmetry group of a generic 4×4 array, might also have some physical significance.

At any rate, the Hawking quotation above suggests the following remarks from T. S. Eliot’s own brief history of time, Four Quartets:

“It seems, as one becomes older,
That the past has another pattern,
and ceases to be a mere sequence….

I sometimes wonder if that is
what Krishna meant—
Among other things—or one way
of putting the same thing:
That the future is a faded song,
a Royal Rose or a lavender spray
Of wistful regret for those who are
not yet here to regret,
Pressed between yellow leaves
of a book that has never been opened.
And the way up is the way down,
the way forward is the way back.”

Related reading:

The Wisdom of Old Age and

Poetry, Language, Thought.

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Sunday, November 9, 2003

Sunday November 9, 2003

Filed under: General,Geometry — Tags: Kummerhenge — m759 @ 5:00 pm

For Hermann Weyl's Birthday:

A Structure-Endowed Entity

"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way."

— Hermann Weyl in Symmetry

Exercise: Apply Weyl's lesson to the following "structure-endowed entity."

4x4 array of dots

What is the order of the resulting group of automorphisms? (The answer will, of course, depend on which aspects of the array's structure you choose to examine. It could be in the hundreds, or in the hundreds of thousands.)

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Wednesday, September 3, 2003

Wednesday September 3, 2003

Filed under: General,Geometry — Tags: Cubehenge, Kummerhenge, Six-Set, Symmetric Generation, The Nutshell — m759 @ 3:00 pm

Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994

Last year's entry on this date:

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

The picture above is of the complete graph K₆… Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M₂₄.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K₆ graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K₆, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See

The Jewel of Arithmetic and

The Golden Theorem.

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Sunday, August 17, 2003

Sunday August 17, 2003

Filed under: General,Geometry — Tags: Cube Coloring, Cube School — m759 @ 6:21 pm

Diamond theory is the theory of affine groups over GF(2) acting on small square and cubic arrays. In the simplest case, the symmetric group of degree 4 acts on a two-colored diamond figure like that in Plato's Meno dialogue, yielding 24 distinct patterns, each of which has some ordinary or color-interchange symmetry .

This symmetry invariance can be generalized to (at least) a group of order approximately 1.3 trillion acting on a 4x4x4 array of cubes.

The theory has applications to finite geometry and to the construction of the large Witt design underlying the Mathieu group of degree 24.

Wednesday, July 30, 2003

Wednesday July 30, 2003

Filed under: General — m759 @ 2:45 am

Transcendental Meditation

This week’s
New Yorker:

Transcendental Man
New books on
Ralph Waldo Emerson
for his bicentennial.
by John Updike

This week’s
Time cover:

The bicentennial of Ralph Waldo Emerson was on May 25, 2003. For a commemoration of Emerson on that date, click on the picture below of Harvard University’s Room 305, Emerson Hall.

This will lead you to a discussion of the properties of a 5×5 array, or matrix, with a symbol of mystical unity at its center. Although this symbol of mystical unity, the number “1,” is not, pace the Shema, a transcendental number, the matrix is, as perhaps a sort of Emersonian compensation, what postmodernists would call phallologocentric. It is possible that Emerson is a saint; if so, his feast day (i.e., date of death), April 27, might reveal to us the sort of miraculous fact hoped for by Fritz Leiber in my previous entry. A check of my April 27 notes shows us, lo and behold, another phallologocentric 5×5 array, this one starring Warren Beatty. This rather peculiar coincidence is, perhaps, the sort of miracle appropriate to a saint who is, as this week’s politically correct New Yorker calls him, a Big Dead White Male.

Leiber’s fiction furnishes “a behind-the-scenes view of the time change wars.”

“It’s quarter to three…” — St. Frank Sinatra

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Monday, April 28, 2003

Monday April 28, 2003

Filed under: General,Geometry — Tags: 4403, Octad, Octad Generator — m759 @ 12:07 am

ART WARS:

Toward Eternity

April is Poetry Month, according to the Academy of American Poets. It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."

Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:

"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.

………………………
Since then–'tis Centuries–and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity– "

Consider the following journal entries from April 7, 2003:

Math Awareness Month

April is Math Awareness Month.
This year's theme is "mathematics and art."

An Offer He Couldn't Refuse

Today's birthday: Francis Ford Coppola is 64.

"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."

— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:

The "horse's head" figure above is from a note I wrote on this date 18 years ago. The following journal entry from April 4, 2003, gives some details:

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight. Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight. Consider eight rectangular cells arranged in an array of four rows and two columns. Let us label these cells with coordinates, then apply a permutation.

Decimal
labeling

Binary
labeling

Algebraic
labeling

Permutation
labeling

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight. This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact. It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M₂₄ (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions. See also "Pieces of Eight," by Robert L. Griess.

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Friday, April 4, 2003

Friday April 4, 2003

Filed under: General,Geometry — Tags: 4403, Octad, Octad Generator — m759 @ 3:33 pm

The Eight

Decimal
labeling

Binary
labeling

Algebraic
labeling

IMAGE- Knight figure for April 4
Permutation
labeling

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Friday, November 8, 2002

Friday November 8, 2002

Filed under: General,Geometry — m759 @ 3:33 am

Religious Symbolism
at Princeton

In memory of Steve McQueen (“The Great Escape” and “The Thomas Crown Affair”… see preceding entry) and of Rudolf Augstein (publisher of Der Spiegel), both of whom died on November 7 (in 1980 and 2002, respectively), in memory of the following residents of

The Princeton Cemetery
of the Nassau Presbyterian Church
Established 1757

SYLVIA BEACH (1887-1962), whose father was pastor of the First Presbyterian Church, founded Shakespeare & Company, a Paris bookshop which became a focus for struggling expatriate writers. In 1922 she published James Joyce’s Ulysses when others considered it obscene, and she defiantly closed her shop in 1941 in protest against the Nazi occupation.

KURT GÖDEL (1906-1978), a world-class mathematician famous for a vast array of major contributions to logic, was a longtime professor at the Institute for Advanced Study, founded in 1930. He was a corecipient of the Einstein Award in 1951.

JOHN (HENRY) O’HARA (1905-1970) was a voluminous and much-honored writer. His novels, Appointment in Samarra (1934) and Ten North Frederick (1955), and his collection of short stories, Pal Joey (1940), are among his best-known works.

and of the long and powerful association of Princeton University with the Presbyterian Church, as well as the theological perspective of Carl Jung in Man and His Symbols, I offer the following “windmill,” taken from the Presbyterian Creedal Standards website, as a memorial:

The background music Les Moulins de Mon Coeur, selected yesterday morning in memory of Steve McQueen, continues to be appropriate.

“A is for Anna.”
— James Joyce

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Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: Change Logos, Eightfold Cube, four-set, Map Systems, Octad, Octad Generator, Strange Change — m759 @ 10:13 pm

ABSTRACT: 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 4×4 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 2×2 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 4×4 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 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 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 4×4 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 a movable JavaScript version of these 4×4 patterns, see The Diamond 16 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.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry