4x4x4 “i ching” « Search Results

Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

Filed under: General — Tags: Aitchison, Master Plan, Octad, Octad Generator, The Crimson Possibilities — m759 @ 11:00 am

The title refers to Calabi-Yau spaces.

T. S. Eliot —

Four Quartets

. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.

A less "cosmic" but still noteworthy code — The Golay code.

This resides in a 12-dimensional space over GF(2).

Related material from Plato and R. T. Curtis —

A related Calabi-Yau "Chinese jar" first described in detail in 1905 —

A figure that may or may not be related to the 4x4x4 cube that
holds the classical Chinese "cosmic code" — the I Ching —

ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf

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Monday, March 11, 2019

Ant-Man Meets Doctor Strange

Filed under: General — m759 @ 1:22 pm

From 'A Wrinkle in Time,' by Madeleine L'Engle

The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .

"Think outside the tesseract."

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Saturday, February 18, 2017

Verbum

Filed under: General,Geometry — Tags: Cubehenge, Cullinane Cube, Solomon's Cube, Würfelspiel — m759 @ 3:00 am

The Log24 version (Nov. 9, 2005, and later posts) —

VERBUM
SAT
SAPIENTI

Escher's 'Verbum'

Escher's Verbum

Solomon's Cube

Geometry of the I Ching

The Warner Brothers version —

The Paramount version —

See also related material in the previous post, Transformers.

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Sunday, January 8, 2017

A Theory of Everything

Filed under: General,Geometry — Tags: Conceptualist Minimalism, Eightfold Cube — m759 @ 7:11 pm

The title refers to the Chinese book the I Ching ,
the Classic of Changes .

The 64 hexagrams of the I Ching may be arranged
naturally in a 4x4x4 cube. The natural form of transformations
("changes") of this cube is given by the diamond theorem.

A related post —

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Thursday, June 13, 2013

Gate

Filed under: General,Geometry — Tags: Cube School, Field Dream, Field Theology, Vanishing Point — m759 @ 2:13 pm

"Eight is a Gate." — Mnemonic rhyme

Today's previous post, Window, showed a version
of the Chinese character for "field"—

This suggests a related image—

The related image in turn suggests…

Unlike linear perspective, axonometry has no vanishing point,
and hence it does not assume a fixed position by the viewer.
This makes axonometry 'scrollable'. Art historians often speak of
the 'moving' or 'shifting' perspective in Chinese paintings.

Axonometry was introduced to Europe in the 17th century by
Jesuits returning from China.

— Jan Krikke

As was the I Ching. A related structure:

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Thursday, September 27, 2012

Kummer and the Cube

Filed under: General,Geometry — Tags: Cube Mine, Kummer-Sept-2012, Kummerhenge — m759 @ 7:11 pm

Denote the d-dimensional hypercube by γ_d .

"… after coloring the sixty-four vertices of γ₆
alternately red and blue, we can say that
the sixteen pairs of opposite red vertices represent
the sixteen nodes of Kummer's surface, while
the sixteen pairs of opposite blue vertices
represent the sixteen tropes."

— From "Kummer's 16₆," section 12 of Coxeter's 1950
"Self-dual Configurations and Regular Graphs"

Just as the 4×4 square represents the 4-dimensional
hypercube γ₄over the two-element Galois field GF(2),
so the 4x4x4 cube represents the 6-dimensional
hypercube γ₆ over GF(2).

For religious interpretations, see
Nanavira Thera (Indian) and
I Ching geometry (Chinese).

See also two professors in The New York Times
discussing images of the sacred in an op-ed piece
dated Sept. 26 (Yom Kippur).

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Monday, September 7, 2009

Monday September 7, 2009

Filed under: General,Geometry — Tags: Cube School, date-090907, Four Color — m759 @ 12:00 am

Magic Boxes

"Somehow it seems to fill my head with ideas– only I don't exactly know what they are!…. Let's have a look at the garden first!"

— A passage from Lewis Carroll's Through the Looking-Glass. The "garden" part– but not the "ideas" part– was quoted by Jacques Derrida in Dissemination in the epigraph to Chapter 7, "The Time before First."

Commentary
on the passage:

Part I "The Magic Box," shown on Turner Classic Movies earlier tonight

Part II: "Mimsy Were the Borogoves," a classic science fiction story:

"… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."

Part III: A Crystal Block —

Four coloring pencils, of four different colors

Image of pencils is by
Diane Robertson Design.

Related material:
"A Four-Color Theorem."

Part IV:

"Click on the Yellow Book."

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Saturday, September 5, 2009

Saturday September 5, 2009

Filed under: General,Geometry — Tags: Cullinane Cube, Kaplan Boo, Solomon's Cube — m759 @ 10:31 pm

For the
Burning Man

'The Stars My Destination,' current edition (with cover slightly changed)

(Cover slightly changed.)

Background —

SAT

Part I:

Sophists (August 20th)

Part II:

VERBUM
SAT
SAPIENTI

Escher's 'Verbum'

Escher's Verbum

Solomon's Cube

Geometry of the I Ching

Part III:

From August 25th —

"Boo, boo, boo,
square root of two."

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Monday, October 6, 2008

Monday October 6, 2008

Filed under: General,Geometry — Tags: Eternal Color — m759 @ 1:26 pm

Leap Day of Faith

Yesterday's entry contained the following unattributed quotation:

"One must join forces with friends of like mind."

As the link to Leap Day indicated, the source of the quotation is the I Ching.

Yesterday's entry also quoted the late Terence McKenna, a confused writer on psychosis and the I Ching. Lest the reader conclude that I consider McKenna or similar authors (for instance, Timothy Leary in Cuernavaca) as "friends of like mind," I would point rather to more sober students of the I Ching (cf. my June 2002 notes on philosophy, religion, and science) and to the late Scottish theologian John Macquarrie:

The Rev. John Macquarrie, Scottish Theologian, Dies at 87

Macquarrie's connection in this journal to the I Ching is, like that book itself, purely coincidental. For details, click on the figure below.

The persistent reader will
find a further link that
leads to an entry titled
"Notes on the I Ching."

McKenna's writing was of value to me for its (garbled) reference to a thought of Alfred North Whitehead:

"A colour is eternal. It haunts time like a spirit. It comes and it goes. But where it comes it is the same colour. It neither survives nor does it live. It appears when it is wanted."

— Science and the Modern World, 1925

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Monday, June 16, 2008

Monday June 16, 2008

Filed under: General,Geometry — Tags: Darkinbad the Brightdayler, Games Theory, Round Square, Square Round — m759 @ 9:00 pm

Bloomsday for Nash:

The Revelation Game

(American Mathematical Society Feb. 2008
review of Steven Brams’s Superior Beings:
If They Exist, How Would We Know?)
(pdf, 15 megabytes)

"Brams does not attempt to prove or disprove God. He uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….

In the 'Revelation Game,' for example, the Person (P) has two options:
1) P can believe in SB's existence
2) P can not believe in SB's existence
The Supreme Being also has two options:
1) SB can reveal Himself
2) SB can not reveal Himself

Each player also has a primary and secondary goal. For the Person, the primary goal is to have his belief (or non-belief) confirmed by evidence (or lack thereof). The secondary goal is to 'prefer to believe in SB’s existence.' For the Supreme Being, the primary goal is to have P believe in His existence, while the secondary goal is to not reveal Himself. These goals allow us to rank all the outcomes for each player from best (4) to worst (1). We end up with a matrix as follows (the first number in the parentheses represents the SB's ranking for that box; the second number represents P's ranking):

Revelation Game payoff matrix

The question we must answer is: what is the Nash equilibrium in this case?"

Analogously:

Lotteries on
Bloomsday,
June 16,
2008

Pennsylvania
(No revelation)

New York
(Revelation)

Mid-day
(No belief)

418

The Exorcist

No belief,
no revelation

064

4x4x4 cube summarizing geometry of the I Ching

Revelation
without belief

Evening
(Belief)

709

Human Conflict Number Five album by The 10,000 Maniacs

Belief without
revelation

198

(A Cheap
Epiphany)

Belief and
revelation

The holy image

denoting belief and revelation
may be interpreted as
a black hole or as a
symbol by James Joyce:

When?

Going to dark bed there was a square round Sinbad the Sailor roc's auk's egg in the night of the bed of all the auks of the rocs of Darkinbad the Brightdayler.

Where?

— Ulysses, conclusion of Chapter 17

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Monday, May 28, 2007

Monday May 28, 2007

Filed under: General,Geometry — Tags: Beadgame Space, Geometry, Glasperlenspiel, Octad, Octad Generator — m759 @ 5:00 pm

Space-Time

and a Finite Model

Notes by Steven H. Cullinane
May 28, 2007

Part I: A Model of Space-Time

The following paper includes a figure illustrating Penrose’s model of “complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space.”

The image “http://www.log24.com/log/pix07/070528-Twistor.jpg” cannot be displayed, because it contains errors.

Click on picture to enlarge.

For some background on the Klein quadric and space-time, see Roger Penrose, “On the Origins of Twistor Theory,” from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.

Part II: A Corresponding Finite Model

The Klein quadric also occurs in a finite model of projective 5-space. See a 1910 paper:

G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.

Conwell discusses the quadric, and the related Klein correspondence, in detail. This is noted in a more recent paper by Philippe Cara:

The image “http://www.log24.com/log/pix07/070528-Quadric.jpg” cannot be displayed, because it contains errors.

As Cara goes on to explain, the Klein correspondence underlies Conwell’s discussion of eight heptads. These play an important role in another correspondence, illustrated in the Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M₂₄.

Related material:

The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.

The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.

There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube. This correspondence leads to a natural way to generate the affine group AGL(6,2). This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.

See

Finite Geometry of
the Square and Cube

A 4x4x4 cube

and

Geometry of the I Ching.

“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game. Elder Brother laughed. ‘Go ahead and try,’ he exclaimed. ‘You’ll see how it turns out. Anyone can create a pretty little bamboo garden in the world. But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”

— Hermann Hesse, The Glass Bead Game,

translated by Richard and Clara Winston

Comments (2)

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