Friday, August 14, 2026

Thursday, August 6, 2026

Wednesday, January 14, 2026

Today Is World Logic Day 2026

Filed under: General — Tags: Art Space, Cullinane AI, Group 384 — m759 @ 12:34 pm

"The date chosen to celebrate World Logic Day, 14 January, corresponds to
the date of death of Kurt Gödel and the date of birth of Alfred Tarski,.
two of the most prominent logicians of the twentieth century." — Wikipedia

Some "presentatonal symbols" — The Gray Lotus.

Diamond Theory
by NotebookLM

94 sources

The Cullinane diamond theorem establishes a profound link between visual 4×4 patterns and advanced mathematical structures such as finite geometry and group theory. By analyzing symmetry-preserving transformations within a group of 322,560 permutations, the research connects simple grid designs to the complex projective space PG(3,2). These geometric frameworks further extend into coding theory, illustrating how "diamond" rings are isomorphic to matrix algebras over the finite field GF(4). The sources also highlight the Miracle Octad Generator (MOG) and its relationship to the Mathieu group M24, which is essential for understanding error-correcting codes and sphere packings. Beyond pure mathematics, the text explores "brick space" and the Klein correspondence, suggesting that these abstract symmetries represent a "universal" order found in both art and nature. Ultimately, this body of work bridges the gap between ancient motifs and modern computational science through the elegant study of structural invariants.

How does the diamond theorem bridge visual tile patterns and abstract group theory?

In what ways does PG(3,2) serve as a foundation for Diamond Theory?

How do symmetry and automorphism groups link 4×4 arrays to the Mathieu group?

__________________________________________________________________

Thursday, November 20, 2025

How are algebraic, combinatorial, and geometric structures unified within the Brick Space model?

The unification of algebraic, combinatorial, and geometric structures within the Brick Space model is achieved through its role as a graphic, isomorphic representation of key finite geometries, particularly the projective spaces PG(5,2) and PG(3,2), and their fundamental relationship via the Klein correspondence.

————————————————————————————.

The term "Brick Space" originates from the

2 \times 4

arrays, or "bricks," coined by R. T. Curtis as component parts of his Miracle Octad Generator (MOG)

. These physical or graphic arrangements serve as a concrete visualization (or model) of highly abstract mathematical structures.

1. Geometric Unification

The Brick Space model provides a geometric foundation for combinatorial objects through direct visualization of finite geometries defined over the two-element field, GF(2)

• Modeling Projective Space: The Brick Space model is the natural geometric setting for the MOG's components, which may be regarded as forming both the 5-dimensional projective space

PG (5, 2)

and the 3-dimensional projective space

PG (3, 2)

• Visualizing PG(5,2) and the Klein Quadric: The model visually represents the

PG (5, 2)

space, which has 63 points, by partitioning these points into 28 points lying off the Klein quadric and 35 points lying on the Klein quadric

. The geometry of $PG (5, 2)$ is linked to the concept of the 8-set

• The Klein Correspondence: The geometric relationship between

PG (3, 2)

and

PG (5, 2)

is formalized by the Klein correspondence

. The $PG (5, 2)$ model, developed as a brick space, is analogous to the "line diagrams" construction of $PG (3, 2)$ found in the Cullinane diamond theorem

• Lines and Points: The Brick Space implicitly models how the 35 lines of

PG (3, 2)

correspond to the 35 points on the Klein quadric in

PG (5, 2)

2. Combinatorial Unification

Combinatorial structures, primarily partitions of sets and block designs, are mapped directly onto geometric entities within the Brick Space framework

• Partitions and Lines: The central combinatorial equivalence involves mapping partitions of sets to geometric objects

. The 35 combinatorial structures arising from permutations of $4 \times 4$ diamond tiles (related to the $2 \times 4$ bricks) are isomorphic to the 35 lines of $PG (3, 2)$ . These 35 structures also mirror the 35 square patterns within the original MOG

• Partitions of the 8-Set: The

PG (3, 2)

lines correspond to the 35 partitions of an 8-set into two 4-sets

. This same framework connects to the 105 partitions of an 8-set into four 2-sets, which are essentially the same as the 105 lines contained in the Klein quadric

• Latin Squares: The combinatorial property of orthogonality of Latin squares (specifically, order-4 Latin squares) is unified geometrically by corresponding exactly to the property of skewness of lines (non-intersecting lines) in

PG (3, 2)

. This provides a geometric dictionary for abstract algebraic combinatorics

3. Algebraic Unification

Algebraic structures, rooted in finite field theory and group theory, govern the behavior and symmetries observed in the combinatorial and geometric models

• Binary Field

GF (2)

: The foundation of the geometry is the Galois field of two elements,

GF (2)

(binary arithmetic:

1 + 1 = 0

)

. The algebraic rules of $GF (2)$ dictate the structure of the affine and projective spaces represented by the bricks

• Binary Addition (XOR): The line diagrams used to formalize the

4 \times 4

patterns (related to the bricks) can be "added" via component-wise binary addition (XOR)

. The closure property of lines in $PG (3, 2)$ is represented algebraically by the fact that the three line diagrams corresponding to a pattern always sum to zero ( $D 1 \oplus D 2 \oplus D 3 = 0$ )

• Group Theory and Symmetry: The massive permutation group

G

(of order 322,560) that transforms the visual patterns is isomorphic to the affine group

AGL (4, 2)

. This group, which explains the pervasive symmetry found in all $G$ -images, is the automorphism group of the underlying finite geometry, establishing the algebraic control over the system's geometric and combinatorial properties

• Ring Theory: The

4 \times 4

patterns can be endowed with algebraic structure to generate an ideal within a ring of symmetric patterns, leading to an infinite family of "diamond rings" isomorphic to rings of matrices over

GF (4)

. This links the geometry to the machinery of abstract algebra

• M24 Connection: The affine group

AGL (4, 2)

is identified as the octad stabilizer subgroup of the Mathieu group

M_{24}

, linking the visual patterns and the underlying geometries to the abstract world of sporadic simple groups, coding theory (Golay code), and lattice theory (Leech lattice)

The Brick Space model, therefore, functions as a visual tapestry where graphic arrangements (geometry/visualization) encode set partitions (combinatorics) that behave according to the rules of binary algebra (

GF (2)

and affine groups)

. The entire structure is unified by powerful isomorphisms relating these three mathematical domains through finite geometries like $PG (3, 2)$ and $PG (5, 2)$

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Thursday, November 20, 2025

On Brick Space

Filed under: General — Tags: Brick Space, Cullinane AI, Group 384 — m759 @ 6:24 pm

A question suggested by NotebookLM in the "Diamond Theory" notebook,
and its answer today by NotebookLM —

How are algebraic, combinatorial, and geometric structures unified within the Brick Space model?

The term "Brick Space" originates from the

2 \times 4

arrays, or "bricks," coined by R. T. Curtis as component parts of his Miracle Octad Generator (MOG)

. These physical or graphic arrangements serve as a concrete visualization (or model) of highly abstract mathematical structures

1. Geometric Unification

The Brick Space model provides a geometric foundation for combinatorial objects through direct visualization of finite geometries defined over the two-element field, GF(2)

• Modeling Projective Space: The Brick Space model is the natural geometric setting for the MOG's components, which may be regarded as forming both the 5-dimensional projective space

PG (5, 2)

and the 3-dimensional projective space

PG (3, 2)

• Visualizing PG(5,2) and the Klein Quadric: The model visually represents the

PG (5, 2)

space, which has 63 points, by partitioning these points into 28 points lying off the Klein quadric and 35 points lying on the Klein quadric

. The geometry of $PG (5, 2)$ is linked to the concept of the 8-set

• The Klein Correspondence: The geometric relationship between

PG (3, 2)

and

PG (5, 2)

is formalized by the Klein correspondence

. The $PG (5, 2)$ model, developed as a brick space, is analogous to the "line diagrams" construction of $PG (3, 2)$ found in the Cullinane diamond theorem

• Lines and Points: The Brick Space implicitly models how the 35 lines of

PG (3, 2)

correspond to the 35 points on the Klein quadric in

PG (5, 2)

2. Combinatorial Unification

Combinatorial structures, primarily partitions of sets and block designs, are mapped directly onto geometric entities within the Brick Space framework

• Partitions and Lines: The central combinatorial equivalence involves mapping partitions of sets to geometric objects

• Partitions of the 8-Set: The

PG (3, 2)

lines correspond to the 35 partitions of an 8-set into two 4-sets

. This same framework connects to the 105 partitions of an 8-set into four 2-sets, which are essentially the same as the 105 lines contained in the Klein quadric

PG (3, 2)

. This provides a geometric dictionary for abstract algebraic combinatorics

3. Algebraic Unification

Algebraic structures, rooted in finite field theory and group theory, govern the behavior and symmetries observed in the combinatorial and geometric models

• Binary Field

GF (2)

: The foundation of the geometry is the Galois field of two elements,

GF (2)

(binary arithmetic:

1 + 1 = 0

)

. The algebraic rules of $GF (2)$ dictate the structure of the affine and projective spaces represented by the bricks

• Binary Addition (XOR): The line diagrams used to formalize the

4 \times 4

patterns (related to the bricks) can be "added" via component-wise binary addition (XOR)

. The closure property of lines in $PG (3, 2)$ is represented algebraically by the fact that the three line diagrams corresponding to a pattern always sum to zero ( $D 1 \oplus D 2 \oplus D 3 = 0$ )

• Group Theory and Symmetry: The massive permutation group

G

(of order 322,560) that transforms the visual patterns is isomorphic to the affine group

AGL (4, 2)

• Ring Theory: The

4 \times 4

patterns can be endowed with algebraic structure to generate an ideal within a ring of symmetric patterns, leading to an infinite family of "diamond rings" isomorphic to rings of matrices over

GF (4)

. This links the geometry to the machinery of abstract algebra

• M24 Connection: The affine group

AGL (4, 2)

is identified as the octad stabilizer subgroup of the Mathieu group

M_{24}

, linking the visual patterns and the underlying geometries to the abstract world of sporadic simple groups, coding theory (Golay code), and lattice theory (Leech lattice)

GF (2)

and affine groups)

. The entire structure is unified by powerful isomorphisms relating these three mathematical domains through finite geometries like $PG (3, 2)$ and $PG (5, 2)$ .

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Monday, January 23, 2023

Saturday, May 12, 2018

Friday, February 14, 2014

Saturday, January 4, 2014

Tuesday, December 24, 2013

Friday, April 20, 2012

Complex Reflection

Filed under: General — Tags: Autist Artist, Group 384 — m759 @ 11:32 am

Yesterday's post in memory of Octavio Paz—

… the free-standing, two-sided “Life-Death Figure,”
carved from stone in Mexico some time between
A.D. 900 and 1250, has multiple personalities.

— Holland Cotter, New York Times

An earlier post yesterday, Fashion Notes, linked to a Sting video—

IMAGE- Sting meets his own reflection in a mirror in 'We'll Be Together' video

From "Loo Ree," by Zenna Henderson

"It's so hard to explain–"

"Oh, foof!" I cried defiantly, taking off my glasses and, smearing the tears across both lenses with a tattered Kleenex. "So I'm a dope, a moron! If I can explain protective coloration to my six-year-olds and the interdependence of man and animals, you can tell me something of what the score is!" I scrubbed the back of my hand across my blurry eyes. "If you have to, start out 'Once upon a time."' I sat down– hard.

Loo Ree smiled and sat down, too. "Don't cry, teacher. Teachers aren't supposed to have tears."

"I know it," I sniffed. "A little less than human-that's us."

"A little more than human, sometimes." Loo Ree corrected gently. "Well then, you must understand that I'll have to simplify. You will have to dress the bare bones of the explanation according to your capabilities.

"Once upon a time there was a classroom. Oh, cosmic in size, but so like yours that you would smile in recognition if you could see it all. And somewhere in the classroom something was wrong. Not the whispering and murmuring– that's usual. Not the pinching and poking and tattling that goes on until you get so you don't even hear it." I nodded. How well I knew.

"It wasn't even the sudden blow across the aisle or the unexpected wrestling match in the back of the room. That happens often, too. But something else was wrong. It was an undercurrent, a stealthy, sly sort of thing that has to be caught early or it disrupts the whole classroom and tarnishes the children with a darkness that will never quite rub off.

"The teacher could feel it –as all good teachers can– and she spoke to the principal. He, being a good principal, immediately saw the urgency of the matter and also saw that it was beyond him, so he called in an Expert." "You?" I asked, feeling quite bright because I had followed the analogy so far.

Loo Ree smiled. "Well, I'm part of the Expert."

"If you have to, start out 'Once upon a time.'"

Yesterday's Paz post was at 6:48 PM EDT.

For the autistic, here is some related mathematics.

Yesterday's Fashion Notes post was at 1:06 PM EDT.

A related chronological note from Rolling Stone yesterday—

"Levon Helm, singer and drummer for the Band,
died on April 19th in New York of throat cancer.
He was 71.

"He passed away peacefully at 1:30 this afternoon…."

Helm and The Band performing "The Weight"—

"I pulled into Nazareth, I was a-feelin' 'bout half past dead…"

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Wednesday, April 11, 2012

Friday, November 18, 2011

Thursday, September 23, 2010

Sunday, August 3, 2008

Sunday August 3, 2008

Filed under: General,Geometry — Tags: Affine Cube, Cubehenge, EGBDF, Geometric Theology, Geometry, Group 384, Kidshapes, Symmetric Generation — m759 @ 3:00 pm

Kindergarten
Geometry

Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein's birthday):

The play's title, "Every Good Boy Deserves Favour," is a mnemonic for the notes of the treble clef EGBDF.

The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.

The “u” in favour is the British way, the Stoppard way, "EGBDF" being "a Play for Actors and Orchestra" by Tom Stoppard (words) and André Previn (music).

And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.

When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard "to write something which had the need of a live full-time orchestra onstage," the 36-year-old playwright jumped at the chance.

One hitch: Stoppard at the time knew "very little about 'serious' music… My qualifications for writing about an orchestra," he says in his introduction to the 1978 Grove Press edition of "EGBDF," "amounted to a spell as a triangle player in a kindergarten percussion band."

— Jerry Tallmer in The Villager, March 12-18, 2008

Review of the same play as presented at Chautauqua Institution on July 24, 2008:

"Stoppard's modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience."

— Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008

"The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul."

— Dan Fogelberg

"He's watching us all the time."

— Lucia Joyce

Finnegans Wake,
Book II, Episode 2, pp. 296-297:

I'll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you'd wheeze whyse Salmonson set his seel on a hexengown.¹ Hissss!, Arrah, go on! Fin for fun!

¹ The chape of Doña Speranza of the Nacion.

Log 24, Sept. 3, 2003:

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, a novel by Michael Kruger, in The 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.

FinnegansWiki:

Salmonson set his seel:

"Finn MacCool ate the Salmon of Knowledge."

Wikipedia:

"George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest."

Related material:

Kindergarten Theology,

Kindergarten Relativity,

Arrangements for
56 Triangles.

For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.

Happy birthday,
Martin Sheen.

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Tuesday, October 31, 2006

Tuesday October 31, 2006

Filed under: General,Geometry — Tags: Galois's Space, Group 384, Halloween 2006, Wrinkle Tesseract — m759 @ 11:00 pm

To Announce a Faith

From 7/07, an art review from The New York Times:

Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College

"The show has an endgame, end-time mood….

I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.

Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

— Roberta Smith

It is doubtful that Smith
would consider the
following "found" art an
example of originality.

It nevertheless does
"announce a faith."

The image “http://www.log24.com/log/pix06A/061031-PAlottery2.jpg” cannot be displayed, because it contains errors.

"First for yourself"

Today's mid-day
Pennsylvania number:
707

See Log24 on 7/07
and the above review.

"Second, perhaps,
for the rest of us"

Today's evening
Pennsylvania number:
384

This number is an
example of what the
reviewer calls "compression"–

"an artistic focus that condenses
experiences, ideas and feelings
into something
whole, committed
and visually comprehensible."

"Experiences"

See (for instance)

Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."

"Ideas"

See Plato.

"Feelings"

See A Wrinkle in Time.

"Whole"

The automorphisms
of the tesseract
form a group
of order 384.

"Committed"

See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order.

"Visually comprehensible"

See "Diamond Theory in 1937,"
an excerpt from which
is shown below.

The image “http://www.log24.com/theory/images/Carmichael440abbrev.gif” cannot be displayed, because it contains errors.

The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:

"There is such a thing
as a tesseract."

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Friday, August 14, 2026

Thursday, August 6, 2026

Wednesday, January 14, 2026

Diamond Theory by NotebookLM

Thursday, November 20, 2025

Monday, January 23, 2023

Saturday, May 12, 2018

Friday, February 14, 2014

Saturday, January 4, 2014

Tuesday, December 24, 2013

Friday, April 20, 2012

Wednesday, April 11, 2012

Friday, November 18, 2011

Thursday, September 23, 2010

Sunday, August 3, 2008

Tuesday, October 31, 2006

Diamond Theory
by NotebookLM