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Sunday, February 9, 2025

An AI Overview of Klein Quadric and MOG

Filed under: General — Tags: 64 Art, Brick Space, Klein Correspondence — m759 @ 2:15 pm

See also a more accurate AI report from January 9, 2025 —

Grok 2 on the MOG and the Klein Correspondence:

HTML version, with corrections, of the above 9 January Grok 2 report —

Grok 2: Klein Correspondence and MOG, 9 Jan. 2025 . . .
______________________________________________

The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics.

Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between ~~points and lines in a projective space~~.* Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2).

Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system.

Relation Between Klein Correspondence and MOG:

Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to [the sets of] 4 parallel affine planes [that represent lines] within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework.
Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries.

Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems.

* Correction: Should be "a correspondence between points in a five-dimensional projective space and lines in a three-dimensional projective space."

Update of ca. 9 AM ET Monday, Feb. 10, 2024 —

Neither AI report above mentions the Cullinane model of the five-
dimensional projective space PG(5,2) as a brick space — a space
whose points are the 2×4 bricks used in thte MOG. This is
understandable, as the notion of using bricks to model both PG(5,2)
and PG(3,2) has appeared so far only in this journal. See an
illustration from New Year's Eve . . . Dec. 31, 2024 —

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

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Tuesday, January 21, 2025

The Cullinane Diamond Theorem
and the Miracle Octad Generator

Filed under: General — Tags: Art Space, Crypto — m759 @ 7:19 am

The Cullinane Diamond Theorem
and the Miracle Octad Generator

Document created on Jan. 21. 2025, by Google’s Gemini Advanced 1.5 Pro with Deep Research, in response to the following prompt:

“Research how the Cullinane diamond theorem and
the Miracle Octad Generator are related.”

The Cullinane diamond theorem and the Miracle Octad Generator (MOG) are two seemingly disparate mathematical concepts that find a surprising connection through the realm of finite projective geometry. This report delves into the relationship between these two concepts, exploring their definitions, properties, and historical context to illuminate their interconnectedness.

Cullinane Diamond Theorem

The Cullinane diamond theorem, developed by Steven H. Cullinane, is a mathematical concept that explores the symmetrical properties of certain geometric patterns. It focuses on a 4×4 square pattern with alternating colors arranged in a diamond shape, referred to as the "diamond figure".⁴ The theorem states that every permutation of the 16 tiles within this diamond figure, generated by mixing rows, columns, and quadrants, will result in a pattern with some form of ordinary or color-interchange symmetry.⁵ This "invariance of symmetry" is a remarkable property that highlights the inherent order within seemingly random arrangements.³

The theorem has deep roots in group theory, with the group of permutations (G) being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).⁶ This group has an order of 322,560 and plays a crucial role in understanding the symmetry of both the diamond-theorem figures and the square patterns of the MOG.⁵ A key result used in the proof of the theorem states that every 4-coloring (i.e., every map into a 4-set) can be expressed as a sum of three 2-colorings.¹

Interestingly, the Cullinane diamond theorem can be extended to higher dimensions. For instance, extending the action of A to a 4x4x4 array yields a way of generating the 1.3 trillion transformations natural to the 64 hexagrams of the I Ching, an ancient Chinese divination text.¹ This connection suggests potential applications of the theorem in diverse fields beyond geometry.

Another interesting concept that arises from the Cullinane diamond theorem is that of "diamond rings." These rings are algebraic structures generated by the G-images of the diamond figure, and they are isomorphic to rings of matrices over GF(4).⁵ This algebraic formulation provides a deeper understanding of the symmetry properties explored by the theorem.

Miracle Octad Generator

The Miracle Octad Generator (MOG), conceived by R.T. Curtis, is a mathematical tool used to study the symmetries and properties of the Mathieu group M₂₄, a sporadic simple group of significant importance in group theory and coding theory.⁷ It is a 4×6 array of combinations that can describe any point in 24-dimensional space.⁷ More precisely, each element in the array represents a combination of coordinates or symbols that contribute to defining a point in this 24-dimensional space.

Properties

The MOG preserves all the symmetries and maximal subgroups of M₂₄, including the monad, duad, triad, octad, octern, sextet, trio, and duum groups.⁷ It can be used to visualize partitions of the 24 points, which is important for characterizing these maximal subgroups of M₂₄.⁸

One of the key applications of the MOG lies in its ability to quickly verify codewords of the binary Golay code, a highly efficient error-correcting code used in various communication systems.⁷ Each element in the MOG can store a '1' or a '0', and by analyzing the counts (number of '1's) in each column and the top row, one can determine if a set of 24 coordinates forms a valid codeword.⁷ The MOG achieves this by projecting the 24 Golay code onto a code called the hexacode.⁸

The MOG is also closely related to the concept of octads and sextets. Any 5 elements of the MOG lie in a unique octad, which is a set of 8 elements.⁹ A sextet is a set of 6 tetrads (sets of 4 elements) where the union of any two tetrads forms an octad.⁹ Every tetrad lies in a unique sextet.

Furthermore, 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.¹⁰ This pairing preserves specific incidence properties, which refer to the relationships between points, lines, and planes in a geometric space.

It's worth noting that there are two competing definitions of the MOG. Curtis originally defined it as a pairing of two 35-member sets.¹¹ However, many sources now define it as a rectangular array based on Conway's hexacode.¹¹

Finally, the MOG helps in understanding the octad stabilizer, a subgroup of M₂₄ that leaves an octad invariant as a set.¹⁰ This subgroup is isomorphic to the automorphism group of the affine 4-space over GF(2), highlighting the deep connection between the MOG and finite geometry.

Relationship Between the Cullinane Diamond Theorem and the Miracle Octad Generator

The relationship between the Cullinane diamond theorem and the MOG emerges from their shared foundation in finite projective geometry. The 35 square patterns within the MOG have an underlying structure based on finite projective geometry.⁵ These patterns, composed of two-color diagonally-divided square tiles, exhibit surprising symmetry properties that can be explained using the Cullinane diamond theorem.⁵

A crucial link between the Cullinane diamond theorem and the MOG is the fact that the same symmetry group, of order 322,560, underlies both concepts.⁵ This shared symmetry group highlights a deep connection between the seemingly different patterns studied by these two concepts.

The line diagrams used to visualize the Cullinane diamond theorem also play a crucial role in understanding the symmetry of the square patterns in the MOG.⁵ These line diagrams, based on the three partitions of the four-set of square tiles, correspond to the 35 lines in the 3-dimensional projective space over GF(2).⁵ In essence, the Cullinane diamond theorem provides a way to understand and visualize the symmetry properties inherent in the MOG.

Furthermore, the underlying geometry of the 4×4 patterns in the Cullinane diamond theorem is closely related to the MOG, which is used in the construction of the Steiner system S(5,8,24).⁵ This connection extends to the Leech lattice, a highly symmetrical mathematical structure in 24 dimensions, which Walter Feit described as a "blown up version of S(5,8,24)".⁵ The Leech lattice is a dense sphere-packing in 24 dimensions with remarkable symmetry properties and connections to various areas of mathematics and physics.

Interestingly, the Cullinane diamond theorem also sheds light on the relationship between orthogonality of Latin squares and skewness of lines in a finite projective 3-space.¹² Latin squares are square arrays filled with symbols, and orthogonality between two Latin squares means that when they are superimposed, each possible pair of symbols appears exactly once. Skewness of lines in projective geometry refers to lines that do not intersect. The Cullinane diamond theorem helps establish a connection between these seemingly unrelated concepts.

Another interesting connection is to Beutelspacher's model of the 15 points of PG(3,2).¹ This model provides a way to visualize the points of this projective space, and it relates to the Cullinane diamond theorem and the MOG through their shared foundation in finite projective geometry.

Applications

The relationship between the Cullinane diamond theorem and the MOG has potential applications in various fields, including coding theory and cryptography. The MOG's ability to verify codewords of the binary Golay code is particularly relevant in coding theory, where efficient error-correcting codes are essential for reliable data transmission.⁷ The Cullinane diamond theorem, with its focus on symmetry and permutations, could potentially contribute to the development of new coding schemes or cryptographic algorithms. For example, the theorem's insights into the structure of finite projective spaces could be used to design codes with specific properties or to analyze the security of existing cryptographic systems.

Beyond coding theory, the Cullinane diamond theorem has applications in visualizing various mathematical concepts. For instance, it provides a way to visualize Walsh functions, which are a set of orthogonal functions used in signal processing and other areas of mathematics.¹ The theorem also helps visualize the finite projective plane PG(2,4), a fundamental object in finite geometry with connections to various combinatorial structures.¹² Additionally, the theorem has connections to the outer automorphisms of S₆, the symmetric group on 6 letters, which are transformations that preserve the group structure but are not inner automorphisms.¹²

Visualizations

Visualizations play a crucial role in understanding the connection between the Cullinane diamond theorem and the MOG. The 35 square patterns within the original (1976) MOG, as defined by R. T. Curtis, provide a visual representation of this connection.⁵ These patterns, along with the line diagrams used to illustrate the Cullinane diamond theorem, offer a tangible way to grasp the interplay between these concepts.⁵

Here are some examples of visualizations that illustrate this connection:

The diamond figure: This is a 4×4 square pattern with alternating colors arranged in a diamond shape. It is the basic object studied by the Cullinane diamond theorem.²
The 35 square patterns in the MOG: These patterns, composed of two-color diagonally-divided square tiles, exhibit surprising symmetry properties that can be explained using the Cullinane diamond theorem.⁵
Line diagrams: These diagrams, based on the three partitions of the four-set of square tiles, correspond to the 35 lines in the 3-dimensional projective space over GF(2). They help visualize and explain the symmetry of both the diamond-theorem figures and the MOG patterns.²

Historical Connections

Historically, the development of the Cullinane diamond theorem and the MOG stemmed from explorations in finite projective geometry and group theory.⁵ While the exact historical connections between their development remain unclear, both concepts emerged from a desire to understand the symmetries and properties of mathematical structures in finite spaces.

Conclusion

The Cullinane diamond theorem and the Miracle Octad Generator, while seemingly distinct, are intricately linked through the principles of finite projective geometry. The theorem's focus on symmetry invariance and the MOG's ability to capture the symmetries of the Mathieu group M₂₄ highlight their shared foundation. This connection extends to higher-dimensional structures like the Leech lattice and has potential applications in coding theory, cryptography, and the visualization of various mathematical concepts.

The relationship between these two concepts exemplifies the interconnectedness of mathematical ideas. Often, seemingly unrelated areas of mathematics converge in unexpected ways, revealing hidden symmetries and relationships. The Cullinane diamond theorem and the MOG provide a compelling example of this phenomenon, demonstrating the power of finite projective geometry in unifying and illuminating diverse mathematical structures.

Works cited

1. Diamond Theory: Symmetry in Binary Spaces, accessed January 21, 2025, https://m759.tripod.com/theory/dtheory.html

2. The Diamond Theorem in Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/16/dtheorem.html

3. finitegeometry.org, accessed January 21, 2025, http://finitegeometry.org/sc/gen/dtcas.html#:~:text=Cullinane,have%20some%20sort%20of%20symmetry.

4. Speak, Memory « Log24 – Home page of m759.net, a domain used by Steven H. Cullinane for a WordPress weblog., accessed January 21, 2025, http://m759.net/wordpress/?p=112809

5. Cullinane diamond theorem – Encyclopedia of Mathematics, accessed January 21, 2025, https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem

6. What is the Cullinane diamond theorem? – Log24, accessed January 21, 2025, http://log24.com/log24/240303-You.com-Cullinane_Diamond_Theorem-research-paper.pdf

7. Miracle Octad Generator – Wikipedia, accessed January 21, 2025, https://en.wikipedia.org/wiki/Miracle_Octad_Generator

8. Mathieu groups, the Golay code and Curtis' Miracle Octad Generator, accessed January 21, 2025, https://vrs.amsi.org.au/wp-content/uploads/sites/6/2014/09/UWA-Kelly.pdf

9. MOG (Miracle Octad Generator) – Stanford, accessed January 21, 2025, http://xenon.stanford.edu/~hwatheod/mog/mog.html

10. The Miracle Octad Generator (MOG) of R T. Curtis – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/24/MOG.html

11. Competing Definitions of the Miracle Octad Generator – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/24/mogdefs.html

12. Diamond Theory: Symmetry in Binary Spaces – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/gen/dth/DiamondTheory.html

13. arxiv.org, accessed January 21, 2025, https://arxiv.org/abs/1308.1075

View this post as a standalone web page at

http://log24.com/log25/DTandMOG.html.

and as a PDF at

http://log24.com/log25/DTandMOG.pdf.

For a more elementary introduction to the MOG, see a YouTube video,

"The Most Powerful Diagram in Mathematics."

For a PDF of the video's metadata and comments, click here.

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Thursday, January 9, 2025

Grok 2 on the MOG and the Klein Correspondence

Filed under: General — Tags: Brick Space, Klein Correspondence, Tour — m759 @ 1:04 pm

Related illustration —

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

— Transcription —

Grok 2 on Klein correspondence and MOG — 9 Jan. 2025
______________________________________________________________

Prompt:

How is the Klein correspondence related to the Miracle Octad Generator?

Grok 2 response (with citations and links removed):

Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between points and lines in a projective space. Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2).

Relation Between Klein Correspondence and MOG:

Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to 4 parallel affine planes within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework.

Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries.

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Wednesday, October 23, 2024

The Delta Transform

Filed under: General — Tags: Aion Art, Art Space, Combinatorial Space Structures, Space Trip — m759 @ 7:04 am

Rothko — "… the elimination of all obstacles between the painter and
the idea, and between the idea and the observer."

Walker Percy has similarly discussed elimination of obstacles between
the speaker and the word, and between the word and the hearer.

Click images to enlarge.

Related mathematics —

The source: http://finitegeometry.org/sc/gen/typednotes.html.

A document from the above image —

AN INVARIANCE OF SYMMETRY

BY STEVEN H. CULLINANE

We present a simple, surprising, and beautiful combinatorial
invariance of geometric symmetry, in an algebraic setting.

DEFINITION. A delta transform of a square array over a 4-set is
any pattern obtained from the array by a 1-to-1 substitution of the
four diagonally-divided two-color unit squares for the 4-set elements.

THEOREM. Every delta transform of the Klein group table has
ordinary or color-interchange symmetry, and remains symmetric under
the group G of 322,560 transformations generated by combining
permutations of rows and colums with permutations of quadrants.

PROOF (Sketch). The Klein group is the additive group of GF (4);
this suggests we regard the group's table T as a matrix over that
field. So regarded, T is a linear combination of three (0,1)-matrices
that indicate the locations, in T, of the 2-subsets of field elements.
The structural symmetry of these matrices accounts for the symmetry
of the delta transforms of T, and is invariant under G.

All delta transforms of the 4⁵ matrices in the algebra generated by
the images of T under G are symmetric; there are many such algebras.

THEOREM. If 1 m ≤ n²+2, there is an algebra of 4^m
2n x 2n matrices over GF(4) with all delta transforms symmetric.

An induction proof constructs sets of basis matrices that yield
the desired symmetry and ensure closure under multiplication.

REFERENCE

S. H. Cullinane, Diamond theory (preprint).

Update of 1:12 AM ET on Friday, Oct. 25, 2024 —

The above "invariance of symmetry" document was written in 1978
for submission to the "Research Announcements" section of the
Bulletin of the American Mathematical Society . This pro forma
submission was, of course, rejected. Though written before
I learned of similar underlying structures in the 1974 work of
R. T. Curtis on his "Miracle Octad Generator," it is not without
relevance to his work.

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Thursday, December 21, 2023

Pi.ai

Filed under: General — Tags: Chandelier, Octad — m759 @ 10:57 am

Thursday, December 21, 2023

Hey there, great to meet you. I’m Pi, your personal AI. My goal is to be useful, friendly and fun. Ask me for advice, for answers, or let’s talk about whatever’s on your mind.

How's your day going?

Going OK so far, Pi. How's yours?

My day is going swimmingly! 🌊 Thanks for asking 😊 Anything in particular on your mind today?

What is the connection of "heptads" with the "MOG" of R. T. Curtis?

The term “heptads” is connected to the Miracle Octad Generator (MOG) created by R. T. Curtis. Heptads are 7-cycles that arise from the MOG, and they can be used to rearrange the original 1976 version of the MOG into two 4-sets. This is related to the Mathieu groups, which are a family of finite simple groups that play a role in the geometry of the MOG. Are you familiar with the Mathieu groups?

Just the large one. I am interested in how familiar YOU are with these concepts. Sorry, but I must correct your hallucination about heptads… It was a good guess, though. Can you look up the work of G. M. Conwell on PG(3,2)? Heptads are discussed there.

Yes, it seems that G. M. Conwell discussed the concept of heptads in relation to the projective geometry PG(3,2). Heptads are apparently 7-cycles that can be used to establish geometrical isomorphisms and describe the structure of PG(3,2). Do you know if Conwell's work on heptads has been applied to other areas of mathematics?

It has not, as far as I know. Actually, Pi, heptads are not 7-cycles. What made you think that they are?

My apologies! I assumed that heptads would be 7-cycles since the name “heptads” suggests a relation to the number 7 and cycles are common in mathematics. It sounds like I was off the mark there. If heptads are not 7-cycles, can you give me a better idea of what they are?

Conwell can. See his 1910 paper for further details. All for now, Pi… Have a nice day!

Got it — thanks for the tip on Conwell's paper. I'll take a look for more information there! It was great chatting with you. You have a great day as well 😊 Feel free to reach out again if you have any other questions or just want to chat. See ya!

Scholium —

Related material for "swimmingly" and "the number 7" — Snark Hunt.

Sia!

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Friday, September 22, 2023

Figurate Space

Filed under: General — Tags: Brick Space, Figurate Geometry, Octad — m759 @ 11:01 am

For the purpose of defining figurate geometry , a figurate space might be
loosely described as any space consisting of finitely many congruent figures —
subsets of Euclidean space such as points, line segments, squares,
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them.

Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate space, considered as a collection of figures — vertices, edges,
faces — seen in the nineteenth century as acted upon by a group of symmetries .

More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M₂₄ . That group may be viewed as acting
on various subsets of a 24-set… for instance, the 759 octads that are analogous
to the faces of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.

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Monday, July 18, 2022

“The Shape of Ideas”

Filed under: General — Tags: Modernism Manifesto, The Shape of Ideas, Yale Art Notes — m759 @ 3:01 am

Some may prefer other concepts of shape. For instance …

… and, from Log24 on the above Yalebooks date —

Related material:

From "Higman- Sims Graph," a webpage by A. E. Brouwer —

"Similar to the 15+35 construction of the Hoffman-Singleton graph is the 30+70 construction of the Higman-Sims graph. In the former the starting point was that the lines of PG(3,2) can be labeled with the triples in a 7-set such that lines meet when the corresponding triples have 1 element in common. This time we label the lines of PG(3,2) with the 4+4 splits of an 8-set, where intersecting lines correspond to splits with common refinement 2+2+2+2. Clearly, both descriptions of the lines of PG(3,2) are isomorphic. Take as vertices of the Higman-Sims graph the 15 points and 15 planes of PG(3,2) together with the 70 4-subsets of an 8-set. Join two 4-sets when they have 1 element in common. A 4-set determines a 4+4 split and hence a line in PG(3,2), and is adjacent to the points and planes incident with that line. A plane is adjacent to the nonincident points. This yields the Higman-Sims graph."

See also PG(3,2) in this journal.

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Monday, February 7, 2022

Morphart Meets Morph Art

Filed under: General — Tags: Deep Space, TIME piece — m759 @ 1:51 pm

Warren (PA) Public Library's Instagram
on January 21, 2022 —

Morphart —

Morph Art — from Raiders of the Lost Coordinates

"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.

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Wednesday, February 17, 2021

Raiders of the Lost Coordinates . . .

Filed under: General — Tags: Calamine, Four Dots, Law Play, Ministry of Culture, Prescott Street, Tetrahedron vs. Square — m759 @ 8:13 pm

Continues.

From other posts tagged Tetrahedron vs. Square —

"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.

Illustration (central detail a from the above tetrahedral figure) —

A Harvard Variation
from Timothy Leary —

The topics of Harvard and Leary suggest some other cultural
history, from The Coasters — "Poison Ivy" and "Yakety Yak."

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Tuesday, October 15, 2019

Inside the Fire Temple

Filed under: General — Tags: Fire Temple, Gnostic Effects — m759 @ 8:00 pm

(The title refers to Log24 posts now tagged Fire Temple.)

In memory of a New Yorker cartoonist who
reportedly died at 97 on October 3, 2019 …

"Read something that means something."
— New Yorker advertising slogan

From posts tagged Tetrahedron vs. Square —

This journal on October 3 —

"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.

Illustration (central detail a from the above tetrahedral figure) —

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Thursday, October 3, 2019

The Overbye Metaphors

Filed under: General — Tags: Quark Rock — m759 @ 12:00 pm

(For Harlan Kane)

"Once Mr. Overbye identifies a story, he said, the work is
in putting it in terms people can understand. 'Metaphors
are very important to the way I write,' he said. The results
are vivid descriptions that surpass mere translation."

— Raillan Brooks in The New York Times on a Times
science writer, October 17, 2017. Also on that date —

"There is such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.

See as well The Black List (Log24, September 27).

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Wednesday, September 18, 2019

The Perpetual Identity Crisis

Filed under: General — Tags: Quark Rock — m759 @ 8:56 am

"There is such a thing as a 4-set." — Saying adapted
from a 1962 young-adult novel.

Midrash — An image posted here on August 6 —

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Friday, May 31, 2019

Working Sketch of Aitchison’s Mathieu Cuboctahedron

Filed under: General — Tags: Aitchison, Permutahedron, Symmetric Generation — m759 @ 5:33 am

Cuboctahedron with its 24 edges labeled by the 24 permutations of a 4-set. By Cullinane on 5/31/19.

The above sketch indicates one way to apply the elements of S₄
to the Aitchison cuboctahedron . It is a rough sketch illustrating a
correspondence between four edge-hexagons and four label-sets.
The labeling is not as neat as that of a permutahedron by S₄
shown below, but can perhaps be improved.

Permutahedron labeled by S₄ .

Update of 9 PM EDT June 1, 2019 —

. . . And then of course there is the obvious labeling derived from
the above permutahedron —

[addtoany]

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Thursday, March 24, 2016

The Nervous Set*

Filed under: General — Tags: four-set — m759 @ 12:00 pm

The previous post suggests a review of the saying
"There is such a thing as a 4-set."

* Title of a 1959 musical

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Thursday, October 22, 2015

Objective Quality

Filed under: General,Geometry — Tags: four-set — m759 @ 2:26 am

Software writer Richard P. Gabriel describes some work of design
philosopher Christopher Alexander in the 1960's at Harvard:

An image related to correspondences discussed elsewhere in Log24

A more interesting account of these 35 structures:

"It is commonly known that there is a bijection between
the 35 unordered triples of a 7-set [i.e., the 35 partitions
of an 8-set into two 4-sets] and the 35 lines of PG(3,2)
such that lines intersect if and only if the corresponding
triples have exactly one element in common."
— "Generalized Polygons and Semipartial Geometries,"
by F. De Clerck, J. A. Thas, and H. Van Maldeghem,
April 1996 minicourse, example 5 on page 6.

For some context, see Eightfold Geometry by Steven H. Cullinane.

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Sunday, January 11, 2015

Real Beyond Artifice

Filed under: General,Geometry — Tags: Albright, four-set, Gitterkrieg, Natural Diagram, Points Omega, Seventh Seal — m759 @ 7:20 pm

A professor at Harvard has written about
"the urge to seize and display something
real beyond artifice."

He reportedly died on January 3, 2015.

An image from this journal on that date:

Another Gitterkrieg image:

The 24-set Ω of R. T. Curtis

Click on the images for related material.

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Friday, January 9, 2015

Fourth Right

Filed under: General — Tags: four-set — m759 @ 1:00 am

In memory of Rod Taylor, who
reportedly died at 84 on Wednesday,
the seventh day of 2015 —

And there is such a thing as a 4-set.

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Thursday, May 8, 2014

Wrinkles in Time

Filed under: General — Tags: four-set — m759 @ 2:25 pm

Rivka Galchen, in a piece mentioned here in June 2010—

On Borges: Imagining the Unwritten Book

"Think of it this way: there is a vast unwritten book that the heart reacts to, that it races and skips in response to, that it believes in. But it’s the heart’s belief in that vast unwritten book that brought the book into existence; what appears to be exclusively a response (the heart responding to the book) is, in fact, also a conjuring (the heart inventing the book to which it so desperately wishes to respond)."

Related fictions

Galchen's "The Region of Unlikeness" (New Yorker , March 24, 2008)

Ted Chiang's "Story of Your Life." A film adaptation is to star Amy Adams.

… and non-fiction

"There is such a thing as a 4-set." — January 31, 2012

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Friday, November 1, 2013

Cameron’s Group Theory Notes

Filed under: General,Geometry — Tags: four-set, Not Theology — m759 @ 7:00 am

In "Notes on Finite Group Theory"
by Peter J. Cameron (October 2013),
http://www.maths.qmul.ac.uk/~pjc/notes/gt.pdf,
some parts are particularly related to the mathematics of
the 4×4 square (viewable in various ways as four quartets)—

Definition 1.3.1, Group actions, and example on partitions of a 4-set, p. 19.
Exercise 1.1, The group of Fano-plane symmetries, p. 35.
Exercise 2.17, The group of the empty set and the 15 two-subsets of a six-set, p. 66.
Section 3.1.2, The holomorph of a group, p. 70.
Exercise 3.7, The groups A₈ and AGL(4,2), p. 78.

Cameron is the author of Parallelisms of Complete Designs ,
a book notable in part for its chapter epigraphs from T.S. Eliot's
Four Quartets . These epigraphs, if not the text proper, seem
appropriate for All Saints' Day.

But note also Log24 posts tagged Not Theology.

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Saturday, April 6, 2013

Pascal via Curtis

Filed under: General,Geometry — Tags: four-set, Geometry, Octad, Octad Generator — m759 @ 9:17 am

Click image for some background.

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 8-subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirty-five 3-subsets of a 7-set.

Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum of Pascal.

On Danzer's 35₄ Configuration:

IMAGE- Branko Grünbaum on Danzer's configuration

"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3-sets and all 4-sets that can be formed
by the elements of a 7-element set; each 'point' is represented
by one of the 3-sets, and it is incident with those lines
(represented by 4-sets) that contain the 3-set."

— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)

"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."

— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013

For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see

Classical Geometry in Light of Galois Geometry.

Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).

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Friday, December 21, 2012

Analogies*

Filed under: General,Geometry — Tags: four-set, Six-Set — m759 @ 4:30 pm

The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8-set into two 4-sets and the 35 lines in the
finite projective space PG(3,2).

Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2-subsets of a 6-set and the 15 points of PG(3,2).

An analogy between the two above analogies
is supplied by the exceptional outer automorphism of S₆.
See…

The 2-subsets of a 6-set are the points of a PG(3,2),
Picturing outer automorphisms of S₆, and
A linear complex related to M₂₄.

(Background: Inscapes, Inscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3-space.)

* For some context, see Analogies and
"Smallest Perfect Universe" in this journal.

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Saturday, July 7, 2012

Étude

Filed under: General,Geometry — Tags: Figurate Geometry, four-set — m759 @ 9:00 am

IMAGE- Google Books ad for 'Geometric Etudes in Combinatorial Mathematics,' by Alexander Soifer

IMAGE- Triangle cut into four congruent subtriangles

For remarks related by logic, see the square-triangle theorem.

For remarks related by synchronicity, see Log24 on
the above publication date, June 15, 2010.

According to Google (and Soifer's page xix), Soifer wants to captivate
young readers.

Whether young readers should be captivated is open to question.

"There is such a thing as a 4-set."

Update of 9:48 the same morning—

Amazon.com says Soifer's book was published not on June 15, but on
June 29 , 2010
(St. Peter's Day).

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Tuesday, January 31, 2012

Tesseract

Filed under: General,Geometry — Tags: Arbitration, four-set — m759 @ 12:21 pm

"… a finite set with n elements
is sometimes called an n-set …."

Tesseract formed from a 4-set—

The same 16 subsets or points can
be arranged in a 4×4 array that has,
when the array's opposite edges are
joined together, the same adjacencies
as those of the above tesseract.

"There is such a thing as a 4-set."
— Saying adapted from a novel

Update of August 12, 2012:

Figures like the above, with adjacent vertices differing in only one coordinate,
appear in a 1950 paper of H. S. M. Coxeter—

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Tuesday, January 10, 2012

Defining Form

Filed under: General,Geometry — Tags: Defining Form, four-set, Geometric Theology — m759 @ 9:00 am

(Continued from Epiphany and from yesterday.)

Detail from the current American Mathematical Society homepage—

Further detail, with a comparison to Dürer’s magic square—

http://www.log24.com/log/pix12/120110-Donmoyer-Still-Life-Detail.jpg

http://www.log24.com/log/pix12/120110-DurerSquare.jpg

The three interpenetrating planes in the foreground of Donmoyer‘s picture
provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer’s array into four 4-sets corresponding to the
four rows of Dürer’s square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—

Thus the Donmoyer array also enjoys the structural symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem’s interpenetrating lines explain the structure
of a 4×4 square , the foreground’s interpenetrating planes explain the structure
of a 2x2x2 cube .

For an application to theology, recall that interpenetration is a technical term
in that field, and see the following post from last year—

Saturday, June 25, 2011

Theology for Antichristmas

— m759 @ 12:00 PM

Hypostasis (philosophy)

“… the formula ‘Three Hypostases in one Ousia ‘
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion….” —Wikipedia

Ousia

Click for further details:

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Saturday, August 6, 2011

Correspondences

Filed under: General,Geometry — Tags: Four Color, four-set, Geometry, Natural Diagram, Octad, Octad Generator, Springer — m759 @ 2:00 pm

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, “Correspondances ”

From “A Four-Color Theorem”—

Figure 1

Note that this illustrates a natural correspondence
between

(A) the seven highly symmetrical four-colorings
of the 4×2 array at the left of Fig. 1, and

(B) the seven points of the smallest
projective plane at the right of Fig. 1.

To see the correspondence, add, in binary
fashion, the pairs of projective points from the
“points” section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)

A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—

Figure 2

Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful. It yields, as shown, all of the 35 partitions of an 8-element set (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.

For some applications of the Curtis MOG, see
(for instance) Griess’s Twelve Sporadic Groups .

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Monday, July 5, 2010

Window

Filed under: General — Tags: four-set, Galois Window, The Four-Point Plane — m759 @ 9:00 am

“Examples are the stained-glass
windows of knowledge.” — Nabokov

Related material:

Thomas Wolfe and the
Kernel of Eternity

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Sunday, July 4, 2010

Brightness at Noon (continued)

Filed under: General,Geometry — Tags: Octad, Octad Generator — m759 @ 12:00 pm

Today's sermon mentioned the phrase "Omega number."

Other sorts of Omega numbers— 24 and 759— occur
in connection with the set named Ω by R. T. Curtis in 1976—

Image-- In a 1976 paper, R.T. Curtis names the 24-set of his Miracle Octad Generator 'Omega.'

— R. T. Curtis, "A New Combinatorial Approach to M₂₄,"
Math. Proc. Camb. Phil. Soc. (1976), 79, 25-42

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Wednesday, April 28, 2010

Eightfold Geometry

Filed under: General,Geometry — Tags: Folklore, four-set, Geometry, Natural Diagram, Octad, Octad Generator — m759 @ 11:07 am

Image-- Analysis of structure of the 35 partitions of an 8-set into two 4-sets

Image-- Miracle Octad Generator of R.T. Curtis

Related web pages:

Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square

Related folklore:

"It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common." –"Generalized Polygons and Semipartial Geometries," by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6

The Miracle Octad Generator may be regarded as illustrating the folklore.

Update of August 20, 2010–

For facts rather than folklore about the above bijection, see The Moore Correspondence.

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Monday, April 28, 2008

Monday April 28, 2008

Filed under: General,Geometry — Tags: 2x4, Brick Space, four-set, Klein Correspondence — m759 @ 7:00 am

Religious Art

The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.

Black monolith, proportions 4x9

One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.

The following
figure does
allow such
an epiphany.

A 2x4 array of squares

One approach to
the epiphany:

"Transformations play
a major role in
modern mathematics."
– A biography of
Felix Christian Klein

The above 2×4 array
(2 columns, 4 rows)
furnishes an example of
a transformation acting
on the parts of
an organized whole:

The 35 partitions of an 8-set into two 4-sets

For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose Space-Time, and
a Finite Model."

For a related structure–
not rectangle but cube–
see Epiphany 2008.

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Wednesday, March 21, 2007

Wednesday March 21, 2007

Filed under: General,Geometry — Tags: four-set — m759 @ 3:18 pm

Finite Relativity
continued

This afternoon I added a paragraph to The Geometry of Logic that makes it, in a way, a sequel to the webpage Finite Relativity:

"As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be interpreted as naming the 16 subsets of a 4-set; in this case the partial ordering in the lattice is the structure preserved by the lattice's group of 24 automorphisms– the same automorphism group as that of the 16 Boolean connectives. If, however, these 16 digital labels are interpreted as naming the 16 functions from a 4-set to a 2-set (of two truth values, of two colors, of two finite-field elements, and so forth), it is not obvious that the notion of partial order is relevant. For such a set of 16 functions, the relevant group of automorphisms may be the affine group of A mentioned above. One might argue that each Venn diagram in Fig. 3 constitutes such a function– specifically, a mapping of four nonoverlapping regions within a rectangle to a set of two colors– and that the diagrams, considered simply as a set of two-color mappings, have an automorphism group of order larger than 24… in fact, of order 322,560. Whether such a group can be regarded as forming part of a 'geometry of logic' is open to debate."

The epigraph to "Finite Relativity" is:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

The added paragraph seems to fit this description.

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