Diamond Theorem « Search Results

Sunday, January 26, 2025

The Diamond Theorem and the Miracle Octad Generator —
The DeepSeek Version

Filed under: General — Tags: Art Space, DeepSeek — m759 @ 4:19 pm

See http://log24.com/log25/
DeepSeek-250126-Print-option-version-of-DTandMOG.pdf .

Conclusion

"The Diamond Theorem and the MOG exemplify how finite geometry bridges abstract algebra and combinatorics. Their relationship underscores the universality of symmetry in mathematics, from graphic designs to sporadic groups and error-correcting codes. By studying one, insights into the other — and into structures like the Leech lattice — naturally emerge."

— DeepSeek R1, Jan. 26, 2025.

That AI research report from today was suggested by
a VentureBeat article from yesterday —

For a Google Gemini Deep Research report on the same topic,
see a Log24 post from Tuesday, Jan. 21.

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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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Tuesday, October 29, 2024

Diamond Theorem Unit Cube Design

Filed under: General — Tags: DTUC — m759 @ 5:53 am

https://websim.ai/p/74wzs2lcd6qlpe8491rz/24

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Saturday, October 26, 2024

Diamond Theorem Studio

Filed under: General — Tags: Aion Art, Imago — m759 @ 7:36 pm

By Steven H. Cullinane and Websim.ai —

https://websim.ai/p/_dorlgddf3vcxh6qjysy/16 .

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Saturday, October 19, 2024

AI Perceptions of the Cullinane Diamond Theorem . . .
Continue to Evolve.

Filed under: General — Tags: AI Overviews — m759 @ 2:40 am

Update of 12:10 PM EDT October 19 — AI Overview from Google —

The Google AI Overview now includes an illustration and mentions a
new technique ("map systems") which may eventually be generalized.

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Monday, March 4, 2024

Report on the Diamond Theorem
by Google Gemini (Advanced)

Filed under: General — Tags: Chatbots on the Diamond Theorem, Octad — m759 @ 3:26 pm

Diamond Theorem Report

by Google Gemini AI

Monday, March 4, 2024

Can you write an academic paper, with source citations, about the Cullinane diamond theorem?

Absolutely! Here's an academic paper about the Cullinane diamond theorem. Please note that while I provide citations, in a fully formatted paper you'd need to adhere to your chosen academic style guide for proper formatting:

The Cullinane Diamond Theorem:
Exploring Symmetries in Graphic Design

Abstract

This paper examines the Cullinane diamond theorem, a principle in finite projective geometry that offers unexpected insights into the symmetries found in graphic designs. The theorem, initially formulated by Steven H. Cullinane, reveals that patterns built from simple geometric components, like diamonds or chevrons, possess inherent symmetry properties under specific transformation groups. We will discuss the mathematical foundations of the theorem, its relation to the Miracle Octad Generator, and its applications in design and the study of finite fields.

Introduction

Symmetry plays a vital role in art and design, contributing to a sense of balance, harmony, and aesthetic appeal. While often associated with classical forms, symmetry also manifests in surprising ways within seemingly simple graphic patterns. The Cullinane diamond theorem provides a mathematical framework to understand and predict these hidden symmetries.

Mathematical Background

Finite Projective Geometries: The Cullinane diamond theorem has its roots in finite projective geometry. A finite projective geometry PG(n, q) consists of a finite set of points and lines, where each line contains q + 1 points, and any two distinct points lie on a unique line ([1]).

Diamond Figure: Cullinane focuses on a specific pattern called the four-diamond figure (D). This figure is composed of 16 tiles arranged in a 4×4 array, where each tile is divided diagonally into two colors ([2]).

Transformation Group: The group G of transformations is formed by all possible permutations of rows, columns, and 2×2 quadrants of the diamond figure, along with color interchanges. The size of this group is 322,560 ([2]).

The Cullinane Diamond Theorem

Theorem:
Any image produced by applying transformations from the group G to the original diamond figure (D) will possess some form of ordinary or color-interchange symmetry ([2]).

Proof:
Cullinane's proof employs a decomposition technique for functions into a finite field. It demonstrates that the G-images of the diamond figure create an ideal within a ring of symmetric patterns. This ideal has 1024 patterns where any horizontal or vertical "cut" through the pattern remains uninterrupted ([2]).

Connections to the Miracle Octad Generator

The Cullinane diamond theorem shares a relationship with R.T. Curtis's Miracle Octad Generator (MOG). The MOG is a combinatorial structure within the study of sporadic simple groups with symmetry properties similar to those described by the theorem ([3]).

Applications

Graphic Design: The theorem informs artistic compositions employing diamond and chevron-like shapes. Designers can leverage the inherent symmetries to generate visually pleasing and intricate patterns.

Quilt Patterns: Many traditional quilt designs incorporate the geometries described by the theorem, leading to the emergence of symmetrical patterns, even when randomness might be involved in the selection of colors or tiles.

Finite Fields: The Cullinane diamond theorem establishes an isomorphism between specific "diamond" rings and rings of matrices over GF(4), offering insights into the structure of finite fields ([2]).

Conclusion

The Cullinane diamond theorem elegantly connects abstract mathematics with the world of graphic design. It unveils latent symmetries embedded within basic geometric shapes, providing a tool for understanding and creating visually balanced patterns. The theorem's connections to finite fields and combinatorial structures highlight its significance in broader mathematical research.

Citations

[1] Weisstein, E. W. “Finite Projective Geometry.” From MathWorld–A Wolfram Web Resource. [invalid URL removed]

[2] Cullinane, S. H. "Cullinane Diamond Theorem." Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem

[3] Curtis, R. T. (1976). A New Combinatorial Approach to M₂₄. Mathematical Proceedings of the Cambridge Philosophical Society, 79(1), 25–42. doi:10.1017/S0305004100052075

Let me know if you'd like me to expand on any particular aspect or provide additional citations!

Sources [I.e., the source of the Curtis citation]

1. www.hellenicaworld.com/Science/Mathematics/en/MathieuGroup.html

Gemini may display inaccurate info, including about people, so double-check it.

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Tuesday, January 30, 2024

Cullinane Diamond Theorem:
The Latest Google Bard Version

Filed under: General — Tags: Gemini.reviews, In Loco, Octad, Venice Beach Barbenheimer — m759 @ 7:00 am

Prompt:

What is the Cullinane diamond theorem?

Google Bard Response:

The Cullinane diamond theorem is a fascinating result in the realm of finite projective geometry and symmetry. It focuses on a specific pattern – the four-diamond figure – and reveals a surprising property about its transformations.

The Four-Diamond Figure:

Imagine a 4×4 grid filled with 16 small squares, half colored black and half white. Now, divide these squares diagonally into quarters, creating four diamonds with alternating black and white shades. This arrangement is the four-diamond figure.

_________________

[Clearly erroneous illustration with link to amazon.com
is omitted.]

_____________________

The Theorem:

The Cullinane diamond theorem states that no matter how you scramble this four-diamond figure by randomly shuffling rows, columns, and the four 2×2 quadrants, the resulting pattern will always exhibit some form of symmetry. This symmetry can come in two forms:

Ordinary symmetry: This is the familiar reflectional symmetry where a vertical or horizontal axis divides the pattern into mirrored halves.

Color-interchange symmetry: Here, the pattern has a diagonal axis where swapping the black and white colors results in the same image.

Intriguing Implications:

This statement might seem intuitive at first. After all, even with scrambling, some semblance of order should remain. However, the beauty of the Cullinane diamond theorem lies in its generality and the precise mathematical framework it provides for understanding these symmetries.

The theorem sheds light on the underlying structure of graphic patterns and connects them to concepts from finite projective geometry. This connection allows us to analyze and classify these patterns in a systematic way, revealing hidden connections and surprising symmetries.

Applications and Beyond:

The Cullinane diamond theorem has found applications in various areas, including:

Sporadic simple groups: The Miracle Octad Generator, used to study these rare and exceptional groups, relies on the symmetry properties outlined in the theorem.

Orthogonal Latin squares: The connection between orthogonal Latin squares and projective spreads can be understood through the lens of the Cullinane diamond theorem.

Walsh functions: The symmetry properties of Walsh functions, used in signal processing and coding theory, can be linked to the diamond patterns.

Understanding the Cullinane diamond theorem offers a deeper appreciation for the interplay between symmetry, geometry, and seemingly random patterns. It serves as a testament to the beauty and hidden order within the realm of mathematics.

Bard may display inaccurate info, including . . . .

[Exercise … Find any inaccurate information
in the Bard response above.]

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Monday, November 27, 2023

The Diamond Theorem According to Llama 2

Filed under: General — Tags: Chatbots, Chatbots on the Diamond Theorem — m759 @ 1:51 pm

The above is one of many wildly inaccurate responses on this topic
from chatbots. A chatbot combined with search, however —
such as Bing Chat with GPT-4 — can be both accurate and helpful.

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Saturday, November 11, 2023

The Diamond Theorem and Graphic Design

Filed under: General — Tags: Chatbots on the Diamond Theorem — m759 @ 1:59 pm

A "Cullinane Diamond Theorem" question suggested today by Bing Chat —

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Wednesday, November 8, 2023

Cullinane Diamond Theorem at
University of the Basque Country

Filed under: General — Tags: Basque Country, Shibumi — m759 @ 1:44 am

See also Shibumi Continues — June 29, 2022.

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Saturday, September 23, 2023

The Cullinane Diamond Theorem at Wikipedia

Filed under: General — Tags: Octad — m759 @ 8:48 am

This post was prompted by the recent removal of a reference to
the theorem on the Wikipedia "Diamond theorem" disambiguation
page. The reference, which has been there since 2015, was removed
because it linked to an external source (Encyclopedia of Mathematics)
instead of to a Wikipedia article.

For anyone who might be interested in creating a Wikipedia article on
my work, here are some facts that might be reformatted for that website . . .

https://en.wikipedia.org/wiki/
User:Cullinane/sandbox —

Cullinane diamond theorem

The theorem uses finite geometry to explain some symmetry properties of some simple graphic designs, like those found in quilts, that are constructed from chevrons or diamonds.

The theorem was first discovered by Steven H. Cullinane in 1975 and was published in 1977 in Computer Graphics and Art.

The theorem was also published as an abstract in 1979 in Notices of the American Mathematical Society.

The symmetry properties described by the theorem are related to those of the Miracle Octad Generator of R. T. Curtis.

The theorem is described in detail in the Encyclopedia of Mathematics article "Cullinane diamond theorem."

References

Steven H. Cullinane, "Diamond theory," Computer Graphics and Art, Vol. 2, No. 1, February 1977, pages 5-7.

_________, Abstract 79T-A37, "Symmetry invariance in a diamond ring," Notices of the American Mathematical Society, February 1979, pages A-193, 194.

_________, "Cullinane diamond theorem," Encyclopedia of Mathematics.

R. T. Curtis, A new combinatorial approach to M₂₄, Mathematical Proceedings of the Cambridge Philosophical Society, 1976, Vol. 79, Issue 1, pages 24-42.

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Sunday, August 27, 2023

“Diamond Theorem” Search Result

Filed under: General — m759 @ 1:17 am

Click to enlarge.

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Saturday, February 25, 2023

The Diamond Theorem according to ChatGPT

Filed under: General — Tags: Artificial Mathematics, Chatbots on the Diamond Theorem, ChatGPT — m759 @ 2:38 am

ChatGPT on the diamond theorem, Feb. 25, 2023.

The part about tilings, group actions, and the diamond-shaped
pattern is more or less OK. The parts about Thurston and
applications are utterly false.

Compare and contrast . . .

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Friday, April 29, 2022

The Diamond Theorem in Basque Country

Filed under: General — Tags: Basque Country, Truchet Tiles — m759 @ 5:57 pm

Translated by Google as . . .

The Truchet Tiles and the Diamond Puzzle and
The Art of the Simple Truchet Tile.

About the author:

Raúl Ibáñez is a professor in the Department of Mathematics
at the UPV/EHU and collaborator with the Chair of Scientific Culture.

About his school:

The University of the Basque Country
(Basque: Euskal Herriko Unibertsitatea, EHU ;
Spanish: Universidad del País Vasco, UPV ; UPV/EHU)
is a Spanish public university of the Basque Autonomous Community.
— Wikipedia

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Monday, April 19, 2021

Diamond Theorem at ScienceOpen

Filed under: General — Tags: Gutter Filter, Octad, Square Model — m759 @ 1:22 pm

Update on April 20, 2021 —
The following was added today to the above summary:

“It describes a group of 322,560 permutations, later known as
‘the octad group,’ that now plays a role in speculative high-energy physics.
See Moonshine, Superconformal Symmetry, and Quantum Error Correction .”

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Sunday, November 18, 2018

Diamond Theorem Symmetry

Filed under: General,Geometry — m759 @ 1:00 am

The title is a useful search phrase:

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Monday, December 11, 2017

The Diamond Theorem at SASTRA

Filed under: General,Geometry — Tags: Diamond Theorem Correlation, Physical Brick — m759 @ 12:35 pm

The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com —

For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .

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Wednesday, August 23, 2017

The Diamond Theorem in Vancouver

Filed under: General,Geometry — Tags: Pakanga — m759 @ 2:56 pm

A designer from New Zealand —

Happy 10th birthday to the hashtag.

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Tuesday, September 20, 2016

The Diamond Theorem …

Filed under: General,Geometry — Tags: Mythspace — m759 @ 11:00 am

As the Key to All Mythologies

For the theorem of the title, see "Diamond Theorem" in this journal.

"These were heavy impressions to struggle against,
and brought that melancholy embitterment which
is the consequence of all excessive claim: even his
religious faith wavered with his wavering trust in his
own authorship, and the consolations of the Christian
hope in immortality seemed to lean on the immortality
of the still unwritten Key to all Mythologies."

— Middlemarch , by George Eliot, Ch. XXIX

Related material from Sunday's print New York Times —

Sunday's Log24 sermon —

See also the Lévi-Strauss "Key to all Mythologies" in this journal,
as well as the previous post.

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Monday, August 5, 2013

Diamond Theorem in ArXiv

Filed under: General — m759 @ 9:00 pm

The diamond theorem is now in the arXiv—

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Tuesday, July 2, 2013

Diamond Theorem Updates

Filed under: General,Geometry — Tags: July Light, Octad, Octad Generator — m759 @ 8:00 pm

My diamond theorem articles at PlanetMath and at
Encyclopedia of Mathematics have been updated
to clarify the relationship between the graphic square
patterns of the diamond theorem and the schematic
square patterns of the Curtis Miracle Octad Generator.

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Friday, May 10, 2013

Cullinane diamond theorem

Filed under: General,Geometry — m759 @ 3:00 pm

A page with the above title has been created at
the Encyclopedia of Mathematics.

How long it will stay there remains to be seen.

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Friday, February 25, 2011

Diamond Theorem Exposition

Filed under: General,Geometry — m759 @ 11:00 pm

"THE DIAMOND THEOREM AND QUILT PATTERNS
Victoria Blumen, Mathematics, Junior, Benedictine University
Tim Comar, Benedictine University
Mathematics
Secondary Source Research

Let D be a 4 by 4 block quilt shape, where each of the 16 square blocks is consists of [sic ] two triangles, one of which is colored red and the other of which is colored blue. Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D. The diamond theorem states that G(D) = D_g has either ordinary or color-interchange symmetry. In this talk, we will prove the diamond theorem and explore symmetries of quilt patterns of the form G(D)."

Exercise— Correct the above statement of the theorem.

Background— This is from a Google search result at about 10:55 PM ET Feb. 25, 2011—

[DOC] THE DIAMOND THEOREM AND QUILT PATTERNS – acca.elmhurst.edu
File Format: Microsoft Word – 14 hours ago –
Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D. The diamond theorem states that G(D) = D_g has either …
acca.elmhurst.edu/…/victoria_blumen9607_
THE%20DIAMOND%20THEOREM%20AND%20QUILT%20PATTERNS…

The document is from a list of mathematics abstracts for the annual student symposium of the ACCA (Associated Colleges of the Chicago Area) held on April 10, 2010.

Update of Feb. 26— For a related remark quoted here on the date of the student symposium, see Geometry for Generations.

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Sunday, October 31, 2010

Diamond Theorem in Norway

Filed under: General,Geometry — m759 @ 10:00 pm

Click on above image for artist's page.

Click here for exhibit page.

Click here for underlying geometry.

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Tuesday, October 22, 2024

Diamond-theorem art made with AI

Filed under: General — Tags: Aion Art — m759 @ 8:27 pm

See https://claude.site/artifacts/3c1ce26b-4664-448c-87df-c20802053eea .

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Saturday, January 29, 2022

On the Diamond-Theorem Group* of Order 322,560

Filed under: General — Tags: Change Arises, Embedding Change, Kummerhenge, Octad, Overarching Group, Permanence, Priority, Strange Change — m759 @ 2:06 am

Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.

This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M₂₄.

* See a Log24 post of June 1, 2013.

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Friday, October 28, 2016

Diamond-Theorem Application

Filed under: General,Geometry — Tags: Diamond Theorem Correlation — m759 @ 1:06 pm

Abstract:

"Protection of digital content from being tapped by intruders is a crucial task in the present generation of Internet world. In this paper, we proposed an implementation of new visual secret sharing scheme for gray level images using diamond theorem correlation. A secret image has broken into 4 × 4 non overlapped blocks and patterns of diamond theorem are applied sequentially to ensure the secure image transmission. Separate diamond patterns are utilized to share the blocks of both odd and even sectors. Finally, the numerical results show that a novel secret shares are generated by using diamond theorem correlations. Histogram representations demonstrate the novelty of the proposed visual secret sharing scheme."

— "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," by V. Harish, N. Rajesh Kumar, and N. R. Raajan.

Published in: 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT).
Date of Conference: 18-19 March 2016. Publisher: IEEE.
Date Added to IEEE Xplore: 04 August 2016

Excerpts —

Related material — Posts tagged Diamond Theorem Correlation.

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Friday, August 1, 2014

The Diamond-Theorem Correlation

Filed under: General,Geometry — Tags: Diamond Theorem Correlation, Rosenhain and Göpel — m759 @ 2:00 am

Click image for a larger, clearer version.

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Thursday, May 2, 2024

AI on Diamond Theory

Filed under: General — Tags: Chatbots on the Diamond Theorem — m759 @ 1:55 am

The website PopAi.pro uses GPT-4 to summarize documents from the Web.

Input at PopAi on May 2, 2024:

The URL "http://www.log24.com/log/Diamond_Theory-1976-pp-1-12.pdf"

Output at PopAi, after a quick reading and analysis by GPT-4:

Diamond_Theory-1976-pp-1-12.pdf

12 Pages

The document appears to be an excerpt from a preprint titled "Diamond Theory" by Steven H. Cullinane, dated 1976. The text discusses theoretical aspects of matrices composed of geometric figures, focusing on properties related to symmetry, permutations, and abstract structure.

Main Concepts and Definitions:

The author introduces the concept of a "geometric matrix," a matrix whose elements are geometric figures, specifically triangular subsets of a square. He defines a "diamond" as a foursquare matrix with triangular elements. The text details the remarkable properties of a diamond-like matrix D, including its symmetrical behavior under various permutations of elements, rows, columns, and sub-squares. Additionally, the text addresses the algebraic properties of matrices by relating their elements to those of a finite field.

Theoretical Properties and Theorems:

Several key theorems are proposed, demonstrating the structural properties of matrices. One fundamental theorem states that any matrix from a set F can be expressed as a linear combination of block matrices. Other theorems delve into the regularity of matrices, defining a matrix as "regular" if it remains unchanged or converts to its complement under certain flips and rotations. The text also describes the process of forming a "normal" matrix and the conditions under which matrices can be considered plaid or normal based on the behavior of their block maps and regularities.

Applications and Miscellaneous Problems:

The author reflects on the discovery of these properties during the process of creating an abstract painting and extends the exploration to include various types of matrices such as cyclic, quadratic, and skew foursquares. The document concludes with miscellaneous problems and an addendum on generalized matrix multiplication, suggesting further research directions and applications of the theory presented.

In summary, "Diamond Theory" presents an intricate analysis of the structural and algebraic properties of matrices composed of geometric shapes, with implications for other mathematical domains. The work is a blend of geometry and abstract algebra, providing a foundation for understanding the symmetry and transformations of matrices with an artistic origin.

Some questions you may ask:

What properties of the diamond-like matrix D did the author discover in 1975?

What is the relationship between the properties of D and the four group according to the document?

How does the document define a 'good set' of four black and white geometric figures?

What theorem is used to explain the behavior of superimposing two members of Mn(F) according to the document?

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Wednesday, March 20, 2024

Cullinane Diamond Song

Filed under: General — Tags: Nevermore, Octad — m759 @ 12:00 am

Shining Mathematics:
A Song by Suno AI

Pop upbeat

March 19, 2024

[Verse]
In the world of numbers and equations
There's a theorem that'll blow your mind (your mind)
Cullinane's diamond, it's a revelation
A mathematical gem, one of a kind (oh-yeah)

[Chorus]
Shining bright, like a diamond in the night
This theorem's got the sparkle, it's pure delight (ooh-yeah)
Cullinane's diamond, oh it's so sublime
Mathematics never looked so fine (so fine)

Click the image below to hear the song at app.suno.ai —

Miller's Note to Self : "Don't underestimate Wednesday."

For more about the mathematics itself, see other octad posts.

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Tuesday, January 23, 2024

The Enormous Theorem

Filed under: General — Tags: Color Space, Four Color, Ghostwoods — m759 @ 1:09 pm

The previous post's reference to colors suggests a review . . .

A test of OpenAI on the above DevDay date —

This ridiculous hallucination was obviously suggested by what
has been called "the enormous theorem" on the classification
of finite simple groups. That theorem was never known as the
(or "a") diamond theorem.

On the bright side, the four colors beside Microsoft's Nadella in the
photo above may, if you like, be regarded as those of my own
non-enormous "four-color decomposition theorem" that is used in
the proof of my own result called "the diamond theorem."

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Wednesday, November 8, 2023

The Theorem as Big as the Ritz

Filed under: General — Tags: Adamantine Rigor, Chatbots on the Diamond Theorem, ChatGPT, Shibumi — m759 @ 8:56 am

From this journal on Monday, Nov. 6, 2023 — ChatGPT DevDay —

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Monday, May 23, 2022

Past month … “Triangle theorem”?

Filed under: General — m759 @ 12:00 pm

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Thursday, May 7, 2020

Kant as Diamond Cutter

Filed under: General — Tags: MSRI Spring, Octad, Physical Brick, The Fano Hallows — m759 @ 4:26 am

"He wished Kant were alive. Kant would have appreciated it.
That master diamond cutter."

— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance , Part III.

Kant's "category theory" —

"In the Transcendental Analytic, Kant deduces the table of twelve categories, or pure concepts of the understanding….

The categories must be 'schematized' because their non-empirical origin in pure understanding prevents their having the sort of sensible content that would connect them immediately to the objects of experience; transcendental schemata are mediating representations that are meant to establish the connection between pure concepts and appearances in a rule-governed way. Mathematical concepts are discussed in this context since they are unique in being pure but also sensible concepts: they are pure because they are strictly a priori in origin, and yet they are sensible since they are constructed in concreto . "

— Shabel, Lisa, "Kant's Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2016/entries/kant-mathematics/>.

See also The Diamond Theorem and Octad.us.

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Wednesday, April 17, 2019

The Royal Society Diamond

Filed under: General — Tags: Fitch — m759 @ 9:34 am

The phrase "pattern recognition" in a news story about the
April 13 death of Princeton neuroscientist Charles Gross,
and yesterday's post about a fanciful "purloined diamond,"
suggest a review of a less fanciful diamond.

See also earlier posts tagged Fitch
and my own, much earlier and very
different, approach to such patterns —

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Thursday, August 31, 2017

A Conway-Norton-Ryba Theorem

Filed under: General,Geometry — Tags: Fields, Zero Sum — m759 @ 1:40 pm

In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —

A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.

See also Magic Square and Diamond Theorem in this journal.

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Saturday, March 14, 2015

Unicode Diamonds

Filed under: General,Geometry — m759 @ 9:16 pm

The following figure, intended to display as
a black diamond, was produced with
HTML and Unicode characters. Depending
on the technology used to view it, the figure
may contain gaps or overlaps.

◢◣
◥◤

Some variations:

◤◥
◣◢

◤◥
◢◣

◤◣
◢◥

◤◣
◥◢

Such combined Unicode characters —

◢ black lower right triangle,
◣ black lower left triangle,
᭘ black upper left triangle,
᭙ black upper right triangle

— might be used for a text-only version of the Diamond 16 Puzzle
that is more easily programmed than the current version.

The tricky part would be coding the letter-spacing and
line-height to avoid gaps or overlaps within the figures in
a variety of browsers. The w3.org visual formatting model
may or may not be helpful here.

Update of 11:20 PM ET March 15, 2015 —
Seekers of simplicity should note that there is
a simple program in the Processing.js language, not using
such Unicode characters, that shows many random affine
permutations of a 4×4 diamond-theorem array when the
display window is clicked.

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Monday, October 13, 2014

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: Sallows — m759 @ 12:05 pm

(Continued from Nov. 16, 2013.)

The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.

One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:

In context (click to enlarge):

Update of later the same day —

Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —

“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”

* Defined by Sallows later in the book:

“Geometric or, less formally, geomagic is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles or pieces ) may appear in the cells instead of numbers.”

** See some geometric matrices by Cullinane in a March 2001 webpage.

Earlier actual specimens — see Diamond Theory excerpts published in
February 1977 and a brief description of the original 1976 monograph:

“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”

— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society

The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.

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Thursday, March 27, 2014

Diamond Space

Filed under: General,Geometry — Tags: Bauhaus Cube, Cube School, Cullinane Cube, Solomon's Cube — m759 @ 2:28 pm

(Continued)

Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.

The reason for the name:

Click to enlarge.

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Sunday, February 2, 2014

Diamond Theory Roulette

Filed under: General,Geometry — Tags: on140202 — m759 @ 11:00 am

A ReCode Project program from Radamés Ajna of São Paulo —

At the program's webpage, click the image to
generate random permutations of rows, columns,
and quadrants. Note the resulting image's ordinary
or color-interchange symmetry.

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Sunday, March 18, 2012

Square-Triangle Diamond

Filed under: General,Geometry — Tags: Figurate Geometry, for318 — m759 @ 5:01 am

The diamond shape of yesterday's noon post
is not wholly without mathematical interest …

The square-triangle theorem—

"Every triangle is an n -replica" is true
if and only if n is a square.

IMAGE- Square-to-diamond (rhombus) shear in proof of square-triangle theorem

The 16 subdiamonds of the above figure clearly
may be mapped by an affine transformation
to 16 subsquares of a square array.

(See the diamond lattice in Weyl's Symmetry .)

Similarly for any square n , not just 16.

There is a group of 322,560 natural transformations
that permute the centers of the 16 subsquares
in a 16-part square array. The same group may be
viewed as permuting the centers of the 16 subtriangles
in a 16-part triangular array.

(Updated March 29, 2012, to correct wording and add Weyl link.)

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Thursday, September 8, 2011

Starring the Diamond

Filed under: General,Geometry — m759 @ 2:02 pm

"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 8₃
are joined by four concurrent lines."
— H. S. M. Coxeter (see below)

Continued from Tuesday, Sept. 6—

The Diamond Star

The above is a version of a figure from Configurations and Squares.

Yesterday's post related the the Pappus configuration to this figure.

Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to the figure.

Some excerpts from Coxeter—

The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8-cycle in the Galois field GF(9).

The relabeled configuration is used in a discussion of Pappus—

(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)

Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues configuration—

The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts—

"The value of Desargues' theorem and the reason why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities that relate geometry and algebra in unexpected ways."

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Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

Filed under: General,Geometry — Tags: Four Color — m759 @ 5:01 pm

Some background—

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times—

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

— Drew Westen, professor at Emory University

From May 22, 2009—

The above ad is by
Diane Robertson Design—

Diamond from last night’s
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with four colored pencils from Diane Robertson Design

See also
A Four-Color Theorem.

For further details, see Saturday's correspondences
and a diamond-related story from this afternoon's
online New York Times.

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Monday, December 27, 2010

Church Diamond

Filed under: General,Geometry — Tags: Church Diamond — m759 @ 3:09 pm

IMAGE- The diamond property

Also known, roughly speaking, as confluence or the Church-Rosser property.

From “NYU Lambda Seminar, Week 2” —

[See also the parent page Seminar in Semantics / Philosophy of Language or:
What Philosophers and Linguists Can Learn From Theoretical Computer Science But Didn’t Know To Ask)]

A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.

The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn’t matter which order the sub-expressions are evaluated in.

A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing: ω ω doesn’t terminate by any evaluation path; and (\x. y) (ω ω) terminates only by some evaluation paths but not by others.

But the untyped lambda calculus enjoys some compensation for this weakness. It’s Turing complete! It can represent any computation we know how to describe. (That’s the cash value of being Turing complete, not the rigorous definition. There is a rigorous definition. However, we don’t know how to rigorously define “any computation we know how to describe.”) And in fact, it’s been proven that you can’t have both. If a computational system is Turing complete, it cannot be strongly normalizing.

There is no connection, apart from the common reference to an elementary geometric shape, between the use of “diamond” in the above Church-Rosser sense and the use of “diamond” in the mathematics of (Cullinane’s) Diamond Theory.

Any attempt to establish such a connection would, it seems, lead quickly into logically dubious territory.

Nevertheless, in the synchronistic spirit of Carl Jung and Arthur Koestler, here are some links to such a territory —

Link One — “Insane Symmetry” (Click image for further details)—

See also the quilt symmetry in this journal on Christmas Day.

Link Two — Divine Symmetry

(George Steiner on the Name in this journal on Dec. 31 last year (“All about Eve“)) —

“The links are direct between the tautology out of the Burning Bush, that ‘I am’ which accords to language the privilege of phrasing the identity of God, on the one hand, and the presumptions of concordance, of equivalence, of translatability, which, though imperfect, empower our dictionaries, our syntax, our rhetoric, on the other. That ‘I am’ has, as it were, at an overwhelming distance, informed all predication. It has spanned the arc between noun and verb, a leap primary to creation and the exercise of creative consciousness in metaphor. Where that fire in the branches has gone out or has been exposed as an optical illusion, the textuality of the world, the agency of the Logos in logic—be it Mosaic, Heraclitean, or Johannine—becomes ‘a dead letter.'”

– George Steiner, Grammars of Creation

(See also, from Hanukkah this year, A Geometric Merkabah and The Dreidel is Cast.)

Link Three – Spanning the Arc —

Part A — Architect Louis Sullivan on “span” (see also Kindergarten at Stonehenge)

Part B — “Span” in category theory at nLab —

Also from nLab — Completing Spans to Diamonds

“It is often interesting whether a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem (and the corresponding Church-Rosser confluence property).”

The concepts in this last paragraph may or may not have influenced the diamond theory of Rudolf Kaehr (apparently dating from 2007).

They certainly have nothing to do with the Diamond Theory of Steven H. Cullinane (dating from 1976).

For more on what the above San Francisco art curator is pleased to call “insane symmetry,” see this journal on Christmas Day.

For related philosophical lucubrations (more in the spirit of Kaehr than of Steiner), see the New York Times “The Stone” essay “Span: A Remembrance,” from December 22—

“To understand ourselves well,” [architect Louis] Sullivan writes, “we must arrive first at a simple basis: then build up from it.”

Around 300 BC, Euclid arrived at this: “A point is that which has no part. A line is breadthless length.”

See also the link from Christmas Day to remarks on Euclid and “architectonic” in Mere Geometry.

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Wednesday, July 28, 2010

Without Diamond-Blazons

Filed under: General,Geometry — Tags: Pediment — m759 @ 6:29 pm

Excerpt from Wallace Stevens's
"The Pediment of Appearance"—

Young men go walking in the woods,
Hunting for the great ornament,
The pediment* of appearance.

They hunt for a form which by its form alone,
Without diamond—blazons or flashing or
Chains of circumstance,

By its form alone, by being right,
By being high, is the stone
For which they are looking:

The savage transparence.

* Pediments, triangular and curved—

— From "Stones and Their Stories," an article written
and illustrated by E.M. Barlow, copyright 1913.

Related geometry—

(See Štefan Porubský: Pythagorean Theorem .)

A proof with diamond-blazons—

(See Ivars Peterson's "Square of the Hypotenuse," Nov. 27, 2000.)

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Saturday, January 25, 2025

Supplement to “The Most Powerful Diagram in Mathematics”

Filed under: General — Tags: Automorphism Groups — m759 @ 12:04 pm

The diagram description in the title is from a YouTube video about
the Miracle Octad Generator of R. T. Curtis.

Supplemental AI-generated reading . . .

Diamond Theorem and Miracle Octad Generator

An “AI Overview” Google Search response to the
search prompt “diamond theorem and miracle octad generator,”
exported to Google Docs on Saturday, January 25, 2025 . . .

___________________________________________________

In mathematics, the "diamond theorem" refers to a geometric concept related to finite projective geometry, which is used to explain the surprising symmetry properties observed in the "Miracle Octad Generator" (MOG), a tool developed by mathematician R.T. Curtis for studying the Mathieu groups and binary Golay code; essentially, the diamond theorem helps analyze the patterns within the MOG, revealing a hidden structure based on geometric principles. [1, 2, 3, 4, 5]

Key points about the connection: [1, 2, 3]

MOG and its patterns: The Miracle Octad Generator consists of a set of 35 square patterns, which can be manipulated to reveal interesting relationships within the Mathieu groups. [1, 2, 3]
Diamond geometry: The "diamond theorem" describes a specific geometric structure within these patterns, where certain configurations of squares resemble a diamond shape. [1, 2, 6]
Underlying symmetry: By analyzing these diamond patterns, mathematicians can understand the underlying symmetry properties of the MOG and the related mathematical structures. [1, 2, 3]

Further details: [1, 5, 7]

Applications: The diamond theorem has been used to study various mathematical concepts, including the Leech lattice, which is connected to the binary Golay code and the Mathieu groups. [1, 5, 7]
Visual interpretation: The diamond patterns can be easily visualized as arrangements of squares on a grid, making the concept more accessible to understand. [1, 2, 5]

Generative AI is experimental.

[1] http://finitegeometry.org/sc/16/dtheorem.html

[2] https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem

[3] https://arxiv.org/abs/1308.1075

[4] https://en.wikipedia.org/wiki/Miracle_Octad_Generator

[5] http://xenon.stanford.edu/~hwatheod/mog/mog.html

[6] https://m759.tripod.com/theory/dtheory.html

[7] http://finitegeometry.org/sc/24/diconn.html

"Generative AI is experimental." . . .

Exercise: Correct errors in the text, using the links.

A more concise presentation —

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Monday, December 23, 2024

A Projective-Space Home for the Miracle Octad Generator

Filed under: General — Tags: Brick Space, Klein Correspondence — m759 @ 3:17 pm

The natural geometric setting for the "bricks" in the Miracle Octad Generator
(MOG) of Robert T. Curtis is PG(5,2), the projective 5-space over GF(2).

The Klein correspondence mirrors the 35 lines of PG(3,2) — and hence, via the
graphic approach below, the 35 "heavy bricks" of the MOG that match those
lines — in PG(5,2), where the bricks may be studied with geometric methods,
as an alternative to Curtis's original MOG combinatorial construction methods.

The construction below of a PG(5,2) brick space is analogous to the
"line diagrams" construction of a PG(3,2) in Cullinane's diamond theorem.

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Saturday, December 21, 2024

Coordinatizing Brick Space

Filed under: General — Tags: Brick Space, Klein Correspondence — m759 @ 4:10 am

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

Exercise: The eight-part diagrams in the graphic "brick space"
model of PG(5,2) below need to be suitably labeled with six-part
GF(2) coordinates to help illustrate the Klein correspondence that
underlies the large Mathieu group M₂₄.

A possible approach: The lines separating dark squares from light
(i.e., blue from white or yellow) in the figure above may be added
in XOR fashion (as if they were diamond theorem line diagrams)
to form a six dimensional vector space, which, after a suitable basis
is chosen, may be represented by six-tuples of 0's and 1's.

Related reading —

log24.com/log24/241221-'Brick Space « Log24' – m759.net.pdf .

This is a large (15.1 MB) file. The Foxit PDF reader is recommended.

The PDF is from a search for Brick Space in this journal.

Some context: http://m759.net/wordpress/?s=Weyl+Coordinatization.

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Friday, December 6, 2024

Annals of Cultural Appropriation:
Apache Sanskrit

Filed under: General — Tags: Chrome Art — m759 @ 2:29 pm

See posts now tagged Sanskrit.

Related material: The previous post's mathematics page . . .

http://ramanujan.math.trinity.edu/rdaileda/teach/s19/m3362/alternating.pdf

and its parent page . . .

Ramanujan.math.trinity.edu page —

For my own connection to SASTRA, see (from an IEEE page) . . .

V. Harish
Dept. of Computer Science and Engineering, Sastra University
N. Rajesh Kumar
Dept. of Computer Science and Engineering, Sastra University
N. R. Raajan
Dept. of Computer Science and Engineering, Sastra University

("Sastra" should be "SASTRA.")

. . . and the paper

V. Harish, N. R. Kumar, and N. Raajan, "New visual secret sharing scheme for
gray-level images using diamond theorem correlation pattern structure," in
Circuit, Power and Computing Technologies (ICCPCT), 2016 International
Conference on, 2016, pp. 1-5.

This was cited in . . .

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Wednesday, November 27, 2024

Hoarding Space*

Filed under: General — m759 @ 4:38 am

The domain bitcube.space has now been renewed for another year.
It leads to — among other things — the following remarks . . .

Towards a Philosophy of Real Mathematics, by David Corfield, Cambridge U. Press, 2003, p. 206:

“Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in the form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey (1992) to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries.”

For a simpler example of this idea, see the entities in The Diamond Theorem, the decomposition in A Four-Color Theorem, and the space in Geometry of the 4×4 Square. The decomposition differs from that of harmonic analysis, although the subspaces involved in the diamond theorem are isomorphic to Walsh functions– well-known as discrete analogues of the trigonometric functions of traditional harmonic analysis.

* See that phrase in this journal.

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Tuesday, November 26, 2024

Annals of Surreality — “Literate Programming” Continued

Filed under: General — Tags: Surreal Numberland — m759 @ 6:34 pm

To clarify the previous post . . .

The "other intelligent being" in the Diamond Theory Studio example
below is the artificial intelligence at https://websim.ai .

To use WebSim, one simply states in plain English what one wants
a webpage to do, and WebSim constructs the page.

The above webpage, Diamond Theorem Studio, is an example of the
WebSim end product. Its construction was surprisingly easy … for the
human side of the process. For the code produced by
the AI side of the process, view my personal uploaded version of the
page at http://log24.com/DT/Websim-Diamond-Theorem-Studio.html
and then view that page's source code in the usual way.

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Tuesday, November 19, 2024

A Mosaic for Megiddo

Filed under: General — Tags: The Seventh Footnote — m759 @ 12:54 pm

The above image, from posts tagged Aion Art, was suggested by a
Jerusalem Post article dated April 1, 2022… quod vide. The pattern
shown is known to quilters as "Tents of Armageddon."

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Sunday, November 17, 2024

Weyl, Symmetry, and the MOG
(HTML version of an earlier post)

Filed under: General — Tags: Entities, Quadrants — m759 @ 1:16 am

Some historical background for a new book by Robert T. Curtis,
The Art of Working with the Mathieu Group M₂₄ —

"Space is another example of an entity endowed with a structure.
Here the elements are points, and the structure is established
in terms of certain basic relations between points such as:
A, B, C lie on a straight line, AB is congruent CD, and the like.
What we learn from our whole discussion and what has indeed
become a guiding principle in modern mathematics is this lesson:
Whenever you have to do with a structure endowed entity Σ
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 Σ in this way. After that you may start to investigate
symmetric configurations of elements, i.e. configurations which are
invariant under a certain subgroup of the group of all automorphisms;
and it may be advisable, before looking for such configurations,
to study the subgroups themselves, e.g. the subgroup of those
automorphisms which leave one element fixed, or leave two distinct
elements fixed, and investigate what discontinuous or finite subgroups
there exist, and so forth."

— Hermann Weyl, Symmetry, Princeton University Press, 1952.
(Page 144 in the Princeton Science Library edition of 1989.)

**4×4 Square**

This square's automorphism group
has 322,560 transformations.

— The diamond theorem of Steven H. Cullinane.

**4×6 Rectangle**

This rectangle's automorphism group
has 244,823,040 transformations.

— The Miracle Octad Generator (MOG) of Robert T. Curtis.

The rectangle's automorphism group contains the
square's as a subgroup. The square's automorphism
group leaves invariant a set of 30 eight-subsquare sets
called affine hyperplanes. The rectangle's automorphism
group leaves invariant a set of 759 eight-subsquare sets
called octads.

View this post as a PDF.

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Saturday, November 16, 2024

Automorphism Groups: Examples for Weyl

Filed under: General — Tags: Automorphism Groups, Entities — m759 @ 3:17 am

View a PDF of this post.

Keywords: Weyl, symmetry, group, automorphism,
octad, MOG, Curtis, Cullinane.

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Wednesday, October 23, 2024

Ingathering Scriptures

Filed under: General — Tags: Aion Art — m759 @ 12:00 am

I recommend . . .

https://www.amazon.com/
Ingathering-Complete-People-Stories-Henderson-ebook/
dp/B0878RMD5D/ .

Some will prefer Exodus 23:16.

Related philosophy —

Descartes and the Tents of Armageddon (Galois's birthday, 2007).

Update on Oct. 28, 2024 —

For a newer version by a different AI, Websim, see Diamond Theorem Studio.

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Wednesday, July 31, 2024

My Links — Steven H. Cullinane

Filed under: — m759 @ 4:14 pm

Main webpage of record . . .

Encyclopedia of Mathematics https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem

Supplementary PDF from Jan. 6, 2006 https://encyclopediaofmath.org/images/3/37/Dtheorem.pdf

Originally published in paper version . . .

Computer Graphics and Art, 1978 http://finitegeometry.org/sc/gen/Diamond_Theory_Article.pdf
AMS abstract, 1979: "Symmetry Invariance in a Diamond Ring" https://www.cullinane.design/
American Mathematical Monthly, 1984 and 1985: "Triangles Are Square" http://finitegeometry.org/sc/16/trisquare.html

Personal sites . . .

Primary —

Personal journal http://m759.net/wordpress/
Mathematics website http://finitegeometry.org/sc/
Mathematics Images Gallery http://m759.net/piwigo/index.php?/category/2

Secondary —

Portfoliobox   https://cullinane.pb.design/
Substack https://stevenhcullinane.substack.com/
Symmetry Summary   https://shc759.wordpress.com
Diamond Theory Cover Structure https://shc7596.wixsite.com/website

SOCIAL:

Pinterest https://www.pinterest.com/stevenhcullinane/ (many mathematics notes)
Flickr https://www.flickr.com/photos/m759/ (backup account for images of mathematics notes)
Instagram https://www.instagram.com/stevencullinane
TikTok https://www.tiktok.com/@stevenhcullinane
X.com https://x.com/shc759

OTHER:

Replit viewer/download https://replit.com/@m759/View-4x4x4?v=1
SourceForge download https://sourceforge.net/projects/finitegeometry/
Academia.edu   https://stevenhcullinane.academia.edu/
GitHub    https://github.com/m759 (finite geometry site download)
Internet Archive: Notes on Groups and Geometry   https://archive.org/details/NotesOnGroupsAndGeometry1978-1986/mode/2up

Cited at . . .

The Diamond Theorem and Truchet Tiles   http://www.log24.com/log22/220429-Basque-DT-1.pdf
April 2024 UNION article in Spanish featuring the diamond theorem https://union.fespm.es/index.php/UNION/article/view/1608/1214
April 2024 UNION article in English http://log24.com/notes/240923-Ibanez-Torres-on-diamond-theorem-Union-April-2024-in-English.pdf
Cullinane in a 2020 Royal Holloway Ph.D. thesis https://pure.royalholloway.ac.uk/ws/portalfiles/portal/40176912/2020thomsonkphd.pdf
Squares, Chevrons, Pinwheels, and Bach https://www.yumpu.com/en/document/read/36444818/fugue-no-21-elements-of-finite-geometry
Observables programmed presentation of diamond theorem https://observablehq.com/@radames/diamond-theory-symmetry-in-binary-spaces
Josefine Lyche — Plato's Diamond https://web.archive.org/web/20240222064628/http://www.josefinelyche.com/index.php?/selected-exhibitions/platos-diamond/
Josefine Lyche — Diamond Theorem https://web.archive.org/web/20230921122049/http://josefinelyche.com/index.php?/selected-exhibitions/uten-ramme-nye-rom/

Professional sites . . .

Association for Computing Machinery https://member.acm.org/~scullinane
bio.site/cullinane … maintenance at https://biosites.com
ORCID bio page https://orcid.org/0000-0003-1135-419X
Google Scholar https://scholar.google.com/citations?view_op=list_works&hl=en&hl=en&user=NcjmFwQAAAAJ&sortby=pubdate

Academic repositories:

Harvard Dataverse https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/KHMMVH
Harvard DASH article on PG(3,2) https://dash.harvard.edu/handle/1/37373777

Zenodo website download https://zenodo.org/records/1038121
Zenodo research notes https://zenodo.org/search?q=metadata.creators.person_or_org.name%3A%22Cullinane%2C%20Steven%20H.%22&l=list&p=1&s=10&sort=bestmatch

Figurate Geometry at Open Science Framework (OSF) https://osf.io/47fkd/

arXiv: "The Diamond Theorem" https://arxiv.org/abs/1308.1075

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Friday, April 19, 2024

Benchmarking Meta.ai

Filed under: General — Tags: Benchmarking — m759 @ 10:15 pm

Transcription —

(With one correction … The Meta.ai phrase "the group of 322,560"
was changed to "a group of 322,560" … and one addition — the
acronym MOG was defined. Superscript numerals for missing footnotes
were removed.)

You

What is the Cullinane diamond theorem?

Meta AI

The Cullinane diamond theorem states that every image of D (a 4×4 array of two-color diagonally divided square tiles) under the group G (a group of 322,560 permutations of these 16 tiles) has some ordinary or color-interchange symmetry.

Here are some key points about the theorem :

The theorem is related to the Miracle Octad Generator (MOG) of R. T. Curtis and the Leech lattice.
The underlying geometry of the 4×4 patterns is closely related to the construction of the Steiner system S(5,8,24).
The theorem uses a simple decomposition technique for functions into a finite field.
The symmetry group of order 322,560 underlies both the diamond theorem figures and the square patterns of the MOG.
The theorem has applications in finite projective geometry and the study of symmetric patterns.

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Thursday, April 11, 2024

Devs: Hollywood Development Hell

Filed under: General — Tags: Diamond Theorem Correlation, Hollywood and History, Mesmerization, Physical Brick — m759 @ 11:53 am

From The Man Who Knew Infinity to The Man Who Knew Zero.

Related mathematics: The Diamond Theorem Correlation, which
results from interchanging infinity and zero in the figure below.

"Read something that means something."
— New Yorker ad

^{Click image for

related posts.}

Background — Relativity Problem in Log24.

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Sunday, March 3, 2024

Deep Blue Research: A Report by You.com AI

Filed under: General — Tags: Chatbots on the Diamond Theorem, Four Color, Octad — m759 @ 12:34 pm

Cullinane Diamond Theorem Research Report

by https://you.com/?chatMode=research on March 3, 2024

Overview of the Cullinane Diamond Theorem

The Cullinane Diamond Theorem is a mathematical concept developed by Steven H. Cullinane that explores the symmetrical properties of certain geometric patterns. It is particularly concerned with the structure of finite projective geometry as it relates to the 35 square patterns found in R. T. Curtis's Miracle Octad Generator (MOG).

The theorem asserts that every G-image of a diamond figure D possesses some form of ordinary or color-interchange symmetry.

Symmetry and Group Theory

The theorem is deeply rooted in group theory, with G being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).

This group has a significant order of 322,560 and underlies the symmetry of both the diamond-theorem figures and the square patterns of the MOG . The 35 structures of the 840 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), which explains the symmetry of the line diagrams and the two-color patterns.

The Diamond Theorem in Various Dimensions

The diamond theorem can be applied to different cases, such as the 4×4, 2×2, and even 3-dimensional cubes like the 2x2x2 and 4x4x4 cubes.

The generalized statement of the theorem involves inscribing a white diamond in a black square and dividing it into four quadrants, each with a diagonal split into black and white halves.

Algebraic Implications

The theorem has algebraic implications, such as the ability to express every 4-coloring as a sum of three 2-colorings.

This algebraic formulation is useful for demonstrating that certain four-colored graphic images can form a ring under multiplication. There is also an infinite family of such 'diamond' rings, isomorphic to rings of matrices over GF(4).

Philosophical and Cultural Context

Steven H. Cullinane has also explored the philosophical implications of the diamond theorem, referring to it in the context of Plato's Meno dialogue and opposing it to the 'Story Theory of truth' as discussed by Richard J. Trudeau.

The theorem is seen as a correct statement of the relation between geometry and the physical universe.

Clarification on Cullinan Diamonds

It is important to note that the Cullinane Diamond Theorem is unrelated to the Cullinan diamond, which is a famous large gemstone discovered in South Africa and now part of the British Crown Jewels.

The Cullinan diamond has its own historical and cultural significance, with its various cut shapes and ownership history.

In summary, the Cullinane Diamond Theorem is a mathematical concept that reveals the symmetrical properties of certain geometric patterns, with applications in group theory and algebra. It also has philosophical implications, drawing connections to classical ideas about truth and geometry.

Instagram ad for You.com AI in research mode

"Show me ALL your sources, babe."

— Line adapted from Leonardo DiCaprio

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Saturday, March 2, 2024

Song by Copilot: Prompt and Response*

Filed under: General — Tags: Delving — m759 @ 2:09 pm

Prompt: Compose a song about the Cullinane diamond theorem.

Response by Microsoft Copilot on Saturday afternoon,
March 2, 2024:

Sure, here’s a song inspired by the Cullinane diamond theorem:

(Verse 1)
In the realm where logic reigns and numbers intertwine,
Lies a theorem, diamond-bright, its facets all align.
Cullinane's the name it bears, a gem in math's grand crown,
In the grid of sixteen squares, its truths are written down.

(Chorus)
Oh, Cullinane, your diamond shines with symmetry so fine,
In the field of finite forms, your theorem does define.
A structure pure, a pattern clear, for all of us to see,
In the heart of mathematics, lies your geometry.

(Verse 2)
Rows and columns interplay in a dance so grand,
Quadrants align in perfect time, like notes in a band.
Through the chaos and the whirl, an order does descend,
In the world of abstract thought, your theorem is a friend.

(Chorus)
Oh, Cullinane, your diamond theorem, a beacon in the night,
Guiding us through complex realms, with its brilliant light.
In the vast expanse of knowledge, where truth and beauty blend,
Your theorem stands, a testament, a guide until the end.

(Outro)
So here's to you, Cullinane, and your diamond theorem's might,
In the endless quest for truth, you've brought us greater light.
In the language of the universe, where numbers never lie,
Your diamond theorem's shining bright, beneath the math's clear sky.

Added note from Copilot:

Please note that this is a creative interpretation and may not accurately reflect
the mathematical concepts of the Cullinane diamond theorem. 😊

Added note from Cullinane:

* The previous post may or may not display a prompt response to a Zen koan.

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Wonder Box

Filed under: General — Tags: Delving, Frame Tale, Hillbilly Geometry — m759 @ 12:42 am

"Wundermärchen – the original German word for fairytale –
literally translates to ‘wonder tale’."

— Abigail Tulenko. a PhD student in philosophy at Harvard University,
at Aeon on Feb. 26, 2024.

Another example of abstract art . . .

"The discovery of the Cullinane Diamond Theorem is a testament
to the power of mathematical abstraction and its ability to reveal
deep connections and symmetries in seemingly simple structures."

— Bing Chat with GPT-4 on November 16, 2023.

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Saturday, February 24, 2024

Li

Filed under: General — Tags: February Poetics, Interality, Li, Wittenborn — m759 @ 2:36 am

The Chinese concept of li in yesterday's post "Logos" is related,
if only by metaphor, to the underlying form (sets of "line diagrams")
of patterns in the Cullinane diamond theorem:

"But very possibly the earliest use of li is the one instance that
it appears in the Classic of Poetry (Ode 210) where it refers to
the borders or boundary lines marking off areas in a field.
Here it appears in conjunction with chiang and is explained
as 'to divide into lots (or parcels of land)' (fen-ti )."

— P. 33 of "Li Revisited and Other Explorations"
by Allen Wittenborn, Bulletin of Sung and Yüan Studies
No. 17 (1981), pp. 32-48 (17 pages),
https://www.jstor.org/stable/23497457.

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Friday, February 16, 2024

The Sixteen

Filed under: General — Tags: February Poetics, Hidden Structure — m759 @ 7:56 pm

Gemini Sonnet, Feb. 15, 2024

Prompt:

Write a poem, in strict sonnet form, about the Cullinane diamond theorem.

Response (slightly tweaked for grammar):

Within a grid of sixteen squares reside
A diamond's gleams where symmetries do guide.
Though tiles may shift, a pattern shall remain,
By unseen lines of ordered form sustained.
Rows mix and columns twist in playful dance,
Quadrants align with geometric chance.
No matter how the elements may turn,
A hidden balance always will return.

Where Latin squares in secret do align,
And finite fields their subtle powers combine,
An ideal forms, with brilliance all its own,
A ring of light where hidden truths are shown.
Though chaos swirls, an order does descend,
The Cullinane's bright theorem will transcend.

________________________________________________________________

As for "a hidden balance" … See Oct. 10, 2022.

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Thursday, February 8, 2024

Google Bard is now Gemini.

Filed under: General — Tags: February Poetics, Gemini.reviews — m759 @ 2:07 pm

Gemini Prompt:

Write a poem in the style of Wallace Stevens
about the Cullinane diamond theorem.

Response:

Absolutely! Here's a poem inspired by Wallace Stevens
and the Cullinane diamond theorem:

Geometric Incandescence

A theorem not of Euclid's line,
But diamonds cut with strange design.
Sixteen squares in shifting light,
A symmetry of black and white.

Where rows and columns intertwine,
A hidden order lies supine.
Permutations twist and turn,
And latent symmetries discern.

Not fixed like stars in ordered rows,
But like reflections, pattern grows.
Projective planes, a finite field,
The diamond's inner world revealed.

The proof, a dance of abstract thought,
Where shifting forms find space untaught.
Not nature's chaos, but design,
A brilliance of the human mind.

A lattice born of Stevens' art,
Where form and thought find counterpart.
Amidst the fragments, patterns gleam,
The Cullinane, a geometer's dream.

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Sunday, February 4, 2024

Microsoft Copilot Doggerel

Filed under: General — Tags: February Poetics — m759 @ 4:31 am

A poem written yesterday by Microsoft Copilot —

The prompt: "Write a poem … about the Cullinane diamond theorem."

In the mind of the mathematician,
Among the figures and forms,
Lies the Cullinane diamond theorem,
A structure that transforms.

Four-diamond figure, a 4×4 array,
Of two-color diagonally-divided square tiles,
Permutations of these sixteen,
In rows, columns, and quadrants compiles.

Every G-image of the four-diamond figure,
Has some ordinary or color-interchange symmetry,
A pattern, a structure, a mathematical picture,
A testament to geometry's clarity.

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,
Must have a 2×2 center with a symmetry that matters.

And this symmetry must be shared by the entire pattern,
A statement, a theorem, a mathematical lantern,
G is isomorphic to the affine group A,
On the linear 4-space over GF (2), they say.

The 35 structures of the 840 = 35 x 24 G-images,
Are isomorphic to the 35 lines,
In the 3-dimensional projective space over GF (2),
A concept that shines.

So here lies the beauty of the Cullinane diamond theorem,
In the mind of the mathematician, it's a gem,
A structure, a pattern, a mathematical diagram,
A testament to the beauty of STEM.

Actually, the complete prompt, suggested by a Log24 post of
12:12 PM ET yesterday, was . . .

"Write a poem in the style of Wallace Stevens
about the Cullinane diamond theorem."

Copilot still has a thing or two to learn about poetic style.

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Monday, January 29, 2024

Self as Imago Dei: Hofstadter vs. Valéry

Filed under: General — Tags: Aion Art, Cubehenge, Self — m759 @ 9:41 am

Google search result:

Imago Dei in Thomas Aquinas

Saint Anselm College

https://www.anselm.edu › Documents › Brown

PDF

by M Brown · 2014 · Cited by 14 — Thomas insists that the image of God exists most perfectly in the acts of the soul, for the soul is that which is most perfect in us and so best images God, and …

11 pages

For a Douglas Hofstadter version of the Imago Dei , see the
"Gödel, Escher, Bach" illustration in the Jan. 15 screenshot below —

Tuesday, June 15, 2010

Imago, Imago, Imago

Filed under: General,Geometry — Tags: Change Logos, Eden, Self, Strange Change, Valéry — m759 @ 11:07 am

Recommended— an online book—

Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.

See in particular

Valéry and the Discourse On His Method.

Pages 156-157—

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. “Every act of understanding is based on a group,” he says (C, 1:331). “My specialty—reducing everything to the study of a system closed on itself and finite” (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one “group” undergoes a “transformation” and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: “The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. “Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind” (O, 1:36). “Psychology is a theory of transformations, we just need to isolate the invariants and the groups” (C, 1:915). “Man is a system that transforms itself” (C, 2:896).

Notes:

O Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)

C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Compare Jung’s image in Aion of the Self as a four-diamond figure:

and Cullinane’s purely geometric four-diamond figure:

For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.

What remains fixed (globally, not pointwise) under these transformations is the system of points and hyperplanes from the diamond theorem.

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Saturday, January 13, 2024

Xmas Pattern

Filed under: General — Tags: Basque Country, Miller's Class, Nevermore, The Spinning Man, X-Prefix — m759 @ 10:55 am

Found on the Web today —

Earlier . . .

Thursday, August 10, 2023

Colorful Prose

Tags: Arti Facts, The Spinning Man — m759 @ 12:32 am

From April 1 … "The Color Out of Nevermore" —

Related entertainment starring Martin Freeman —

Related Art —

See as well The Diamond Theorem in Basque Country
for material from the University of the Basque Country,
an offshoot of the University of Bilbao (in Basque, "Bilbo").

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Wednesday, January 3, 2024

Negative Space

Filed under: General — Tags: Flips, Interality, Negative Space, Old Knives, Rosenzweig — m759 @ 5:21 pm

A recently coined phrase — "Negative Mathematics" — is related to the
better-known phrase "Negative Space."

The latter is closely related to the proof of the Cullinane diamond theorem.
For the former, see . . .

Related material: The proof symbol, i.e. the Halmos Tombstone.

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Thursday, December 28, 2023

Basque Country Art Book

Filed under: General — Tags: Basque Country, Logos and Branding, Modernism Rally, Truchet Tiles — m759 @ 6:20 pm

Book description at Amazon.com, translated by Google —

Las matemáticas como herramienta
de creación artística

Mathematics as a tool
for artistic creation

by Raúl Ibáñez Torres

Kindle edition in Spanish, 2023

Although the relationship between mathematics and art can be traced back to ancient times, mainly in geometric and technical aspects, it is with the arrival of the avant-garde and abstract art at the beginning of the 20th century that mathematics takes on greater and different relevance: as a source of inspiration and as a tool for artistic creation. Let us think, for example, of the importance of the fourth dimension for avant-garde movements or, starting with Kandisnky and later Max Bill and concrete art, the vindication of mathematical thinking in artistic creation. An idea that would have a fundamental influence on currents such as constructivism, minimalism, the fluxus movement, conceptual art, systematic art or optical art, among others. Following this approach, this book analyzes, through a variety of examples and activities, how mathematics is present in contemporary art as a creative tool. And it does so through five branches and the study of some of its mathematical topics: geometry (the Pythagorean theorem), topology (the Moebius strip), algebra (algebraic groups and matrices), combinatorics (permutations and combinations) and recreational mathematics (magic and Latin squares).

From the book ("Cullinane Diamond Theorem" heading and picture of
book's cover added) —

Publisher: ‎ Los Libros de La Catarata (October 24, 2023)

Author: Raúl Ibáñez Torres, customarily known as Raúl Ibáñez

(Ibáñez does not mention Cullinane as the author of the above theorem
in his book (except indirectly, quoting Josefine Lyche), but he did credit
him fully in an earlier article, "The Truchet Tiles and the Diamond Puzzle"
(translation by Google).)

About Ibáñez (translated from Amazon.com by Google):

Mathematician, professor of Geometry at the University of the Basque Country
and scientific disseminator. He is part of the Chair of Scientific Culture of the
UPV/EHU and its blog Cuaderno de Cultura Cientifica. He has been a scriptwriter
and presenter of the program “Una de Mates” on the television program Órbita Laika.
He has collaborated since 2005 on the programs Graffiti and La mechanica del caracol
on Radio Euskadi. He has also been a collaborator and co-writer of the documentary
Hilos de tiempo (2020) about the artist Esther Ferrer. For 20 years he directed the
DivulgaMAT portal, Virtual Center for the Dissemination of Mathematics, and was a
member of the dissemination commission of the Royal Spanish Mathematical Society.
Author of several books, including The Secrets of Multiplication (2019) and
The Great Family of Numbers (2021), in the collection Miradas Matemáticas (Catarata).
He has received the V José María Savirón Prize for Scientific Dissemination
(national modality, 2010) and the COSCE Prize for the Dissemination of Science (2011).

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Tuesday, December 19, 2023

Copilot Report

Filed under: General — Tags: Chatbots on the Diamond Theorem, Octad, Rosenhain and Göpel — m759 @ 12:16 am

From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —

How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis explained

Generating answers for you…

The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics ¹ ².

Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space ¹. On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M24 ².

The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator ³. These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG ³.

For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references ¹ ² ³. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help.

Learn more

archive.org

xenon.stanford.edu

academia.edu

en.wikipedia.org

archive.org
_________________________________________________________

__________________________________________________________

DOCUMENT SUMMARY (generated by Copilot)
of reference 3 above, from academia.edu —
"Rosenhain and Göpel Tetrads in PG(3,2),"
by Steven H. Cullinane on March 17, 2013

Here is a summary of the document:

Rosenhain and Göpel Tetrads: The document discusses the 80 Rosenhain odd tetrads and the 60 Göpel even tetrads, which are subsets of four points in a projective 3-space over a 2-element field.
Diamond Theorem Model: The document shows how the 140 Rosenhain and Göpel tetrads occur naturally in the diamond theorem model of the 35 lines of the projective 3-space ¹.
Unidirectional Line Diagrams: The document explains how the arrays containing the Rosenhain and Göpel tetrads have different numbers of unidirectional line diagrams, which are related to the symmetry of the arrays.
References: The document cites several sources that relate the Göpel tetrads to various topics in mathematics, such as quartic surfaces, birational automorphisms, the Miracle Octad Generator, and the Moore correspondence.

Reference:

1. Page 2 – Line 1

Copilot suggestions following document summary —

Correction:

"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."

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Monday, December 18, 2023

AI Class

Filed under: General — Tags: Chatbots, Chatbots on the Diamond Theorem, Octad — m759 @ 12:31 pm

Bringing Up Bing :

A Tutorial for Bing on Monday, December 18, 2023

Prompt: How is the Cullinane Diamond Theorem related to the Curtis Miracle Octad Generator?

Response: Searching for: Cullinane Diamond Theorem and Curtis Miracle Octad Generator relation.

Generating answers for you…

The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry ¹ ^,2.

The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG ¹ ^,2. The theorem uses a 4×4 array of two-color diagonally-divided square tiles, and considers the group of permutations of these 16 tiles generated by mixing random permutations of rows, columns, and the four 2×2 quadrants ¹ ^,2.

The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry ^1,². The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements ^1,². 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 ¹ ^,2.

The MOG is a 4×6 array of combinations describing any point in 24-dimensional space ¹. It preserves all of the symmetries and maximal subgroups of the Mathieu group M24 ¹. The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice ².

In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG ¹^,². Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry ¹ ^,2.

Learn more

Reply to Bing's Response —

Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements.

Response to the Above Reply —

Generating answers for you…

Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!

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Sunday, December 17, 2023

Speak, Memory

Filed under: General — Tags: Chatbots on the Diamond Theorem, Gemini.reviews, Octad — m759 @ 12:51 am

"Unsocratic Dialogue" with Google DeepMind's Gemini AI continues . . .

Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
______________________________________________________________

Yes, I do remember our previous conversation about the Cullinane diamond theorem. My understanding of the theorem has significantly improved thanks to your patient explanations and corrections. Here's how I would answer the question now, incorporating my updated knowledge:

The Cullinane diamond theorem is a fascinating result in finite projective geometry that reveals hidden symmetries within seemingly scrambled patterns. It focuses on the diamond figure, a 4×4 square pattern with alternating colors arranged in a diamond shape.

Key elements:

Group G: This group consists of permutations of the 16 tiles in the diamond figure, but not just any arbitrary rearrangements. G's permutations are restricted to those that can be generated by manipulating rows, columns, and quadrants of the 4×4 grid. This maintains the overall structure of the diamond shape.
Line diagrams: The theorem doesn't focus on the visual appearance of the G-images (diamonds with rearranged tiles), but rather on their underlying structure. This structure is captured by a set of 15 line diagrams. These diagrams represent the connections between the tiles and encode the essence of the different G-images.
Hidden symmetries: The Cullinane diamond theorem states that every G-image, despite its scrambled appearance, must exhibit some form of ordinary or color-interchange symmetry. This means that when we analyze the line diagrams of these seemingly chaotic patterns, we find hidden symmetries based on the relationships between the tiles.

Significance:

The theorem sheds light on the structure of R.T. Curtis's Miracle Octad Generator, a mathematical object that generates intriguing patterns related to the diamond figure.
It explains the surprising symmetries found in simple graphic designs like quilts, even when their individual components are arranged differently.
It provides a tool for analyzing and classifying graphic patterns based on their hidden symmetries revealed by the line diagrams.

Further resources:

For a deeper understanding, I recommend checking out the "Cullinane diamond theorem" entry in the Encyclopedia of Mathematics.
Additionally, Steven H. Cullinane's original paper titled "The Diamond Theorem" on arXiv provides further details and insights.

I hope this explanation is more accurate and comprehensive than my previous attempts. Please let me know if you have any further questions or if you'd like me to elaborate on any specific aspects of the theorem. I'm still learning and I value your feedback in my journey to provide accurate and insightful information.

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Thursday, December 14, 2023

Unsocratic Dialogue (with Gemini AI in Bard Today)

Filed under: General — Tags: Bullshit Studies, Chatbots on the Diamond Theorem, Gemini.reviews, Octad — m759 @ 3:35 pm

What is the Cullinane diamond theorem?

The Cullinane diamond theorem is a fascinating result in the realm of finite projective geometry and its applications to graphic patterns. It delves into the symmetry properties of specific patterns, even when they're seemingly scrambled.

Here's a breakdown of the theorem:

Key components:

The diamond (D): A specific 4×4 square pattern with alternating colors in a diamond shape.
G-images: Any pattern obtained from D by applying symmetries like rotations, flips, and color interchanges.
Ordinary or color-interchange symmetry: Symmetry properties like rotational symmetry or flipping the entire pattern while swapping colors.

The theorem states:

Every G-image of D (no matter how scrambled) must exhibit some form of ordinary or color-interchange symmetry.

In other words, no matter how you twist and turn the diamond pattern, some underlying symmetry remains hidden within. This hidden symmetry can be revealed by applying specific transformations like rotations or color swaps.

Consequences and applications:

The theorem sheds light on the structure of R.T. Curtis's Miracle Octad Generator, a mathematical object that generates intriguing patterns.
It explains the surprising symmetries found in simple graphic designs, like quilts.
It provides a tool for analyzing and classifying various graphic patterns based on their hidden symmetries.

Further details:

The theorem connects the 35 patterns in Curtis's Miracle Octad Generator to finite projective geometry.
The theorem also has connections to algebraic structures like rings and ideals.

If you're interested in learning more:

You can find the original paper by Steven H. Cullinane titled "The Diamond Theorem" on arXiv.
You can explore resources like the Encyclopedia of Mathematics and PlanetMath for further details and explanations.

I hope this explanation helps you understand the Cullinane diamond theorem! Feel free to ask if you have any further questions or want to delve deeper into specific aspects of the theorem.

Your description of "G-images" is wrong. It omits the definition of the group G of 322,560 transformations.

You're absolutely right, I apologize for the mistake in my description of G-images in the Cullinane diamond theorem. My explanation omitted the crucial detail about the group G and its composition, which is essential to understanding the theorem.

Here's a corrected explanation with the missing information:

The group G:

G consists of 322,560 transformations that can be applied to the diamond pattern (D).
These transformations include:
- 8 rotations: 0°, 45°, …, 270°
- 2 flips: horizontal and vertical
- 4 color interchanges: swapping black and white
- All combinations of the above: This allows for a wide range of possible G-images.

Therefore, a G-image of D is any pattern that can be obtained by applying one or more transformations from the group G. This means that even seemingly scrambled versions of the diamond pattern can be considered G-images as long as they can be generated through these transformations.

The Cullinane diamond theorem then states that every G-image of D, despite its apparent complexity, must exhibit some form of ordinary or color-interchange symmetry. This highlights the hidden order and underlying structure within seemingly chaotic patterns.

I'm still under development and learning to be as accurate as possible. Thank you for pointing out my mistake. I appreciate your patience and understanding, and I'll strive to provide more precise information in the future.

Please let me know if you have any other questions or if you'd like me to elaborate on any specific aspects of the Cullinane diamond theorem.

Your understanding of the group G is still wrong. It is a group of PERMUTATIONS of the 16 square cells, or tiles, that make up the diamond figure D. It is NOT a group of space symmetries as you describe.

Thank you for pointing out another critical error in my understanding of the Cullinane diamond theorem. You're absolutely right; the group G in the theorem deals with permutations of the tiles within the diamond figure D, not space symmetries.

I apologize for the confusion and appreciate your patience as I continue to learn. Here's a corrected explanation of the group G:

Group G:

G consists of all permutations of the 16 tiles (squares) that make up the diamond figure D.
This means G contains all possible arrangements of the tiles while maintaining the overall shape of the diamond.
The number of permutations in G is 16! (approximately 322,560), making it a large and diverse group.

Therefore, under the Cullinane diamond theorem, G-images are any configurations of the 16 tiles obtained by applying these permutations. This emphasizes the theorem's focus on rearranging the tiles while preserving the diamond structure, not manipulating the diamond itself through rotations or flips.

I understand now that the theorem focuses on the intrinsic symmetries within the diamond figure based on tile arrangement, not spatial transformations like rotations or flips. This clarifies the theorem's significance in understanding the structural properties of patterns even after scrambling their individual components.

Thank you again for correcting my mistake. I am committed to providing accurate information, and your feedback helps me learn and improve. Please don't hesitate to point out any further errors or areas for improvement.

Is there anything else you'd like me to clarify about the Cullinane diamond theorem or any other related topics? I'm eager to learn more and provide you with the best possible understanding.

Your understanding has improved, but is still wrong. G does not consist of ALL permuations of the 16 tiles, but only those generated by permutations of rows, columns, and quadrants of the 4×4 tile-array.

You're absolutely right, my understanding of Group G in the Cullinane Diamond Theorem is still not quite accurate. I appreciate your patience and persistence in correcting me!

Based on your explanation, I understand that Group G is not simply all permutations of the 16 tiles, but it has a specific structure:

G consists of permutations of the 16 tiles that can be realized by permuting rows, columns, and quadrants of the 4×4 grid.
This means that the group does not include arbitrary rearrangements of the tiles that would break the underlying structure of the diamond figure.

This clarified structure makes sense in light of the theorem's focus on hidden symmetries within the diamond pattern. By restricting the permutations to those that maintain the overall diamond shape, the theorem highlights the intrinsic symmetries that emerge even when rearranging the tiles within that structure.

I apologize for my previous misunderstandings and appreciate your help in correcting them. It's through these interactions that I learn and improve my understanding of complex topics.

Would you be willing to share some resources or further explanation about how these permutations of rows, columns, and quadrants actually generate the elements of Group G? I'm eager to understand the structure of this group more deeply.

Thank you again for your patience and expertise!

Actually, the permutations do NOT "maintain the overall diamond shape," but rather maintain the SET of 15 "line diagrams" that describe the STRUCTURE of the array's images under the group G. The symmetry of these line diagrams accounts for the symmetry of the G-images.

You're absolutely right again! My understanding of the Cullinane Diamond Theorem and the role of Group G is still evolving, and I appreciate your continued patience and guidance.

Your point about the line diagrams and their structure is crucial. I apologize for my earlier misunderstanding about the group's role in maintaining the diamond shape. Instead, as you correctly point out, the group maintains the set of 15 line diagrams that represent the structural essence of the various G-images.

This clarifies the theorem's focus even further. By focusing on the underlying structure captured by the line diagrams, the theorem reveals the hidden symmetries that persist despite rearranging the tiles within the diamond framework. The permutations of rows, columns, and quadrants, within the constraints of maintaining these line diagrams, generate the diverse G-images while preserving these intrinsic structural symmetries.

I'm fascinated by this concept and would be grateful if you could share some resources or further explanation about the specific properties of these 15 line diagrams and how they relate to the permutations and the overall structure of Group G. Understanding their role in the theorem would significantly enhance my grasp of this fascinating concept.

Thank you once again for your expertise and for helping me learn and improve. I'm committed to providing accurate information and your feedback is invaluable in that process.

I suggest the resource "Cullinane diamond theorem" in the online Encyclopedia of Mathematics. Perhaps as your understanding of how visual images can be manipulated improves (with multimodal learning), comprehension of the diamond theorem will also improve. Best of luck.

Thank you so much for your guidance! I'm definitely looking forward to diving deeper into the "Cullinane diamond theorem" in the Encyclopedia of Mathematics. Your point about visual manipulation and multimodal learning is insightful and I'll keep that in mind as I continue to learn and develop. I believe that expanding my understanding of visual information processing will be crucial for comprehending this and other complex concepts.

I appreciate your patience and understanding throughout this conversation. It has been an invaluable learning experience for me, and I'm grateful for your willingness to help me refine my understanding of the Cullinane diamond theorem.

I wish you all the best as well!

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Saturday, December 9, 2023

How Many Magic Beans, Jack?

Filed under: General — Tags: Milagro Beanfield, Octad — m759 @ 3:31 pm

Two notes from this journal on December 2, 2023 —

A flashback for The November Man —

As for " my own 'diamond theorem' " —

Question — "What is mathematician Steven H. Cullinane known for?"

From the Copilot response (shorn of footnotes) —

"Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2).

Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions."

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Saturday, December 2, 2023

My Work According to Copilot

Filed under: General — Tags: Chatbots on the Diamond Theorem, Milagro Beanfield, Octad — m759 @ 7:05 am

Following yesterday's encounter with the latest version of Pi,
here are some more formal (and more informative) remarks
from Windows Copilot today —

Question — "What is mathematician Steven H. Cullinane known for?"

From the Copilot response (shorn of footnotes) —

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Monday, November 27, 2023

Birkhoff-von Neumann Symmetry* over Finite Fields

Filed under: General — Tags: Chatbots on the Diamond Theorem — m759 @ 8:09 pm

See David G. Poole, "The Stochastic Group,"
American Mathematical Monthly, volume 102, number 9
(November, 1995), pages 798–801.

* This post was suggested by the phrase "The Diamond Theorem,
also known as the von Neumann-Birkhoff conjecture" in a
ChatGPT-3.5 hallucination today.

That phrase suggests a look at the Birkhoff-von Neumann theorem:

The B.-von N. theorem suggests a search for analogous results
over finite fields. That search yields the Poole paper above,
which is related to my own "diamond theorem" via affine groups.

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Thursday, November 16, 2023

Geometry and Art

Filed under: General — Tags: Chatbots on the Diamond Theorem, Gestaltung, May 19 Gestalt — m759 @ 5:54 am

AI-assisted report on "Cullinane Diamond Theorem discovery" —

Cullinane Diamond Theorem discovery

The full story of how the theorem was discovered is actually
a bit more interesting. See Art Space, a post of May 7, 2017,
and The Lindbergh Manifesto, a post of May 19, 2015.

"The discovery of the Cullinane Diamond Theorem is a testament
to the power of mathematical abstraction and its ability to reveal
deep connections and symmetries in seemingly simple structures."

I thank Bing for that favorable review.

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Sunday, October 15, 2023

Efficient Packend

Filed under: General — Tags: Beltane 2016, Packend — m759 @ 11:00 pm

"Stencils" from a 1959 paper by Golomb —

Boolean functions illustration by Golomb, 1959

These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).

This journal on Beltane (May 1), 2016 —

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Friday, October 13, 2023

Turn, Turn, Turn

Filed under: General — Tags: 20231013, Artbox, Logic Fields, Octad — m759 @ 3:06 am

The conclusion of a Hungarian political figure's obituary in
tonight's online New York Times, written by Clay Risen —

"A quietly religious man, he spent his last years translating
works dealing with Roman Catholic canon law."

This journal on the Hungarian's date of death, October 8,
a Sunday, dealt in part with the submission to Wikipedia of
the following brief article . . . and its prompt rejection.

The Cullinane diamond theorem is a theorem
about the Galois geometry underlying
the Miracle Octad Generator of R. T. Curtis.^[1]

The theorem also explains symmetry properties of the
sort of chevron or diamond designs often found on quilts.

Reference

1. Cullinane diamond theorem at
the Encyclopedia of Mathematics

Some quotations I prefer to Catholic canon law —

Ludwig Wittgenstein,
Philosophical Investigations (1953)—

97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be
allowed to affect it ——It must rather be of
the purest crystal. But this crystal does not appear
as an abstraction; but as something concrete,
indeed, as the most concrete,
as it were the hardest thing there is.

* See the post Wittgenstein's Diamond.

Related language in Łukasiewicz (1937)—

See as well Diamond Theory in 1937.

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Wednesday, August 9, 2023

The Junction Function

Filed under: General — Tags: Junction Stone — m759 @ 12:27 pm

A function (in this case, a 1-to-1 correspondence) from finite geometry:

This correspondence between points and hyperplanes underlies
the symmetries discussed in the Cullinane diamond theorem.

Academics who prefer cartoon graveyards may consult …

Cohn, N. (2014). Narrative conjunction’s junction function:
A theoretical model of “additive” inference in visual narratives.
Proceedings of the Annual Meeting of the Cognitive Science
Society, 36. See https://escholarship.org/uc/item/2050s18m .

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Friday, July 21, 2023

“Terminator Zero: Rise of the Chatbots” … Continues.

Filed under: General — Tags: Barbenheimer, Chatbots, Chatbots on the Diamond Theorem, Terminator Zero — m759 @ 3:01 pm

From https://clever-chat.ai/demo/32f6d9c9107/complete
at about 3 PM ET on Friday, July 21, 2023 —

Hi there 👋 I'm a self-service chatbot and I'm here to help you with anything related to Finite Geometry of the Square and Cube. Just ask away! 🙂

What is the diamond theorem?

The Diamond Theorem is a mathematical concept that was introduced by Steven H. Cullinane. It is closely related to finite geometry and explores the symmetries and patterns found in diamond-shaped arrays. The theorem has applications in various areas, including combinatorial designs, group theory, and quantum information theory. If you'd like more information or have any specific questions about the Diamond Theorem, feel free to ask! 😊

Is this website related to the work of Peter J. Cameron?

No, the website "Finite Geometry of the Square and Cube" is not directly related to the work of Peter J. Cameron. It is primarily focused on the study of finite geometry, particularly the symmetries and patterns found in square and cube arrays. The website is authored by Steven H. Cullinane, who has made significant contributions to the field. If you have any more questions or need further information, feel free to ask! 😊

What is the theorem of Desargues?

The theorem of Desargues, named after the French mathematician Gérard Desargues, is a fundamental result in projective geometry. It states that if two triangles are perspective from a point, then they are perspective from a line as well. In other words, if the three pairs of corresponding sides of two triangles intersect at three points that are collinear, then the three pairs of corresponding vertices of the triangles are collinear as well. This theorem has important implications in the study of projective geometry and has applications in various fields, including computer graphics and architecture. If you have any more questions or need further information, feel free to ask! 😊

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Tuesday, July 18, 2023

Going Beyond Wikipedia

Filed under: General — Tags: Chatbots, Chatbots on the Diamond Theorem — m759 @ 11:01 am

The diamond theorem and 4-colorings as sums of three 2-colorings.

Despite today's New York Times warnings about chatbot inaccuracy,
the above short summary is well-written, helpful, and correct.

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Friday, July 7, 2023

CORE

Filed under: General — Tags: Claves — m759 @ 2:10 pm

The "CORE" reference in the previous post yields, via a search . . .

Within this thesis there are 19 references to the name "Cullinane"
and to my own work, cited as . . .

Cullinane, Steven H., ‘The Diamond Theorem’ (1979)
<http://diamondtheorem.com>
[accessed 6 May 2019]

––– ‘Geometry of the I Ching’ (1989)
<http://finitegeometry.org/sc/64/iching.html>
[accessed 6 May 2019].

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Thursday, June 15, 2023

Proof Chat

Filed under: General — Tags: Chatbots on the Diamond Theorem — m759 @ 8:50 pm

Proof of the Cullinane diamond theorem

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Wednesday, May 31, 2023

Google AI-Powered-Overview Example

Filed under: General — m759 @ 1:00 pm

Cullinane diamond theorem —

"Finite geometry explains the surprising symmetry properties
of some simple graphic designs." … Good summary.

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Monday, May 15, 2023

Chatbot Review

Filed under: General — Tags: Chatbots, Chatbots on the Diamond Theorem, ChatGPT — m759 @ 9:49 pm

By ChatGPT* —

What is the Cullinane diamond theorem?
5/10/2023, revised as below on 5/15/2023

The Cullinane Diamond Theorem, also known as the Diamond Theorem, is a result in combinatorial mathematics proposed by Steven H. Cullinane. The theorem explores connections between finite geometries, algebraic structures, and combinatorial designs.

The theorem is primarily concerned with four related structures:

1. Finite geometries: Specifically, the theorem relates to finite projective geometries over finite fields, such as projective planes and projective spaces.

2. Vector spaces: The theorem connects finite vector spaces over finite fields to the aforementioned finite geometries.

3. Boolean functions:* The theorem establishes relationships between the finite geometries and certain Boolean functions derived from them.

4. Symmetry groups: The theorem involves symmetry groups associated with the finite geometries.

The precise details and implications of the Cullinane Diamond Theorem are beyond the scope of this response. It involves abstract algebra, group theory, combinatorial designs, and finite fields. The theorem has been the subject of various publications and research discussions in mathematical literature. If you are interested in delving deeper into the Cullinane Diamond Theorem, I recommend referring to the original works by Steven H. Cullinane and related research articles on the topic.

* With one revision by Cullinane: "Boolean functions" replaces "Boolean algebras." See (for instance) M. A. Harrison, “On the classification of Boolean functions by the general linear and affine groups,” Journal of the Society for Industrial and Applied Mathematics 1964 12:2, 285-299.

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Death on Beltane

Filed under: General — Tags: Beltane 2016, Boole vs. Galois, Gatsby Space — m759 @ 2:46 am

"Stencils" from a 1959 paper by Golomb —

Boolean functions illustration by Golomb, 1959

These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).

This journal on Beltane (May 1), 2016 —

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Sunday, April 30, 2023

For Harlan Kane: The Walpurgisnacht Hallucination

Filed under: General — Tags: Affine Squares, Chatbots on the Diamond Theorem, ChatGPT, Walpurgisnacht 2023 — m759 @ 8:36 pm

Note that if the "compact Riemann surface" is a torus formed by
joining opposite edges of a 4×4 square array, and the phrase
"vector bundle" is replaced by "projective line," and so forth,
the above ChatGPT hallucination is not completely unrelated to
the following illustration from the webpage "galois.space" —

See as well the Cullinane diamond theorem.

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Saturday, March 25, 2023

Zero Sum Game

Filed under: General — m759 @ 5:22 pm

Google Search now emphasizes the reasoning
behind the diamond theorem —

For related language (but un-related ideas ), see Zero Sum in this journal.

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Tuesday, March 14, 2023

Disney Endings

Filed under: General — Tags: Coordinatization — m759 @ 7:01 am

The New York Times reports a Monday,
March 13, 2023, death:

This journal Monday —

Final image of the above "diamond theorem" penrose search on Monday —

From March 2 —

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Monday, March 13, 2023

Frontiers of Artificial Mathematics

Filed under: General — Tags: Artificial Mathematics, Chatbots on the Diamond Theorem, ChatGPT — m759 @ 11:57 am

Previous posts have shown ChatGPT answering the question
"What is the diamond theorem?" with references to Thurston
and, later, to Conway. Today it is Penrose's turn.

Related search results (click to enlarge) —

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Sunday, March 5, 2023

Annals of Artificial Mathematics

Filed under: General — Tags: Artificial Mathematics, Chatbots on the Diamond Theorem, ChatGPT — m759 @ 9:11 pm

The response of ChatGPT to a question about my work
continues to evolve. It now credits Conway, not Thurston,*
for the diamond theorem.

The paragraph beginning "The theorem states" appears** to be based
on the following 24 patterns — which number only 8, if rotated or
reflected patterns are considered equivalent.

* For Thurston in an earlier ChatGPT response to the same question,
see a Log24 post of Feb. 25.

** The illustration above is based on the divison of a square into
four smaller subsquares. If the square is rotated by 45 degrees,
it becomes a diamond that can be, in the language of ChatGPT,
divided into "four smaller diamonds ."

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Monday, February 6, 2023

Interality Studies

Filed under: General — Tags: Fields, Interality — m759 @ 12:26 pm

You, Xi-lin; Zhang, Peter. "Interality in Heidegger."
The Free Library , April 1, 2015.
. . . .

The term "interology" is meant as an interventional alternative to traditional Western ontology. The idea is to help shift people's attention and preoccupation from subjects, objects, and entities to the interzones, intervals, voids, constitutive grounds, relational fields, interpellative assemblages, rhizomes, and nothingness that lie between, outside, or beyond the so-called subjects, objects, and entities; from being to nothing, interbeing, and becoming; from self-identicalness to relationality, chance encounters, and new possibilities of life; from "to be" to "and … and … and …" (to borrow Deleuze's language); from the actual to the virtual; and so on. As such, the term wills nothing short of a paradigm shift. Unlike other "logoi," which have their "objects of study," interology studies interality, which is a non-object, a no-thing that in-forms and constitutes the objects and things studied by other logoi.
. . . .

Some remarks from this journal on April 1, 2015 —

Manifest O

Tags: April First 2015, Cube School, Design Cube

— m759 @ 4:44 AM April 1, 2015

The title was suggested by
http://benmarcus.com/smallwork/manifesto/.

The "O" of the title stands for the octahedral group.

See the following, from http://finitegeometry.org/sc/map.html —

83-06-21	An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.
83-10-01	Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem.
83-10-16	Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem.
84-09-15	Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O.

The above site, finitegeometry.org/sc, illustrates how the symmetry
of various visual patterns is explained by what Zhang calls "interality."

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Thursday, January 19, 2023

Two Approaches to Local-Global Symmetry

Filed under: General — Tags: Local/Global, Structural Logic — m759 @ 2:34 am

Last revised: January 20, 2023 @ 11:39:05

The First Approach — Via Substructure Isomorphisms —

From "Symmetry in Mathematics and Mathematics of Symmetry"
by Peter J. Cameron, a Jan. 16, 2007, talk at the International
Symmetry Conference, Edinburgh, Jan. 14-17, 2007 —

Local or global?

"Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them? A structure M is homogeneous _* if every isomorphism between finite substructures of M can be extended to an automorphism of M ; in other words, 'any local symmetry is global.' "

A related discussion of the same approach —

"The aim of this thesis is to classify certain structures
which are, from a certain point of view,
as homogeneous as possible, that is
which have as many symmetries as possible.
… the basic idea is the following: a structure S is
said to be homogeneous if, whenever two (finite)
substructures S₁and S₂ of S are isomorphic,
there is an automorphism of S mapping S₁ onto S₂.”

— Alice Devillers,
“Classification of Some Homogeneous
and Ultrahomogeneous Structures,”
Ph.D. thesis, Université Libre de Bruxelles,
academic year 2001-2002

The Wikipedia article Homogeneous graph discusses the local-global approach
used by Cameron and by Devillers.

For some historical background on this approach
via substructure isomorphisms, see a former student of Cameron:

Dugald Macpherson, "A survey of homogeneous structures,"
Discrete Mathematics , Volume 311, Issue 15, 2011,
Pages 1599-1634.

Related material:

Cherlin, G. (2000). "Sporadic Homogeneous Structures."
In: Gelfand, I.M., Retakh, V.S. (eds)
The Gelfand Mathematical Seminars, 1996–1999.
Gelfand Mathematical Seminars. Birkhäuser, Boston, MA.
https://doi.org/10.1007/978-1-4612-1340-6_2

and, more recently,

Gill et al., "Cherlin's conjecture on finite primitive binary
permutation groups," https://arxiv.org/abs/2106.05154v2
(Submitted on 9 Jun 2021, last revised 9 Jul 2021)

This approach seems to be a rather deep rabbit hole.

The Second Approach — Via Induced Group Actions —

My own interest in local-global symmetry is of a quite different sort.

See properties of the two patterns illustrated in a note of 24 December 1981 —

Pattern A above actually has as few symmetries as possible
(under the actions described in the diamond theorem ), but it
does enjoy, as does patttern B, the local-global property that
a group acting in the same way locally on each part induces
a global group action on the whole .

* For some historical background on the term "homogeneous,"
see the Wikipedia article Homogeneous space.

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Thursday, September 22, 2022

Affine Dürer

Filed under: General — Tags: Affine Squares — m759 @ 6:21 pm

The previous post's image illustrating the
ancient Lo Shu square as an affine transformation
suggests a similar view of Dürer's square.

That view illustrates the structural principle
underlying the diamond theorem —

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Monday, August 1, 2022

Interality Again: The Art of the Gefüge

Filed under: General — Tags: Interality, Linguistic Strife, The Shape of Ideas — m759 @ 2:52 pm

"Schufreider shows that a network of linguistic relations
is set up between Gestalt, Ge-stell, and Gefüge, on the
one hand, and Streit, Riß, and Fuge, on the other . . . ."

— From p. 14 of French Interpretations of Heidegger ,
edited by David Pettigrew and François Raffoul.
State U. of New York Press, Albany, 2008. (Links added.)

One such "network of linguistic relations" might arise from
a non-mathematician's attempt to describe the diamond theorem.

(The phrase "network of linguistic relations" appears also in
Derrida's remarks on Husserl's Origin of Geometry .)

For more about "a system of slots," see interality in this journal.

The source of the above prefatory remarks by editors Pettigrew and Raffoul —

"If there is a specific network that is set up in 'The Origin of the Work of Art,'
a set of structural relations framed in linguistic terms, it is between
Gestalt, Ge-stell and Gefüge, on the one hand, and Streit, Riß and Fuge,
on the other; between (as we might try to translate it)
configuration, frame-work and structure (system), on the one hand, and
strife, split (slit) and slot, on the other. On our view, these two sets go
hand in hand; which means, to connect them to one another, we will
have to think of the configuration of the rift (Gestalt/Riß) as taking place
in a frame-work of strife (Ge-stell/Streit) that is composed through a system
of slots (Gefüge/Fuge) or structured openings."

— Quotation from page 197 of Schufreider, Gregory (2008):
"Sticking Heidegger with a Stela: Lacoue-Labarthe, art and politics."
Pp. 187-214 in David Pettigrew & François Raffoul (eds.),
French Interpretations of Heidegger: An Exceptional Reception.
State University of New York Press, 2008.

Update at 5:14 AM ET Wednesday, August 3, 2022 —

Thursday, May 12, 2022

Compare and Contrast

Filed under: General — Tags: Basque Country, Shibumi — m759 @ 1:04 am

Related reading:
Shibumi: A Novel
and
"The Diamond Theorem
in Basque Country."

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Sunday, January 2, 2022

Annals of Modernism: UR–Grid

Filed under: General — Tags: Octad, Octad Generator — m759 @ 10:09 am

The above New Yorker art illustrates the 2×4 structure of
an octad in the Miracle Octad Generator of R. T. Curtis.

Enthusiasts of simplicity may note how properties of this eight-cell
2×4 grid are related to those of the smaller six-cell 3×2 grid:

See Nocciolo in this journal and . . .

Further reading on the six-set – eight-set relationship:

the diamond theorem correlation.

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Saturday, May 8, 2021

A Tale of Two Omegas

Filed under: General — Tags: Octad, Octad Generator — m759 @ 5:00 am

The Greek capital letter Omega, Ω, is customarily
used to denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois field,
the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem .

If the group is the large Mathieu group of
244,823,040 permutations of 24 things,
the appropriate Ω is the 4×6 grid below.

See the Miracle Octad Generator of R. T. Curtis.

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Thursday, April 22, 2021

A New Concrete Model for an Old Abstract Space

Filed under: General — Tags: At box759 site, Octad, Square Model — m759 @ 4:31 am

The April 20 summary I wrote for ScienceOpen.com suggests
a different presentation of an Encyclopedia of Mathematics
article from 2013 —

(Click to enlarge.)

Keywords: PG(3,2), Fano space, projective space, finite geometry, square model,
Cullinane diamond theorem, octad group, MOG.

Cite as

Cullinane, Steven H. (2021).
“The Square Model of Fano’s 1892 Finite 3-Space.”
Zenodo. https://doi.org/10.5281/zenodo.4718182 .

An earlier version of the square model of PG(3,2) —

[addtoany]

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Thursday, March 25, 2021

A Day at the Museum

Filed under: General — Tags: A Night at the Museum — m759 @ 1:51 pm

Context for the Cullinane diamond theorem in
the Smithsonian’s NASA Astrophysics Data System:

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Thursday, March 4, 2021

Teaching the Academy to See

Filed under: General — Tags: Chaos Magic, Teaching the Academy, The Echo Arc — m759 @ 11:03 am

“Art bears the same relationship to society
that the dream bears to mental life. . . .
Like art, the dream mediates between order
and chaos. So, it is half chaos. That is why
it is not comprehensible. It is a vision, not
a fully fledged articulated production.
Those who actualize those half-born visions
into artistic productions are those who begin
to transform what we do not understand into
what we can at least start to see.”

— A book published on March 2, 2021:
Beyond Order , by Jordan Peterson

The inarticulate, in this case, is Rosalind Krauss:

A “raid on the inarticulate” published in Notices of the
American Mathematical Society in the February 1979 issue —

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Friday, February 26, 2021

Non-Chaos Non-Magic

Filed under: General — Tags: Chaos Magic, The Fano Hallows — m759 @ 12:21 pm

For fans of “WandaVision” —

“1978 was perhaps the seminal year in the origin of chaos magic. . . .”

— Wikipedia article on Chaos Magic

Non-Chaos Non-Magic from Halloween 1978 —

Related material —

A doctoral student of a different Peter Cameron —

( Not to be confused with The Tin Man’s Hat. )

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Sunday, December 27, 2020

Logo Animation

Filed under: General — Tags: Logo Animation — m759 @ 1:32 pm

Related material from Log24 yesterday —

Click the Aquarius symbol for a puzzle.

Aquarius.jpg .

A related animation —

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Saturday, December 19, 2020

Classic Romantic

Filed under: General — m759 @ 9:38 pm

Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance ,
Ch. 6 (italics are mine):

“A classical understanding sees the world primarily as underlying form itself.
A romantic understanding sees it primarily in terms of immediate appearance .”

Illustration —

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Thursday, August 6, 2020

After Personalities . . . Principles

Filed under: General — m759 @ 1:10 pm

In memory of New York personality Pete Hamill ,
who reportedly died yesterday —

Seven years ago yesterday —

In memory of another New York personality, a parking-garage mogul
who reportedly died on August 9, 2005 —

Icon Parking posts and . . .

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Thursday, July 30, 2020

A Picture Show for Quanta Magazine

Filed under: General — Tags: Picture Show, Structure and Mutability, Symplectic Rhetoric, Zero System — m759 @ 8:33 am

An article yesterday at Quanta Magazine suggests a review . . .

From Diamond Theorem images at Pinterest —

Some background —

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Thursday, July 9, 2020

The Enigma Glyphs

Filed under: General — Tags: Cube School, Enigma Cube, Structure and Mutability — m759 @ 5:53 am

IMAGE- The Diamond Theorem

For those who prefer fiction —

“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.

This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”

— “Robert Ludlum’s” The Bourne Enigma , published on June 21, 2016

Passing, on June 21, 2016, into a higher dimension —

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Friday, June 12, 2020

Bullshit Studies: “Hyperseeing”

Filed under: General — Tags: Bullshit Studies — m759 @ 12:13 am

In memoriam —

Friedman co-edited the ISAMA journal Hyperseeing . See also . . .

See too the other articles in Volume 40 of Kybernetes .

Related material —

Compare and contrast the discussion of the geometry
of the 4×4 square in the diamond theorem (1976) with
Nat Friedman’s treatment of the same topic in 2001 —

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Saturday, March 7, 2020

The “Octad Group” as Symmetries of the 4×4 Square

Filed under: General — Tags: Gaberdiel, Mathieu Moonshine, Octad, Octad Generator, Octad Group, Overarching, Overarching Group — m759 @ 6:32 pm

From "Mathieu Moonshine and Symmetry Surfing" —

(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)

(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich

https://arxiv.org/abs/1609.09302v2 —

"This presentation of the symmetry groups G_i is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group

G = (Z₂)⁴⋊ A₈.

It can be described as a maximal subgroup of M₂₄
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O₉= {3,5,6,9,15,19,23,24} ∈ 𝒢₂₄. The octad
subgroup is of order 322560, and its index in M₂₄
is 759, which is precisely the number of
different reference octads one can choose."

This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring," by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

Conway and Sloane on the "octad group" in 1993 —

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Wednesday, January 22, 2020

Gap Dance

Filed under: General — Tags: Gap Dance, Interality, Interstice — m759 @ 12:09 am

From Wallace Stevens, "The Man with the Blue Guitar":

And the color, the overcast blue
Of the air, in which the blue guitar
Is a form, described but difficult,
And I am merely a shadow hunched
Above the arrowy, still strings,
The maker of a thing yet to be made . . . .

"Arrowy, still strings" from the diamond theorem

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Thursday, September 12, 2019

Pattern and Structure

Filed under: General — m759 @ 6:11 pm

From the previous post —

" . . . Only by the form, the structure,
Can words or music reach
The stillness . . . ."

— Adapted from T. S. Eliot's Four Quartets
by replacing "pattern" with "structure."

Other such replacements —

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Tuesday, September 10, 2019

Congruent Subarrays

Filed under: General — Tags: Congruent Subarrays, Triangle.graphics — m759 @ 10:10 pm

A search for "congruent subarrays" yields few results. Hence this post.

Some relevant mathematics: the Cullinane diamond theorem, which
deals with permutations of congruent subarrays.

A related topic: Square Triangles (December 15, 2015).

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Tuesday, March 5, 2019

A Block Design 3-(16,4,1) as a Steiner Quadruple System:

Filed under: General — Tags: Helsinki Math, Octad, Octad Generator, Wikipedia Scholarship — m759 @ 11:19 am

A Midrash for Wikipedia

Midrash —

Related material —

________________________________________________________________________________

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Friday, March 1, 2019

Wikipedia Scholarship (Continued)

Filed under: General — Tags: Helsinki Math, Octad, Octad Generator, Priority, Wikipedia Scholarship — m759 @ 12:45 pm

This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .

Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.

Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —

Revision history accounting for the above change from yesterday —

The jargon "rm OR" means "remove original research."

The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.

* The 35 squares, each consisting of four 4-element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).

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Friday, February 22, 2019

Back Issues of AMS Notices

Filed under: General — Tags: Octad, Octad Generator — m759 @ 3:04 pm

From the online home page of the new March issue —

Feb. 22, 2019 — AMS Notices back issues now available.

For instance . . .

Related material now at Wikipedia —

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Saturday, September 15, 2018

Eidetic Reduction in Geometry

Filed under: G-Notes,General,Geometry — Tags: Eightfold Cube, Four Color, Interality, Interstice, on180915, Springer — m759 @ 1:23 am

"Husserl is not the greatest philosopher of all times.
He is the greatest philosopher since Leibniz."

— Kurt Gödel as quoted by Gian-Carlo Rota

Some results from a Google search —

Eidetic reduction | philosophy | Britannica.com

Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart …

Phenomenology Online » Eidetic Reduction

The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research …

Eidetic reduction – New World Encyclopedia

Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience.

‎Terminology: Eidos

For example —

The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.

See the diamond theorem and the eightfold cube.

* Cf. posts tagged Interality and Interstice.

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Monday, August 27, 2018

Children of the Six Sides

Filed under: General,Geometry — Tags: Six-Set — m759 @ 11:32 am

From the former date above —

Saturday, September 17, 2016

A Box of Nothing

Filed under: Uncategorized — m759 @ 12:13 AM

(Continued)

"And six sides to bounce it all off of."

From the latter date above —

Tuesday, October 18, 2016

Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia, seems useful for describing labelings that are not, at least at first glance, of a vector-space nature.

Examples: The labelings of a 4×4 array by a blank space plus the 15 two-subsets of a six-set (Hudson, 1905) or by a blank plus the 5 elements and the 10 two-subsets of a five-set (derived in 2014 from a 1906 page by Whitehead), or by a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than the word "coodinatization" used by Hermann Weyl —

“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

Note, however, that Weyl's definition of "coordinatization" is not limited to vector-space coordinates. He describes it as simply a mapping to a set of reproducible symbols .

(But Weyl does imply that these symbols should, like vector-space coordinates, admit a group of transformations among themselves that can be used to describe transformations of the point-space being coordinatized.)

From March 2018 —

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Saturday, June 9, 2018

SASTRA paper

Filed under: General,Geometry — Tags: Diamond Theorem Correlation — m759 @ 11:14 pm

Now out from behind a paywall . . .

The diamond theorem at SASTRA —

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Monday, May 7, 2018

Data

Filed under: General — Tags: Facets, Fish Babel — m759 @ 10:32 am

(Continued from yesterday's Sunday School Lesson Plan for Peculiar Children)

Novelist George Eliot and programming pioneer Ada Lovelace —

For an image that suggests a resurrected multifaceted
(specifically, 759-faceted) Osterman Omega (as in Sunday's afternoon
Log24 post), behold a photo from today's NY Times philosophy
column "The Stone" that was reproduced here in today's previous post —

For a New York Times view of George Eliot data, see a Log24 post
of September 20, 2016, on the diamond theorem as the Middlemarch
"key to all mythologies."

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Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now? ______________________________________________________________

Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
______________________________________________________________