Log24

Wednesday, December 11, 2019

Miracle Octad Generator Structure

Miracle Octad Generator — Analysis of Structure

(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:24 pm

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M24," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis's 35  4×6  1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not  by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.

* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Monday, March 4, 2024

Report on the Diamond Theorem
by Google Gemini (Advanced)

Filed under: General — Tags: , — 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.

Sunday, March 3, 2024

Deep Blue Research: A Report by You.com AI

Filed under: General — Tags: , — 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

Tuesday, January 30, 2024

Cullinane Diamond Theorem:
The Latest Google Bard Version

Filed under: General — Tags: , , , — 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.]

Thursday, December 21, 2023

Pi.ai

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

Thursday, December 21, 2023

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

How's your day going?

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

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

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

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

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

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

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

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

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

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

Scholium —

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

Sia!

Tuesday, December 19, 2023

Copilot Report

Filed under: General — Tags: , , — 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 mathematics12.

Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242.

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 Generator3These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3.

For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. 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

5 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:

Reference:
 
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."

Monday, December 18, 2023

AI Class

Filed under: General — Tags: , , — 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 geometry1,2.

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 MOG1,2The 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 quadrants1,2.

The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry1,2The 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 elements1,2This 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-sets1,2.

The MOG is a 4×6 array of combinations describing any point in 24-dimensional space1It preserves all of the symmetries and maximal subgroups of the Mathieu group M241The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice2.

In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG1,2Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry1,2.

Learn more

  1. arxiv.org
  2. encyclopediaofmath.org
  3. arxiv.org
  4. m759.net

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!

Sunday, December 17, 2023

Speak, Memory

Filed under: General — Tags: , , — m759 @ 12:51 am

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

profile picture

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.

Thursday, December 14, 2023

Unsocratic Dialogue (with Gemini AI in Bard Today)

Filed under: General — Tags: , , , — m759 @ 3:35 pm

profile picture

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.

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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.

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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.

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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!

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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.

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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!

Saturday, December 9, 2023

How Many  Magic Beans, Jack?

Filed under: General — Tags: , — 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."

Saturday, December 2, 2023

My Work According to Copilot

Filed under: General — Tags: , , — 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) —

"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."

Saturday, November 18, 2023

“Don’t solicit for your sister,* it’s not nice.” — Tom Lehrer

Filed under: General — Tags: — m759 @ 11:36 am

André Weil to his sister:

From this journal at 1:51 AM  ET Thursday, September 8, 2022

"The pleasure comes from the illusion" . . .

Exercise:

Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).

http://www.log24.com/log/pix11B/110805-The24.jpg

Note that the 4-row-2-column "brick" at left is quite 
different from the other two bricks, which together
show chevron variations within a Galois tesseract —

.

Further Weil remarks . . .

A Slew of Prayers

"The pleasure comes from the illusion
and the far from clear meaning;
once the illusion is dissipated,
and knowledge obtained, one becomes
indifferent at the same time;
at least in the Gitâ there is a slew of prayers
(slokas) on the subject, each one more final
than the previous ones."

*

Friday, October 13, 2023

Turn, Turn, Turn

Filed under: General — Tags: , , — 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)—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

See as well Diamond Theory in 1937.

Saturday, September 23, 2023

The Cullinane Diamond Theorem at Wikipedia

Filed under: General — Tags: — 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.

Wednesday, May 3, 2023

Folklore

Filed under: General — Tags: , — m759 @ 8:58 am

"Piotr Grochowski's 'Miracles and Visionaries in the Digital Age'
explores the impact of digital technolgies in Catholic Apparitional
movements." (links added) — Journal of American Folklore
Fall issue, October 2022 preview.

As for "Miracles and Visionaries," I prefer the literature associated
with the 1974 Miracle Octad Generator  of R. T. Curtis.

Friday, October 28, 2022

The Lexeme “Deploy”

Filed under: General — Tags: , , , — m759 @ 2:22 pm

From an Oct. 26 elegy in the form of a book review …

From a search in this  journal for "deploy" —

Related art —

As for "Miracles and Visionaries," I prefer the literature associated
with the 1974 Miracle Octad Generator  of R. T. Curtis.

Thursday, September 29, 2022

The 4×6 Problem*

Filed under: General — Tags: , — m759 @ 4:03 pm

The exercise posted here on Sept. 11, 2022, suggests a 
more precisely stated problem . . .

The 24 coordinate-positions of the 4096 length-24 words of the 
extended binary Golay code G24 can be arranged in a 4×6 array
in, of course, 24! ways.

Some of these ways are more geometrically natural than others.
See, for instance, the Miracle Octad Generator of R. T. Curtis.
What is the size of the largest subcode C of G24 that can be 
arranged in a 4×6 array in such a way that the set  of words of C 
is invariant under the symmetry group of the rectangle itself, i.e. the
four-group of the identity along with horizontal and vertical reflections
and 180-degree rotation.

Recent Log24 posts tagged Bitspace describe the structure of
an 8-dimensional (256-word) code in a 4×6 array that has such
symmetry, but it is not yet clear whether that "cube-motif" code
is a Golay subcode. (Its octads are Golay, but possibly not all its
dodecads; the octads do not quite generate the entire code.) 
Magma may have an answer, but I have had little experience in
its use.

* Footnote of 30 September 2022.  The 4×6 problem is a
special case of a more general symmetric embedding problem.
Given a linear code C and a mapping of C to parts of a geometric
object A with symmetry group G, what is the largest subcode of C
invariant under G? What is the largest such subcode under all
such mappings from C to A?

Thursday, September 8, 2022

Analogy in Mathematics: Chevron Variations

André Weil in 1940 on analogy in mathematics —

. "Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one. In this sense, around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle (Gauss’s problem) and the division of elliptic functions. Today, we can easily show that both problems have a place in the theory of abelian equations; we have the theory (I am speaking of a purely algebraic theory, so it is not a matter of number theory in this case) of abelian extensions. Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time; at least in the Gitâ there is a slew of prayers (slokas) on the subject, each one more final than the previous ones."

"The pleasure comes from the illusion" . . .

Exercise:

Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).

http://www.log24.com/log/pix11B/110805-The24.jpg

Note that the 4-row-2-column "brick" at left is quite 
different from the other two bricks, which together
show chevron variations within a Galois tesseract —

Sunday, September 4, 2022

Dice and the Eightfold Cube

Filed under: General — Tags: , , — m759 @ 4:47 pm

At Hiroshima on March 9, 2018, Aitchison discussed another 
"hexagonal array" with two added points… not at the center, but
rather at the ends  of a cube's diagonal axis of symmetry.

See some related illustrations below. 

Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.

Iain Aitchison's 'dice-labelled' cuboctahedron at Hiroshima, March 2018

The exceptional role of  0 and  in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a 
Miracle Octad Generator octad —

Transposition of  0 and  in the knight coordinatization 
induces the symplectic polarity of PG(3,2) discussed by 
(for instance) Anne Duncan in 1968.

Friday, August 19, 2022

The Guy Embedding

Filed under: General — Tags: , , — m759 @ 3:43 pm

M. J. T. Guy discovered that the lexicographic  version
of the Golay code contains, embedded within it, the
Miracle Octad Generator  (MOG)  of R. T. Curtis.

For 12 basis vectors of the lexicographic version, see below.

Basis vectors for the lexicographic version of the binary Golay code

For some context, click the embedded guy.

For a closely related, but simpler, mathematical
structure, see posts tagged The Omega Matrix.

Thursday, February 3, 2022

Through the Asian Looking Glass

Filed under: General — Tags: , , , — m759 @ 10:59 pm

The Miracle Octad Generator (MOG) —
The Conway-Sloane version of 1988:

Embedding Change, Illustrated

See also the 1976 R. T. Curtis version, of which the Conway-Sloane version
is a mirror reflection —

“There is a correspondence between the two systems
of 35 groups, which is illustrated in Fig. 4 (the MOG or
Miracle Octad Generator).”
—R.T. Curtis, “A New Combinatorial Approach to M24,” 
Mathematical Proceedings of the Cambridge Philosophical
Society
 (1976), 79: 25-42

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

Curtis’s 1976 Fig. 4. (The MOG.)

The Guy Embedding (named for M.J.T., not Richard K., Guy) states that
the MOG is naturally embedded in the codewords of the extended binary
Golay code, if those codewords are generated in lexicographic order.

MOG in LOG embedding

The above reading order for the MOG 4×6 array —
down the columns, from left to right — yields the Conway-Sloane MOG.

Since that is a mirror image of the original Curtis MOG, the reading order
yielding that  MOG is down the columns, from right to left.

"Traditionally, ChineseJapaneseVietnamese and Korean are written vertically
in columns going from top to bottom and ordered from right to left, with each
new column starting to the left of the preceding one." — Wikipedia

The Asian reading order has certain artistic advantages:

Four-Color Structures (Review)

Filed under: General — Tags: , , — m759 @ 1:30 pm

Four-color decomposition applied to the 8-point binary affine space

Miracle Octad Generator — Analysis of Structure

For those who prefer art that is less abstract — Heartland Sutra.

Sunday, January 2, 2022

Annals of Modernism:  URGrid

Filed under: General — Tags: , — 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

Thursday, December 30, 2021

Antidote to Chaos?

Filed under: General — Tags: , , , , — m759 @ 3:57 pm

Some formal symmetry —

"… each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array
from Turyn's 1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields
the 2x2x2 eightfold cube ."

— Steven H. Cullinane on April 19, 2016 — The Folding.

Related art-historical remarks:

The Shape of Time  (Kubler, Yale U.P., 1962).

See yesterday's post The Thing 

Saturday, May 8, 2021

A Tale of Two Omegas

Filed under: General — Tags: , — 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.

Sunday, March 21, 2021

Mathematics and Narrative: The Unity

Filed under: General — Tags: , , , — m759 @ 10:25 pm

“To conquer, three boxes* have to synchronize and join together into the Unity.”

―Wonder Woman in Zack Snyder’s Justice League

See also The Unity of Combinatorics  and The Miracle Octad Generator.

* Cf.  Aitchison’s Octads

Thursday, February 25, 2021

Compare and Contrast

Filed under: General — Tags: , , , — m759 @ 12:31 pm

“… What is your dream—your ideal?  What is your News from Nowhere,
or, rather, What is the result of the little shake your hand has given to
the old pasteboard toy with a dozen bits of colored glass for contents?
And, most important of all, can you present it in a narrative or romance
which will enable me to pass an idle hour not disagreeably? How, for instance,
does it compare in this respect with other prophetic books on the shelf?”

— Hudson, W. H.. A Crystal Age  (p. 2). Open Road Media. Kindle Edition.

See as well . . .

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

Wednesday, October 7, 2020

Between Pyramid and Pentagon

Filed under: General — Tags: , , — m759 @ 12:34 am

The 35 small squares between the pyramid and the pentagon
in the above search result illustrate the role of finite geometry
in the Miracle Octad Generator  of R. T. Curtis.

'Then a miracle occurs' cartoon

Cartoon by S. Harris

Wednesday, September 2, 2020

Space Wars:  Sith Pyramid vs. Jedi Cube

Filed under: General — Tags: , — m759 @ 11:18 am

For the Sith Pyramid, see posts tagged Pyramid Game.
For the Jedi Cube, see posts tagged Enigma Cube
and cube-related remarks by Aitchison at Hiroshima.

This  post was suggested by two events of May 16, 2019 —
A weblog post by Frans Marcelis on the Miracle Octad
Generator of R. T. Curtis (illustrated with a pyramid),
and the death of I. M. Pei, architect of the Louvre pyramid.

That these events occurred on the same date is, of course,
completely coincidental.

Perhaps Dan Brown can write a tune to commemorate
the coincidence.

Thursday, August 27, 2020

The Complete Extended Binary Golay Code

Filed under: General — Tags: , , , , , — m759 @ 12:21 pm

All 4096 vectors in the code are at . . .

http://neilsloane.com/oadir/oa.4096.12.2.7.txt.

Sloane’s list* contains the 12 generating vectors
listed in 2011 by Adlam —

As noted by Conway in Sphere Packings, Lattices and Groups ,
these 4096 vectors, constructed lexicographically, are exactly
the same vectors produced by using the Conway-Sloane version
of the Curtis Miracle Octad Generator (MOG). Conway says this
lexico-MOG equivalence was first discovered by M. J. T. Guy.

(Of course, any  permutation of the 24 columns above produces
a version of the code qua  code. But because the lexicographic and
the MOG constructions yield the same result, that result is in
some sense canonical.)

See my post of July 13, 2020 —

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

For some related results, Google the twelfth generator:

* Sloane’s list is of the codewords as the rows of  an orthogonal array

See also http://neilsloane.com/oadir/.

Tuesday, July 28, 2020

For 7/28

Filed under: General — Tags: , — m759 @ 8:33 am

Miracle Octad Generator — Analysis of Structure

Monday, July 13, 2020

The Lexicographic Octad Generator (LOG)*

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

By Steven H. Cullinane, July 13, 2020

Background —


The Miracle Octad Generator (MOG)
of R. T. Curtis (Conway-Sloane version) —

Embedding Change, Illustrated

A basis for the Golay code, excerpted from a version of
the code generated in lexicographic order, in

"Constructing the Extended Binary Golay Code"
Ben Adlam
Harvard University
August 9, 2011:

000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100

Below, each vector above has been reordered within
a 4×6 array, by Steven H. Cullinane, to form twelve
independent Miracle Octad Generator  vectors
(as in the Conway-Sloane SPLAG version above, in
which Curtis's earlier heavy bricks are reflected in
their vertical axes) —

01 02 03 04 05 . . . 20 21 22 23 24 -->

01 05 09 13 17 21
02 06 10 14 18 22
03 07 11 15 19 23
04 08 12 16 20 24

0000 0000 0000 0000 1111 1111 -->

0000 11
0000 11
0000 11
0000 11 as in the MOG.

0000 0000 0000 1111 0000 1111 -->

0001 01
0001 01
0001 01
0001 01 as in the MOG.

0000 0000 0011 0011 0011 0011 -->

0000 00
0000 00
0011 11
0011 11 as in the MOG.

0000 0000 0101 0101 0101 0101 -->

0000 00
0011 11
0000 00
0011 11 as in the MOG.

0000 0000 1001 0110 0110 1001 -->

0010 01
0001 10
0001 10
0010 01 as in the MOG.

0000 0011 0000 0011 0101 0110 -->

0000 00
0000 11
0101 01
0101 10 as in the MOG.

0000 0101 0000 0101 0110 0011 -->

0000 00
0101 10
0000 11
0101 01 as in the MOG.

0000 1001 0000 0110 0011 1010 -->

0100 01
0001 00
0001 11
0100 10 as in the MOG.

0001 0001 0001 0001 0111 1000 -->

0000 01
0000 10
0000 10
1111 10 as in the MOG.

0010 0001 0001 0010 0001 1101 -->

0000 01
0000 01
1001 00
0110 11 as in the MOG.

0100 0001 0001 0100 0100 1110 -->

0000 01
1001 11
0000 01
0110 00 as in the MOG.

1000 0001 0001 0111 0010 0100 -->

10 00 00
00 01 01
00 01 10
01 11 00 as in the MOG (heavy brick at center).

Update at 7:41 PM ET the same day —
A check of SPLAG shows that the above result is not new:
MOG in LOG embedding

And at 7:59 PM ET the same day —
Conway seems to be saying that at some unspecified point in the past,
M.J.T. Guy, examining the lexicographic Golay code,  found (as I just did)
that weight-8 lexicographic Golay codewords, when arranged naturally
in 4×6 arrays, yield certain intriguing visual patterns. If the MOG existed
at the time of his discovery, he would have identified these patterns as
those of the MOG.  (Lexicographic codes have apparently been
known since 1960, the MOG since the early 1970s.)

* Addendum at 4 AM ET  the next day —
See also Logline  (Walpurgisnacht 2013).

Saturday, May 2, 2020

Turyn’s Octad Theorem: The Next Level*

Filed under: General — Tags: , , — m759 @ 11:24 am

From  the obituary of a game inventor  who reportedly died
on Monday, February 25, 2013 —

” ‘He was hired because of the game,’ Richard Turyn,
a mathematician who worked at Sylvania, told
the Washington Post in a 2004 feature on Diplomacy.”

* For the theorem, see Wolfram Neutsch,  Coordinates .
(Published by de Gruyter, 1996. See pp. 761-766.)

Having defined (pp. 751-752) the Miracle Octad Generator (MOG)
as a 4×6 array to be used with Conway’s “hexacode,” Neutsch says . . .

“Apart from the three constructions of the Golay codes
discussed at length in this book (lexicographic and via
the MOG or the projective line), there are literally
dozens of alternatives. For lack of space, we have to
restrict our attention to a single example. It has been
discovered by Turyn and can be connected in a very
beautiful way with the Miracle Octad Generator….

To this end, we consider the natural splitting of the MOG into
three disjoint octads L, M, R (‘left’, ‘middle’, and ‘right’ octad)….”

— From page 761

“The theorem of Turyn”  is on page 764

Wednesday, April 29, 2020

Curtis at Pilsen, Thursday, July 5, 2018

Filed under: General — Tags: , , , , — m759 @ 11:48 am

For an account by R. T. Curtis of how he discovered the Miracle Octad Generator,
see slides by Curtis, “Graphs and Groups,” from his talk on July 5, 2018, at the
Pilsen conference on algebraic graph theory, “Symmetry vs. Regularity: The first
50 years since Weisfeiler-Leman stabilization” (WL2018).

See also “Notes to Robert Curtis’s presentation at WL2018,” by R. T. Curtis.

Meanwhile, here  on July 5, 2018

Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oft-repeated and ever-different metaphor of identity.

— Paz, Octavio. The Monkey Grammarian
(Kindle Locations 1185-1191).
Arcade Publishing. Kindle Edition.

The 2018 Log24 post containing the above Paz quote goes on to quote
remarks by Lévi-Strauss. Paz’s phrase “series of transparent strata”
suggests a review of other remarks by Lévi-Strauss in the 2016 post
Key to  All Mythologies.

Wednesday, February 19, 2020

Aitchison’s Octads

Filed under: General — Tags: , , , , , — m759 @ 11:36 am

The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.

A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a  cuboctahedron :

The Curtis octads are related to symmetries of the square.

See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —

Aitchison's  octads are instead related to symmetries of the cube.

Note that essentially the same model as Aitchison's can be pictured 
by using, instead of the 24 edges of a cuboctahedron, the 24 outer 
faces of subcubes in the eightfold cube .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

http://www.math.sci.hiroshima-u.ac.jp/branched/files/2018/
presentations/Aitchison-Hiroshima-2-2018.pdf
.
See also Aitchison in this journal.

 
 

Saturday, December 14, 2019

Colorful Tale

Filed under: General — Tags: , , , — m759 @ 9:00 pm

(Continued)

Four-color correspondence in an eightfold array (eightfold cube unfolded)

The above image is from 

"A Four-Color Theorem:
Function Decomposition Over a Finite Field,"
http://finitegeometry.org/sc/gen/mapsys.html.

These partitions of an 8-set into four 2-sets
occur also in Wednesday night's post
Miracle Octad Generator Structure.

This  post was suggested by a Daily News
story from August 8, 2011, and by a Log24
post from that same date, "Organizing the
Mine Workers
" —

http://www.log24.com/log/pix11B/110808-DwarfsParade500w.jpg

Thursday, October 31, 2019

56 Triangles

The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.

Related material — A search in this journal for “56 Triangles.”

Saturday, September 21, 2019

Annals of Random Fandom

Filed under: General — Tags: , , , — m759 @ 5:46 pm

For Dan Brown fans …

… and, for fans of The Matrix, another tale
from the above death date: May 16, 2019 —

An illustration from the above
Miracle Octad Generator post:

Related mathematics — Tetrahedron vs. Square.

Wednesday, July 24, 2019

The Batty Farewell

Filed under: General — Tags: , , — m759 @ 8:40 pm

Adam Rogers today on "Rutger Hauer in Blade Runner , playing
the artificial* person Roy Batty in his death scene." 

* See the word "Artifice" in this  journal,
   as well as Tears in Rain . . .

Game Over

The film "The Matrix," illustrated
Coordinates for generating the Miracle Octad Generator

and Adam Rogers in
    the previous post.

 

Thursday, March 7, 2019

In Reality

Filed under: General — Tags: , — m759 @ 11:45 am

The previous post, quoting a characterization of the R. T. Curtis
Miracle Octad Generator , describes it as a "hand calculator ."

Other views 
 

A "natural diagram " —


 

A geometric object

Counting symmetries with the orbit-stabilizer theorem.

Tuesday, March 5, 2019

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

Filed under: General — Tags: , , , — m759 @ 11:19 am

A Midrash for Wikipedia 

Midrash —

Related material —


________________________________________________________________________________

The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Friday, March 1, 2019

Wikipedia Scholarship (Continued)

Filed under: General — Tags: , , , , — 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.

The Cullinane diamond theorem, AMS Notices, Feb. 1979, pp. A-193-194

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)).

Friday, February 22, 2019

Back Issues of AMS Notices

Filed under: General — Tags: , — 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 . . .

The Cullinane diamond theorem, AMS Notices, Feb. 1979, pp. A-193-194

Related material now at Wikipedia

The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

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

From the series of posts tagged Kummerhenge

A Wikipedia article relating the above 4×4 square to the work of Kummer —

A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis.  Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24  with the aid of the MOG.

Sunday, December 2, 2018

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: , , , , — m759 @ 6:43 am

A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018

http://www.math.sci.hiroshima-u.ac.jp/
branched/files/2018/abstract/Aitchison.txt

 

Iain AITCHISON

Title:

Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II

Abstract:

Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness.

Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles.

In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'.

Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set.

Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered.

Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective.

Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve.

See also yesterday morning's post, "Character."

Update: For a followup, see the next  Log24 post.

Wednesday, October 3, 2018

Adamantine Meditation

Filed under: General,Geometry — Tags: — m759 @ 12:24 pm

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

A Catholic philosopher —

Related art —

Image result for mog miracle octad bricks

Sunday, September 23, 2018

Three Times Eight

Filed under: General,Geometry — Tags: , , — m759 @ 9:21 am

The New York Times 's Sunday School today —

I prefer the three bricks of the Miracle Octad Generator —

Image result for mog miracle octad bricks

Wednesday, July 18, 2018

Doodle

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

From "The Educated Imagination: A Website Dedicated
to Northrop Frye
" —

"In one of the notebooks for his first Bible book Frye writes,

'For at least 25 years I’ve been preoccupied by
the notion of a key to all mythologies.' . . . .

Frye made a valiant effort to provide a key to all mythology,
trying to fit everything into what he called the Great Doodle. . . ."

From a different page at the same website —

Here Frye provides a diagram of four sextets.

I prefer the Miracle Octad Generator of R. T. Curtis —

Counting symmetries with the orbit-stabilizer theorem.

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: , , , , — m759 @ 1:05 pm

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Wednesday, December 7, 2016

Spreads and Conwell’s Heptads

Filed under: General,Geometry — Tags: , , , — m759 @ 7:11 pm

For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of 

Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).

This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .

Update, morning of the following day (7:07 ET) — related material:

See also “56 spreads” in this  journal.

Friday, December 2, 2016

A Small Witt Design*

Filed under: General — Tags: , , — m759 @ 2:00 pm

The New York Times 's  online T Magazine  yesterday —

"A version of this article appears in print on December 4, 2016, on page
M263 of T Magazine with the headline: The Year of Magical Thinking."

* Thanks to Emily Witt for inadvertently publicizing the
   Miracle Octad Generator  of R. T. Curtis, which
   summarizes the 759 octads found in the large Witt design.

Tuesday, September 13, 2016

Parametrizing the 4×4 Array

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

The previous post discussed the parametrization of 
the 4×4 array as a vector 4-space over the 2-element 
Galois field GF(2).

The 4×4 array may also be parametrized by the symbol
0  along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface

Hudson in 1905:

These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets —  were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator,"  what turned out to be 15 of Hudson's
1905 "Göpel tetrads":

A recap by Cullinane in 2013:

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Click images for further details.

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface
.

"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least, 
as an everlasting symbol of mathematical culture."

— Werner Kleinert, Mathematical Reviews ,
     as quoted at Amazon.com

Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).

Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .

Some related work of my own (click images for related posts)—

Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)

IMAGE- Desargues's theorem in light of Galois geometry

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Anticommuting Dirac matrices as spreads of projective lines

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Tuesday, April 19, 2016

The Folding

Filed under: General,Geometry — Tags: , , , — m759 @ 2:00 pm

(Continued

A recent post about the eightfold cube  suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields the 2x2x2 eightfold cube .

Those who prefer an entertainment  approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space." 

Friday, April 8, 2016

Ogdoads by Curtis

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

As was previously noted here, the construction of the Miracle Octad Generator
of R. T. Curtis in 1974 may have involved his "folding" the 1×8 octads constructed
in 1967 by Turyn into 2×4 form.

This results in a way of picturing a well-known correspondence (Conwell, 1910)
between partitions of an 8-set and lines of the projective 3-space PG(3,2).

For some background related to the "ogdoads" of the previous post, see
A Seventh Seal (Sept. 15, 2014).

Monday, February 1, 2016

Religious Note

Filed under: General — Tags: , — m759 @ 1:00 pm

See also the previous post.

Saturday, September 19, 2015

Geometry of the 24-Point Circle

Filed under: General,Geometry — Tags: , , — m759 @ 1:13 am

The latest Visual Insight  post at the American Mathematical
Society website discusses group actions on the McGee graph,
pictured as 24 points arranged in a circle that are connected
by 36 symmetrically arranged edges.

Wikipedia remarks that

"The automorphism group of the McGee graph
is of order 32 and doesn't act transitively upon
its vertices: there are two vertex orbits of lengths
8 and 16."

The partition into 8 and 16 points suggests, for those familiar
with the Miracle Octad Generator and the Mathieu group M24,
the following exercise:

Arrange the 24 points of the projective line
over GF(23) in a circle in the natural cyclic order
, 1, 2, 3,  , 22, 0 ).  Can the McGee graph be
modeled by constructing edges in any natural way?

Image that may or may not be related to the extended binary Golay code and the large Witt design

In other words, if the above set of edges has no
"natural" connection with the 24 points of the
projective line over GF(23), does some other 
set of edges in an isomorphic McGee graph
have such a connection?

Update of 9:20 PM ET Sept. 20, 2015:

Backstory: A related question by John Baez
at Math Overflow on August 20.

Thursday, August 27, 2015

Tears in the Rain

Filed under: General — Tags: , , — m759 @ 12:21 pm

For a Norwegian historian

Game Over

The film "The Matrix," illustrated
Coordinates for generating the Miracle Octad Generator

Monday, January 12, 2015

Points Omega*

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

The previous post displayed a set of
24 unit-square “points” within a rectangular array.
These are the points of the
Miracle Octad Generator  of R. T. Curtis.

The array was labeled  Ω
because that is the usual designation for
a set acted upon by a group:

* The title is an allusion to Point Omega , a novel by
Don DeLillo published on Groundhog Day 2010.
See “Point Omega” in this journal.

Thursday, January 8, 2015

Gitterkrieg

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

(Continued)

From the abstract of a talk, "Extremal Lattices," at TU Graz
on Friday, Jan. 11, 2013, by Prof. Dr. Gabriele Nebe
(RWTH Aachen) —

"I will give a construction of the extremal even
unimodular lattice Γ of dimension 72  I discovered
in summer 2010. The existence of such a lattice
was a longstanding open problem. The
construction that allows to obtain the
minimum by computer is similar to the one of the
Leech lattice from E8 and of the Golay code from
the Hamming code (Turyn 1967)."

On an earlier talk by Nebe at Oberwolfach in 2011 —

"Exciting new developments were presented by
Gabriele Nebe (Extremal lattices and codes ) who
sketched the construction of her recently found
extremal lattice in 72 dimensions…."

Nebe's Oberwolfach slides include one on 
"The history of Turyn's construction" —

Nebe's list omits the year 1976. This was the year of
publication for "A New Combinatorial Approach to M24"
by R. T. Curtis, the paper that defined Curtis's 
"Miracle Octad Generator."

Turyn's 1967 construction, uncredited by Curtis, may have
been the basis for Curtis's octad-generator construction.

See Turyn in this journal.

Saturday, October 25, 2014

Foundation Square

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

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean  geometry or of Galois  geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids.  This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

Monday, September 15, 2014

The Eight

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

The image at the end of today’s previous post A Seventh Seal
suggests a review of posts on Katherine Neville’s The Eight .

Update of 1:25 PM ET on Sept. 15, 2014:

Neville’s longtime partner is neurosurgeon and cognitive theorist
Karl H. Pribram. A quote from one of his books:

See also Sense and Sensibility.

Sunday, August 31, 2014

Sunday School

Filed under: General,Geometry — Tags: , , , — m759 @ 9:00 am

The Folding

Cynthia Zarin in The New Yorker , issue dated April 12, 2004—

“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”

The geometry of the 4×4 square array is that of the
3-dimensional projective Galois space PG(3,2).

This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc.  on
15 June 1974).  Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.

Some history: 

Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.

[Rewritten for clarity on Sept. 3, 2014.]

Sunday, August 24, 2014

Symplectic Structure…

In the Miracle Octad Generator (MOG):

The above details from a one-page note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

From R. T. Curtis (1976). A new combinatorial approach to M24,
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42. doi:10.1017/S0305004100052075.

The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.

Note that the interchange of the two squares in the top row of each
heavy brick induces the diamond-theorem correlation.

Note also that the 20 pictured 3-subsets of a 6-set in the 1986 note
occur as paired complements  in two pictures, each showing 10 of the
3-subsets.

This pair of pictures corresponds to the 20 Rosenhain tetrads  among
the 35 lines of PG(3,2), while the picture showing the 2-subsets
corresponds to the 15 Göpel tetrads  among the 35 lines.

See Rosenhain and Göpel tetrads in PG(3,2). Some further background:

Tuesday, June 17, 2014

Finite Relativity

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

Continued.

Anyone tackling the Raumproblem  described here
on Feb. 21, 2014 should know the history of coordinatizations
of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis
and J. H. Conway. Some documentation:

The above two images seem to contradict a statement by R. T. Curtis
in a 1989 paper.  Curtis seemed in that paper to be saying, falsely, that
his original 1973 and 1976 MOG coordinates were those in array M below—

This seemingly false statement involved John H. Conway's supposedly
definitive and natural canonical coordinatization of the 4×6 MOG
array by the symbols for the 24 points of the projective line over GF(23)—
{∞, 0, 1, 2, 3… , 21, 22}:

An explanation of the apparent falsity in Curtis's 1989 paper:

By "two versions of the MOG" Curtis seems to have meant merely that the
octads , and not the projective-line coordinates , in his earlier papers were
mirror images of the octads  that resulted later from the Conway coordinates,
as in the images below.

Thursday, April 24, 2014

The Inscape of 24

Filed under: General,Geometry — Tags: , — m759 @ 9:29 am

“The more intellectual, less physical, the spell of contemplation
the more complex must be the object, the more close and elaborate
must be the comparison the mind has to keep making between
the whole and the parts, the parts and the whole.”

— The Journals and Papers of Gerard Manley Hopkins ,
edited by Humphry House, 2nd ed. (London: Oxford
University Press, 1959), p. 126, as quoted by Philip A.
Ballinger in The Poem as Sacrament 

Related material from All Saints’ Day in 2012:

Talk pointing out that R. T. Curtis's 1974 construction of the Steiner system S(5,8,24) is taken from Turyn.

Thursday, April 3, 2014

Better Late…

Filed under: General — Tags: , — m759 @ 4:00 pm

Last Sunday’s sermon from Princeton’s Nassau Presbyterian
Church is now online. It reveals the answer to the “One Thing”
riddle posted at the church site Sunday:

IMAGE- Sermon topic 'One Thing Do I Know'

The online sermon has been retitled “One Thing I Do Know.”
A related search yields a relevant example of the original
Yoda-like word order:

IMAGE- 'One thing do I know' in a religious book from 1843

From the online sermon —

“What comes into view is the bombarding cynicism,
the barrage of mistrust and questions, and the
flat out trial of the man born blind. The
interrogation coming not because of the miracle
that gave the man sight….”

Related material — “Then a miracle occurs.”

Sunday, March 9, 2014

The Story Creeps Up

Filed under: General,Geometry — Tags: , , — m759 @ 11:01 pm

For Women’s History Month —

Conclusion of “The Storyteller,” a story
by Cynthia Zarin about author Madeleine L’Engle—

See also the exercise on the Miracle Octad Generator (MOG) at the end of
the previous post, and remarks on the MOG by Emily Jennings (non -fiction)
on All Saints’ Day, 2012 (the date the L’Engle quote was posted here).

Hesse’s Table

Filed under: General,Geometry — Tags: , , , — m759 @ 9:00 pm

From “Quartic Curves and Their Bitangents,” by
Daniel Plaumann, Bernd Sturmfels, and Cynthia Vinzant,
arXiv:1008.4104v2  [math.AG] 10 Jan 2011 —

The table mentioned (from 1855) is…

Exercise: Discuss the relationship, if any, to
the Miracle Octad Generator of R. T. Curtis.

Friday, February 21, 2014

Raumproblem*

Despite the blocking of Doodles on my Google Search
screen, some messages get through.

Today, for instance —

"Your idea just might change the world.
Enter Google Science Fair 2014"

Clicking the link yields a page with the following image—

IMAGE- The 24-triangle hexagon

Clearly there is a problem here analogous to
the square-triangle coordinatization problem,
but with the 4×6 rectangle of the R. T. Curtis
Miracle Octad Generator playing the role of
the square.

I once studied this 24-triangle-hexagon
coordinatization problem, but was unable to
obtain any results of interest. Perhaps
someone else will have better luck.

* For a rather different use of this word,
see Hermann Weyl in the Stanford
Encyclopedia of Philosophy.

Friday, December 20, 2013

For Emil Artin

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

(On His Dies Natalis )

An Exceptional Isomorphism Between Geometric and
Combinatorial Steiner Triple Systems Underlies 
the Octads of the M24 Steiner System S(5, 8, 24).

This is asserted in an excerpt from… 

"The smallest non-rank 3 strongly regular graphs
​which satisfy the 4-vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,
BAYREUTHER MATHEMATISCHE SCHRIFTEN 73 (2005), 152-212—

(Click for clearer image)

Note that Theorem 46 of Klin et al.  describes the role
of the Galois tesseract  in the Miracle Octad Generator
of R. T. Curtis (original 1976 version). The tesseract
(a 4×4 array) supplies the geometric  part of the above
exceptional geometric-combinatorial isomorphism.

Saturday, December 14, 2013

Beautiful Mathematics

Filed under: General,Geometry — Tags: , , , , — m759 @ 7:59 pm

The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.

Some material relevant to the title adjective:

"For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books

Some relevant links—

The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links.  See also a post of
​Jan. 31, 2014.

Update of March 9, 2014 —

The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare  the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).

Wednesday, September 4, 2013

Moonshine

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the 
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.) 

A Google search documents the moonshine
relating Rosenhain's and Göpel's 19th-century work
in complex analysis to M24  via the book of Hudson and
the geometry of the 4×4 square.

Monday, August 12, 2013

Form

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

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The Galois tesseract is the basis for a representation of the smallest
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday’s post.

The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—

IMAGE- Steven H. Cullinane, diamond theorem, from 'Diamond Theory,' Computer Graphics and Art, Vol. 2 No. 1, Feb. 1977, pp. 5-7

As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator  (MOG) of
R. T. Curtis.

Monday, August 5, 2013

Wikipedia Updates

Filed under: General,Geometry — Tags: , — m759 @ 12:30 pm

I added links today in the following Wikipedia articles:

The links will probably soon be deleted,
but it seemed worth a try.

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: , , — m759 @ 4:30 am

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Tuesday, July 2, 2013

Diamond Theorem Updates

Filed under: General,Geometry — Tags: , , — 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.

Tuesday, May 28, 2013

Codes

The hypercube  model of the 4-space over the 2-element Galois field GF(2):

IMAGE- A hyperspace model of the 4D vector space over GF(2)

The phrase Galois tesseract  may be used to denote a different model
of the above 4-space: the 4×4 square.

MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

The thirty-five 4×4 structures within the MOG:

IMAGE- The 35 square patterns within the Curtis MOG

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

A later book co-authored by Sloane, first published in 1988,
did  recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.

Between the 1977 and 1988 Sloane books came the diamond theorem.

Update of May 29, 2013:

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Sunday, May 19, 2013

Priority Claim

From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):

"By our construction, this vector space is the dual
of our hypercube F24 built on I \ O9. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis
in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O9."

[Cur89] reference:
 R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 
32 (1989), 345-353 (received on
July 20, 1987).

— Anne Taormina and Katrin Wendland,
    "The overarching finite symmetry group of Kummer
      surfaces in the Mathieu group 24 ,"
     arXiv.org > hep-th > arXiv:1107.3834

"First mentioned by Curtis…."

No. I claim that to the best of my knowledge, the 
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.

Update of the above paragraph on July 6, 2013—

No. The vector space structure was described by
(for instance) Peter J. Cameron in a 1976
Cambridge University Press book —
Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pages 59 and 60.

The vector space structure as it occurs in a 4×4 array
of the sort that appears in the Curtis Miracle Octad
Generator may first have been pointed out by me,
Steven H. Cullinane,
 in an AMS abstract submitted in
October 1978, some nine years before the Curtis article.

See Notes on Finite Geometry for some background.

See in particular The Galois Tesseract.

For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).

Sunday, April 28, 2013

The Octad Generator

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

… And the history of geometry  
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points in space.

(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)

Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:

"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point."

Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black  points and dashed  lines indicate the
complete space five-point and lines connecting it to the plane section
containing the Desargues configuration.

In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six -point to construct a configuration of
15 points and 20 lines in the context not of Desargues '  theorem, but
rather of Brianchon 's theorem and of the Pascal  hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five -point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6-point in space can  be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large  Desargues configuration. See Classical Geometry in Light of 
Galois Geometry
.)

For this large  Desargues configuration see April 19.
For Henderson's complete six –point, see The Six-Set (April 23).
That post ends with figures relating the large  Desargues configuration
to the Galois  geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator  and the large Mathieu group M24 —

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

See also Note on the MOG Correspondence from April 25, 2013.

That correspondence was also discussed in a note 28 years ago, on this date in 1985.

Thursday, April 25, 2013

Rosenhain and Göpel Revisited

Filed under: General,Geometry — Tags: , , , — m759 @ 5:24 pm

Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):

IMAGE- Bateman in 1906 on Rosenhain and Göpel tetrads

The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M24.

For some related material that is more up-to-date, search the Web
for Mathieu + Kummer .

Saturday, April 6, 2013

Pascal via Curtis

Filed under: General,Geometry — Tags: , , , — m759 @ 9:17 am

Click image for some background.

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

Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)

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

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

On Danzer's 354 Configuration:

IMAGE- Branko Grünbaum on Danzer's configuration
 

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

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

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

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

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

Classical Geometry in Light of Galois Geometry.

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

Wednesday, February 13, 2013

Form:

Filed under: General,Geometry — Tags: , , , — m759 @ 9:29 pm

Story, Structure, and the Galois Tesseract

Recent Log24 posts have referred to the 
"Penrose diamond" and Minkowski space.

The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—

IMAGE- The Penrose diamond and the Klein quadric

The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties 
of the R. T. Curtis Miracle Octad Generator  (MOG), hence of
the large Mathieu group M24. These properties are also 
relevant to the 1976 "Diamond Theory" monograph.

For some background on the quadric, see (for instance)

IMAGE- Stroppel on the Klein quadric, 2008

See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model
.

Related material:

"… one might crudely distinguish between philosophical
and mathematical motivation. In the first case one tries
to convince with a telling conceptual story; in the second
one relies more on the elegance of some emergent
mathematical structure. If there is a tradition in logic
it favours the former, but I have a sneaking affection for
the latter. Of course the distinction is not so clear cut.
Elegant mathematics will of itself tell a tale, and one with
the merit of simplicity. This may carry philosophical
weight. But that cannot be guaranteed: in the end one
cannot escape the need to form a judgement of significance."

– J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Those who prefer story to structure may consult 

  1. today's previous post on the Penrose diamond
  2. the remarks of Scott Aaronson on August 17, 2012
  3. the remarks in this journal on that same date
  4. the geometry of the 4×4 array in the context of M24.

Monday, December 24, 2012

All Over Again

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

Octavio Paz —

"… the movement of analogy
begins all over once again."

See A Reappearing Number in this journal.

Illustrations:

Figure 1 —

Background: MOG in this journal.

Figure 2 —

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

Background —

Image-- Google search on 'miracle octad'-- top 3 results

Monday, November 19, 2012

Poetry and Truth

From today's noon post

"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."

— Wallace Stevens at Bard College, March 30, 1951

Stevens also said at Bard that

"When Joan of Arc said: 

Have no fear: what I do, I do by command.
My brothers of Paradise tell me what I have to do.

these words were the words of an hallucination.
No matter what her brothers of Paradise drove her to do,
what she did was never a poetic act of faith in reality
because it could not be."

There are those who would dispute this.

Some related material:

"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—

A poetic approach to geometry—

"A surface" and "the rock," from All Saints' Day, 2012

Spaces as Hypercubes

— and from 1981—

http://www.log24.com/log/pix09/090217-SolidSymmetry.jpg

Some mathematical background for poets in Purgatory—

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

Saturday, September 3, 2011

The Galois Tesseract (continued)

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

http://www.log24.com/log/pix11B/110903-Carmichael-Conway-Curtis.jpg

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

See also a 1987 article by R. T. Curtis—

Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Update of Sept. 4— This post is now a page at finitegeometry.org.

Saturday, August 6, 2011

Correspondences

Filed under: General,Geometry — Tags: , , , , , — m759 @ 2:00 pm

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

— Baudelaire, “Correspondances

From “A Four-Color Theorem”

http://www.log24.com/log/pix11B/110806-Four_Color_Correspondence.gif

Figure 1

Note that this illustrates a natural correspondence
between

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

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

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

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

http://www.log24.com/log/pix11B/110806-Analysis_of_Structure.gif

Figure 2

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

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

Wednesday, July 6, 2011

Nordstrom-Robinson Automorphisms

Filed under: General,Geometry — Tags: , , , , , — m759 @ 1:01 am

A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—

"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."

— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22

A statement by Bierbrauer from 2004 has an error that doubles the above figure—

The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order |M24| = 24 × 23 × 22 × 21 × 20 × 48 in its 5-transitive action on the 24 coordinates. As M24 is transitive on octads, the stabilizer of an octad has order |M24|/759 [=322,560]. The stabilizer of NR has index 8 in this group. It follows that NR admits an automorphism group of order |M24| / (759 × 8 ) = [?] 16 × 7! [=80,640]. This is a huge symmetry group. Its structure can be inferred from the embedding in G as well. The automorphism group of NR is a semidirect product of an elementary abelian group of order 16 and the alternating group A7.

— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170

The error is corrected (though not detected) later in the same 2004 paper—

In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).

For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—

Click to enlarge:

IMAGE - The 112 hexads of the Nordstrom-Robinson code

For some context, see the group of order 322,560 in Geometry of the 4×4 Square.

Sunday, June 19, 2011

Abracadabra (continued)

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

Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator  of R.T. Curtis.

A search tonight for related material on the Web yielded…

(Click to enlarge.)

IMAGE- Art by Steven H. Cullinane displayed as his own in Steve Richards's Piracy Project contribution

   Weblog post by Steve Richards titled "The Search for Invariants:
   The Diamond Theory of Truth, the Miracle Octad Generator
   and Metalibrarianship." The artwork is by Steven H. Cullinane.
   Richards has omitted Cullinane's name and retitled the artwork.

The author of the post is an artist who seems to be interested in the occult.

His post continues with photos of pages, some from my own work (as above), some not.

My own work does not  deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.

The artist's post concludes with the following (note also the beginning of the preceding  post)—

http://www.log24.com/log/pix11A/110619-MOGsteverichards.jpg

"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.

Sunday, June 5, 2011

Edifice Complex

Filed under: General,Geometry — Tags: , , , — m759 @ 7:00 pm

"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."

— Wallace Stevens, "To an Old Philosopher in Rome"

The following edifice may be lacking in grandeur,
and its properties as a configuration  were known long
before I stumbled across a description of it… still…

"What we do may be small, but it has
 a certain character of permanence…."
 — G.H. Hardy, A Mathematician's Apology

The Kummer 166 Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)

IMAGE-- 16_6 configuration from '2-Transitive Symmetric Designs,' by William M. Kantor (AMS Transactions, 1969)

For some background, see Configurations and Squares.

For some quite different geometry of the 4×4 square that  is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do  claim credit
for discovering some geometric properties of the 4×4 square
that constitutes two-thirds of the MOG as originally defined .)

Related material— The Schwartz Notes of June 1.

Wednesday, June 1, 2011

The Schwartz Notes

Filed under: General,Geometry — Tags: , , , , — m759 @ 2:00 pm

A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)

The same search limited to results from only the past month yielded,
in addition, the following—

http://www.log24.com/log/pix11A/110601-Search.jpg

This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.

Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.

See Schwartz’s page 12, page 13, and page 14.

Compare to the web pages from today’s earlier search.

There are no references at the end of the Schwartz document,
but there is this at the beginning—

These are some notes on error correcting codes. Two good sources for
this material are
From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
Sphere Packings, Lattices, and Simple Groups  by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.

It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.

Thursday, May 26, 2011

Life’s Persistent Questions

Filed under: General — Tags: , — m759 @ 4:01 pm

This afternoon's online New York Times  reviews "The Tree of Life," a film that opens tomorrow.

With disarming sincerity and daunting formal sophistication “The Tree of Life” ponders some of the hardest and most persistent questions, the kind that leave adults speechless when children ask them. In this case a boy, in whispered voice-over, speaks directly to God, whose responses are characteristically oblique, conveyed by the rustling of wind in trees or the play of shadows on a bedroom wall. Where are you? the boy wants to know, and lurking within this question is another: What am I doing here?

Persistent answers… Perhaps conveyed by wind, perhaps by shadows, perhaps by the New York Lottery.

For the nihilist alternative— the universe arose by chance out of nothing and all is meaningless— see Stephen Hawking and Jennifer Ouellette.

Update of 10:30 PM EDT May 26—

Today's NY Lottery results: Midday 407, Evening 756. The first is perhaps about the date April 7, the second about the phrase "three bricks shy"— in the context of the number 759 and the Miracle Octad Generator. (See also Robert Langdon and The Poetics of Space.)

Tuesday, May 24, 2011

Noncontinuous (or Non-Continuous) Groups

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

The web page has been updated.

An example, the action of the Mathieu group M24
on the Miracle Octad Generator of R.T. Curtis,
was added, with an illustration from a book cover—

http://www.log24.com/log/pix11A/110524-TwelveSG.jpg

Wednesday, March 2, 2011

Labyrinth of the Line

Filed under: General,Geometry — Tags: , , — m759 @ 11:24 am

“Yo sé de un laberinto griego que es una línea única, recta.”
—Borges, “La Muerte y la Brújula”

“I know of one Greek labyrinth which is a single straight line.”
—Borges, “Death and the Compass”

Another single-line labyrinth—

Robert A. Wilson on the projective line with 24 points
and its image in the Miracle Octad Generator (MOG)—

IMAGE- Robert Wilson on the projective line with 24 points and its image in the MOG

Related material —

The remarks of Scott Carnahan at Math Overflow on October 25th, 2010
and the remarks at Log24 on that same date.

A search in the latter for miracle octad is updated below.

http://www.log24.com/log/pix11/110302-MOGsearch.jpg

This search (here in a customized version) provides some context for the
Benedictine University discussion described here on February 25th and for
the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”

Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.

Monday, October 25, 2010

The Embedding*

Filed under: General,Geometry — Tags: , — m759 @ 4:04 pm

A New York Times  "The Stone" post from yesterday (5:15 PM, by John Allen Paulos) was titled—

Stories vs. Statistics

Related Google searches—

"How to lie with statistics"— about 148,000 results

"How to lie with stories"— 2 results

What does this tell us?

Consider also Paulos's phrase "imbedding the God character."  A less controversial topic might be (with the spelling I prefer) "embedding the miraculous." For an example, see this journal's "Mathematics and Narrative" entry on 5/15 (a date suggested, coincidentally, by the time of Paulos's post)—

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

Image-- Google search on 'miracle octad'-- top 3 results

 

* Not directly  related to the novel The Embedding  discussed at Tenser, said the Tensor  on April 23, 2006 ("Quasimodo Sunday"). An academic discussion of that novel furnishes an example of narrative as more than mere entertainment. See Timothy J. Reiss, "How can 'New' Meaning Be Thought? Fictions of Science, Science Fictions," Canadian Review of Comparative Literature , Vol. 12, No. 1, March 1985, pp. 88-126. Consider also on this, Picasso's birthday, his saying that "Art is a lie that makes us realize truth…."

Saturday, July 24, 2010

Playing with Blocks

"Many of the finite simple groups can be described as symmetries of finite geometries, and it remains a hot topic in group theory to expand our knowledge of the Classification of Finite Simple Groups using finite geometry."

Finite geometry page at the Centre for the Mathematics of
   Symmetry and Computation at the University of Western Australia
   (Alice Devillers, John Bamberg, Gordon Royle)

For such symmetries, see Robert A. WIlson's recent book The Finite Simple Groups.

The finite simple groups are often described as the "building blocks" of finite group theory.

At least some of these building blocks have their own building blocks. See Non-Euclidean Blocks.

For instance, a set of 24 such blocks (or, more simply, 24 unit squares) appears in the Miracle Octad Generator (MOG) of R.T. Curtis, used in the study of the finite simple group M24.

(The octads  of the MOG illustrate yet another sort of mathematical blocks— those of a block design.)

Sunday, July 4, 2010

Brightness at Noon (continued)

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

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

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

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

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

Saturday, May 15, 2010

Mathematics and Narrative continued…

Filed under: General,Geometry — Tags: — m759 @ 4:16 pm

Step Two

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

Image-- Google search on 'miracle octad'-- top 3 results

Friday, May 14, 2010

Competing MOG Definitions

Filed under: General,Geometry — Tags: , , , — m759 @ 9:00 pm

A recently created Wikipedia article says that  “The Miracle Octad Generator [MOG] is an array of coordinates, arranged in four rows and six columns, capable of describing any point in 24-dimensional space….” (Clearly any  array with 24 parts is so capable.) The article ignores the fact that the MOG, as defined by R.T. Curtis in 1976, is not  an array of coordinates, but rather a picture of a correspondence between two sets, each containing 35 structures. (As a later commentator has remarked, this correspondence is a well-known one that preserves a certain incidence property. See Eightfold Geometry.)

From the 1976 paper defining the MOG—

“There is a correspondence between the two systems of 35 groups, which is illustrated in Fig. 4 (the MOG or Miracle Octad Generator).” —R.T. Curtis, “A New Combinatorial Approach to M24,” Mathematical Proceedings of the Cambridge Philosophical Society  (1976), 79: 25-42

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

Curtis’s 1976 Fig. 4. (The MOG.)

The Wikipedia article, like a similar article at PlanetMath, is based on a different definition, from a book first published in 1988—

http://www.log24.com/log/pix10A/100514-SpherePack.jpg

I have not seen the 1973 Curtis paper, so I do not know whether it uses the 35-sets correspondence definition or the 6×4 array definition. The remarks of Conway and Sloane on page 312 of the 1998 edition of their book about “Curtis’s original way of finding octads in the MOG [Cur2]” indicate that the correspondence definition was the one Curtis used in 1973—

http://www.log24.com/log/pix10A/100514-ConwaySloaneMOG.jpg

Here the picture of  “the 35 standard sextets of the MOG”
is very like (modulo a reflection) Curtis’s 1976 picture
of the MOG as a correspondence between two 35-sets.

A later paper by Curtis does  use the array definition. See “Further Elementary Techniques Using the Miracle Octad Generator,” Proceedings of the Edinburgh Mathematical Society  (1989) 32, 345-353.

The array definition is better suited to Conway’s use of his hexacode  to describe octads, but it obscures the close connection of the MOG with finite geometry. That connection, apparent in the phrases “vector space structure in the standard square” and “parallel 2-spaces” (Conway and Sloane, third ed., p. 312, illustrated above), was not discussed in the 1976 Curtis paper.  See my own page on the MOG at finitegeometry.org.

Wednesday, April 28, 2010

Eightfold Geometry

Filed under: General,Geometry — Tags: , , , , , — m759 @ 11:07 am

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

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

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

Related web pages:

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

Related folklore:

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

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

Update of August 20, 2010–

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

Thursday, August 6, 2009

Thursday August 6, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 1:44 pm
A Fisher of Men
 
 
Cover, Schulberg's novelization of 'Waterfront,' Bantam paperback
Update: The above image was added
at about 11 AM ET Aug. 8, 2009.

 
Dove logo, First United Methodist Church of Bloomington, Indiana

From a webpage of the First United Methodist Church of Bloomington, Indiana–

 

Dr. Joe Emerson, April 24, 2005–

"The Ultimate Test"

— Text: I Peter 2:1-9

Dr. Emerson falsely claims that the film "On the Waterfront" was based on a book by the late Budd Schulberg (who died yesterday). (Instead, the film's screenplay, written by Schulberg– similar to an earlier screenplay by Arthur Miller, "The Hook"–  was based on a series of newspaper articles by Malcolm Johnson.)

"The movie 'On the Waterfront' is once more in rerun. (That’s when Marlon Brando looked like Marlon Brando.  That’s the scary part of growing old when you see what he looked like then and when he grew old.)  It is based on a book by Budd Schulberg."

 

Emerson goes on to discuss the book, Waterfront, that Schulberg wrote based on his screenplay–

"In it, you may remember a scene where Runty Nolan, a little guy, runs afoul of the mob and is brutally killed and tossed into the North River.  A priest is called to give last rites after they drag him out."

 

Hook on cover of Budd Schulberg's novel 'Waterfront' (NY Times obituary, detail)

New York Times today

Dr. Emerson flunks the test.

 

Dr. Emerson's sermon is, as noted above (Text: I Peter 2:1-9), not mainly about waterfronts, but rather about the "living stones" metaphor of the Big Fisherman.

My own remarks on the date of Dr. Emerson's sermon

The 4x6 array used in the Miracle Octad Generator of R. T. Curtis

Those who like to mix mathematics with religion may regard the above 4×6 array as a context for the "living stones" metaphor. See, too, the five entries in this journal ending at 12:25 AM ET on November 12 (Grace Kelly's birthday), 2006, and today's previous entry.

Wednesday, May 20, 2009

Wednesday May 20, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 4:00 pm
From Quilt Blocks to the
Mathieu Group
M24

Diamonds

(a traditional
quilt block):

Illustration of a diamond-theorem pattern

Octads:

Octads formed by a 23-cycle in the MOG of R.T. Curtis

 

Click on illustrations for details.

The connection:

The four-diamond figure is related to the finite geometry PG(3,2). (See "Symmetry Invariance in a Diamond Ring," AMS Notices, February 1979, A193-194.) PG(3,2) is in turn related to the 759 octads of the Steiner system S(5,8,24). (See "Generating the Octad Generator," expository note, 1985.)

The relationship of S(5,8,24) to the finite geometry PG(3,2) has also been discussed in–
  • "A Geometric Construction of the Steiner System S(4,7,23)," by Alphonse Baartmans, Walter Wallis, and Joseph Yucas, Discrete Mathematics 102 (1992) 177-186.

Abstract: "The Steiner system S(4,7,23) is constructed from the geometry of PG(3,2)."

  • "A Geometric Construction of the Steiner System S(5,8,24)," by R. Mandrell and J. Yucas, Journal of Statistical Planning and Inference 56 (1996), 223-228.

Abstract: "The Steiner system S(5,8,24) is constructed from the geometry of PG(3,2)."

For the connection of S(5,8,24) with the Mathieu group M24, see the references in The Miracle Octad Generator.

Tuesday, May 19, 2009

Tuesday May 19, 2009

Filed under: General,Geometry — Tags: , , , — m759 @ 7:20 pm
Exquisite Geometries

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

"Block Designs," 1995, by Andries E. Brouwer

"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."

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

"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."

William M. Kantor, 1981

The 1931 paper of Carmichael is now available online from the publisher for $10.
 

Tuesday, January 6, 2009

Tuesday January 6, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 am
Archetypes, Synchronicity,
and Dyson on Jung

The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:

“… we do not need to accept Jung’s theory as true in order to find it illuminating.”

The same is true of Jung’s remarks on synchronicity.

For example —

Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s four-diamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:

R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.

Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.

On these dates, the entries in this journal discussed…

Oct. 24:
Cube Space, 1984-2003

Material related to that entry:

Dec. 19:
Art and Religion: Inside the White Cube

That entry discusses a book by Mark C. Taylor:

The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).

In Chapter 3, “Sutures of Structures,” Taylor asks —

“What, then, is a frame, and what is frame work?”

One possible answer —

Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.

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

Monday, January 5, 2009

Monday January 5, 2009

Filed under: General,Geometry — Tags: , , , , — m759 @ 9:00 pm

A Wealth of
Algebraic Structure

A 4x4 array (part of chessboard)

A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):

Further elementary techniques using the miracle octad generator
, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

 

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.

(Published online by Cambridge University Press 19 Dec 2008.)

In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2).  (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)

For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):

A new combinatorial approach to M24, by R. T. Curtis. Abstract:

“In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”

 

(Received June 15 1974)

Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.

(Published online by Cambridge University Press 24 Oct 2008.)

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Click for details.

Monday, November 24, 2008

Monday November 24, 2008

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

Frame Tale

'Brick' octads in the Miracle Octad Generator (MOG) of R. T. Curtis

Click on image for details.

Thursday, July 31, 2008

Thursday July 31, 2008

Filed under: General,Geometry — Tags: , — m759 @ 12:00 pm
Symmetry in Review

“Put bluntly, who is kidding whom?”

Anthony Judge, draft of
“Potential Psychosocial Significance
of Monstrous Moonshine:
An Exceptional Form of Symmetry
as a Rosetta Stone for
Cognitive Frameworks,”
dated September 6, 2007.

Good question.

Also from
September 6, 2007 —
the date of
Madeleine L’Engle‘s death —

 
Pavarotti takes a bow
Related material:

1. The performance of a work by
Richard Strauss,
Death and Transfiguration,”
(Tod und Verklärung, Opus 24)
by the Chautauqua Symphony
at Chautauqua Institution on
July 24, 2008

2. Headline of a music review
in today’s New York Times:

Welcoming a Fresh Season of
Transformation and Death

3. The picture of the R. T. Curtis
Miracle Octad Generator
on the cover of the book
Twelve Sporadic Groups:

Cover of 'Twelve Sporadic Groups'

4. Freeman Dyson’s hope, quoted by
Gorenstein in 1986, Ronan in 2006,
and Judge in 2007, that the Monster
group is “built in some way into
the structure of the universe.”

5. Symmetry from Plato to
the Four-Color Conjecture

6. Geometry of the 4×4 Square

7. Yesterday’s entry,
Theories of Everything

Coda:

There is such a thing

Tesseract
     as a tesseract.

— Madeleine L’Engle

Cover of The New Yorker, April 12, 2004-- Roz Chast, Easter Eggs

For a profile of
L’Engle, click on
the Easter eggs.

Monday, October 1, 2007

Monday October 1, 2007

Filed under: General,Geometry — Tags: , , — m759 @ 7:20 am
Bright as Magnesium

"Definitive"

— The New York Times,  
Sept. 30, 2007, on
Blade Runner:
The Final Cut

Institute for Advanced Study, Princeton, N.J.

"The art historian Kirk Varnedoe died on August 14, 2003, after a long and valiant battle with cancer. He was 57. He was a faculty member in the Institute for Advanced Study’s School of Historical Studies, where he was the fourth art historian to hold this prestigious position, first held by the German Renaissance scholar Erwin Panofsky in the 1930s."

Hal Crowther

"His final lecture was an eloquent, prophetic flight of free association….

Varnedoe chose to introduce his final lecture with the less-quoted last words of the android Roy Batty (Rutger Hauer) in Ridley Scott's film Blade Runner: 'I've seen things you people wouldn't believe– attack ships on fire off the shoulder of Orion, bright as magnesium; I rode on the back decks of a blinker and watched C-beams glitter in the dark near the Tannhauser Gate. All those moments will be lost in time, like tears in the rain. Time to die.'"


Related material: 
tears in the rain–

Game Over
(Nov. 5, 2003):
 

The film "The Matrix," illustrated

Coordinates for generating the Miracle Octad Generator

Monday, May 28, 2007

Monday May 28, 2007

Filed under: General,Geometry — Tags: , , , , — m759 @ 5:00 pm
Space-Time

and a Finite Model

Notes by Steven H. Cullinane
May 28, 2007

Part I: A Model of Space-Time

The following paper includes a figure illustrating Penrose’s model of  “complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space.”
 
The image “http://www.log24.com/log/pix07/070528-Twistor.jpg” cannot be displayed, because it contains errors.

Click on picture to enlarge.

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


Part II: A Corresponding Finite Model

 

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

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

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

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

 

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


Related material:

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

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

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

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

Wednesday, February 28, 2007

Wednesday February 28, 2007

Filed under: General,Geometry — Tags: , , , — m759 @ 7:59 am

Elements
of Geometry

The title of Euclid’s Elements is, in Greek, Stoicheia.

From Lectures on the Science of Language,
by Max Muller, fellow of All Souls College, Oxford.
New York: Charles Scribner’s Sons, 1890, pp. 88-90 –

Stoicheia

“The question is, why were the elements, or the component primary parts of things, called stoicheia by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist.

Stoichos, from which stoicheion, means a row or file, like stix and stiches in Homer. The suffix eios is the same as the Latin eius, and expresses what belongs to or has the quality of something. Therefore, as stoichos means a row, stoicheion would be what belongs to or constitutes a row….

Hence stoichos presupposes a root stich, and this root would account in Greek for the following derivations:–

  1. stix, gen. stichos, a row, a line of soldiers
  2. stichos, a row, a line; distich, a couplet
  3. steichoestichon, to march in order, step by step; to mount
  4. stoichos, a row, a file; stoichein, to march in a line

In German, the same root yields steigen, to step, to mount, and in Sanskrit we find stigh, to mount….

Stoicheia are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order.”

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: , , , — m759 @ 9:26 am

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

— G. H. Hardy, A Mathematician's Apology

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines,  sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

Saturday, July 29, 2006

Saturday July 29, 2006

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

Big Rock

Thanks to Ars Mathematicaa link to everything2.com:

“In mathematics, a big rock is a result which is vastly more powerful than is needed to solve the problem being considered. Often it has a difficult, technical proof whose methods are not related to those of the field in which it is applied. You say ‘I’m going to hit this problem with a big rock.’ Sard’s theorem is a good example of a big rock.”

Another example:

Properties of the Monster Group of R. L. Griess, Jr., may be investigated with the aid of the Miracle Octad Generator, or MOG, of R. T. Curtis.  See the MOG on the cover of a book by Griess about some of the 20 sporadic groups involved in the Monster:

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

The MOG, in turn, illustrates (via Abstract 79T-A37, Notices of the American Mathematical Society, February 1979) the fact that the group of automorphisms of the affine space of four dimensions over the two-element field is also the natural group of automorphisms of an arbitrary 4×4 array.

This affine group, of order 322,560, is also the natural group of automorphisms of a family of graphic designs similar to those on traditional American quilts.  (See the diamond theorem.)

This top-down approach to the diamond theorem may serve as an illustration of the “big rock” in mathematics.

For a somewhat simpler, bottom-up, approach to the theorem, see Theme and Variations.

For related literary material, see Mathematics and Narrative and The Diamond as Big as the Monster.

“The rock cannot be broken.
It is the truth.”

Wallace Stevens,
“Credences of Summer”

 

Sunday, May 7, 2006

Sunday May 7, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 3:00 am

Bagombo Snuff Box
 
(in memory of
Burt Kerr Todd)


“Well, it may be the devil
or it may be the Lord
But you’re gonna have to
serve somebody.”

— “Bob Dylan”
(pseudonym of Robert Zimmerman),
quoted by “Bob Stewart”
on July 18, 2005

“Bob Stewart” may or may not be the same person as “crankbuster,” author of the “Rectangular Array Theorem” or “RAT.”  This “theorem” is intended as a parody of the “Miracle Octad Generator,” or “MOG,” of R. T. Curtis.  (See the Usenet group sci.math, “Steven Cullinane is a Crank,” July 2005, messages 51-60.)

“Crankbuster” has registered at Math Forum as a teacher in Sri Lanka (formerly Ceylon).   For a tall tale involving Ceylon, see the short story “Bagombo Snuff Box” in the book of the same title by Kurt Vonnegut, who has at times embodied– like Martin Gardner and “crankbuster“– “der Geist, der stets verneint.”

Here is my own version (given the alleged Ceylon background of “crankbuster”) of a Bagombo snuff box:

Related material:

Log24 entries of
April 16-30, 2005,

and the 5 Log24 entries
ending on Friday,
April 28, 2006.

Friday, April 28, 2006

Friday April 28, 2006

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

Exercise

Review the concepts of integritas, consonantia,  and claritas in Aquinas:

"For in respect to beauty three things are essential: first of all, integrity or completeness, since beings deprived of wholeness are on this score ugly; and [secondly] a certain required design, or patterned structure; and finally a certain splendor, inasmuch as things are called beautiful which have a certain 'blaze of being' about them…."

Summa Theologiae Sancti Thomae Aquinatis, I, q. 39, a. 8, as translated by William T. Noon, S.J., in Joyce and Aquinas, Yale University Press, 1957

Review the following three publications cited in a note of April 28, 1985 (21 years ago today):

(1) Cameron, P. J.,
     Parallelisms of Complete Designs,
     Cambridge University Press, 1976.

(2) Conwell, G. M.,
     The 3-space PG(3,2) and its group,
     Ann. of Math. 11 (1910) 60-76.

(3) Curtis, R. T.,
     A new combinatorial approach to M24,
     Math. Proc. Camb. Phil. Soc.
    
79 (1976) 25-42.

Discuss how the sextet parallelism in (1) illustrates integritas, how the Conwell correspondence in (2) illustrates consonantia, and how the Miracle Octad Generator in (3) illustrates claritas.
 

Wednesday, November 30, 2005

Wednesday November 30, 2005

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

For St. Andrew’s Day

The miraculous enters…. When we investigate these problems, some fantastic things happen….”

— John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)

The remarkable Mathieu group M24, a group of permutations on 24 elements, may be studied by picturing its action on three interchangeable 8-element “octads,” as in the “Miracle Octad Generator” of R. T. Curtis.

A picture of the Miracle Octad Generator, with my comments, is available online.


 Cartoon by S.Harris

Related material:
Mathematics and Narrative.

Tuesday, January 6, 2004

Tuesday January 6, 2004

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

720 in the Book

Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of January-February 2004.

An article titled On Mathematical Imagination concludes by looking forward to

“a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation….”

Seeking the source of the phrase “new instauration,” I found it was due to Francis Bacon, who “conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of ‘the seal of God on things,’ ” according to a web page by Nieves Mathews.

Hmm.

The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:

Abel’s Proof:  An Essay
on the Sources and Meaning
of Mathematical Unsolvability

by Peter Pesic,
MIT Press, 2003

From a review:

“… the book is about the idea that polynomial equations in general cannot be solved exactly in radicals….

Pesic concludes his account after Abel and Galois… and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations.  The reader is left with little clarity on this sequel to the story….”

— Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242-244

Here, it seems, is my epiphany:

“Elliptic modular functions suffice to solve all polynomial equations.”


Incidental Remarks
on Synchronicity,
Part I

Those who seek a star
on this Feast of the Epiphany
may click here.


Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higher-degree equations.

Just how such equations can be solved is a less familiar story.  I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.

The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions.  Modular functions are also distantly related, via the topic of “moonshine” and via the  “Happy Family” of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4×4 matrices.


Incidental Remarks
on Synchronicity,
Part II

There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.


Here is what I was able to find on the Web about Pesic’s claim:

From Wolfram Research:

From Solving the Quintic —

“Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein’s approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.”

From Siegel Theta Function —

“Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)”

From Polynomial

“… the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable.  Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron.  Klein’s method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or ‘Siegel functions’ must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be ‘natural’ generalizations of the elliptic functions.”

Belardinelli, G. “Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales.” Mémoral des Sci. Math. 145, 1960.

King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.

Chow, T. Y. “What is a Closed-Form Number.” Amer. Math. Monthly 106, 440-448, 1999. 

From Angel Zhivkov,

Preprint series,
Institut für Mathematik,
Humboldt-Universität zu Berlin:

“… discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker:  in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function….  Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist.  This hypothesis was realized in [a] few cases by F. Klein… Jordan… showed that any algebraic equation is solvable by modular functions.  In 1984 Umemura realized the Kronecker idea in his appendix to Mumford’s book… deducing from a formula of Thomae… a root of [an] arbitrary algebraic equation by Siegel modular forms.”  

— “Resolution of Degree Less-than-or-equal-to Six Algebraic Equations by Genus Two Theta Constants


Incidental Remarks
on Synchronicity,
Part III

From Music for Dunne’s Wake:

Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”

— Carrie Fisher,
Postcards from the Edge

     

720 in  
the Book”

and
Paradise

“The group Sp4(F2) has order 720,”
as does S6. — Angel Zhivkov, op. cit.

Those seeking
“a rediscovery of
‘the seal of God on things,’ “
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.

For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see

Log24, June 2003.

For the relevance of the time
of this entry, 10:10, see

  1. the reference to Paradise
    on the “postcard” above, and
  2. Storyline (10/10, 2003).

Related recreational reading:

Labyrinth



The Shining

Shining Forth

Monday, April 28, 2003

Monday April 28, 2003

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

ART WARS:

Toward Eternity

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

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

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

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

 

Consider the following journal entries from April 7, 2003:
 

Math Awareness Month

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


 

An Offer He Couldn't Refuse

Today's birthday:  Francis Ford Coppola is 64.

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


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

 

From a website titled simply Sinatra:

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

From a note on geometry of April 28, 1985:

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

The Eight

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

 


 Decimal 
labeling

 
Binary
labeling


Algebraic
labeling


Permutation
labeling

 

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

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

Friday, April 4, 2003

Friday April 4, 2003

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

The Eight

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


Decimal 
labeling


Binary
labeling


Algebraic
labeling

IMAGE- Knight figure for April 4
Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.  (See the 4:36 PM entry.)

 

 

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

Saturday, July 20, 2002

Saturday July 20, 2002

 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

 
Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

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

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)



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