Log24

Friday, May 7, 2021

Through the Miracle Looking Glass

Filed under: General — Tags: — m759 @ 11:30 am

(This post was suggested by the order of reading characters in
traditional Chinese calligraphy — top to bottom, right to left .)

. . . the Horses’ Heads
Were toward Eternity”
— Emily Dickinson

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.)

Sunday, December 25, 2016

Miracle Eight Generator

Filed under: General — m759 @ 9:00 am

Merry Xmas to Katherine Neville.

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 — 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 — 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.

Friday, September 22, 2023

Figurate Space

Filed under: General — Tags: — m759 @ 11:01 am

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

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

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

Counting symmetries with the orbit-stabilizer theorem

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.

Sunday, April 23, 2023

Muse Variations

Filed under: General — Tags: — m759 @ 12:24 am

The logo of MUSE, the band —

A logo I prefer . . .

Related material from a post of October 2020

Related material from a post on the above Reddit date

IMAGE- Donna Reed and Montgomery Clift in 'From Here to Eternity'

A Story That Works

“There is the dark, eternally silent, unknown universe;
there are the friend-enemy minds shouting and whispering
their tales and always seeking the three miracles —

  • that minds should really touch, or
  • that the silent universe should speak, tell minds a story,
  • or (perhaps the same thing) that there should be a story
    that works, that is all hard facts, all reality, with
    no illusions and no fantasy;

and lastly, there is lonely, story-telling, wonder-questing,
mortal me.”

– Fritz Leiber in “The Button Molder

Wednesday, November 16, 2022

“A wondrous woven magic” — Song lyric

Filed under: General — Tags: — m759 @ 12:17 pm

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?

Wednesday, September 28, 2022

Bitspace Note

Filed under: General — Tags: — m759 @ 5:28 pm

Update of 5:20 AM ET on Sept. 29. 2022 —

The octads of the [24, 8, 8] cube-motif code
can be transformed by the permutation below
into octads recognizable, thanks to the Miracle
Octad Generator (MOG) of R. T. Curtis, as
belonging to the Golay code.

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.

Wednesday, May 11, 2022

A Story for the TENET Director

Filed under: General — Tags: , — m759 @ 9:14 am

IMAGE- Donna Reed and Montgomery Clift in 'From Here to Eternity'

A Story That Works

“There is the dark, eternally silent, unknown universe;
there are the friend-enemy minds shouting and whispering
their tales and always seeking the three miracles —

  • that minds should really touch, or
  • that the silent universe should speak, tell minds a story,
  • or (perhaps the same thing) that there should be a story
    that works, that is all hard facts, all reality, with
    no illusions and no fantasy;

and lastly, there is lonely, story-telling, wonder-questing,
mortal me.”

– Fritz Leiber in “The Button Molder

"Will the record be unbroken . . . ?"

— Adapted song lyric

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 30, 2022

Rock Music

Filed under: General — Tags: , — m759 @ 1:38 pm

"All I need is a miracle" — Song from "Spencer" (2021)

Sandringham MOG

"Risin' up to the challenge of our rival" — "Rocky III" (1982)

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

Wednesday, March 3, 2021

Out on the Cutting Edge*

Filed under: General — Tags: — m759 @ 1:45 pm
That edge, where artists are always transforming chaos into order, can be a very rough and dangerous place. Living there, an artist constantly risks falling fully into the chaos, instead of transforming it. But artists have always lived there, on the border of human understanding. Art bears the same relationship to society that the dream bears to mental life. You are very creative when you are dreaming. That is why, when you remember a dream, you think, “Where in the world did that come from?” It is very strange and incomprehensible that something can happen in your head, and you have no idea how it got there or what it means. It is a miracle: nature’s voice manifesting itself in your psyche. And it happens every night. Like art, the dream mediates between order and chaos. So, it is half chaos. That is why it is not comprehensible. It is a vision, not a fully fledged articulated production. Those who actualize those half-born visions into artistic productions are those who begin to transform what we do not understand into what we can at least start to see. That is the role of the artist, occupying the vanguard. That is their biological niche. They are the initial civilizing agents.

— Peterson, Jordan B., Beyond Order  (p. 215),
published on March 2, 2021.
Penguin Publishing Group. Kindle Edition.

See also . . .

 

* Title credit

Click for a clearer version —

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

Thursday, April 30, 2020

Walpurgisnacht Geometry

Filed under: General — Tags: — m759 @ 11:59 pm

A version more explicitly connected to finite geometry —

For the six synthematic totals , see The Joy of Six.

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

Filed under: General — Tags: , , — m759 @ 8:09 am

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.

 

Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

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

The title refers to Calabi-Yau spaces.

T. S. Eliot —

Four Quartets

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

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

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

Related material from Plato and R. T. Curtis

Counting symmetries with the orbit-stabilizer theorem

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

Illustration of K3 surface related to Mathieu moonshine

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

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

Wednesday, May 1, 2019

The Medium and the Message

Filed under: General — Tags: , — m759 @ 6:45 pm

In memory of Quentin Fiore — from a Log24 search for McLuhan,
an item related to today's previous post . . .

Related material from Log24 on the above-reported date of death —

See also, from a search for Analogy in this journal . . .

 .

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.

Wednesday, March 6, 2019

A Hand Calculator

Filed under: General — m759 @ 10:28 pm

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 — 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.

Monday, June 25, 2018

The Trials of Device

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

"A blank underlies the trials of device."

Wallace Stevens

"Designing with just a blank piece of paper is very quiet."

Kate Cullinane

Related material —

An image posted at 12 AM ET December 25, 2014:

The image stands for the
phrase "five by five,"
meaning "loud and clear."

Other posts featuring the above 5×5 square with some added structure:

Saturday, June 16, 2018

For June 16

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

"But perhaps the desire for story
is what gets us into trouble to begin with."

Sarah Marshall on June 5, 2018

"Beckett wrote that Joyce believed fervently in
the significance of chance events and of
random connections. ‘To Joyce reality was a paradigm,
an illustration of a possibly unstateable rule
According to this rule, reality, no matter how much
we try to manipulate it, can only shift about
in continual movement, yet movement
limited in its possibilities’ giving rise to
‘the notion of the world where unexpected simultaneities
are the rule.’ In other words, a coincidence  is actually
just part of a continually moving pattern, like a kaleidoscope.
Or Joyce likes to put it, a ‘collideorscape’."

— Gabrielle Carey, "Breaking Up with James Joyce,"
Sydney Review of Books , 15 June 2018

Carey's carelessness with quotations suggests a look at another
author's quoting of Ellmann on Joyce

Friday, March 2, 2018

A Snowflake on the Iceberg

Filed under: General — m759 @ 11:00 am

A scholium on the previous post, "Mother Ship Meets Mother Church" —

"Consider Saint Hedwig of Silesia (1174–1243), for example,
who is elegantly depicted in this catalogue, striking a pose
with her prayer book, rosary, and Virgin and child statue
(a reminder of the legitimacy of her sex), along with boots
slung over her elbow so she could walk barefoot like
the apostles. Between miracles, she was also known to 
supervise con­struction of new convents. Hedwig is but
a snowflake on the iceberg  of the extra­ordinary role of
actual women in the Middle Ages, to which more evidence
is added continually thanks the Feminae database."

— From "Wonder Women," by Matthew J. Milliner

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

Thursday, March 30, 2017

The Internet Abyss

Filed under: General — Tags: — m759 @ 3:09 pm

Suggested by the previous post, The Crimson Abyss

“Wer mit Ungeheuern kämpft, mag zusehn, dass er nicht dabei
zum Ungeheuer wird. Und wenn du lange in einen Abgrund blickst,
blickt der Abgrund auch in dich hinein.”

“He who fights with monsters should look to it that he himself
does not become a monster. And if you gaze long into an abyss,
the abyss also gazes into you.”

—  Nietzsche, Beyond Good and Evil , Aphorism 146

From the Internet Abyss on Red October Day, October 25, 2010 —

An image reproduced in this  journal on that same day

Image-- 'Then a miracle occurs' cartoon

Cartoon by S.Harris

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.

Friday, August 19, 2016

Ex Machina

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

For the Symmetry Dancers of CERN

1947 Opening

Filed under: General,Geometry — Tags: — m759 @ 1:06 pm
 
1. e4  See also Geometry of the I Ching
and "Miracle on 34th Street."

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." 

Monday, April 11, 2016

Combinatorial Spider

Filed under: General,Geometry — Tags: , , — m759 @ 1:16 pm

“Chaos is order yet undeciphered.”

— The novel The Double , by José Saramago,
on which the film "Enemy" was based

Some background for the 2012 Douglas Glover
"Attack of the Copula Spiders" book
mentioned in Sunday's Synchronicity Check

  • "A vision of Toronto as Hell" — Douglas Glover in the
    March 25, 2011, post Combinatorial Delight
  • For Louise Bourgeois — a post from the date of Galois's death—

http://www.log24.com/log/pix11B/110715-GaloisMemorial-Lg.jpg

  • For Toronto — Scene from a film that premiered there
    on Sept. 8, 2013:

Sunday, April 10, 2016

In Memoriam

Filed under: General — Tags: — m759 @ 8:48 pm

A great cartoonist died on Friday.

Related religious art — Ogdoads and Miracle Cartoon.

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.

Historical Note

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

Possible title

A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24

Friday, November 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group



Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation
 
). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.

References:

Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —

Background reading:

Ron Shaw on finite geometry, Clifford algebras, and Dirac groups 
(undated compilation of publications from roughly 1994-1995)—

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, May 18, 2015

Ex Nihilo

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

Yesterday morning's quote from The Politics of Experience :

"Creation ex nihilo  has been pronounced impossible even for
God. But we are concerned with miracles."

Related material:

Curtis + Turyn in this journal and Sidney Harris on theology —
 

Sunday, May 17, 2015

Metaphysics for Sunday

Filed under: General — m759 @ 8:00 am

From Chapter 1 of R. D. Laing's The Politics of Experience —

"An activity has to be understood in terms of the experience
from which it emerges. These arabesques that mysteriously
embody mathematical truths only glimpsed by a very few —
how beautiful, how exquisite — no matter that they were
the threshing and thrashing of a drowning man.

We are here beyond all questions except those of being
and nonbeing, incarnation, birth, life and death.

Creation ex nihilo  has been pronounced impossible even for
God. But we are concerned with miracles. We must hear the
music of those Braque guitars (Lorca*)."

See also Christmas Day, 2009.

*  Update of Sunday afternoon: A search for the Lorca quote yields
   no result, but Cocteau wrote that "Mon rêve, en musique, serait
   d'entendre la musique des guitares de Picasso. "
   (Oeuvres complètes , Vol. 10, p. 107)

   See also Stevens + "Blue Guitar" in this journal.

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.

Monday, November 3, 2014

Gods and Giants

Filed under: General — Tags: — m759 @ 2:01 pm

A weblog reports Chris Rock's remarks
on Saturday Night Live this past weekend:

"It’s America, we commercialize everything.
Look at what we did to Christmas.
Christmas.  Christmas is Jesus’ birthday.
It’s Jesus’ birthday.  Now, I don’t know Jesus
but from what I’ve read, Jesus is the least
materialistic person to ever roam the earth.
No bling on Jesus.
Jesus kept a low profile and we turned his
birthday into the most materialistic day of the
year.  Matter of fact, we have the Jesus birthday
season.  It’s a whole season of materialism.
Then, at the end of the Jesus birthday season
we have the nerve to have an economist come
on TV and tell you how horrible the Jesus birthday
season was this year.  Oh, we had a horrible Jesus’
birthday this year.  Hopefully, business will pick up
by his Crucifixion.”

Related music and image:

"Show us the way to the next little girl …"

Natalie Wood in "Miracle on 34th Street" (1947)

Related non-materialistic meditations:
The Rhetoric of Abstract Concepts and Gods and Giants.

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, May 15, 2014

Mathematics as Religion

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

On Edward Frenkel:

"Math is, for him, 'a narrative' of human endeavor
that shares much with art, music and religion.

For instance, he describes new mathematical insights
as 'revelations,' and the utterly unchanging truths of
mathematical ideas are 'nothing short of a miracle.'"

Uh-huh.

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.”

Friday, March 28, 2014

Blazing Thule

Filed under: General — Tags: , — m759 @ 10:20 am

The title is suggested by a new novel (see cover below),
and by an unwritten book by Nabokov —

Siri Hustvedt, 'The Blazing World'.

Related material:

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.

Saturday, July 13, 2013

Circles

Filed under: General,Geometry — Tags: , — m759 @ 2:22 am

A sort of poem
by Gauss and Weyl —

Click the circle for the context in Weyl's Symmetry .

For related remarks, see the previous post.

A literary excursus—

Brad Leithauser in a New Yorker  post of July 11, 2013:

Reading Poems Backward

If a poet determines that a poem should begin at point A and conclude at point D, say, the mystery of how to get there—how to pass felicitously through points B and C—strikes me as an artistic task both genuine and enlivening. There are fertile mysteries of transition, no less than of termination.

And I’d like to suppose that Frost himself would recognize that any ingress into a poem is better than being locked out entirely. His little two-liner, “The Secret,” suggests as much: “We dance round in a ring and suppose / But the Secret sits in the middle and knows.” Most truly good poems might be said to contain a secret: the little sacramental miracle by which you connect, intimately, with the words of a total stranger. And whether you come at the poem frontward, or backward, or inside out—whether you approach it deliberately, word by word and line by line, or you parachute into it borne on a sudden breeze from the island of Serendip—surely isn’t the important thing. What matters is whether you achieve entrance into its inner ring, and there repose companionably beside the Secret.

One should try, of course, to avoid repose in an inner circle of Hell .

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."

Speech

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

Speech by Wallace Stevens upon accepting
an honorary degree from Bard College in 1951

(Click to enlarge.)

Transcription of conclusion:

"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

Friday, August 17, 2012

Hidden

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

Detail from last night's 1.3 MB image
"Search for the Lost Tesseract"—

The lost tesseract appears here on the cover of Wittgenstein's
Zettel  and, hidden in the form of a 4×4 array, as a subarray 
of the Miracle Octad Generator on the cover of Griess's
Twelve Sporadic Groups  and in a figure illustrating
the geometry of logic.

Another figure—

IMAGE- Serbian chess innovator Svetozar Gligoric dies at 89

Gligoric died in Belgrade, Serbia, on Tuesday, August 14.

From this journal on that date

"Visual forms, he thought, were solutions to 
specific problems that come from specific needs."

— Michael Kimmelman in The New York Times
    obituary of E. H. Gombrich (November 7th, 2001)

Monday, July 9, 2012

Taps for Fatso

Filed under: General — m759 @ 7:00 am
 
IMAGE- Donna Reed and Montgomery Clift in 'From Here to Eternity'

A Story That Works

“There is the dark, eternally silent, unknown universe;
there are the friend-enemy minds shouting and whispering
their tales and always seeking the three miracles —

  • that minds should really touch, or
  • that the silent universe should speak, tell minds a story,
  • or (perhaps the same thing) that there should be a story
    that works, that is all hard facts, all reality, with
    no illusions and no fantasy;

and lastly, there is lonely, story-telling, wonder-questing,
mortal me.”

Dichtung ist Stiftung.

Sunday, September 18, 2011

Midnight in Dostoevsky

Filed under: General — m759 @ 12:00 am

"But Thou knewest not, it seems, that no sooner would man reject
miracle than he would reject God likewise, for he seeketh less
God than 'a sign' from Him." —The Grand Inquisitor

Update of 1:44 AM—

Seek and Ye Shall Find …

(Click images for larger context)

http://www.log24.com/log/pix11B/110918-DostoevskyDetail.jpg

  The Holy Office —

http://www.log24.com/log/pix11B/110918-ThinkfreeOffice3Setup.jpg

Saturday, September 3, 2011

The Galois Tesseract (continued)

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

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.

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