MOG « Search Results

Sunday, November 17, 2024

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

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

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

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

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

**4×4 Square**

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

— The diamond theorem of Steven H. Cullinane.

**4×6 Rectangle**

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

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

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

View this post as a PDF.

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Tuesday, April 7, 2020

Moonshine, the MOG, and the Hexacode

Filed under: General — m759 @ 9:26 pm

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Thursday, April 25, 2013

Note on the MOG Correspondence

Filed under: General,Geometry — Tags: Geometry, Six-Set — m759 @ 4:15 pm

In light of the April 23 post "The Six-Set,"
the caption at the bottom of a note of April 26, 1986
seems of interest:

"The R. T. Curtis correspondence between the 35 lines and the
2-subsets and 3-subsets of a 6-set. This underlies M₂₄."

A related note from today:

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Friday, May 14, 2010

Competing MOG Definitions

Filed under: General,Geometry — Tags: Geometry, Natural Diagram, Octad, Octad Generator — m759 @ 9:00 pm

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

From the 1976 paper defining the MOG—

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

Curtis’s 1976 Fig. 4. (The MOG.)

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

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

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

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

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

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Saturday, November 16, 2024

Automorphism Groups: Examples for Weyl

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

View a PDF of this post.

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

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Tuesday, September 17, 2024

Geometry for Jews:
“Kickin’ Down the Cobblestones…” — Song lyric

Filed under: General — Tags: Brick Space — m759 @ 2:58 am

Some less stressful material . . .

"a medley cobbled together" —

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Thursday, July 18, 2024

Brick Space

Filed under: General — Tags: Brick Space, Bricks and Bats, Chess Bricks — m759 @ 1:45 am

Compare and Contrast

A rearranged illustration from . . .

R. T. Curtis, "A New Combinatorial Approach to M₂₄,"
Mathematical Proceedings of the Cambridge Philosophical Society ,
Volume 79 , Issue 1 , January 1976 , pp. 25 – 42
DOI: https://doi.org/10.1017/S0305004100052075 —

The image “MOGCurtis03.gif” cannot be displayed, because it contains errors.

The "Brick Space" model of PG(5,2) —

Brick space: The 2x4 model of PG(5,2)

Background: See "Conwell heptads" on the Web.

See as well Nocciolo in this journal and . . .

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Wednesday, July 3, 2024

The Nutshell Miracle

Filed under: General — Tags: Brick Space — m759 @ 10:42 pm

'Then a miracle occurs' cartoon

— Cartoon by S. Harris

From a search in this journal for nocciolo —

From a search in this journal for PG(5,2) —

From a search in this journal for Curtis MOG —

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.)

From a search in this journal for Klein Correspondence —

Philippe Cara on the Klein correspondence

The picture of PG(5,2) above as an expanded nocciolo
shows that the Miracle Octad Generator illustrates
the Klein correspondence.

Update of 10:33 PM ET Friday, July 5, 2024 —

See the July 5 post "De Bruyn on the Klein Quadric."

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Wednesday, June 26, 2024

From the Latin Club Gang

Filed under: General — Tags: Almonds, Cartoon Graveyards, Pathmark — m759 @ 3:08 pm

"We put the sex in sextets."

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Tuesday, May 14, 2024

Octad Space

Filed under: General — Tags: Bitspace, Brick Mosaic, Brick Space, Octad — m759 @ 4:45 am

From "Geometry of 6 and 8," Dec. 8, 2019 —

Brick mosaic.

Compare and contrast —

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Monday, April 22, 2024

For Harlan Kane: The Brick Trick

Filed under: General — m759 @ 12:47 am

From a transcript of the film —

— You could be making a killing out there consulting, but you’re laying brick. Why?

— Because when I hold a brick in my hand, I know exactly what it is and what it will do.

Every single time.

Its form is its function.

That gives me peace.

The film is based on a novel of the same title, by an author,
Paul Lindsay, who reportedly died on September 1, 2011.

For some bricks of a different sort, see the "Miracle Octad Generator"
(MOG) of R. T. Curtis in this journal on Sept. 1, 2011 — September Morn.

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Friday, April 19, 2024

Benchmarking Meta.ai

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

Transcription —

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

You

What is the Cullinane diamond theorem?

Meta AI

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

Here are some key points about the theorem :

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

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Thursday, March 14, 2024

For Aestheticians: Seeds, Good and Not So Good

Filed under: General — Tags: Dakota Aesthetics, February Poetics, Quantum Poetics, Whanganui — m759 @ 7:08 pm

In memory of . . .

Good Seed: Sydell.

Not So Good: Pearl.

This post was suggested by Sam Levinson's work in Whanganui NZ
and Alan D. Perlis's work in Birmingham AL —

From South Dakota, related material for Bible fans —

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Monday, March 4, 2024

Report on the Diamond Theorem
by Google Gemini (Advanced)

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

Diamond Theorem Report

by Google Gemini AI

Monday, March 4, 2024

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

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

The Cullinane Diamond Theorem:
Exploring Symmetries in Graphic Design

Abstract

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

Introduction

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

Mathematical Background

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

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

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

The Cullinane Diamond Theorem

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

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

Connections to the Miracle Octad Generator

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

Applications

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

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

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

Conclusion

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

Citations

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

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

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

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

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

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

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

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Sunday, March 3, 2024

Deep Blue Research: A Report by You.com AI

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

Cullinane Diamond Theorem Research Report

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

Overview of the Cullinane Diamond Theorem

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

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

Symmetry and Group Theory

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

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

The Diamond Theorem in Various Dimensions

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

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

Algebraic Implications

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

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

Philosophical and Cultural Context

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

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

Clarification on Cullinan Diamonds

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

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

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

Instagram ad for You.com AI in research mode

"Show me ALL your sources, babe."

— Line adapted from Leonardo DiCaprio

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Monday, January 29, 2024

Self as Imago Dei: Hofstadter vs. Valéry

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

Google search result:

Imago Dei in Thomas Aquinas

Saint Anselm College

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

PDF

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

11 pages

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

Tuesday, June 15, 2010

Imago, Imago, Imago

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

Recommended— an online book—

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

See in particular

Valéry and the Discourse On His Method.

Pages 156-157—

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

Notes:

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

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

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

and Cullinane’s purely geometric four-diamond figure:

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

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

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Monday, January 15, 2024

Foursquare Variations

Filed under: General — Tags: The Didion Logo — m759 @ 12:00 am

From Encyclopedia of Mathematics —

The above images from the history of mathematics might be
useful at some future point for illustrating academic hurly-burly.

Tuesday, December 26, 2023

Color Box

Filed under: General — Tags: Color Space — m759 @ 10:44 am

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Sunday, December 24, 2023

Let Us Now Praise Famous Omega*

Filed under: General — Tags: Famous Omega — m759 @ 2:35 am

* The title is of course a reference to the Knoxville of the previous post.

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Thursday, December 21, 2023

Pi.ai

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

Thursday, December 21, 2023

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

How's your day going?

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

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

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

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

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

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

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

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

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

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

Scholium —

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

Sia!

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Tuesday, December 19, 2023

Copilot Report

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

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

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

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

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

Generating answers for you…

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

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

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

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

Learn more

archive.org

xenon.stanford.edu

academia.edu

en.wikipedia.org

archive.org
_________________________________________________________

__________________________________________________________

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

Here is a summary of the document:

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

Reference:

1. Page 2 – Line 1

Copilot suggestions following document summary —

Correction:

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

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Monday, December 18, 2023

AI Class

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

Bringing Up Bing :

A Tutorial for Bing on Monday, December 18, 2023

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

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

Generating answers for you…

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

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

The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry ^1,². The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements ^1,². This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets ¹ ^,2.

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

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

Learn more

Reply to Bing's Response —

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

Response to the Above Reply —

Generating answers for you…

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

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Saturday, November 18, 2023

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

Filed under: General — Tags: Octad — 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).

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."

_* ^—

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Monday, October 16, 2023

A Harlan Kane Rite Aid Special: Chapter 11

Filed under: General — Tags: Artfield, Iconology 2023, Priority — m759 @ 11:01 pm

From a search in this journal for "Chapter 11" —

Inner structure —

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

The above three images share the same
vector-space structure —

0	c	d	c + d
a	a + c	a + d	a + c + d
b	b + c	b + d	b + c + d
a + b	a + b + c	a + b + d	a + b + c + d

(This vector-space a b c d  diagram is from
    Chapter 11 of Sphere Packings, Lattices
    and Groups , by John Horton Conway and
    N. J. A. Sloane, first published by Springer
    in 1988.)

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Friday, March 24, 2023

For the Latin Club Gang*

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

Something else —

* "We put the sex in sextets."

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Thursday, January 19, 2023

Two Approaches to Local-Global Symmetry

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

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

The First Approach — Via Substructure Isomorphisms —

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

Local or global?

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

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

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

A related discussion of the same approach —

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

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

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

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

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

Related material:

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

and, more recently,

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

This approach seems to be a rather deep rabbit hole.

The Second Approach — Via Induced Group Actions —

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

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

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

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

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Thursday, November 3, 2022

Dazzled or Baffled?

Filed under: General — m759 @ 11:06 pm

"If you can't dazzle 'em with brilliance,
baffle 'em with bullshit." — Folk saying

Brilliance —

Bullshit —

A New York Times book review on All Saints' Day 2002 —

"Like every villain, Sill has an origin story. This one involves
the murder of his father and the assassination of Martin Luther King Jr.
As vengeance, Sill resolves to destroy America using a weapon
called a 'complex projective plane orbiter.' What does that mean?
Don’t worry about it."

Wednesday, September 28, 2022

Bitspace Note

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

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Sunday, September 11, 2022

Raiders of the Lost Space

Filed under: General — Tags: Bitspace, Brick Space — m759 @ 2:29 pm

From 1981 —

From today —

Update —

A Magma check of the motif-generated space shows that
its dimension is only 8, not 12 as with the MOG space.
Four more basis vectors can be added to the 24 motifs to
bring the generated space up to 12 dimensions: the left
brick, the middle brick, the top half (2×6), the left half (4×3).
I have not yet checked the minimum weight in the resulting
12-dimensional 4×6 bit-space.

— SHC 4 PM ET, Sept. 12, 2022.

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Thursday, September 8, 2022

Analogy in Mathematics: Chevron Variations

Filed under: General — Tags: Analogy and Mysticism, Chevron Variations, Circle Square Gauss, Maltese V, Octad, The 24 Variations — m759 @ 1:51 am

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).

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 —

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Friday, August 19, 2022

The Guy Embedding

Filed under: General — Tags: Octad, on220819, The Guy Embedding — 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.

For some context, click the embedded guy.

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

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Wednesday, May 18, 2022

“Make it new.”

Filed under: General — Tags: Math vs. Narrative — m759 @ 10:41 am

"Literature is not demography, nor is it politics, even if it is quite often political. Progress, at least when it comes to cultural production, becomes lasting not when one is trying to join the reigning establishment, or lamenting how exclusionary it is (it so often is!), but, to quote that anti-Semite Ezra Pound, when one seeks, in the first place, to make it new ."

— Mordechai Levy-Eichel and Daniel Scheinerman on
May 17, 2022, in The Chronicle of Higher Education —
"Digital humanists need to learn how to count:
A prominent recent book in the field
suffers serious methodological pitfalls."

And then there are methodological sinkholes —

See Log24 posts tagged Sinkhole and . . .

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Saturday, February 5, 2022

Mathieu Cube Labeling

Filed under: General — Tags: Aitchison, Cube Labeling, Geometric Theology, Mathieu Cube, Octad — m759 @ 2:08 pm

Shown below is an illustration from "The Puzzle Layout Problem" —

September 2003
Lecture Notes in Computer Science 2912:500-501
DOI:10.1007/978-3-540-24595-7_50
Source: DBLP
Conference: Graph Drawing, 11th International Symposium,
GD 2003, Perugia, Italy, September 21-24, 2003, Revised Papers
Authors: Kozo Sugiyama, Seok-Hee Hong, Atsuhiko Maeda

Exercise: Using the above numerals 1 through 24
(with 23 as 0 and 24 as ∞) to represent the points
∞, 0, 1, 2, 3 … 22 of the projective line over GF(23),
reposition the labels 1 through 24 in the above illustration
so that they appropriately* illustrate the cube-parts discussed
by Iain Aitchison in his March 2018 Hiroshima slides on
cube-part permutations by the Mathieu group M₂₄.

A note for Northrop Frye —

Interpenetration in the eightfold cube — the three midplanes —

A deeper example of interpenetration:

Aitchison has shown that the Mathieu group M₂₄ has a natural
action on the 24 center points of the subsquares on the eightfold
cube's six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.

* "Appropriately" — I.e. , so that the Aitchison cube octads correspond
exactly, via the projective-point labels, to the Curtis MOG octads.

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Thursday, February 3, 2022

Through the Asian Looking Glass

Filed under: General — Tags: Knight Move, Octad, Octad Generator, The Guy Embedding — 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

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.

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, Chinese, Japanese, Vietnamese 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:

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Four-Color Structures (Review)

Filed under: General — Tags: Four Color, Heartland Sutra, Octad, Octad Generator — m759 @ 1:30 pm

Miracle Octad Generator — Analysis of Structure

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

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Sunday, January 30, 2022

Rock Music

Filed under: General — Tags: Glitter for Stoppard, Music for Steiner — m759 @ 1:38 pm

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

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

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Thursday, December 16, 2021

Veritas

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

"You get right down to the naked truth
With those dirty, dirty looks" — Juice Newton (1983)

Search result for "Sandringham juice octads" —

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Saturday, September 11, 2021

Reality for Academia

Filed under: General — Tags: Matrix 4 — m759 @ 4:16 pm

The title of the previous post, "Ground Omega," suggests a related nightmare . . .

A writer of fiction in the previous post —

"When we say a thing is unreal, we mean it is too real…."

Old joke —

"What you mean 'we,' paleface?"

At Ground Omega in the above My Hero Academia site —

"The twenty-four students are split into six groups of four…."

I prefer the similar splittings of the Curtis Omega —

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Friday, May 7, 2021

Through the Miracle Looking Glass

Filed under: General — Tags: History Museum — 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

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Thursday, April 22, 2021

A New Concrete Model for an Old Abstract Space

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

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

(Click to enlarge.)

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

Cite as

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

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

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Wednesday, April 21, 2021

The Spielvogel Conundrum (Attn: Harlan Kane*)

Filed under: General — m759 @ 7:31 pm

In memory of an advertising mogul who reportedly died today:

The above Altmetric report is apparently thanks to
my registering with ScienceOpen.com on April 19.

* Author of The Abacus Conundrum.

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Saturday, April 17, 2021

Note

Filed under: General — Tags: Richter — m759 @ 5:12 pm

Analyze this.

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Monday, March 15, 2021

The Abstract Signature

Filed under: General — Tags: Abstract Signature, IDE of March, Kentucky Gambit, Penetrability, Scully Signature — m759 @ 1:40 pm

Caption: "I notice the signatures are never abstract." —

Abstract Art

Abstract Signature

From Encyclopedia of Mathematics —

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Saturday, November 28, 2020

Star Quality: “Mirror Guy”

Filed under: General — Tags: The Lindelof Watchmen — m759 @ 12:30 pm

Related material from pure mathematics — M. J. T. Guy —

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Saturday, October 24, 2020

Switchin’ the Positions*

Filed under: General — Tags: Code Girl, Switchin' the Positions — m759 @ 11:27 am

“C24 is the list of codewords of the extended
binary Golay code C₂₄. Each codeword is expressed
by a subset of the set M of the positions [1, . . . , 24]
of MOG.”

— From Shimada’s notes on computational data at
http://www.math.sci.hiroshima-u.ac.jp/~shimada/
preprints/Edge/PaperEdge/compdataEdge.pdf .

* Related material — A new Ariana Grande video and . . .

a recent digital artwork, “Code Girl,” with accompanying story —

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Wednesday, September 2, 2020

Space Wars: Sith Pyramid vs. Jedi Cube

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

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Thursday, August 27, 2020

The Complete Extended Binary Golay Code

Filed under: General — Tags: Canonicity, MJT Guy, Octad, Octad Generator, Springer, The Guy Embedding — 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 —

Thursday, August 6, 2020

After Personalities . . . Principles

Filed under: General — m759 @ 1:10 pm

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

Seven years ago yesterday —

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

Icon Parking posts and . . .

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Tuesday, July 28, 2020

For 7/28

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

Miracle Octad Generator — Analysis of Structure

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Wednesday, July 22, 2020

Card

Filed under: General — Tags: Geometric Theology, MJT Guy, Structure and Mutability, The Guy Embedding — m759 @ 2:48 am

“The pattern of the thing precedes the thing.
I fill in the gaps of the crossword at any spot
I happen to choose. These bits I write on
index cards until the novel is done.”

— Vladimir Nabokov, interview,
Paris Review No. 41 (Summer-Fall 1967).

Another story —

Related material: Mathematics as a Black Art.

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Monday, July 13, 2020

The Lexicographic Octad Generator (LOG)*

Filed under: General — Tags: Canonicity, Lexico.graphics, MJT Guy, Natural Diagram, Octad, Octad Generator, The Guy Embedding, The Log — m759 @ 5:43 pm

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:

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).

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Monday, May 25, 2020

The Shimada Documents

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

(For Harlan Kane)

From Shimada’s notes on computational data at
http://www.math.sci.hiroshima-u.ac.jp/~shimada/
preprints/Edge/PaperEdge/compdataEdge.pdf —

“C24 is the list of codewords of the extended
binary Golay code C₂₄. Each codeword is expressed
by a subset of the set M of the positions [1, . . . , 24]
of MOG.”

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Thursday, May 14, 2020

Art Issue*

Filed under: General — Tags: Art Issue, May Play, Richter — m759 @ 11:25 am

"… the beautiful object
that stood in
for something else.”

— Holland Cotter quoting an art historian
in The New York Times on May 13

From a post of April 27, 2020 —

“The yarns of seamen have a direct simplicity,
the whole meaning of which lies within the shell
of a cracked nut. But Marlow was not typical
(if his propensity to spin yarns be excepted),
and to him the meaning of an episode was not inside
like a kernel but outside….”

— Joseph Conrad in Heart of Darkness

The beautiful object —

Something else —

* The title is a reference to other posts now also tagged Art Issue.

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Saturday, May 2, 2020

Turyn’s Octad Theorem: The Next Level*

Filed under: General — Tags: Octad, Octad Generator, The Next Level — 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 —

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Monday, April 27, 2020

The Cracked Nut

Filed under: General — Tags: Art Issue, Octad, The Nutshell Suite — m759 @ 1:25 pm

“At that instant he saw, in one blaze of light, an image of unutterable
conviction, the reason why the artist works and lives and has his being –
the reward he seeks –the only reward he really cares about, without which
there is nothing. It is to snare the spirits of mankind in nets of magic,
to make his life prevail through his creation, to wreak the vision of his life,
the rude and painful substance of his own experience, into the congruence
of blazing and enchanted images that are themselves the core of life, the
essential pattern whence all other things proceed, the kernel of eternity.”

— Thomas Wolfe, Of Time and the River

“… the stabiliser of an octad preserves the affine space structure on its
complement, and (from the construction) induces AGL(4,2) on it.
(It induces A₈ on the octad, the kernel of this action being the translation
group of the affine space.)”

— Peter J. Cameron,
The Geometry of the Mathieu Groups (pdf)

“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside…."

— Joseph Conrad in Heart of Darkness

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Friday, April 24, 2020

Art at Cologne

Filed under: General — Tags: Art Issue, Richter, The Cologne Sextet — m759 @ 10:53 pm

This post was suggested by a New York Review of Books article
on Cologne artist Gerhard Richter in the May 14, 2020, issue —

“The Master of Unknowing,” by Susan Tallman.

Some less random art —

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Thursday, April 23, 2020

Octads and Geometry

Filed under: General — Tags: Art Issue, Brick Space, Brick Symmetry, Brick Wall, Devs, Octad, Octad Generator, Octad Group, Overarching Group — m759 @ 10:11 pm

See the web pages octad.group and octad.us.

Related geometry (not the 759 octads, but closely related to them) —

The 4×6 rectangle of R. T. Curtis
illustrates the geometry of octads —

Curtis splits the 4×6 rectangle into three 4×2 "bricks" —

"In fact the construction enables us to describe the octads
in a very revealing manner. It shows that each octad,
other than Λ₁, Λ₂, Λ₃, intersects at least one of these ' bricks' —
the 'heavy brick' – in just four points." . . . .

— R. T. Curtis (1976). "A new combinatorial approach to M₂₄,"
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42.

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Wednesday, February 19, 2020

Aitchison’s Octads

Filed under: General — Tags: Aitchison, Eightfold Cube, Fro Zen, Mathieu Cube, Octad, Octad Generator — 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 .

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.

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Monday, December 23, 2019

Orbit

Filed under: General — Tags: Inner-Space Variations, Obit et Orbit, Octad — m759 @ 7:34 pm

"December 22, the birth anniversary of India’s famed mathematician
Srinivasa Ramanujan, is celebrated as National Mathematics Day."
— Indian Express yesterday

"Orbits and stabilizers are closely related." — Wikipedia

Symmetries by Plato and R. T. Curtis —

In the above, 322,560 is the order
of the octad stabilizer group .

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Wednesday, December 11, 2019

Miracle Octad Generator Structure

Filed under: General — Tags: Design Notes Dec. 11, Geometry of Even Subsets, Natural Diagram, Octad, Octad Generator — m759 @ 11:44 pm

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.)

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Tuesday, October 29, 2019

Triangles, Spreads, Mathieu

Filed under: General — Tags: Cubehenge, Geometry of Even Subsets, Triangles-Spreads-Mathieu — m759 @ 8:04 pm

There are many approaches to constructing the Mathieu
group M₂₄. The exercise below sketches an approach that
may or may not be new.

Exercise:

It is well-known that …

There are 56 triangles in an 8-set.
There are 56 spreads in PG(3,2).
The alternating group A_nis generated by 3-cycles.
The alternating group A₈is isomorphic to GL(4,2).

Use the above facts, along with the correspondence
described below, to construct M₂₄.

Some background —

A Log24 post of May 19, 2013, cites …

Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.

See also a Google search for "56 triangles" "56 spreads" Mathieu.

Update of October 31, 2019 — A related illustration —

Update of November 2, 2019 —

See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel (Academic Press, 1991).
That page is from a paper published in 1970.

Update of December 20, 2019 —

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Saturday, September 21, 2019

Annals of Random Fandom

Filed under: General — Tags: Octad, Octad Generator, Pyramid Game, Tetrahedron vs. Square — 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.

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Wednesday, March 6, 2019

A Hand Calculator

Filed under: General — m759 @ 10:28 pm

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Tuesday, March 5, 2019

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

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

A Midrash for Wikipedia

Midrash —

Related material —

________________________________________________________________________________

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Friday, March 1, 2019

Wikipedia Scholarship (Continued)

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

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

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

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

Revision history accounting for the above change from yesterday —

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

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

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

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Friday, February 22, 2019

Back Issues of AMS Notices

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

From the online home page of the new March issue —

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

For instance . . .

Related material now at Wikipedia —

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Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

Filed under: General — Tags: Kummerhenge, Octad, Octad Generator, The Omega Matrix — 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 M₂₄with the aid of the MOG.

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Sunday, December 2, 2018

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: Aitchison, Geometric Theology, Mathieu Cube, Octad, Octad Generator — 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.

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Wednesday, October 3, 2018

Adamantine Meditation

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

A Catholic philosopher —

Related art —

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Sunday, September 23, 2018

Three Times Eight

Filed under: General,Geometry — Tags: ABC Art 2018, Octad, Octad Generator — m759 @ 9:21 am

The New York Times 's Sunday School today —

I prefer the three bricks of the Miracle Octad Generator —

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Wednesday, July 11, 2018

Titans

Filed under: General — Tags: Clash of the Titans — m759 @ 5:01 pm

"By The Boston Globe

July 10, 2018

The private jets have begun clogging the jetways
in Sun Valley, Idaho, which can only mean one thing:
'Billionaire summer camp’' has begun.

The annual Allen & Company conference, the investment
firm’s invite-only gathering of some of the world’s most
powerful corporate titans, officially begins on Wednesday."

In other news —

"NASHVILLE, Tenn. –

Get ready to see the Titans in training camp."

See also another post now tagged "Clash of the Titans."

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Friday, May 4, 2018

Entropy

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

A more serious note in memory of Anatole Katok:

"Entropy measures the unpredictability
of a system that evolves over time."

— Alex Wright, BULLETIN (New Series)
OF THE AMERICAN MATHEMATICAL SOCIETY

Volume 53, Number 1, January 2016, Pages 41–56

http://dx.doi.org/10.1090/bull/1513

Article electronically published on September 8, 2015:

FROM RATIONAL BILLIARDS
TO DYNAMICS ON MODULI SPACES

Abstract:

"This short expository note gives an elementary
introduction to the study of dynamics on certain
moduli spaces and, in particular, the recent
breakthrough result of Eskin, Mirzakhani,
and Mohammadi. We also discuss the context
and applications of this result, and its connections
to other areas of mathematics, such as algebraic
geometry, Teichmüller theory, and ergodic theory
on homogeneous spaces."

See also the lives of Ratner and Mirzakhani.

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Wednesday, April 25, 2018

An Idea

Filed under: General,Geometry — Tags: 2x4 — m759 @ 11:45 am

"There was an idea . . ." — Nick Fury in 2012

". . . a calm and objective work that has no special
dance excitement and whips up no vehement
audience reaction. Its beauty, however, is extraordinary.
It’s possible to trace in it terms of arithmetic, geometry,
dualism, epistemology and ontology, and it acts as
a demonstration of art and as a reflection of
life, philosophy and death."

— New York Times dance critic Alastair Macaulay,
quoted here in a post of August 20, 2011.

Illustration from that post —

A 2x4 array of squares

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Thursday, November 16, 2017

A Line at Infinity

Filed under: G-Notes,General,Geometry — m759 @ 12:00 pm

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Tuesday, September 12, 2017

Think Different

Filed under: General,Geometry — Tags: Cube Bricks 1984, Cube School — m759 @ 11:00 pm

The New York Times online this evening —

"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."

— James Poniewozik

Review —

Thursday, September 1, 2011

How It Works

Filed under: Uncategorized — Tags: Geometry, Priority — m759 @ 11:00 AM

"Design is how it works." — Steven Jobs (See Symmetry and Design.)

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
— "Block Designs," by Andries E. Brouwer

. . . .

See also 1984 Bricks in this journal.

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Saturday, September 9, 2017

How It Works

Filed under: General,Geometry — Tags: Aqua, Bullshit Studies — m759 @ 8:48 pm

Del Toro and the History of Mathematics ,
Or: Applied Bullshit Continues

For del Toro —

For the history of mathematics —

Thursday, September 1, 2011

How It Works

Filed under: Uncategorized — Tags: Geometry, Priority — m759 @ 11:00 AM

"Design is how it works." — Steven Jobs (See Symmetry and Design.)

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
— "Block Designs," by Andries E. Brouwer

. . . .

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Sunday, September 3, 2017

Broomsday Revisited

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

Ivars Peterson in 2000 on a sort of conceptual art —

" Brill has tried out a variety of grid-scrambling transformations
to see what happens. Aesthetic sensibilities govern which
transformation to use, what size the rectangular grid should be,
and which iteration to look at, he says. 'Once a fruitful
transformation, rectangle size, and iteration number have been
found, the artist is in a position to create compelling imagery.' "

— "Scrambled Grids," August 28, 2000

Or not.

If aesthetic sensibilities lead to a 23-cycle on a 4×6 grid, the results
may not be pretty —

From "Geometry of the 4×4 Square."

See a Log24 post, Noncontinuous Groups, on Broomsday 2009.

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Wednesday, August 23, 2017

Pakanga

Filed under: General — Tags: Pakanga — m759 @ 4:44 am

("Every Picture Tells a Story," continued from August 15 )

Related material — Laughing-Academy Cartography.

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Saturday, January 28, 2017

Cranking It Up

Filed under: General — m759 @ 12:17 pm

From "Core," a post of St. Lucia's Day, Dec. 13, 2016 —

In related news yesterday —

California yoga mogul’s mysterious death:
Trevor Tice’s drunken last hours detailed

"Police found Tice dead on the floor in his home office,
blood puddled around his head. They also found blood
on walls, furniture, on a sofa and on sheets in a nearby
bedroom, where there was a large bottle of Grey Goose
vodka under several blood-stained pillows on the floor."

See as well an image from "The Stone," a post of March 18, 2016 —

Some backstory —

“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”

—Many Dimensions (1931), by Charles Williams

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Wednesday, January 18, 2017

An Associative Function …

Filed under: General — m759 @ 2:02 pm

Quoted here on December 16, 2006 —

See also …

"Ryan Reynolds Named Hasty Pudding’s
2017 Man of the Year"
Reviews of Reynolds' 2015 film "Self/Less" and of
the earlier similar film "Seconds" —

Point/Less Thriller

4 July 2015 9:00 AM, PDT | The Wrap |

"Ben Kingsley plays a New York real estate mogul who
pays big bucks to have his consciousness microwaved
into Ryan Reynolds' body in 'Self/less,' but the real
reheating of leftovers has already occurred: this new
science-fiction thriller borrows the foundation of a much
better film — John Frankenheimer’s 1966 'Seconds' —
and strips it of any larger meaning."

The date of the "Seconds" review above, 16 Dec. 2006, was
the reason for the requotation in the first paragraph above.

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Sunday, June 19, 2016

In Memoriam

Filed under: General — m759 @ 10:30 pm

For those who prefer the red pill to the blue pill —

See as well this afternoon's related Vanity Fair piece.

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Tuesday, June 7, 2016

Art and Space…

Filed under: General,Geometry — Tags: Barth Art, St. Bridget's Day — m759 @ 6:00 am

Continues, in memory of chess grandmaster Viktor Korchnoi,
who reportedly died at 85 yesterday in Switzerland —

IMAGE- Spielfeld (1982-83), by Wolf Barth

The coloring of the 4×4 "base" in the above image
suggests St. Bridget's cross.

From this journal on St. Bridget's Day this year —

"Possible title:

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

The narrative leap from image to date may be regarded as
an example of "knight's move" thinking.

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Saturday, March 19, 2016

Two-by-Four

Filed under: General,Geometry — Tags: 2x4, Philosopher's Stone — m759 @ 11:27 am

For an example of "anonymous content" (the title of the
previous post), see a search for "2×4" in this journal.

A 2x4 array of squares

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Monday, February 1, 2016

Historical Note

Filed under: General,Geometry — Tags: Natural Diagram, Priority — 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

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Monday, January 12, 2015

Points Omega*

Filed under: General,Geometry — Tags: Natural Diagram, Octad, Octad Generator, Points Omega — 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.

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Sunday, January 11, 2015

Real Beyond Artifice

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

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

He reportedly died on January 3, 2015.

An image from this journal on that date:

Another Gitterkrieg image:

The 24-set Ω of R. T. Curtis

Click on the images for related material.

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Thursday, January 8, 2015

Gitterkrieg

Filed under: General,Geometry — Tags: Gitterkrieg, Octad, Octad Generator — 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.

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Tuesday, December 2, 2014

Models

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

Continued from November 30, 2014

"Number right → Everything right." — Burkard Polster.

See also the six posts of November 30, St. Andrew's Day.

Related material —

Peter J. Cameron today discussing Julia Kristeva on poetry …

"This seems to be saying that the Kolmogorov
complexity of poetry is very low: the entire poem
can be generated from a small amount of information."

… and this journal on St. Andrew's day :

From "A Piece of the Storm,"
by the late poet Mark Strand —

A snowflake, a blizzard of one….

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Saturday, October 25, 2014

Foundation Square

Filed under: General,Geometry — Tags: 5x5, Fields, Octad, Octad Generator — 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…

"The Hoffman-Singleton Graph and its Automorphisms," by
Paul R. Hafner, Journal of Algebraic Combinatorics , 18 (2003), 7–12, and
the Web pages "Hoffman-Singleton Graph" and "Higman-Sims Graph"
of A. E. Brouwer.

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).)

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Monday, October 6, 2014

Mysterious Correspondences

Filed under: General,Geometry — m759 @ 9:36 am

(Continued from Beautiful Mathematics, Dec. 14, 2013)

“Seemingly unrelated structures turn out to have
mysterious correspondences.” — Jim Holt, opening
paragraph of a book review in the Dec. 5, 2013, issue
of The New York Review of Books

One such correspondence:

For bibliographic information and further details, see
the March 9, 2014, update to “Beautiful Mathematics.”

See as well posts from that same March 9 now tagged “Story Creep.”

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Tuesday, September 23, 2014

Meanwhile, Back at Harvard…

Filed under: General — Tags: Academy Map — m759 @ 10:18 am

"William Deresiewicz argued his claim that students of elite universities
are growingly risk-averse, homogeneous, and career-focused with a
panel of faculty members and students on Monday evening.

Hosted by Harvard’s Mahindra Humanities Center, the question-and-
answer-style forum involved a panel…. The panel was moderated by
Homi K. Bhabha, director of the Mahindra Center."

— Alexander H. Patel in today's online Harvard Crimson

Sunday, August 31, 2014

Sunday School

Filed under: General,Geometry — Tags: Galois's Space, Octad, Octad Generator, Springer — 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.]

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Tuesday, August 26, 2014

Lux et Veritas

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

Omega by Lux:

Mathieu and Omega Steps

Omega by Curtis:

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Sunday, August 24, 2014

Symplectic Structure…

Filed under: General,Geometry — Tags: Diamond Theorem Correlation, Octad, Octad Generator, Picture Show, Rosenhain and Göpel, Six-Set — m759 @ 12:00 pm

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:

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:

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Tuesday, June 17, 2014

Finite Relativity

Filed under: General,Geometry — Tags: Finite Relativity, Mathieu Raumproblem, Octad, Octad Generator — 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.

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Sunday, June 15, 2014

Aaron Eckhart Strikes Deep

Filed under: General — m759 @ 12:00 am

“Even paranoids have real enemies.”

— Attributed to Delmore Schwartz

“There is a difference as to whether you are describing paranoia
or whether you in fact are paranoid yourself.”

— The late Frank Schirrmacher, dw.de , July 2, 2013.

Schirrmacher reportedly died on Thursday, June 12, 2014.
See that date in this journal.

Paranoia is, of course, a fertile field for politicians and filmmakers:

Related material in this journal:

I, Frankenstein (May 15, 2014) and, for the Eckhart film “Erased,”
Hour of the Wolf (Nov. 9, 2006).

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Thursday, April 24, 2014

The Inscape of 24

Filed under: General,Geometry — Tags: Octad, Octad Generator — 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:

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Friday, March 28, 2014

Blazing Thule

Filed under: General — Tags: Octad, Octad Generator, Rune Art — m759 @ 10:20 am

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

Related material:

An artists' book scheduled to be released on March 21, 2014
A piece by Josefine Lyche in the artists' book
The original by Borges on which Lyche's piece was based
A solar image from a March 13 post echoing
that on the Blazing World cover above
A Tune for Josefine
The circular blazing image from last midnight's post Symbol
From March 21, the scheduled date of the Oslo
artists' book release, some remarks on the mathematics of the
Golay code, "Three Constructions of the Miracle Octad Generator"
Backstory: Duelle in this journal.

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Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: Brick Space, Depth, Geometry, Octad, Octad Generator, Priority — m759 @ 12:24 pm

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 Relativity, Galois 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 —

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Sunday, March 9, 2014

The Story Creeps Up

Filed under: General,Geometry — Tags: Octad, Octad Generator, Story Creep — 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).

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Friday, February 21, 2014

Raumproblem*

Filed under: General,Geometry — Tags: Aitchison, Figurate Geometry, Finite Relativity, Mathieu Raumproblem, Octad, Octad Generator — m759 @ 7:01 pm

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.

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Wednesday, December 25, 2013

Rotating the Facets

Filed under: General,Geometry — Tags: Cube School, Facets, Objects of Beauty — m759 @ 12:00 pm

Previous post—

“… her mind rotated the facts….”

Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:

IMAGE- Count rotational symmetries by rotating facets. Illustrated with 'Plato's Dice.'

“I’ve heard of affairs that are strictly Platonic”

— Song lyric by Leo Robin

* Footnote added on Dec. 26, 2013 —

See Arnold Emch, “Triple and Multiple Systems, Their Geometric
Configurations and Groups,” Trans. Amer. Math. Soc. 31 (1929),
No. 1, 25–42.

On page 42, Emch describes the above method of rotating a
hypercube’s 8 facets (i.e., three-dimensional cubes) to count
rotational symmetries —

Friday, December 20, 2013

For Emil Artin

Filed under: General,Geometry — Tags: Geometry, Octad, Octad Generator, Priority — 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.

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Saturday, December 14, 2013

Beautiful Mathematics

Filed under: General,Geometry — Tags: Depth, Gitterkrieg, Octad, Octad Generator, on131214 — 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).

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Saturday, November 23, 2013

Frame Tale (continued)

Filed under: General — Tags: Yale Weekend — m759 @ 10:30 am

See The X-Men Tree, another tree, and Trinity MOG.

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Monday, August 12, 2013

Form

Filed under: General,Geometry — Tags: Diamond-Theory-tesseract, Natural Diagram, Octad, Octad Generator — 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—

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—

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.

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Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: Fields, Octad, Octad Generator — 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 .

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Tuesday, May 28, 2013

Codes

Filed under: General,Geometry — Tags: Coding Theory, Diamond-Theory-tesseract, Geometry, Natural Diagram, Octad, Octad Generator, Priority — m759 @ 12:00 pm

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):

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Sunday, April 28, 2013

The Octad Generator

Filed under: General,Geometry — Tags: Dots, Geometry, Octad, Octad Generator, Six-Set — 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 —

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.

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Thursday, April 25, 2013

Rosenhain and Göpel Revisited

Filed under: General,Geometry — Tags: Geometry, Octad, Octad Generator, Rosenhain and Göpel — m759 @ 5:24 pm

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

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 .

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Saturday, April 6, 2013

Pascal via Curtis

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

Click image for some background.

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

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

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

On Danzer's 35₄ Configuration:

IMAGE- Branko Grünbaum on Danzer's configuration

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

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

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

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

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

Classical Geometry in Light of Galois Geometry.

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

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Wednesday, February 13, 2013

Form:

Filed under: General,Geometry — Tags: Geometry, Octad, Octad Generator, Penrose diamond — 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

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

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

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Monday, December 24, 2012

All Over Again

Filed under: General,Geometry — Tags: Gitterkrieg, Octad — 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 —

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Saturday, November 24, 2012

Reappearing All Over Again

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

For the title, see the phrase "reappearing number" in this journal.

Some related mathematics—

the Greek labyrinth of Borges, as well as…

Note that "0" here stands for "23," while _∞ corresponds to today's date.

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Monday, November 19, 2012

Poetry and Truth

Filed under: General,Geometry — Tags: Bard Rock, Geometry, Octad, Octad Generator, Poetry, Stevens Rock, The Poet's Pineapple — m759 @ 7:59 pm

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—

— and from 1981—

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."

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Sunday, October 14, 2012

Crossroads

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

"Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself."

— A translated remark by Hermann Weyl, p. 136, "The Current Epistemogical Situation in Mathematics" in Paolo Mancosu (ed.) From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s , Oxford University Press, 1998, pp. 123-142, as cited by David Corfield

Corfield once wrote that he would like to know the original German of Weyl's remark. Here it is:

"Die Mathematik ist nicht das starre und Erstarrung bringende Schema, als das der Laie sie so gerne ansieht; sondern wir stehen mit ihr genau in jenem Schnittpunkt von Gebundenheit und Freiheit, welcher das Wesen des Menschen selbst ist."

— Hermann Weyl, page 533 of "Die heutige Erkenntnislage in der Mathematik" (Symposion 1, 1-32, 1925), reprinted in Gesammelte Abhandlungen, Band II (Springer, 1968), pages 511-542

For some context, see a post of January 23, 2006.

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Friday, October 5, 2012

The Elegant Fowl

Filed under: General — m759 @ 6:29 pm

For the late Helen Nicoll—

A scene from a video starring Nicoll's characters
Meg, Mog, and Owl. The video was uploaded on
November 2, 2008— All Souls' Day.
A different cartoon from All Souls' Day, 2008
A rhyme from 1871—

The Owl looked up to the stars above,
And sang to a small guitar,
"O lovely Pussy! O Pussy, my love,
What a beautiful Pussy you are, you are, you are,
What a beautiful Pussy you are."
Pussy said to the Owl "You elegant fowl,
How charmingly sweet you sing.
O let us be married, too long we have tarried;
But what shall we do for a ring?"

— Edward Lear

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Thursday, October 4, 2012

Kids Grow Up

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

From an obituary for Helen Nicoll, author
of a popular series of British children's books—

"They feature Meg, a witch whose spells
always seem to go wrong, her cat Mog,
and their friend Owl."

For some (very loosely) related concepts that
have been referred to in this journal, see…

Meg, Mog, and Owl.

See, too, "Kids grow up" (Feb. 13, 2012).

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Sunday, September 9, 2012

Grid Compass

Filed under: General — Tags: Immersion — m759 @ 10:31 pm

Related material: The Empty Chair Award.

For a different sort of grid compass, see February 3, 2011.

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Sunday, July 29, 2012

The Galois Tesseract

Filed under: General,Geometry — Tags: Defining Form, Galois's Space, Springer — m759 @ 11:00 pm

(Continued)

The three parts of the figure in today's earlier post "Defining Form"—

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

— share the same vector-space structure:

0	c	d	c + d
a	a + c	a + d	a + c + d
b	b + c	b + d	b + c + d
a + b	a + b + c	a + b + d	a + b + c + d

(This vector-space a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)

The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)

A 1982 discussion of a more abstract form of AG(4, 2):

Source:

The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

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