https://websim.ai/p/74wzs2lcd6qlpe8491rz/24
Tuesday, October 29, 2024
Diamond Theorem Unit Cube Design
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Saturday, October 26, 2024
Diamond Theorem Studio
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Saturday, October 19, 2024
AI Perceptions of the Cullinane Diamond Theorem . . .
Continue to Evolve.
Update of 12:10 PM EDT October 19 — AI Overview from Google —
The Google AI Overview now includes an illustration and mentions a
new technique ("map systems") which may eventually be generalized.
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Continue to Evolve.
Continue to Evolve.
Monday, March 4, 2024
Report on the Diamond Theorem
by Google Gemini (Advanced)
|
by Google Gemini AI Monday, March 4, 2024
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: 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:
Proof: 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!
1. www.hellenicaworld.com/Science/Mathematics/en/MathieuGroup.html
|
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by Google Gemini (Advanced)
by Google Gemini (Advanced)
Tuesday, January 30, 2024
Monday, November 27, 2023
The Diamond Theorem According to Llama 2
The above is one of many wildly inaccurate responses on this topic
from chatbots. A chatbot combined with search, however —
such as Bing Chat with GPT-4 — can be both accurate and helpful.
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Saturday, November 11, 2023
The Diamond Theorem and Graphic Design
A "Cullinane Diamond Theorem" question suggested today by Bing Chat —
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Wednesday, November 8, 2023
Cullinane Diamond Theorem at
University of the Basque Country
See also Shibumi Continues — June 29, 2022.
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University of the Basque Country
University of the Basque Country
Saturday, September 23, 2023
The Cullinane Diamond Theorem at Wikipedia
This post was prompted by the recent removal of a reference to
the theorem on the Wikipedia "Diamond theorem" disambiguation
page. The reference, which has been there since 2015, was removed
because it linked to an external source (Encyclopedia of Mathematics)
instead of to a Wikipedia article.
For anyone who might be interested in creating a Wikipedia article on
my work, here are some facts that might be reformatted for that website . . .
https://en.wikipedia.org/wiki/
User:Cullinane/sandbox —
|
Cullinane diamond theorem The theorem uses finite geometry to explain some symmetry properties of some simple graphic designs, like those found in quilts, that are constructed from chevrons or diamonds. The theorem was first discovered by Steven H. Cullinane in 1975 and was published in 1977 in Computer Graphics and Art. The theorem was also published as an abstract in 1979 in Notices of the American Mathematical Society. The symmetry properties described by the theorem are related to those of the Miracle Octad Generator of R. T. Curtis. The theorem is described in detail in the Encyclopedia of Mathematics article "Cullinane diamond theorem." References Steven H. Cullinane, "Diamond theory," Computer Graphics and Art, Vol. 2, No. 1, February 1977, pages 5-7. _________, Abstract 79T-A37, "Symmetry invariance in a diamond ring," Notices of the American Mathematical Society, February 1979, pages A-193, 194. _________, "Cullinane diamond theorem," Encyclopedia of Mathematics. R. T. Curtis, A new combinatorial approach to M24, Mathematical Proceedings of the Cambridge Philosophical Society, 1976, Vol. 79, Issue 1, pages 24-42. |
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Sunday, August 27, 2023
Saturday, February 25, 2023
The Diamond Theorem according to ChatGPT
The part about tilings, group actions, and the diamond-shaped
pattern is more or less OK. The parts about Thurston and
applications are utterly false.
Compare and contrast . . .
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Friday, April 29, 2022
The Diamond Theorem in Basque Country
Translated by Google as . . .
The Truchet Tiles and the Diamond Puzzle and
The Art of the Simple Truchet Tile.
About the author:
Raúl Ibáñez is a professor in the Department of Mathematics
at the UPV/EHU and collaborator with the Chair of Scientific Culture.
About his school:
The University of the Basque Country
(Basque: Euskal Herriko Unibertsitatea, EHU ;
Spanish: Universidad del País Vasco, UPV ; UPV/EHU)
is a Spanish public university of the Basque Autonomous Community.
— Wikipedia
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Monday, April 19, 2021
Diamond Theorem at ScienceOpen
Update on April 20, 2021 —
The following was added today to the above summary:
“It describes a group of 322,560 permutations, later known as
‘the octad group,’ that now plays a role in speculative high-energy physics.
See Moonshine, Superconformal Symmetry, and Quantum Error Correction .”
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Sunday, November 18, 2018
Diamond Theorem Symmetry
The title is a useful search phrase:
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Monday, December 11, 2017
The Diamond Theorem at SASTRA
The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com —
For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .
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Wednesday, August 23, 2017
The Diamond Theorem in Vancouver
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Tuesday, September 20, 2016
The Diamond Theorem …
As the Key to All Mythologies
For the theorem of the title, see "Diamond Theorem" in this journal.
"These were heavy impressions to struggle against,
and brought that melancholy embitterment which
is the consequence of all excessive claim: even his
religious faith wavered with his wavering trust in his
own authorship, and the consolations of the Christian
hope in immortality seemed to lean on the immortality
of the still unwritten Key to all Mythologies."
— Middlemarch , by George Eliot, Ch. XXIX
Related material from Sunday's print New York Times —
Sunday's Log24 sermon —
See also the Lévi-Strauss "Key to all Mythologies" in this journal,
as well as the previous post.
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Monday, August 5, 2013
Diamond Theorem in ArXiv
The diamond theorem is now in the arXiv—
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Tuesday, July 2, 2013
Diamond Theorem Updates
My diamond theorem articles at PlanetMath and at
Encyclopedia of Mathematics have been updated
to clarify the relationship between the graphic square
patterns of the diamond theorem and the schematic
square patterns of the Curtis Miracle Octad Generator.
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Friday, May 10, 2013
Cullinane diamond theorem
A page with the above title has been created at
the Encyclopedia of Mathematics.
How long it will stay there remains to be seen.
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Friday, February 25, 2011
Diamond Theorem Exposition
"THE DIAMOND THEOREM AND QUILT PATTERNS
Victoria Blumen, Mathematics, Junior, Benedictine University
Tim Comar, Benedictine University
Mathematics
Secondary Source Research
Let D be a 4 by 4 block quilt shape, where each of the 16 square blocks is consists of [sic ] two triangles, one of which is colored red and the other of which is colored blue. Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D. The diamond theorem states that G(D) = D_g has either ordinary or color-interchange symmetry. In this talk, we will prove the diamond theorem and explore symmetries of quilt patterns of the form G(D)."
Exercise— Correct the above statement of the theorem.
Background— This is from a Google search result at about 10:55 PM ET Feb. 25, 2011—
[DOC] THE DIAMOND THEOREM AND QUILT PATTERNS – acca.elmhurst.edu
File Format: Microsoft Word – 14 hours ago –
Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D. The diamond theorem states that G(D) = D_g has either …
acca.elmhurst.edu/…/victoria_blumen9607_
THE%20DIAMOND%20THEOREM%20AND%20QUILT%20PATTERNS…
The document is from a list of mathematics abstracts for the annual student symposium of the ACCA (Associated Colleges of the Chicago Area) held on April 10, 2010.
Update of Feb. 26— For a related remark quoted here on the date of the student symposium, see Geometry for Generations.
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Sunday, October 31, 2010
Diamond Theorem in Norway
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Tuesday, October 22, 2024
Diamond-theorem art made with AI
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Saturday, January 29, 2022
On the Diamond-Theorem Group* of Order 322,560
Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.
This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M24.
* See a Log24 post of June 1, 2013.
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Friday, October 28, 2016
Diamond-Theorem Application
|
Abstract: "Protection of digital content from being tapped by intruders is a crucial task in the present generation of Internet world. In this paper, we proposed an implementation of new visual secret sharing scheme for gray level images using diamond theorem correlation. A secret image has broken into 4 × 4 non overlapped blocks and patterns of diamond theorem are applied sequentially to ensure the secure image transmission. Separate diamond patterns are utilized to share the blocks of both odd and even sectors. Finally, the numerical results show that a novel secret shares are generated by using diamond theorem correlations. Histogram representations demonstrate the novelty of the proposed visual secret sharing scheme." — "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," by V. Harish, N. Rajesh Kumar, and N. R. Raajan.
Published in: 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT). |
Excerpts —
Related material — Posts tagged Diamond Theorem Correlation.
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Friday, August 1, 2014
The Diamond-Theorem Correlation
Click image for a larger, clearer version.
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Thursday, May 2, 2024
AI on Diamond Theory
The website PopAi.pro uses GPT-4 to summarize documents from the Web.
Input at PopAi on May 2, 2024:
The URL "http://www.log24.com/log/Diamond_Theory-1976-pp-1-12.pdf"
Output at PopAi, after a quick reading and analysis by GPT-4:
|
Diamond_Theory-1976-pp-1-12.pdf 12 Pages The document appears to be an excerpt from a preprint titled "Diamond Theory" by Steven H. Cullinane, dated 1976. The text discusses theoretical aspects of matrices composed of geometric figures, focusing on properties related to symmetry, permutations, and abstract structure. Main Concepts and Definitions: The author introduces the concept of a "geometric matrix," a matrix whose elements are geometric figures, specifically triangular subsets of a square. He defines a "diamond" as a foursquare matrix with triangular elements. The text details the remarkable properties of a diamond-like matrix D, including its symmetrical behavior under various permutations of elements, rows, columns, and sub-squares. Additionally, the text addresses the algebraic properties of matrices by relating their elements to those of a finite field. Theoretical Properties and Theorems: Several key theorems are proposed, demonstrating the structural properties of matrices. One fundamental theorem states that any matrix from a set F can be expressed as a linear combination of block matrices. Other theorems delve into the regularity of matrices, defining a matrix as "regular" if it remains unchanged or converts to its complement under certain flips and rotations. The text also describes the process of forming a "normal" matrix and the conditions under which matrices can be considered plaid or normal based on the behavior of their block maps and regularities. Applications and Miscellaneous Problems: The author reflects on the discovery of these properties during the process of creating an abstract painting and extends the exploration to include various types of matrices such as cyclic, quadratic, and skew foursquares. The document concludes with miscellaneous problems and an addendum on generalized matrix multiplication, suggesting further research directions and applications of the theory presented. In summary, "Diamond Theory" presents an intricate analysis of the structural and algebraic properties of matrices composed of geometric shapes, with implications for other mathematical domains. The work is a blend of geometry and abstract algebra, providing a foundation for understanding the symmetry and transformations of matrices with an artistic origin. Some questions you may ask: What properties of the diamond-like matrix D did the author discover in 1975? What is the relationship between the properties of D and the four group according to the document? How does the document define a 'good set' of four black and white geometric figures? What theorem is used to explain the behavior of superimposing two members of Mn(F) according to the document? |
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Wednesday, March 20, 2024
Cullinane Diamond Song
|
Shining Mathematics: Pop upbeat V2 March 19, 2024
[Verse]
|
Click the image below to hear the song at app.suno.ai —
Miller's Note to Self : "Don't underestimate Wednesday."
For more about the mathematics itself, see other octad posts.
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Tuesday, January 23, 2024
The Enormous Theorem
The previous post's reference to colors suggests a review . . .
A test of OpenAI on the above DevDay date —
This ridiculous hallucination was obviously suggested by what
has been called "the enormous theorem" on the classification
of finite simple groups. That theorem was never known as the
(or "a") diamond theorem.
On the bright side, the four colors beside Microsoft's Nadella in the
photo above may, if you like, be regarded as those of my own
non-enormous "four-color decomposition theorem" that is used in
the proof of my own result called "the diamond theorem."
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Wednesday, November 8, 2023
The Theorem as Big as the Ritz
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Monday, May 23, 2022
Thursday, May 7, 2020
Kant as Diamond Cutter
"He wished Kant were alive. Kant would have appreciated it.
That master diamond cutter."
— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance , Part III.
Kant's "category theory" —
|
"In the Transcendental Analytic, Kant deduces the table of twelve categories, or pure concepts of the understanding….
The categories must be 'schematized' because their non-empirical origin in pure understanding prevents their having the sort of sensible content that would connect them immediately to the objects of experience; transcendental schemata are mediating representations that are meant to establish the connection between pure concepts and appearances in a rule-governed way. Mathematical concepts are discussed in this context since they are unique in being pure but also sensible concepts: they are pure because they are strictly a priori in origin, and yet they are sensible since they are constructed in concreto . " — Shabel, Lisa, "Kant's Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2016/entries/kant-mathematics/>. |
See also The Diamond Theorem and Octad.us.
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Wednesday, April 17, 2019
The Royal Society Diamond
The phrase "pattern recognition" in a news story about the
April 13 death of Princeton neuroscientist Charles Gross,
and yesterday's post about a fanciful "purloined diamond,"
suggest a review of a less fanciful diamond.
See also earlier posts tagged Fitch
and my own, much earlier and very
different, approach to such patterns —
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Thursday, August 31, 2017
A Conway-Norton-Ryba Theorem
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —
A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.
See also Magic Square and Diamond Theorem in this journal.
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Saturday, March 14, 2015
Unicode Diamonds
The following figure, intended to display as
a black diamond, was produced with
HTML and Unicode characters. Depending
on the technology used to view it, the figure
may contain gaps or overlaps.
◢◣
◥◤
Some variations:
◤◥
◣◢
◤◥
◢◣
◤◣
◢◥
◤◣
◥◢
Such combined Unicode characters —
◢ black lower right triangle,
◣ black lower left triangle,
᭘ black upper left triangle,
᭙ black upper right triangle
— might be used for a text-only version of the Diamond 16 Puzzle
that is more easily programmed than the current version.
The tricky part would be coding the letter-spacing and
line-height to avoid gaps or overlaps within the figures in
a variety of browsers. The w3.org visual formatting model
may or may not be helpful here.
Update of 11:20 PM ET March 15, 2015 —
Seekers of simplicity should note that there is
a simple program in the Processing.js language, not using
such Unicode characters, that shows many random affine
permutations of a 4×4 diamond-theorem array when the
display window is clicked.
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Monday, October 13, 2014
Raiders of the Lost Theorem
(Continued from Nov. 16, 2013.)
The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.
One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:
In context (click to enlarge):
Update of later the same day —
Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —
“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”
* Defined by Sallows later in the book:
“Geometric or, less formally, geomagic is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles or pieces ) may appear in the cells instead of numbers.”
** See some geometric matrices by Cullinane in a March 2001 webpage.
Earlier actual specimens — see Diamond Theory excerpts published in
February 1977 and a brief description of the original 1976 monograph:
“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”
— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society
The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.
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Thursday, March 27, 2014
Diamond Space
Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.
The reason for the name:
Click to enlarge.
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Sunday, February 2, 2014
Diamond Theory Roulette
A ReCode Project program from Radamés Ajna of São Paulo —
At the program's webpage, click the image to
generate random permutations of rows, columns,
and quadrants. Note the resulting image's ordinary
or color-interchange symmetry.
|
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Sunday, March 18, 2012
Square-Triangle Diamond
The diamond shape of yesterday's noon post
is not wholly without mathematical interest …
"Every triangle is an n -replica" is true
if and only if n is a square.
The 16 subdiamonds of the above figure clearly
may be mapped by an affine transformation
to 16 subsquares of a square array.
(See the diamond lattice in Weyl's Symmetry .)
Similarly for any square n , not just 16.
There is a group of 322,560 natural transformations
that permute the centers of the 16 subsquares
in a 16-part square array. The same group may be
viewed as permuting the centers of the 16 subtriangles
in a 16-part triangular array.
(Updated March 29, 2012, to correct wording and add Weyl link.)
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Thursday, September 8, 2011
Starring the Diamond
"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 83
are joined by four concurrent lines."
— H. S. M. Coxeter (see below)
Continued from Tuesday, Sept. 6—
The Diamond Star
The above is a version of a figure from Configurations and Squares.
Yesterday's post related the the Pappus configuration to this figure.
Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to the figure.
Some excerpts from Coxeter—
The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8-cycle in the Galois field GF(9).
The relabeled configuration is used in a discussion of Pappus—
(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)
Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues configuration—
The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts—
"The value of Desargues' theorem and the reason why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities that relate geometry and algebra in unexpected ways."
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Monday, August 8, 2011
Diamond Theory vs. Story Theory (continued)
Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.
(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)
From yesterday's Sunday morning New York Times—
"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."
— Drew Westen, professor at Emory University
From May 22, 2009—
|
The above ad is by
Diamond from last night’s
|
|
For further details, see Saturday's correspondences |
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Monday, December 27, 2010
Church Diamond
Also known, roughly speaking, as confluence or the Church-Rosser property.
From “NYU Lambda Seminar, Week 2” —
[See also the parent page Seminar in Semantics / Philosophy of Language or:
What Philosophers and Linguists Can Learn From Theoretical Computer Science But Didn’t Know To Ask)]
A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.
The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn’t matter which order the sub-expressions are evaluated in.
A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing: ω ω doesn’t terminate by any evaluation path; and (\x. y) (ω ω) terminates only by some evaluation paths but not by others.
But the untyped lambda calculus enjoys some compensation for this weakness. It’s Turing complete! It can represent any computation we know how to describe. (That’s the cash value of being Turing complete, not the rigorous definition. There is a rigorous definition. However, we don’t know how to rigorously define “any computation we know how to describe.”) And in fact, it’s been proven that you can’t have both. If a computational system is Turing complete, it cannot be strongly normalizing.
There is no connection, apart from the common reference to an elementary geometric shape, between the use of “diamond” in the above Church-Rosser sense and the use of “diamond” in the mathematics of (Cullinane’s) Diamond Theory.
Any attempt to establish such a connection would, it seems, lead quickly into logically dubious territory.
Nevertheless, in the synchronistic spirit of Carl Jung and Arthur Koestler, here are some links to such a territory —
Link One — “Insane Symmetry” (Click image for further details)—
See also the quilt symmetry in this journal on Christmas Day.
Link Two — Divine Symmetry
(George Steiner on the Name in this journal on Dec. 31 last year (“All about Eve“)) —
“The links are direct between the tautology out of the Burning Bush, that ‘I am’ which accords to language the privilege of phrasing the identity of God, on the one hand, and the presumptions of concordance, of equivalence, of translatability, which, though imperfect, empower our dictionaries, our syntax, our rhetoric, on the other. That ‘I am’ has, as it were, at an overwhelming distance, informed all predication. It has spanned the arc between noun and verb, a leap primary to creation and the exercise of creative consciousness in metaphor. Where that fire in the branches has gone out or has been exposed as an optical illusion, the textuality of the world, the agency of the Logos in logic—be it Mosaic, Heraclitean, or Johannine—becomes ‘a dead letter.'”
– George Steiner, Grammars of Creation
(See also, from Hanukkah this year, A Geometric Merkabah and The Dreidel is Cast.)
Link Three – Spanning the Arc —
Part A — Architect Louis Sullivan on “span” (see also Kindergarten at Stonehenge)
Part B — “Span” in category theory at nLab —
Also from nLab — Completing Spans to Diamonds
“It is often interesting whether a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem (and the corresponding Church-Rosser confluence property).”
The concepts in this last paragraph may or may not have influenced the diamond theory of Rudolf Kaehr (apparently dating from 2007).
They certainly have nothing to do with the Diamond Theory of Steven H. Cullinane (dating from 1976).
For more on what the above San Francisco art curator is pleased to call “insane symmetry,” see this journal on Christmas Day.
For related philosophical lucubrations (more in the spirit of Kaehr than of Steiner), see the New York Times “The Stone” essay “Span: A Remembrance,” from December 22—
“To understand ourselves well,” [architect Louis] Sullivan writes, “we must arrive first at a simple basis: then build up from it.”
Around 300 BC, Euclid arrived at this: “A point is that which has no part. A line is breadthless length.”
See also the link from Christmas Day to remarks on Euclid and “architectonic” in Mere Geometry.
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Wednesday, July 28, 2010
Without Diamond-Blazons
Excerpt from Wallace Stevens's
"The Pediment of Appearance"—
Young men go walking in the woods,
Hunting for the great ornament,
The pediment* of appearance.
They hunt for a form which by its form alone,
Without diamond—blazons or flashing or
Chains of circumstance,
By its form alone, by being right,
By being high, is the stone
For which they are looking:
The savage transparence.
* Pediments, triangular and curved—
— From "Stones and Their Stories," an article written
and illustrated by E.M. Barlow, copyright 1913.
Related geometry—
(See Štefan Porubský: Pythagorean Theorem .)
A proof with diamond-blazons—
(See Ivars Peterson's "Square of the Hypotenuse," Nov. 27, 2000.)
Comments Off on Without Diamond-Blazons
Tuesday, November 19, 2024
A Mosaic for Megiddo
Sunday, November 17, 2024
Weyl, Symmetry, and the MOG
(HTML version of an earlier post)
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.)
This square's automorphism group
has 322,560 transformations.
— The diamond theorem of Steven H. Cullinane.
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.
Saturday, November 16, 2024
Automorphism Groups: Examples for Weyl
Keywords: Weyl, symmetry, group, automorphism,
octad, MOG, Curtis, Cullinane.
Wednesday, October 23, 2024
Ingathering Scriptures
I recommend . . .
https://www.amazon.com/
Ingathering-Complete-People-Stories-Henderson-ebook/
dp/B0878RMD5D/ .
Some will prefer Exodus 23:16.
Related philosophy —
Descartes and the Tents of Armageddon (Galois's birthday, 2007).
Related art —
https://claude.site/artifacts/3c1ce26b-4664-448c-87df-c20802053eea .
Update on Oct. 28, 2024 —
For a newer version by a different AI, Websim, see Diamond Theorem Studio.
Comments Off on Ingathering Scriptures
Wednesday, July 31, 2024
My Links — Steven H. Cullinane
Main webpage of record . . .
Encyclopedia of Mathematics https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem
Supplementary PDF from Jan. 6, 2006 https://encyclopediaofmath.org/images/3/37/Dtheorem.pdf
Originally published in paper version . . .
Computer Graphics and Art, 1978 http://finitegeometry.org/sc/gen/Diamond_Theory_Article.pdf
AMS abstract, 1979: "Symmetry Invariance in a Diamond Ring" https://www.cullinane.design/
American Mathematical Monthly, 1984 and 1985: "Triangles Are Square" http://finitegeometry.org/sc/16/trisquare.html
Personal sites . . .
Primary —
Personal journal http://m759.net/wordpress/
Mathematics website http://finitegeometry.org/sc/
Mathematics Images Gallery http://m759.net/piwigo/index.php?/category/2
Secondary —
Portfoliobox https://cullinane.pb.design/
Substack https://stevenhcullinane.substack.com/
Symmetry Summary https://shc759.wordpress.com
Diamond Theory Cover Structure https://shc7596.wixsite.com/website
SOCIAL:
Pinterest https://www.pinterest.com/stevenhcullinane/ (many mathematics notes)
Flickr https://www.flickr.com/photos/m759/ (backup account for images of mathematics notes)
Instagram https://www.instagram.com/stevencullinane
TikTok https://www.tiktok.com/@stevenhcullinane
X.com https://x.com/shc759
OTHER:
Replit viewer/download https://replit.com/@m759/View-4x4x4?v=1
SourceForge download https://sourceforge.net/projects/finitegeometry/
Academia.edu https://stevenhcullinane.academia.edu/ GitHub https://github.com/m759 (finite geometry site download)
Internet Archive: Notes on Groups and Geometry https://archive.org/details/NotesOnGroupsAndGeometry1978-1986/mode/2up
Cited at . . .
The Diamond Theorem and Truchet Tiles http://www.log24.com/log22/220429-Basque-DT-1.pdf
April 2024 UNION article in Spanish featuring the diamond theorem https://union.fespm.es/index.php/UNION/article/view/1608/1214
April 2024 UNION article in English http://log24.com/notes/240923-Ibanez-Torres-on-diamond-theorem-Union-April-2024-in-English.pdf
Cullinane in a 2020 Royal Holloway Ph.D. thesis https://pure.royalholloway.ac.uk/ws/portalfiles/portal/40176912/2020thomsonkphd.pdf
Squares, Chevrons, Pinwheels, and Bach https://www.yumpu.com/en/document/read/36444818/fugue-no-21-elements-of-finite-geometry
Observables programmed presentation of diamond theorem https://observablehq.com/@radames/diamond-theory-symmetry-in-binary-spaces
Josefine Lyche — Plato's Diamond https://web.archive.org/web/20240222064628/http://www.josefinelyche.com/index.php?/selected-exhibitions/platos-diamond/
Josefine Lyche — Diamond Theorem https://web.archive.org/web/20230921122049/http://josefinelyche.com/index.php?/selected-exhibitions/uten-ramme-nye-rom/
Professional sites . . .
Association for Computing Machinery https://member.acm.org/~scullinane
bio.site/cullinane … maintenance at https://biosites.com
ORCID bio page https://orcid.org/0000-0003-1135-419X
Google Scholar https://scholar.google.com/citations?view_op=list_works&hl=en&hl=en&user=NcjmFwQAAAAJ&sortby=pubdate
Academic repositories:
Harvard Dataverse https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/KHMMVH
Harvard DASH article on PG(3,2) https://dash.harvard.edu/handle/1/37373777
Zenodo website download https://zenodo.org/records/1038121
Zenodo research notes https://zenodo.org/search?q=metadata.creators.person_or_org.name%3A%22Cullinane%2C%20Steven%20H.%22&l=list&p=1&s=10&sort=bestmatch
Figurate Geometry at Open Science Framework (OSF) https://osf.io/47fkd/
arXiv: "The Diamond Theorem" https://arxiv.org/abs/1308.1075
Comments Off on My Links — Steven H. Cullinane
Friday, April 19, 2024
Benchmarking Meta.ai
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 :
|
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Thursday, April 11, 2024
Devs: Hollywood Development Hell
From The Man Who Knew Infinity to The Man Who Knew Zero.
Related mathematics: The Diamond Theorem Correlation, which
results from interchanging infinity and zero in the figure below.
"Read something that means something."
— New Yorker ad
.
Click image for
related posts.
Background — Relativity Problem in Log24.
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Sunday, March 3, 2024
Deep Blue Research: A Report by You.com AI
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Cullinane Diamond Theorem Research Report by https://you.com/?chatMode=research on March 3, 2024 Overview of the Cullinane Diamond Theorem The Cullinane Diamond Theorem is a mathematical concept developed by Steven H. Cullinane that explores the symmetrical properties of certain geometric patterns. It is particularly concerned with the structure of finite projective geometry as it relates to the 35 square patterns found in R. T. Curtis's Miracle Octad Generator (MOG). The theorem asserts that every G-image of a diamond figure D possesses some form of ordinary or color-interchange symmetry. Symmetry and Group Theory The theorem is deeply rooted in group theory, with G being isomorphic to the affine group A on the linear 4-space over the finite field GF(2). This group has a significant order of 322,560 and underlies the symmetry of both the diamond-theorem figures and the square patterns of the MOG . The 35 structures of the 840 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), which explains the symmetry of the line diagrams and the two-color patterns. The Diamond Theorem in Various Dimensions The diamond theorem can be applied to different cases, such as the 4×4, 2×2, and even 3-dimensional cubes like the 2x2x2 and 4x4x4 cubes. The generalized statement of the theorem involves inscribing a white diamond in a black square and dividing it into four quadrants, each with a diagonal split into black and white halves. Algebraic Implications The theorem has algebraic implications, such as the ability to express every 4-coloring as a sum of three 2-colorings. This algebraic formulation is useful for demonstrating that certain four-colored graphic images can form a ring under multiplication. There is also an infinite family of such 'diamond' rings, isomorphic to rings of matrices over GF(4). Philosophical and Cultural Context Steven H. Cullinane has also explored the philosophical implications of the diamond theorem, referring to it in the context of Plato's Meno dialogue and opposing it to the 'Story Theory of truth' as discussed by Richard J. Trudeau. The theorem is seen as a correct statement of the relation between geometry and the physical universe. Clarification on Cullinan Diamonds It is important to note that the Cullinane Diamond Theorem is unrelated to the Cullinan diamond, which is a famous large gemstone discovered in South Africa and now part of the British Crown Jewels. The Cullinan diamond has its own historical and cultural significance, with its various cut shapes and ownership history. In summary, the Cullinane Diamond Theorem is a mathematical concept that reveals the symmetrical properties of certain geometric patterns, with applications in group theory and algebra. It also has philosophical implications, drawing connections to classical ideas about truth and geometry. |
Instagram ad for You.com AI in research mode
"Show me ALL your sources, babe."
— Line adapted from Leonardo DiCaprio
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Saturday, March 2, 2024
Song by Copilot: Prompt and Response*
Prompt: Compose a song about the Cullinane diamond theorem.
Response by Microsoft Copilot on Saturday afternoon,
March 2, 2024:
Sure, here’s a song inspired by the Cullinane diamond theorem:
(Verse 1)
In the realm where logic reigns and numbers intertwine,
Lies a theorem, diamond-bright, its facets all align.
Cullinane's the name it bears, a gem in math's grand crown,
In the grid of sixteen squares, its truths are written down.
(Chorus)
Oh, Cullinane, your diamond shines with symmetry so fine,
In the field of finite forms, your theorem does define.
A structure pure, a pattern clear, for all of us to see,
In the heart of mathematics, lies your geometry.
(Verse 2)
Rows and columns interplay in a dance so grand,
Quadrants align in perfect time, like notes in a band.
Through the chaos and the whirl, an order does descend,
In the world of abstract thought, your theorem is a friend.
(Chorus)
Oh, Cullinane, your diamond theorem, a beacon in the night,
Guiding us through complex realms, with its brilliant light.
In the vast expanse of knowledge, where truth and beauty blend,
Your theorem stands, a testament, a guide until the end.
(Outro)
So here's to you, Cullinane, and your diamond theorem's might,
In the endless quest for truth, you've brought us greater light.
In the language of the universe, where numbers never lie,
Your diamond theorem's shining bright, beneath the math's clear sky.
Added note from Copilot:
Please note that this is a creative interpretation and may not accurately reflect
the mathematical concepts of the Cullinane diamond theorem. 😊
Added note from Cullinane:
* The previous post may or may not display a prompt response to a Zen koan.
Comments Off on Song by Copilot: Prompt and Response*
Wonder Box
"Wundermärchen – the original German word for fairytale –
literally translates to ‘wonder tale’."
— Abigail Tulenko. a PhD student in philosophy at Harvard University,
at Aeon on Feb. 26, 2024.
Another example of abstract art . . .
"The discovery of the Cullinane Diamond Theorem is a testament
to the power of mathematical abstraction and its ability to reveal
deep connections and symmetries in seemingly simple structures."
Comments Off on Wonder Box
Saturday, February 24, 2024
Li
The Chinese concept of li in yesterday's post "Logos" is related,
if only by metaphor, to the underlying form (sets of "line diagrams")
of patterns in the Cullinane diamond theorem:
"But very possibly the earliest use of li is the one instance that
it appears in the Classic of Poetry (Ode 210) where it refers to
the borders or boundary lines marking off areas in a field.
Here it appears in conjunction with chiang and is explained
as 'to divide into lots (or parcels of land)' (fen-ti )."
— P. 33 of "Li Revisited and Other Explorations"
by Allen Wittenborn, Bulletin of Sung and Yüan Studies
No. 17 (1981), pp. 32-48 (17 pages),
https://www.jstor.org/stable/23497457.
Comments Off on Li
Friday, February 16, 2024
The Sixteen
Prompt:
Write a poem, in strict sonnet form, about the Cullinane diamond theorem.
Response (slightly tweaked for grammar):
Within a grid of sixteen squares reside
A diamond's gleams where symmetries do guide.
Though tiles may shift, a pattern shall remain,
By unseen lines of ordered form sustained.
Rows mix and columns twist in playful dance,
Quadrants align with geometric chance.
No matter how the elements may turn,
A hidden balance always will return.
Where Latin squares in secret do align,
And finite fields their subtle powers combine,
An ideal forms, with brilliance all its own,
A ring of light where hidden truths are shown.
Though chaos swirls, an order does descend,
The Cullinane's bright theorem will transcend.
________________________________________________________________
As for "a hidden balance" … See Oct. 10, 2022.
Comments Off on The Sixteen
Thursday, February 8, 2024
Google Bard is now Gemini.
Gemini Prompt:
Write a poem in the style of Wallace Stevens
about the Cullinane diamond theorem.
Response:
Absolutely! Here's a poem inspired by Wallace Stevens
and the Cullinane diamond theorem:
A theorem not of Euclid's line,
But diamonds cut with strange design.
Sixteen squares in shifting light,
A symmetry of black and white.
Where rows and columns intertwine,
A hidden order lies supine.
Permutations twist and turn,
And latent symmetries discern.
Not fixed like stars in ordered rows,
But like reflections, pattern grows.
Projective planes, a finite field,
The diamond's inner world revealed.
The proof, a dance of abstract thought,
Where shifting forms find space untaught.
Not nature's chaos, but design,
A brilliance of the human mind.
A lattice born of Stevens' art,
Where form and thought find counterpart.
Amidst the fragments, patterns gleam,
The Cullinane, a geometer's dream.
Comments Off on Google Bard is now Gemini.
Sunday, February 4, 2024
Microsoft Copilot Doggerel
A poem written yesterday by Microsoft Copilot —
The prompt: "Write a poem … about the Cullinane diamond theorem."
|
In the mind of the mathematician,
Four-diamond figure, a 4×4 array,
Every G-image of the four-diamond figure,
The symmetry of the line diagrams,
And this symmetry must be shared by the entire pattern,
The 35 structures of the 840 = 35 x 24 G-images,
So here lies the beauty of the Cullinane diamond theorem, |
Actually, the complete prompt, suggested by a Log24 post of
12:12 PM ET yesterday, was . . .
"Write a poem in the style of Wallace Stevens
about the Cullinane diamond theorem."
Copilot still has a thing or two to learn about poetic style.
Comments Off on Microsoft Copilot Doggerel
Monday, January 29, 2024
Self as Imago Dei: Hofstadter vs. Valéry
|
Google search result: Saint Anselm College https://www.anselm.edu › Documents › Brown 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
|
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Saturday, January 13, 2024
Xmas Pattern
Found on the Web today —
Earlier . . .
Thursday, August 10, 2023
|
Related entertainment starring Martin Freeman —
Related Art —
See as well The Diamond Theorem in Basque Country
for material from the University of the Basque Country,
an offshoot of the University of Bilbao (in Basque, "Bilbo").
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Wednesday, January 3, 2024
Negative Space
A recently coined phrase — "Negative Mathematics" — is related to the
better-known phrase "Negative Space."
The latter is closely related to the proof of the Cullinane diamond theorem.
For the former, see . . .
Related material: The proof symbol, i.e. the Halmos Tombstone.
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Thursday, December 28, 2023
Basque Country Art Book
Book description at Amazon.com, translated by Google —
|
Las matemáticas como herramienta
Mathematics as a tool by Raúl Ibáñez Torres Kindle edition in Spanish, 2023 Although the relationship between mathematics and art can be traced back to ancient times, mainly in geometric and technical aspects, it is with the arrival of the avant-garde and abstract art at the beginning of the 20th century that mathematics takes on greater and different relevance: as a source of inspiration and as a tool for artistic creation. Let us think, for example, of the importance of the fourth dimension for avant-garde movements or, starting with Kandisnky and later Max Bill and concrete art, the vindication of mathematical thinking in artistic creation. An idea that would have a fundamental influence on currents such as constructivism, minimalism, the fluxus movement, conceptual art, systematic art or optical art, among others. Following this approach, this book analyzes, through a variety of examples and activities, how mathematics is present in contemporary art as a creative tool. And it does so through five branches and the study of some of its mathematical topics: geometry (the Pythagorean theorem), topology (the Moebius strip), algebra (algebraic groups and matrices), combinatorics (permutations and combinations) and recreational mathematics (magic and Latin squares). |
From the book ("Cullinane Diamond Theorem" heading and picture of
book's cover added) —
Publisher: Los Libros de La Catarata (October 24, 2023)
Author: Raúl Ibáñez Torres, customarily known as Raúl Ibáñez
(Ibáñez does not mention Cullinane as the author of the above theorem
in his book (except indirectly, quoting Josefine Lyche), but he did credit
him fully in an earlier article, "The Truchet Tiles and the Diamond Puzzle"
(translation by Google).)
About Ibáñez (translated from Amazon.com by Google):
Mathematician, professor of Geometry at the University of the Basque Country
and scientific disseminator. He is part of the Chair of Scientific Culture of the
UPV/EHU and its blog Cuaderno de Cultura Cientifica. He has been a scriptwriter
and presenter of the program “Una de Mates” on the television program Órbita Laika.
He has collaborated since 2005 on the programs Graffiti and La mechanica del caracol
on Radio Euskadi. He has also been a collaborator and co-writer of the documentary
Hilos de tiempo (2020) about the artist Esther Ferrer. For 20 years he directed the
DivulgaMAT portal, Virtual Center for the Dissemination of Mathematics, and was a
member of the dissemination commission of the Royal Spanish Mathematical Society.
Author of several books, including The Secrets of Multiplication (2019) and
The Great Family of Numbers (2021), in the collection Miradas Matemáticas (Catarata).
He has received the V José María Savirón Prize for Scientific Dissemination
(national modality, 2010) and the COSCE Prize for the Dissemination of Science (2011).
Comments Off on Basque Country Art Book
Tuesday, December 19, 2023
Copilot Report
|
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:
Searching for: Generating answers for you… The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics12. Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1. On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242. The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator3. These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3. For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help. Learn more
5
_________________________________________________________
__________________________________________________________
Here is a summary of the document:
Reference:
|
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."
Comments Off on Copilot Report
Monday, December 18, 2023
AI Class
|
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: Generating answers for you… The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry1,2. The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG1,2. 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 quadrants1,2. The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry1,2. 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 elements1,2. 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-sets1,2. The MOG is a 4×6 array of combinations describing any point in 24-dimensional space1. It preserves all of the symmetries and maximal subgroups of the Mathieu group M241. The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice2. In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG1,2. Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry1,2. Learn more Reply to Bing's Response — Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements. Response to the Above Reply — Generating answers for you… Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!
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Sunday, December 17, 2023
Speak, Memory
"Unsocratic Dialogue" with Google DeepMind's Gemini AI continues . . .
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Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
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Thursday, December 14, 2023
Unsocratic Dialogue (with Gemini AI in Bard Today)
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What is the Cullinane diamond theorem?
|
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Saturday, December 9, 2023
How Many Magic Beans, Jack?
Two notes from this journal on December 2, 2023 —
A flashback for The November Man —
As for " my own 'diamond theorem' " —
|
Question — "What is mathematician Steven H. Cullinane known for?" From the Copilot response (shorn of footnotes) — "Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2). Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions." |
Comments Off on How Many Magic Beans, Jack?
Saturday, December 2, 2023
My Work According to Copilot
Following yesterday's encounter with the latest version of Pi,
here are some more formal (and more informative) remarks
from Windows Copilot today —
|
Question — "What is mathematician Steven H. Cullinane known for?" From the Copilot response (shorn of footnotes) — "Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2). Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions." |
Comments Off on My Work According to Copilot
Monday, November 27, 2023
Birkhoff-von Neumann Symmetry* over Finite Fields
See David G. Poole, "The Stochastic Group,"
American Mathematical Monthly, volume 102, number 9
(November, 1995), pages 798–801.
* This post was suggested by the phrase "The Diamond Theorem,
also known as the von Neumann-Birkhoff conjecture" in a
ChatGPT-3.5 hallucination today.
That phrase suggests a look at the Birkhoff-von Neumann theorem:
The B.-von N. theorem suggests a search for analogous results
over finite fields. That search yields the Poole paper above,
which is related to my own "diamond theorem" via affine groups.
Comments Off on Birkhoff-von Neumann Symmetry* over Finite Fields
Thursday, November 16, 2023
Geometry and Art
AI-assisted report on "Cullinane Diamond Theorem discovery" —
The full story of how the theorem was discovered is actually
a bit more interesting. See Art Space, a post of May 7, 2017,
and The Lindbergh Manifesto, a post of May 19, 2015.
"The discovery of the Cullinane Diamond Theorem is a testament
to the power of mathematical abstraction and its ability to reveal
deep connections and symmetries in seemingly simple structures."
I thank Bing for that favorable review.
Comments Off on Geometry and Art
Sunday, October 15, 2023
Efficient Packend
"Stencils" from a 1959 paper by Golomb —
These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).
This journal on Beltane (May 1), 2016 —
Comments Off on Efficient Packend
Friday, October 13, 2023
Turn, Turn, Turn
The conclusion of a Hungarian political figure's obituary in
tonight's online New York Times, written by Clay Risen —
"A quietly religious man, he spent his last years translating
works dealing with Roman Catholic canon law."
This journal on the Hungarian's date of death, October 8,
a Sunday, dealt in part with the submission to Wikipedia of
the following brief article . . . and its prompt rejection.
|
The Cullinane diamond theorem is a theorem
The theorem also explains symmetry properties of the Reference
1. Cullinane diamond theorem at |
Some quotations I prefer to Catholic canon law —
|
Ludwig Wittgenstein,
97. Thought is surrounded by a halo. * See the post Wittgenstein's Diamond. Related language in Łukasiewicz (1937)—
|
See as well Diamond Theory in 1937.
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Wednesday, August 9, 2023
The Junction Function
A function (in this case, a 1-to-1 correspondence) from finite geometry:
This correspondence between points and hyperplanes underlies
the symmetries discussed in the Cullinane diamond theorem.
Academics who prefer cartoon graveyards may consult …
Cohn, N. (2014). Narrative conjunction’s junction function:
A theoretical model of “additive” inference in visual narratives.
Proceedings of the Annual Meeting of the Cognitive Science
Society, 36. See https://escholarship.org/uc/item/2050s18m .
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Friday, July 21, 2023
Tuesday, July 18, 2023
Going Beyond Wikipedia
Despite today's New York Times warnings about chatbot inaccuracy,
the above short summary is well-written, helpful, and correct.
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Friday, July 7, 2023
CORE
The "CORE" reference in the previous post yields, via a search . . .
Within this thesis there are 19 references to the name "Cullinane"
and to my own work, cited as . . .
Cullinane, Steven H., ‘The Diamond Theorem’ (1979)
<http://diamondtheorem.com>
[accessed 6 May 2019]
––– ‘Geometry of the I Ching’ (1989)
<http://finitegeometry.org/sc/64/iching.html>
[accessed 6 May 2019].
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Thursday, June 15, 2023
Wednesday, May 31, 2023
Google AI-Powered-Overview Example
"Finite geometry explains the surprising symmetry properties
of some simple graphic designs." … Good summary.
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Monday, May 15, 2023
Chatbot Review
|
By ChatGPT* —
What is the Cullinane diamond theorem? The Cullinane Diamond Theorem, also known as the Diamond Theorem, is a result in combinatorial mathematics proposed by Steven H. Cullinane. The theorem explores connections between finite geometries, algebraic structures, and combinatorial designs.
The theorem is primarily concerned with four related structures: 2. Vector spaces: The theorem connects finite vector spaces over finite fields to the aforementioned finite geometries. 3. Boolean functions:* The theorem establishes relationships between the finite geometries and certain Boolean functions derived from them.
4. Symmetry groups: The theorem involves symmetry groups associated with the finite geometries. * With one revision by Cullinane: "Boolean functions" replaces "Boolean algebras." See (for instance) M. A. Harrison, “On the classification of Boolean functions by the general linear and affine groups,” Journal of the Society for Industrial and Applied Mathematics 1964 12:2, 285-299. |
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Death on Beltane
"Stencils" from a 1959 paper by Golomb —
These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).
This journal on Beltane (May 1), 2016 —
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Sunday, April 30, 2023
For Harlan Kane: The Walpurgisnacht Hallucination
Note that if the "compact Riemann surface" is a torus formed by
joining opposite edges of a 4×4 square array, and the phrase
"vector bundle" is replaced by "projective line," and so forth,
the above ChatGPT hallucination is not completely unrelated to
the following illustration from the webpage "galois.space" —
See as well the Cullinane diamond theorem.
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Saturday, March 25, 2023
Zero Sum Game
Google Search now emphasizes the reasoning
behind the diamond theorem —
For related language (but un-related ideas ), see Zero Sum in this journal.
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Tuesday, March 14, 2023
Disney Endings
The New York Times reports a Monday,
March 13, 2023, death:
This journal Monday —
Final image of the above "diamond theorem" penrose search on Monday —
From March 2 —
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Monday, March 13, 2023
Frontiers of Artificial Mathematics
Previous posts have shown ChatGPT answering the question
"What is the diamond theorem?" with references to Thurston
and, later, to Conway. Today it is Penrose's turn.
Related search results (click to enlarge) —
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Sunday, March 5, 2023
Annals of Artificial Mathematics
The response of ChatGPT to a question about my work
continues to evolve. It now credits Conway, not Thurston,*
for the diamond theorem.
The paragraph beginning "The theorem states" appears** to be based
on the following 24 patterns — which number only 8, if rotated or
reflected patterns are considered equivalent.
* For Thurston in an earlier ChatGPT response to the same question,
see a Log24 post of Feb. 25.
** The illustration above is based on the divison of a square into
four smaller subsquares. If the square is rotated by 45 degrees,
it becomes a diamond that can be, in the language of ChatGPT,
divided into "four smaller diamonds ."
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Monday, February 6, 2023
Interality Studies
|
You, Xi-lin; Zhang, Peter. "Interality in Heidegger."
The term "interology" is meant as an interventional alternative to traditional Western ontology. The idea is to help shift people's attention and preoccupation from subjects, objects, and entities to the interzones, intervals, voids, constitutive grounds, relational fields, interpellative assemblages, rhizomes, and nothingness that lie between, outside, or beyond the so-called subjects, objects, and entities; from being to nothing, interbeing, and becoming; from self-identicalness to relationality, chance encounters, and new possibilities of life; from "to be" to "and … and … and …" (to borrow Deleuze's language); from the actual to the virtual; and so on. As such, the term wills nothing short of a paradigm shift. Unlike other "logoi," which have their "objects of study," interology studies interality, which is a non-object, a no-thing that in-forms and constitutes the objects and things studied by other logoi. |
Some remarks from this journal on April 1, 2015 —
Manifest O
|
| 83-06-21 | An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. |
| 83-10-01 | Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. |
| 83-10-16 | Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. |
| 84-09-15 | Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. |
The above site, finitegeometry.org/sc, illustrates how the symmetry
of various visual patterns is explained by what Zhang calls "interality."
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Thursday, January 19, 2023
Two Approaches to Local-Global Symmetry
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; 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
— Alice Devillers, |
The Wikipedia article Homogeneous graph discusses the local-global approach
used by Cameron and by Devillers.
For some historical background on this approach
via substructure isomorphisms, see a former student of Cameron:
Dugald Macpherson, "A survey of homogeneous structures,"
Discrete Mathematics , Volume 311, Issue 15, 2011,
Pages 1599-1634.
Related material:
Cherlin, G. (2000). "Sporadic Homogeneous Structures."
In: Gelfand, I.M., Retakh, V.S. (eds)
The Gelfand Mathematical Seminars, 1996–1999.
Gelfand Mathematical Seminars. Birkhäuser, Boston, MA.
https://doi.org/10.1007/978-1-4612-1340-6_2
and, more recently,
Gill et al., "Cherlin's conjecture on finite primitive binary
permutation groups," https://arxiv.org/abs/2106.05154v2
(Submitted on 9 Jun 2021, last revised 9 Jul 2021)
This approach seems to be a rather deep rabbit hole.
The Second Approach — Via Induced Group Actions —
My own interest in local-global symmetry is of a quite different sort.
See properties of the two patterns illustrated in a note of 24 December 1981 —
Pattern A above actually has as few symmetries as possible
(under the actions described in the diamond theorem ), but it
does enjoy, as does patttern B, the local-global property that
a group acting in the same way locally on each part induces
a global group action on the whole .
* For some historical background on the term "homogeneous,"
see the Wikipedia article Homogeneous space.
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Thursday, September 22, 2022
Affine Dürer
The previous post's image illustrating the
ancient Lo Shu square as an affine transformation
suggests a similar view of Dürer's square.
That view illustrates the structural principle
underlying the diamond theorem —
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Monday, August 1, 2022
Interality Again: The Art of the Gefüge
"Schufreider shows that a network of linguistic relations
is set up between Gestalt, Ge-stell, and Gefüge, on the
one hand, and Streit, Riß, and Fuge, on the other . . . ."
— From p. 14 of French Interpretations of Heidegger ,
edited by David Pettigrew and François Raffoul.
State U. of New York Press, Albany, 2008. (Links added.)
One such "network of linguistic relations" might arise from
a non-mathematician's attempt to describe the diamond theorem.
(The phrase "network of linguistic relations" appears also in
Derrida's remarks on Husserl's Origin of Geometry .)
For more about "a system of slots," see interality in this journal.
The source of the above prefatory remarks by editors Pettigrew and Raffoul —
"If there is a specific network that is set up in 'The Origin of the Work of Art,'
a set of structural relations framed in linguistic terms, it is between
Gestalt, Ge-stell and Gefüge, on the one hand, and Streit, Riß and Fuge,
on the other; between (as we might try to translate it)
configuration, frame-work and structure (system), on the one hand, and
strife, split (slit) and slot, on the other. On our view, these two sets go
hand in hand; which means, to connect them to one another, we will
have to think of the configuration of the rift (Gestalt/Riß) as taking place
in a frame-work of strife (Ge-stell/Streit) that is composed through a system
of slots (Gefüge/Fuge) or structured openings."
— Quotation from page 197 of Schufreider, Gregory (2008):
"Sticking Heidegger with a Stela: Lacoue-Labarthe, art and politics."
Pp. 187-214 in David Pettigrew & François Raffoul (eds.),
French Interpretations of Heidegger: An Exceptional Reception.
State University of New York Press, 2008.
Update at 5:14 AM ET Wednesday, August 3, 2022 —
See also "six-set" in this journal.
"There is such a thing as a six-set."
— Saying adapted from a 1962 young-adult novel.
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Thursday, May 12, 2022
Compare and Contrast
Related reading:
Shibumi: A Novel
and
"The Diamond Theorem
in Basque Country."
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Sunday, January 2, 2022
Annals of Modernism: UR–Grid
The above New Yorker art illustrates the 2×4 structure of
an octad in the Miracle Octad Generator of R. T. Curtis.
Enthusiasts of simplicity may note how properties of this eight-cell
2×4 grid are related to those of the smaller six-cell 3×2 grid:
See Nocciolo in this journal and . . .
Further reading on the six-set – eight-set relationship:
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Saturday, May 8, 2021
A Tale of Two Omegas
The Greek capital letter Omega, Ω, is customarily
used to denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois field,
the appropriate Ω is the 4×4 grid above.
See the Cullinane diamond theorem .
If the group is the large Mathieu group of
244,823,040 permutations of 24 things,
the appropriate Ω is the 4×6 grid below.
See the Miracle Octad Generator of R. T. Curtis.
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Thursday, April 22, 2021
A New Concrete Model for an Old Abstract Space
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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Thursday, March 25, 2021
A Day at the Museum
Context for the Cullinane diamond theorem in
the Smithsonian’s NASA Astrophysics Data System:
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Thursday, March 4, 2021
Teaching the Academy to See
“Art bears the same relationship to society
that the dream bears to mental life. . . .
Like art, the dream mediates between order
and chaos. So, it is half chaos. That is why
it is not comprehensible. It is a vision, not
a fully fledged articulated production.
Those who actualize those half-born visions
into artistic productions are those who begin
to transform what we do not understand into
what we can at least start to see.”
— A book published on March 2, 2021:
Beyond Order , by Jordan Peterson
The inarticulate, in this case, is Rosalind Krauss:
A “raid on the inarticulate” published in Notices of the
American Mathematical Society in the February 1979 issue —
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Friday, February 26, 2021
Non-Chaos Non-Magic
For fans of “WandaVision” —
“1978 was perhaps the seminal year in the origin of chaos magic. . . .”
— Wikipedia article on Chaos Magic
Non-Chaos Non-Magic from Halloween 1978 —
Related material —
A doctoral student of a different Peter Cameron —
( Not to be confused with The Tin Man’s Hat. )
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Sunday, December 27, 2020
Logo Animation
Related material from Log24 yesterday —
Click the Aquarius symbol for a puzzle.
A related animation —
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Saturday, December 19, 2020
Classic Romantic
Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance ,
Ch. 6 (italics are mine):
“A classical understanding sees the world primarily as underlying form itself.
A romantic understanding sees it primarily in terms of immediate appearance .”
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Thursday, August 6, 2020
After Personalities . . . Principles
In memory of New York personality Pete Hamill ,
who reportedly died yesterday —
Seven years ago yesterday —
In memory of another New York personality, a parking-garage mogul
who reportedly died on August 9, 2005 —
Icon Parking posts and . . .
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Thursday, July 30, 2020
A Picture Show for Quanta Magazine
An article yesterday at Quanta Magazine suggests a review . . .
From Diamond Theorem images at Pinterest —
Some background —
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Thursday, July 9, 2020
The Enigma Glyphs
For those who prefer fiction —
“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.
This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”
— “Robert Ludlum’s” The Bourne Enigma , published on June 21, 2016
Passing, on June 21, 2016, into a higher dimension —
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Friday, June 12, 2020
Bullshit Studies: “Hyperseeing”
In memoriam —
Friedman co-edited the ISAMA journal Hyperseeing . See also . . .
See too the other articles in Volume 40 of Kybernetes .
Related material —
Compare and contrast the discussion of the geometry
of the 4×4 square in the diamond theorem (1976) with
Nat Friedman’s treatment of the same topic in 2001 —
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Saturday, March 7, 2020
The “Octad Group” as Symmetries of the 4×4 Square
From "Mathieu Moonshine and Symmetry Surfing" —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
"This presentation of the symmetry groups Gi is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group
G = (Z2)4 ⋊ A8 .
It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O9 = {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose."
This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring," by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A-193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the "octad group" in 1993 —
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Wednesday, January 22, 2020
Gap Dance
From Wallace Stevens, "The Man with the Blue Guitar":
IX
And the color, the overcast blue
Of the air, in which the blue guitar
Is a form, described but difficult,
And I am merely a shadow hunched
Above the arrowy, still strings,
The maker of a thing yet to be made . . . .
"Arrowy, still strings" from the diamond theorem
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Thursday, September 12, 2019
Pattern and Structure
From the previous post —
" . . . Only by the form, the structure,
Can words or music reach
The stillness . . . ."
— Adapted from T. S. Eliot's Four Quartets
by replacing "pattern" with "structure."
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Tuesday, September 10, 2019
Tuesday, March 5, 2019
A Block Design 3-(16,4,1) as a Steiner Quadruple System:
A Midrash for Wikipedia
Midrash —
Related material —
________________________________________________________________________________
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Friday, March 1, 2019
Wikipedia Scholarship (Continued)
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)).
Comments Off on Wikipedia Scholarship (Continued)
Friday, February 22, 2019
Back Issues of AMS Notices
From the online home page of the new March issue —
For instance . . .
Related material now at Wikipedia —
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Saturday, September 15, 2018
Eidetic Reduction in Geometry
|
"Husserl is not the greatest philosopher of all times. — Kurt Gödel as quoted by Gian-Carlo Rota Some results from a Google search — Eidetic reduction | philosophy | Britannica.com Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart … Phenomenology Online » Eidetic Reduction
The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research Eidetic reduction – New World Encyclopedia Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience. |
For example —
The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.
See the diamond theorem and the eightfold cube.
* Cf. posts tagged Interality and Interstice.
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Monday, August 27, 2018
Children of the Six Sides
From the former date above —
|
Saturday, September 17, 2016 |
From the latter date above —
|
Tuesday, October 18, 2016
Parametrization
|
From March 2018 —
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Saturday, June 9, 2018
SASTRA paper
Now out from behind a paywall . . .
The diamond theorem at SASTRA —
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Monday, May 7, 2018
Data
(Continued from yesterday's Sunday School Lesson Plan for Peculiar Children)
Novelist George Eliot and programming pioneer Ada Lovelace —
For an image that suggests a resurrected multifaceted
(specifically, 759-faceted) Osterman Omega (as in Sunday's afternoon
Log24 post), behold a photo from today's NY Times philosophy
column "The Stone" that was reproduced here in today's previous post —
For a New York Times view of George Eliot data, see a Log24 post
of September 20, 2016, on the diamond theorem as the Middlemarch
"key to all mythologies."
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Tuesday, April 24, 2018
Illustrators of the Word
Tom Wolfe in The Painted Word (1975) —
“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”
The above group of 322,560 permutations appears also in a 2011 book —
— and in 2013-2015 papers by Anne Taormina and Katrin Wendland:
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Wednesday, April 11, 2018
Behance Post
I just came across the 2010 web page
https://pantone.ccnsite.com/gallery/HELDER-visual-identity/603956,
associated with the Adobe site "Behance.net." That page suggested
I too should have a Behance web presence.
And so the diamond theorem now appears at . . .
https://www.behance.net/gallery/64334249/The-Diamond-Theorem.
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Thursday, March 29, 2018
“Before Creation Itself . . .”
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
"Is this Time Cube?"
Notes toward an answer —
And from Six-Set Geometry in this journal . . .
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Wednesday, March 28, 2018
On Unfairly Excluding Asymmetry
A comment on the the Diamond Theorem Facebook page —
Those who enjoy asymmetry may consult the "Expert's Cube" —
For further details see the previous post.
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Friday, February 16, 2018
Two Kinds of Symmetry
The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:
This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.
In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —
". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)
The adjective "modular" might aptly be applied to . . .
The adjective "affine" might aptly be applied to . . .
The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.
Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —
— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —
For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."
For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.
Comments Off on Two Kinds of Symmetry
Monday, December 18, 2017
Wheelwright and the Dance
The page preceding that of yesterday's post Wheelwright and the Wheel —
See also a Log24 search for
"Four Quartets" + "Four Elements".
A graphic approach to this concept:
"The Bounded Space" —
"The Fire, Air, Earth, and Water" —
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Thursday, November 30, 2017
The Matrix for Quantum Mystics
Scholia on the title — See Quantum + Mystic in this journal.
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976
I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —
.
From a Log24 post of June 26-27, 2017:
A work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
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Tuesday, October 24, 2017
Visual Insight
The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017 —
A visually related concept — See Solomon's Cube in this journal.
Chronologically related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem —
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