Compare and contrast . . .
Saturday, February 15, 2025
Note from the Underground
(The Apple Entertainment Version)
Comments Off on Note from the Underground
(The Apple Entertainment Version)
(The Apple Entertainment Version)
Friday, February 14, 2025
Roots Rising: Archetypes, Figurations, Icons
|
"Entering into us, the painting, the sonata, the poem brings us into reach of our own nativity of consciousness. It does so at a depth inaccessible in any other way. Literature and the arts are the perceptible witness to that freedom of coming into being of which history can give us no account. Hence the utterly arresting congruence between Aristotle’s assertion in the Poetics that fiction is 'truer and more universal than history' and Jung’s inference of archetypes, of inherited figurations and narrative icons, at the roots of human consciousness. I find this congruence seductive. But there is, I repeat, no external evidence for it whatever. It may be pure fantastication."
— Steiner, George. Real Presences (p. 182). |
See also "divine universals" in this journal
and a "three coins" meditation from 2003 . . .
From Philip K. Dick’s The Man in the High Castle :
Juliana said, “Oracle, why did you write The Grasshopper Lies Heavy?
What are we supposed to learn?”
“You have a disconcertingly superstitious way of phrasing your question,”
Hawthorne said. But he had squatted down to witness the coin throwing.
“Go ahead,” he said; he handed her three Chinese brass coins with
holes in the center. “I generally use these.”
This suggests … The Man in the High Tower —
[Actually “The Gorge” script by Zach Dean.]
Cue the Bernstein.
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Steiner Date
The above cello date — January 30, 2022 — in this journal . . .
From the new Apple TV Plus film "The Gorge" —
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Lyrics Game for Gretel —
“Lester, Lana … Lana, Lester.”
Related reading:
- The Log24 post "Patterns" of Jan. 3, 2025
- A new Stephen King Hansel and Gretel, due out Sept. 2
- The "Breakthrough" hexagram (43) from the box-style I Ching
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“Lester, Lana … Lana, Lester.”
“Lester, Lana … Lana, Lester.”
Thursday, February 13, 2025
The Lyrics Game . . .
Comments Off on The Lyrics Game . . .
The Exploitation of Symmetry . . . Continues.
Illustration of a July 1980 title by George Mackey —
Exploitation of Symmetry in 1981 . . .
See also the tetrahedra* in my "square triangles" letter
(1985), as well as "Senechal" in this journal.
"And we both know what memories can bring…" Do we?
Comments Off on The Exploitation of Symmetry . . . Continues.
Wednesday, February 12, 2025
Decompiling Wolfenstein . . . Continues.
Comments Off on Decompiling Wolfenstein . . . Continues.
Annals of Apple Entertainment . . .
Katherine Neville’s The Eight — Enhanced!
Applying the "Go back 10" symbol above . . .
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Katherine Neville’s The Eight — Enhanced!
Katherine Neville’s The Eight — Enhanced!
64 Artworks
The previous two posts suggest a look at an earlier post
on the theme of artworks related to the number 64 —
"The Yarrow Stalker," from a Log24 search for "Chinatown."
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Tuesday, February 11, 2025
Compiling BASIC, Decompiling* Wolfenstein
Yesterday, Peter Woit posted on Bill Gates's new autobiography.
An excerpt from his post:
In other Harvard-related news . . .
A computer-related fantasy film — "The Net" — and the above
headline from February 6 suggest a look at . . .
* Vide "Paranoia Strikes Deep" (Log24, Dec. 1, 2011).
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Monday, February 10, 2025
Brick Space: Points with Parts
This post's "Points with Parts" title may serve as an introduction to
what has been called "the most powerful diagram in mathematics" —
the "Miracle Octad Generator" (MOG) of Robert T. Curtis.
Curtis himself has apparently not written on the geometric background
of his diagram — the finite projective spaces PG(5,2) and PG(3,2), of
five and of three dimensions over the two-element Galois field GF(2).
The component parts of the MOG diagram, the 2×4 Curtis "bricks,"
may be regarded* as forming both PG(5,2) and PG(3,2) . . .
Pace Euclid, points with parts. For more on the MOG's geometric
background, see the Klein correspondence in the previous post.
For a simpler example of "points with parts, see
http://m759.net/wordpress/?s=200229.
* Use the notions of Galois (XOR, or "symmetric-difference") addition
of even subsets, and such addition "modulo complementation," to
decrease the number of dimensions of the spaces involved.
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Sunday, February 9, 2025
An AI Overview of Klein Quadric and MOG
See also a more accurate AI report from January 9, 2025 —
Grok 2 on the MOG and the Klein Correspondence:
HTML version, with corrections, of the above 9 January Grok 2 report —
|
Grok 2: Klein Correspondence and MOG, 9 Jan. 2025 . . . The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics.
Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system. Relation Between Klein Correspondence and MOG:
Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems. * Correction: Should be "a correspondence between points in a five-dimensional projective space and lines in a three-dimensional projective space." |
Update of ca. 9 AM ET Monday, Feb. 10, 2024 —
Neither AI report above mentions the Cullinane model of the five-
dimensional projective space PG(5,2) as a brick space — a space
whose points are the 2×4 bricks used in thte MOG. This is
understandable, as the notion of using bricks to model both PG(5,2)
and PG(3,2) has appeared so far only in this journal. See an
illustration from New Year's Eve . . . Dec. 31, 2024 —
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Saturday, February 8, 2025
Friday, February 7, 2025
Thursday, February 6, 2025
For the Cambridge Philosophical Society:
The Office Supplies Folder
Courtesy of . . .
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The Office Supplies Folder
The Office Supplies Folder
Rooftop Romance . . . Continues.
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Dramarama Continues.
Alternative to the Starbrick of the previous post —
♫ "The lights are much brighter there
You can forget all your troubles, forget all your cares
So go downtown, where all the lights are bright
Downtown, waiting for you tonight "
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The Message from Dark Helmet . . . Immerse Yourself!
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Wednesday, February 5, 2025
Axiom Attics: Ars Longa
At about 37:28 —
Okay. What's the operating system?
Um…
Is there a logo, an extension? Anything?
Go to the top left and open system settings.
( breathes heavily )
Uh, it says AXI .
I know that system, but it's US government only.
The software's designed by Axiorn. ( sighs )
They're a private security firm.
Read more at: https://tvshowtranscripts.ourboard.org/
viewtopic.php?f=2457&t=72920
&sid=37ef753cee8a0baf2bab3e2e4f32967c
From this journal on January 10, 2025, a cartoon from
Axiomatics: Mathematical Thought and High Modernism —
Okay. What's the operating system? Um… Is there a logo, an extension? Anything? Go to the top left and open system settings. ( breathes heavily ) Uh, it says AXI. I know that system, but it's US government only. The software's designed by Axiorn. ( sighs ) They're a private security firm.
Read more at: https://tvshowtranscripts.ourboard.org/viewtopic.php?f=2457&t=72920&sid=37ef753cee8a0baf2bab3e2e4f32967c
Okay. What's the operating system? Um… Is there a logo, an extension? Anything? Go to the top left and open system settings. ( breathes heavily ) Uh, it says AXI. I know that system, but it's US government only. The software's designed by Axiorn. ( sighs ) They're a private security firm.
Read more at: https://tvshowtranscripts.ourboard.org/viewtopic.php?f=2457&t=72920&sid=37ef753cee8a0baf2bab3e2e4f32967c
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The Samuel Butler Memorial Obit* Continues:
♫ “Erewhon Man, Please Listen . . .”
Comments Off on The Samuel Butler Memorial Obit* Continues:
♫ “Erewhon Man, Please Listen . . .”
♫ “Erewhon Man, Please Listen . . .”
Romance of the Avatars: The Ascension of Rosenberg
Investigations and Fantasies weblog on August 21, 2023 —
Meanwhile, in this journal on that date . . .
The dies natalis, in the Catholic sense, of Rosenberg was reportedly . . .
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Tuesday, February 4, 2025
Mergers and Acquisitions Dept. —In Memoriam:
Samuel Butler, former NY Public Library Chair
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Samuel Butler, former NY Public Library Chair
Samuel Butler, former NY Public Library Chair
The Day of the Clercs
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Diagrams
"The most powerful diagram in mathematics" —
The YouTube lecturer is not referring to the Fano plane diagram cited
in the AI Overview below, but to a much more sophisticated figure,
the Miracle Octad Generator (MOG) of R. T. Curtis.
Some context —
A rearrangement of the Miracle Octad Generator —
The diagram below may be less powerful , but it illustrates a result that,
although less miraculous , is perhaps more historically significant —
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Monday, February 3, 2025
The Gombrich Cartoon . . . Continues.
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Space Hoard
"Minimalists are actually extreme hoarders:
they hoard space." — Douglas Coupland
This post was suggested by the date August 23, 2022,
in this journal and elsewhere.
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Spaceballs at Stillwater
New country song . . .
♫ "Here's to swimmin' with high-maintenance women."
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Sunday, February 2, 2025
Again, for Shadowhunters . . .
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Eric Temple Bell on Solomon’s Seal
From pp. 322 ff. of The Development of Mathematics, by Eric Temple Bell, Second Edition, McGraw-Hill, 1945, at https://archive.org/stream/in.ernet.dli.2015.133966/2015.133966. The-Development-Of-Mathematics-Second-Edition_djvu.txt — Rising to a considerably higher level of difficulty, we may instance what the physicist Maxwell called “Solomon’s seal in space of three dimensions,” the twenty-seven real or imaginary straight lines which lie wholly on the general cubic surface, and the forty-five triple tangent planes to the surface, all so curiously related to the twenty-eight bitangents of the general plane quartic curve. If ever there was a fascinating snarl of interlaced theories, Solomon’s seal is one. Synthetic and analytic geometry, the Galois theory of equations, the trisection of hyperelliptic functions, the algebra of invariants and covariants, geometric-algebraic algorithms specially devised to render the tangled configurations of Solomon’s seal more intuitive, the theory of finite groups — all were applied during the second half of the nineteenth century by scores of geometers who sought to break the seal. Some of the most ingenious geometers and algebraists in history returned again and again to this highly special topic. The result of their labors is a theory even richer and more elaborately developed than Klein’s (1884) of the icosahedron. Yet it was said by competent geometers in 1945 that a serious student need never have heard of the twenty-seven lines, the forty-five triple tangent planes, and the twenty-eight bitangents in order to be an accomplished and productive geometer; and it was a fact that few in the younger generation of creative CONTRIBUTIONS FROM GEOMETRY 323 geometers had more than a hazy notion that such a thing as tiie Solomon’s seal of the nineteenth century ever existed. Those rvho could recall from personal experience the last glow of living appreciation that lighted this obsolescent master- piece of geometry and others in the same fading tradition looked back with regret on the dying past, and wished that mathe- matical progress were not always so ruthless as it is. They also sympathized with those who still found the modern geometry of the triangle and the circle worth cultivating. For the differ- ence between the geometry of the twenty-seven lines and that of, say, Tucker, Lemoine, and Brocard circles, is one of degree, not of kind. The geometers of the twentieth century long since piously removed all these treasures to the museum of geometry, where the dust of history quickly dimmed their luster. For those who may be interested in the unstable esthetics rather than the vitality of geometry, we cite a concise modern account1 (exclusive of the connection with hyperclliptic func- tions) of Solomon’s seal. The twenty-seven lines were discovered in 1849 by Cayley and G. Salmon2 (1819-1904, Ireland); the application of transcendental methods originated in Jordan’s work (1869-70) on groups and algebraic equations. Finally, in the 1870’s L. Cremona (1830-1903), founder of the Italian school of geometers, observed a simple connection between the twenty-one distinct straight lines which lie on a cubic surface with a node and the ‘cat’s cradle’ configuration of fifteen straight lines obtained by joining six points on a conic in all possible ways. The ‘mystic hexagram’ of Pascal and its dual (1806) in C. J. Brianchon’s (1783-1864, French) theorem were thus related to Solomon’s seal; and the seventeenth century met the nineteenth in the simple, uniform deduc- tion of the geometry of the plane configuration from that of a corresponding configuration in space by the method of projection. The technique here had an element of generality that was to prove extremely powerful in the discovery and proof of cor- related theorems by projection from space of a given number of dimensions onto a space of lower dimensions. Before Cremona applied this technique to the complete Pascal hexagon, his countryman G. Veronese had investigated the Pascal configura- tion at great length by the methods of plane geometry, as had also several others, including Steiner, Cayley, Salmon, and Kirkman. All of these men were geometers of great talent; 324 THE DEVELOPMENT OF MATHEMATICS Cremona’s flash of intuition illuminated the massed details of all his predecessors and disclosed their simple connections. That enthusiasm for this highly polished masterwork of classical geometry is by no means extinct is evident from the appearance as late as 1942 of an exhaustive monograph (xi + 180 pages) by B. Segre (Italian, England) on The nonsingular cubic surface. Solomon’s seal is here displayed in all its “complicated and many-sided symmetry” — in Cayley’s phrase — as never before. The exhaustive enumeration of special configurations provides an unsurpassed training ground or ‘boot camp’ for any who may wish to strengthen their intuition in space of three dimensions. The principle of continuity, ably seconded by the method of degeneration, consistently applied, unifies the multi- tude of details inherent in the twenty-seven lines, giving the luxuriant confusion an elusive coherence which was lacking in earlier attempts to “bind the sweet influences” of the thirty- six possible double sixes (or ‘double sixers,’ as they were once called) into five types of possible real cubic surfaces, containing respectively 27, 15, 7, 3, 3 real lines. A double six is two sextuples of skew lines such that each line of one is skew to precisely one corresponding line of the other. A more modern touch appears in the topology of these five species. Except for one of the three-line surfaces, all are closed, connected manifolds, while the other three-line is two connected pieces, of which only one is ovoid, and the real lines of the surface are on this second piece. The decompositions of the nonovoid piece into generalized polyhedra by the real lines of the surface are painstakingly classified with respect to their number of faces and other char- acteristics suggested by the lines. The nonovoid piece of one three-line surface is homeomorphic to the real projective plane, as also is the other three-line surface. The topological interlude gives way to a more classical theme in space of three dimensions, which analyzes the group in the complex domain of the twenty- seven lines geometrically, either through the intricacies of the thirty-six double sixes, or through the forty triads of com- plementary Steiner sets. A Steiner set of nine lines is three sets of three such that each line of one set is incident with precisely two lines of each other set. The geometrical significance of permutability of operations in the group is rather more com- plicated than its algebraic equivalent. The group is of order 51840. There is an involutorial transformation in the group for each double six; the transformation permutes corresponding CONTRIBUTIONS FROM GEOMETRY 325 lines of the complementary sets of six of the double six, and leaves each of the remaining fifteen lines invariant. If the double sixes corresponding to two such transformations have four common lines, the transformations are permutable. If the transformations are not permutable, the corresponding double sixes have six common lines, and the remaining twelve lines form a third double six. Although the geometry of the situation may be perspicuous to those gifted with visual imagination, others find the underlying algebraic identities, among even so impressive a number of group operations as 51840, somewhat easier to see through. But this difference is merely one of ac- quired taste or natural capacity, and there is no arguing about it. However, it may be remembered that some of this scintillating pure geometry was subsequent, not antecedent, to many a dreary page of laborious algebra. The group of the twenty- seven lines alone has a somewhat forbidding literature in the tradition of the late nineteenth and early twentieth centuries which but few longer read, much less appreciate. So long as geometry — of a rather antiquated kind, it may be — can clothe the outcome of intricate calculations in visualizable form, the Solomon’s seal of the nineteenth century will attract its de- votees, and so with other famous classics of the geometric imagination. But in the meantime, the continually advancing front of creative geometry will have moved on to unexplored territory of fresher and perhaps wider interest. The world some- times has sufficient reason to be weary of the past in mathe- matics as in everything else. |
See as well a figure from yesterday's Matrix Geometry post —
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Saturday, February 1, 2025
A Symposium for Plato
From the previous post —
"Presented by invitation at the Symposium for Combinatorial Mathematics,
sponsored by the Office of Naval Research…."
— and from a post last night:
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Matrix Geometry: The Schläfli Double-Six
Compare and contrast:
A larger, but in some sense simpler, illustration of the corresponding
double-six combinatorial configuration * —
See also this method of illustration in the article "Configuration"
of the Encyclopedia of Mathematics.
* See Ryser, "Combinatorial Configurations,"
SIAM Journal on Applied Mathematics
Vol. 17, No. 3 (May, 1969), pp. 593-602 —
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Annals of Cultural Appropriation:
♫ “Let’s Hear It for the Buoy” Continues . . .
The previous post suggets a flashback to earlier remarks . . .
Bell-buoy Meets AI Overview . . .
For the bell-buoy itself, see a Log24 search for "Buoy."
Comments Off on Annals of Cultural Appropriation:
♫ “Let’s Hear It for the Buoy” Continues . . .
♫ “Let’s Hear It for the Buoy” Continues . . .
Friday, January 31, 2025
♪ “By these festival rites from the age that is past . . .” ♪
Elvis, 1958, "Spearhead" Division . . .
Kristen, January 2024 Sundance photo by Mariah Tauger:
publicity photo for "Love Me" . . .
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Sigillum Veri: Maori Date
A date from the above Google search for Whanganui meaning —
September 22, 2019.
See that date in other posts now tagged Simplex.
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Thursday, January 30, 2025
Low-Hanging Fruit for Seinfeld
(and the St. Louis Wolves)
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(and the St. Louis Wolves)
(and the St. Louis Wolves)
Show Tune for Conspiracy Theorists:
♫ “Doin’ what comes naturally!”
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♫ “Doin’ what comes naturally!”
♫ “Doin’ what comes naturally!”
Wednesday, January 29, 2025
Kindergarten Shapes
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Where Credit is Due: Chrome A*
* See other posts tagged Chrome Art in this journal.
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Niche Worlds: Motley and Perplexed
|
There is nothing more irritating to enthusiasts than when the mainstream tries to portray their niche world and gets it wrong. And The Brutalist gets an awful lot wrong. Just as Gladiator II recently vexed classicists with its inaccurate portrayal of the emperors and its anachronistic scenes of people reading the newspaper and drinking at cafes (neither of which, apparently, existed at the time), so too has director Brady Corbet riled the architecture world by playing fast and loose with his interpretation of brutalism, the Bauhaus, postwar immigration and the basic process of architecture itself.
— Oliver Wainwright in The Guardian, |
Chrysler Building Niche Worlds:
See also Alec Baldwin in "The Aviator" . . .
" 'There is a game of puzzles,' he resumed,
'which is played upon a map. One party playing
requires another to find a given word — the name
of town, river, state, or empire —
any word, in short, upon the motley and perplexed
surface of the chart.' " — Edgar Allan Poe
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“Accomplished in Steps” Continues …
Previously . . .
Today . . .
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April 1 Art
The date April 1, 2023, from the previous post, and a
Substack illustration from yesterday, January 28, suggest
some art from this journal on the former date —
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Tuesday, January 28, 2025
For Greg Egan Fans . . . Cha-ching!
Related material from April 1, 2023 . . .
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Establishing a Perimeter
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Monday, January 27, 2025
Sunday, January 26, 2025
The Diamond Theorem and the Miracle Octad Generator —
The DeepSeek Version
See http://log24.com/log25/
DeepSeek-250126-Print-option-version-of-DTandMOG.pdf .
|
Conclusion "The Diamond Theorem and the MOG exemplify how finite geometry bridges abstract algebra and combinatorics. Their relationship underscores the universality of symmetry in mathematics, from graphic designs to sporadic groups and error-correcting codes. By studying one, insights into the other — and into structures like the Leech lattice — naturally emerge." — DeepSeek R1, Jan. 26, 2025. |
That AI research report from today was suggested by
a VentureBeat article from yesterday —
For a Google Gemini Deep Research report on the same topic,
see a Log24 post from Tuesday, Jan. 21.
Comments Off on The Diamond Theorem and the Miracle Octad Generator —
The DeepSeek Version
The DeepSeek Version
Words and Things
Peter J. Cameron today presented in his weblog an excerpt
from Aldous Huxley's The Perennial Philosophy .
Another such excerpt —
"Taking words as the measure of things, instead of using
things as the measure of words, Hume imposed the discrete
and, so to say, pointilliste pattern of language upon the
continuum of actual experience — with the impossibly
paradoxical results with which we are all familiar. Most human
beings are not philosophers and care not at all for consistency
in thought or action. Thus, in some circumstances they take it
for granted that events are not 'loose and separate,' but coexist
or follow one another within the organized and organizing field
of a cosmic whole. But on other occasions, where the opposite
view is more nearly in accord with their passions or interests,
they adopt, all unconsciously, the Humian position and treat events
as though they were as independent of one another and the rest
of the world as the words by which they are symbolized. This is
generally true of all occurrences involving 'I,' 'me,' 'mine.'
Reifying the 'loose and separate' names, we regard the things as
also loose and separate — not subject to law, not involved in the
network of relationships, by which in fact they are so obviously
bound up with their physical, social and spiritual environment."
— Aldous Huxley, The Perennial Philosophy , on p. 156 of the
1947 Chatto & Windus (London) edition.
Those in search of an "organized and organizing field" might
consider the Galois field GF(64) — as embodied in the Chinese
classic I Ching … or, more recently, in the finite geometry PG(5,2) —
the natural habitat of the R. T. Curtis Miracle Octad Generator.
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Saturday, January 25, 2025
Supplement to “The Most Powerful Diagram in Mathematics”
The diagram description in the title is from a YouTube video about
the Miracle Octad Generator of R. T. Curtis.
Supplemental AI-generated reading . . .
|
Diamond Theorem and Miracle Octad Generator
An “AI Overview” Google Search response to the ___________________________________________________ In mathematics, the "diamond theorem" refers to a geometric concept related to finite projective geometry, which is used to explain the surprising symmetry properties observed in the "Miracle Octad Generator" (MOG), a tool developed by mathematician R.T. Curtis for studying the Mathieu groups and binary Golay code; essentially, the diamond theorem helps analyze the patterns within the MOG, revealing a hidden structure based on geometric principles. [1, 2, 3, 4, 5] Key points about the connection: [1, 2, 3]
Generative AI is experimental. [1] http://finitegeometry.org/sc/16/dtheorem.html [2] https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem [3] https://arxiv.org/abs/1308.1075 [4] https://en.wikipedia.org/wiki/Miracle_Octad_Generator [5] http://xenon.stanford.edu/~hwatheod/mog/mog.html |
"Generative AI is experimental." . . .
Exercise: Correct errors in the text, using the links.
A more concise presentation —
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“Diamond Eye” Art
For your consideration . . .
And from my browser today, there is . . .
#meetcute The Diamond Eye —
For the more sophisticated reader —
other posts now tagged Diamond Eye.
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Friday, January 24, 2025
Dutch Treat
A midrash for Kantor —
The time of this post, 9:01, suggests a look at the prime factors of 901:
Related entertainment: the new thriller "Prime Target."
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Thursday, January 23, 2025
The Gotham/Getty problem: “Da hats ein Eck”
This post was suggested by Dudeney's "Stonemason's Problem," which
in turn was suggested by the number "204" in the opening episode of
the new Apple TV Plus thriller "Prime Target."
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Wednesday, January 22, 2025
Corner Store Double Date
Below: a "Corner Store" location I prefer. The store itself is now long gone.
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♫ “The Holly Bears a Prickle” — Song Lyric*
" I divide mathematics into discrete and continuous
(prickles and goo, as Alan Watts put it) . . . ."
— Peter J. Cameron on 8 December 2024
* See a YouTube rendition (dated Oct. 28, 2013).
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Tuesday, January 21, 2025
Night at the Museum … of Entertainment —
Sever-ance!
On Friday, Jan. 17, at 3:51 a.m. ET . . .
Comments Off on Night at the Museum … of Entertainment —
Sever-ance!
Sever-ance!
The Cullinane Diamond Theorem
and the Miracle Octad Generator
|
The Cullinane Diamond Theorem Document created on Jan. 21. 2025, by Google’s Gemini Advanced 1.5 Pro with Deep Research, in response to the following prompt:
“Research how the Cullinane diamond theorem and The Cullinane diamond theorem and the Miracle Octad Generator (MOG) are two seemingly disparate mathematical concepts that find a surprising connection through the realm of finite projective geometry. This report delves into the relationship between these two concepts, exploring their definitions, properties, and historical context to illuminate their interconnectedness. Cullinane Diamond Theorem The Cullinane diamond theorem, developed by Steven H. Cullinane, is a mathematical concept that explores the symmetrical properties of certain geometric patterns. It focuses on a 4×4 square pattern with alternating colors arranged in a diamond shape, referred to as the "diamond figure".4 The theorem states that every permutation of the 16 tiles within this diamond figure, generated by mixing rows, columns, and quadrants, will result in a pattern with some form of ordinary or color-interchange symmetry.5 This "invariance of symmetry" is a remarkable property that highlights the inherent order within seemingly random arrangements.3 The theorem has deep roots in group theory, with the group of permutations (G) being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).6 This group has an order of 322,560 and plays a crucial role in understanding the symmetry of both the diamond-theorem figures and the square patterns of the MOG.5 A key result used in the proof of the theorem states that every 4-coloring (i.e., every map into a 4-set) can be expressed as a sum of three 2-colorings.1 Interestingly, the Cullinane diamond theorem can be extended to higher dimensions. For instance, extending the action of A to a 4x4x4 array yields a way of generating the 1.3 trillion transformations natural to the 64 hexagrams of the I Ching, an ancient Chinese divination text.1 This connection suggests potential applications of the theorem in diverse fields beyond geometry. Another interesting concept that arises from the Cullinane diamond theorem is that of "diamond rings." These rings are algebraic structures generated by the G-images of the diamond figure, and they are isomorphic to rings of matrices over GF(4).5 This algebraic formulation provides a deeper understanding of the symmetry properties explored by the theorem. Miracle Octad Generator The Miracle Octad Generator (MOG), conceived by R.T. Curtis, is a mathematical tool used to study the symmetries and properties of the Mathieu group M24, a sporadic simple group of significant importance in group theory and coding theory.7 It is a 4×6 array of combinations that can describe any point in 24-dimensional space.7 More precisely, each element in the array represents a combination of coordinates or symbols that contribute to defining a point in this 24-dimensional space. Properties The MOG preserves all the symmetries and maximal subgroups of M24, including the monad, duad, triad, octad, octern, sextet, trio, and duum groups.7 It can be used to visualize partitions of the 24 points, which is important for characterizing these maximal subgroups of M24.8 One of the key applications of the MOG lies in its ability to quickly verify codewords of the binary Golay code, a highly efficient error-correcting code used in various communication systems.7 Each element in the MOG can store a '1' or a '0', and by analyzing the counts (number of '1's) in each column and the top row, one can determine if a set of 24 coordinates forms a valid codeword.7 The MOG achieves this by projecting the 24 Golay code onto a code called the hexacode.8 The MOG is also closely related to the concept of octads and sextets. Any 5 elements of the MOG lie in a unique octad, which is a set of 8 elements.9 A sextet is a set of 6 tetrads (sets of 4 elements) where the union of any two tetrads forms an octad.9 Every tetrad lies in a unique sextet. Furthermore, the MOG is a pairing of the 35 partitions of an 8-set into two 4-sets with the 35 partitions of AG(4,2) (the affine 4-space over GF(2)) into 4 affine planes.10 This pairing preserves specific incidence properties, which refer to the relationships between points, lines, and planes in a geometric space. It's worth noting that there are two competing definitions of the MOG. Curtis originally defined it as a pairing of two 35-member sets.11 However, many sources now define it as a rectangular array based on Conway's hexacode.11 Finally, the MOG helps in understanding the octad stabilizer, a subgroup of M24 that leaves an octad invariant as a set.10 This subgroup is isomorphic to the automorphism group of the affine 4-space over GF(2), highlighting the deep connection between the MOG and finite geometry. Relationship Between the Cullinane Diamond Theorem and the Miracle Octad Generator The relationship between the Cullinane diamond theorem and the MOG emerges from their shared foundation in finite projective geometry. The 35 square patterns within the MOG have an underlying structure based on finite projective geometry.5 These patterns, composed of two-color diagonally-divided square tiles, exhibit surprising symmetry properties that can be explained using the Cullinane diamond theorem.5 A crucial link between the Cullinane diamond theorem and the MOG is the fact that the same symmetry group, of order 322,560, underlies both concepts.5 This shared symmetry group highlights a deep connection between the seemingly different patterns studied by these two concepts. The line diagrams used to visualize the Cullinane diamond theorem also play a crucial role in understanding the symmetry of the square patterns in the MOG.5 These line diagrams, based on the three partitions of the four-set of square tiles, correspond to the 35 lines in the 3-dimensional projective space over GF(2).5 In essence, the Cullinane diamond theorem provides a way to understand and visualize the symmetry properties inherent in the MOG. Furthermore, the underlying geometry of the 4×4 patterns in the Cullinane diamond theorem is closely related to the MOG, which is used in the construction of the Steiner system S(5,8,24).5 This connection extends to the Leech lattice, a highly symmetrical mathematical structure in 24 dimensions, which Walter Feit described as a "blown up version of S(5,8,24)".5 The Leech lattice is a dense sphere-packing in 24 dimensions with remarkable symmetry properties and connections to various areas of mathematics and physics. Interestingly, the Cullinane diamond theorem also sheds light on the relationship between orthogonality of Latin squares and skewness of lines in a finite projective 3-space.12 Latin squares are square arrays filled with symbols, and orthogonality between two Latin squares means that when they are superimposed, each possible pair of symbols appears exactly once. Skewness of lines in projective geometry refers to lines that do not intersect. The Cullinane diamond theorem helps establish a connection between these seemingly unrelated concepts. Another interesting connection is to Beutelspacher's model of the 15 points of PG(3,2).1 This model provides a way to visualize the points of this projective space, and it relates to the Cullinane diamond theorem and the MOG through their shared foundation in finite projective geometry. Applications The relationship between the Cullinane diamond theorem and the MOG has potential applications in various fields, including coding theory and cryptography. The MOG's ability to verify codewords of the binary Golay code is particularly relevant in coding theory, where efficient error-correcting codes are essential for reliable data transmission.7 The Cullinane diamond theorem, with its focus on symmetry and permutations, could potentially contribute to the development of new coding schemes or cryptographic algorithms. For example, the theorem's insights into the structure of finite projective spaces could be used to design codes with specific properties or to analyze the security of existing cryptographic systems. Beyond coding theory, the Cullinane diamond theorem has applications in visualizing various mathematical concepts. For instance, it provides a way to visualize Walsh functions, which are a set of orthogonal functions used in signal processing and other areas of mathematics.1 The theorem also helps visualize the finite projective plane PG(2,4), a fundamental object in finite geometry with connections to various combinatorial structures.12 Additionally, the theorem has connections to the outer automorphisms of S6, the symmetric group on 6 letters, which are transformations that preserve the group structure but are not inner automorphisms.12 Visualizations Visualizations play a crucial role in understanding the connection between the Cullinane diamond theorem and the MOG. The 35 square patterns within the original (1976) MOG, as defined by R. T. Curtis, provide a visual representation of this connection.5 These patterns, along with the line diagrams used to illustrate the Cullinane diamond theorem, offer a tangible way to grasp the interplay between these concepts.5 Here are some examples of visualizations that illustrate this connection:
Historical Connections Historically, the development of the Cullinane diamond theorem and the MOG stemmed from explorations in finite projective geometry and group theory.5 While the exact historical connections between their development remain unclear, both concepts emerged from a desire to understand the symmetries and properties of mathematical structures in finite spaces. Conclusion The Cullinane diamond theorem and the Miracle Octad Generator, while seemingly distinct, are intricately linked through the principles of finite projective geometry. The theorem's focus on symmetry invariance and the MOG's ability to capture the symmetries of the Mathieu group M24 highlight their shared foundation. This connection extends to higher-dimensional structures like the Leech lattice and has potential applications in coding theory, cryptography, and the visualization of various mathematical concepts. The relationship between these two concepts exemplifies the interconnectedness of mathematical ideas. Often, seemingly unrelated areas of mathematics converge in unexpected ways, revealing hidden symmetries and relationships. The Cullinane diamond theorem and the MOG provide a compelling example of this phenomenon, demonstrating the power of finite projective geometry in unifying and illuminating diverse mathematical structures. Works cited 1. Diamond Theory: Symmetry in Binary Spaces, accessed January 21, 2025, https://m759.tripod.com/theory/dtheory.html 2. The Diamond Theorem in Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/16/dtheorem.html 3. finitegeometry.org, accessed January 21, 2025, http://finitegeometry.org/sc/gen/dtcas.html#:~:text=Cullinane,have%20some%20sort%20of%20symmetry. 4. Speak, Memory « Log24 – Home page of m759.net, a domain used by Steven H. Cullinane for a WordPress weblog., accessed January 21, 2025, http://m759.net/wordpress/?p=112809 5. Cullinane diamond theorem – Encyclopedia of Mathematics, accessed January 21, 2025, https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem 6. What is the Cullinane diamond theorem? – Log24, accessed January 21, 2025, http://log24.com/log24/240303-You.com-Cullinane_Diamond_Theorem-research-paper.pdf 7. Miracle Octad Generator – Wikipedia, accessed January 21, 2025, https://en.wikipedia.org/wiki/Miracle_Octad_Generator 8. Mathieu groups, the Golay code and Curtis' Miracle Octad Generator, accessed January 21, 2025, https://vrs.amsi.org.au/wp-content/uploads/sites/6/2014/09/UWA-Kelly.pdf 9. MOG (Miracle Octad Generator) – Stanford, accessed January 21, 2025, http://xenon.stanford.edu/~hwatheod/mog/mog.html 10. The Miracle Octad Generator (MOG) of R T. Curtis – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/24/MOG.html 11. Competing Definitions of the Miracle Octad Generator – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/24/mogdefs.html 12. Diamond Theory: Symmetry in Binary Spaces – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/gen/dth/DiamondTheory.html 13. arxiv.org, accessed January 21, 2025, https://arxiv.org/abs/1308.1075 |
View this post as a standalone web page at
http://log24.com/log25/DTandMOG.html.
and as a PDF at
http://log24.com/log25/DTandMOG.pdf.
For a more elementary introduction to the MOG, see a YouTube video,
"The Most Powerful Diagram in Mathematics."
For a PDF of the video's metadata and comments, click here.
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and the Miracle Octad Generator
and the Miracle Octad Generator
Monday, January 20, 2025
Sunday, January 19, 2025
Saturday, January 18, 2025
The Oregon Adjective
"Exquisite" — William Kantor, University of Oregon
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Requiem for a Fixture
Pure Mathematics as a "Relaxed Field" —
Not So Relaxed —
See as well "Devil's Night Art Notes" in other news of October 2020.
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Friday, January 17, 2025
A Figure from Design Theory …
(The Visual Kind, Not the Purely Mathematical*)
In memory of a meeting of the Philomorphs at Sever Hall,
Harvard University, in 1978 . . .
Loeb died on July 19, 2002. Vide this journal on the next day.
* The purely mathematical kind —
“By far the most important structure in design theory
is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer
(Ch. 14 (pp. 693-746) of Handbook of Combinatorics,
Vol. I, MIT Press, 1995, edited by Ronald L. Graham,
Martin Grötschel, and László Lovász, Section 16 (p. 716))
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(The Visual Kind, Not the Purely Mathematical*)
(The Visual Kind, Not the Purely Mathematical*)
Harlan Kane Goes for the Gold:
The Case of the Baffled Prime
See also Gauss in this journal.
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The Case of the Baffled Prime
The Case of the Baffled Prime
Thursday, January 16, 2025
Faustus Revisited
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Field Dreams
Related reading:
Michael Harris on the Spielraum and the "relaxed field" —
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AI Overview for the Twilight Zone . . . I Ching Space
Ron Shaw in "Configurations of planes in PG(5,2)" . . .
"There are some rather weird things happening here."
Related entertainment — The Yarrow Stalker .
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Wednesday, January 15, 2025
Making a Textured Rotatable Solomon’s Cube
Click image to see program at WebSim. To enlarge image click here.
The data file for the above cube is at
log24.com/log/pix25/Websim-SolCube-data.json.
(Right-click the above json link and choose "Save Link As.")
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Tuesday, January 14, 2025
Proofs
A phrase by Aitchison at Hiroshima . . .
"The proof of the above is a relabelling of the Klein quartic . . . ."
Related art — A relabelling of the Klein quadric by Curtis bricks:
Update of 12:26 PM EST Wednesday, January 15, 2025 —
Here is a large (17.5 MB) PDF file containing all posts touching upon
the concept underlying the above illustration — the Klein correspondence.
(A PDF reader such as Foxit is recommended for such large files.)
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For the Dies Natalis of Gödel
World Logic Day: 14 January
"On 26 November 2019, the 40th General Conference
of UNESCO proclaimed 14 January to be World Logic Day,
a global day of supporting the development of logic through
teaching and research, as well as to [sic] public dissemination of
the discipline. The date chosen to celebrate World Logic Day,
14 January, corresponds to the date of death of Kurt Gödel . . . ."
— https://wld.cipsh.international
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Monday, January 13, 2025
From the Dies Natalis of
a Tolkien Mapmaker
This post was suggested by a New York Times
"Overlooked No More" obituary today.
See also Durer Knight in this journal.
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a Tolkien Mapmaker
a Tolkien Mapmaker
Alphabet vs. Numbers: 23 and Me
— R. T. Curtis, "A New Combinatorial Approach to M 24 ,"
Mathematical Proceedings of the Cambridge Philosophical Society (1976),
Related material —
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For Fans of Hallucinatory Fiction:
Angel Eyes
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Angel Eyes
Angel Eyes
Moon News
Related art —
From part two of the recent film triptych "Kinds of Kindness" . . .
Window with Couch and Cat —
From last night's TV —
Schrödinger's Box
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Sunday, January 12, 2025
Art Notes: Moody Blue Meets Deep Blue
|
Oh, moody blue Tell me, am I gettin' through? I keep hangin' on Tryin' to learn your song But I never do Oh, moody blue Tell me who I'm talking to You're like night and day And it's hard to say Which one is you |
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Sontag Variations: I Sing the Body Electromagnetic
See as well the Diamond Cube at Pinterest —
See also news from the day I turned 11 …
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Sontag Variations: A Canticle for Leibovitz
|
Oh, moody blue Tell me, am I gettin' through? I keep hangin' on Tryin' to learn your song But I never do Oh, moody blue Tell me who I'm talking to You're like night and day And it's hard to say Which one is you |
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Saturday, January 11, 2025
Synchronology Meets Sin Chronology
For the former, see
A synchronology check of the date June 7, 2023, in this journal yields
an account of a sin allegedly committed by author Susan Sontag . . .
A biographer accuses her of falsifying, by two years, the date of a
meeting with Thomas Mann in his Pacific Palisades home. There may,
of course, have been more than one such meeting — once in 1947,
as described much later in detail by Sontag in a New Yorker piece, and
again in 1949, as noted privately at the time by Sontag in her journal
and by Mann himself in his diary. But the thoroughness of the biographer's
research makes this seem unlikely.
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Octads within a PG(5,2) Addition Table
-extended.bmp)
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Friday, January 10, 2025
Annals of Academic Prose: The Gombrich Cartoon
In memory of related remarks in a book I think of as
the Black Hole of Seattle —
Friday, January 8, 2016
|
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Thursday, January 9, 2025
For the Church of Synchronology
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Grok 2 on the MOG and the Klein Correspondence
Related illustration —
|
— Transcription —
Grok 2 on Klein correspondence and MOG — 9 Jan. 2025 Prompt: How is the Klein correspondence related to the Miracle Octad Generator? Grok 2 response (with citations and links removed): The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics. Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between points and lines in a projective space. Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2). Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system. Relation Between Klein Correspondence and MOG: Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to 4 parallel affine planes within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework. Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries. Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems. |
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Wednesday, January 8, 2025
Show Tune from the Garden of Good and Evil
From posts tagged EGBDF —
♫ “And we may see the meadow in December,
icy white* and crystalline” — Johnny Mercer
“The devil likes metamorphoses.“
Related geography . . .
* See a related (if only verbally) obituary.
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For the Seattle Mathematics Meeting:
The Song of the Ancient Mariner
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The Song of the Ancient Mariner
The Song of the Ancient Mariner
Tuesday, January 7, 2025
Monday, January 6, 2025
Tea Party
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“Deux Mots” *
* https://www.google.com/search?q=
Boufflers+%22il+faut+dire+en+deux+mots%22
Bluntly . . .
In this case, the "deux mots" are "subscribe" and "tinker,"
the latter written beneath the former.
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Sunday, January 5, 2025
Precursor to Dürer’s 1514 Square
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Saturday, January 4, 2025
For a Contemporary Faustus
A contemporary minimalist composer whose work resembles that of
Thomas Mann's Doctor Faustus reportedly died at 85 Tuesday
in Paris on New Year's Eve (December 31, 2024). The phrase
"mathematical clarity" in his obituary in today's New York Times
suggests a synchronology check —
Compare and contrast.
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Graphics for Avatars
(New “Love Me” Trailer)
"The upcoming film Love Me has an intriguing concept.
In a post-apocalyptic world in which humans have gone extinct,
a buoy falls in love with a satellite. To be together, they review
historical accounts of humanity and create avatars of themselves,
played by Steven Yeun and Kristen Stewart."
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(New “Love Me” Trailer)
(New “Love Me” Trailer)