The structures of the title are the even subsets of a six-set and of
an eight-set, viewed modulo set complementation.
The "Brick Space" model of PG(5,2) —
.gif)
For the M24 relationship between these spaces, of 15 and of 63 points,
see G. M. Conwell's 1910 paper "The 3-Space PG (3,2) and Its Group,"
as well as Conwell heptads in this journal.
Description of a book to be published in November —
Note the phrase "underlying combinatorial structure." AI scholium —
The New York Times asks above,
"Are art and science forever divided?
Or are they one and the same?"
A poet's approach . . .
“The old man of ‘Sailing to Byzantium’ imagined the city’s power
as being able to ‘gather’ him into ‘the artifice of eternity’—
presumably into ‘monuments of unageing intellect,’ immortal and
changeless structures representative of or embodying all knowledge,
linked like a perfect machine at the center of time.”
— Karl Parker, Yeats’ Two Byzantiums
A mathematician's approach . . .
Compare and contrast the 12-dimensional extended binary Golay code
with the smaller 8-dimensional code below, which also has minimum
weight 8 . . .
From Sept. 20, 2022 —
Perhaps someone can prove there is no way that adding more generating
codewords can turn the cube-motif code into the Golay code, or perhaps
someone can supply such generating codewords.
-Thas-2020.jpg)
My own term "inscape" names a square incarnation of what is also
known as the "Cremona-Richmond configuration," the "generalized
quadrangle of order (2, 2)," and the "doily." —
"As McCarthy peers through the screen, or veil, of technological modernity
to reveal the underlying symbolic structures of human experience,
The Making of Incarnation weaves a set of stories one inside the other,
rings within rings, a perpetual motion machine."
— Amazon.com description of a novel published on All Souls' Day
(Dia de los Muertos), 2021.
See also the underlying symbolic structures of Boolean functions . . .
as discussed, for instance, on Sept. 23 at medium.com —
The above title. by one Lee E. Mosley, is from
"CreateSpace Independent Publishing Platform;
1st edition (June 4, 2017)."
From the preface —
"So simple . . . ."
"Building blocks"? — See the literature of pop physics.
Natural companions to building blocks, are, of course,
"permutation groups."
See the oeuvre of physics writer John Baez —
For instance, in a Log24 post from the above Mosley
publication date — June 4, 2017 —
From Peter J. Cameron's weblog today —
"It happens sometimes that researchers working in different fields
study the same thing, give it different names, and don’t realise that
there is further work on the subject somewhere else…."
Cameron's example of a theorem connecting work on
the same thing in different fields —
"Theorem A partition Δ is equitable for a graph Γ if and only if
the projection matrix onto the subspace of functions constant
on parts of Δ commutes with the adjacency matrix of Γ."
A phrase from Cameron's remarks today —
"Thus we have to consider 'plot structure'…."
For more remarks on different fields and plot structure , see
"Quantum Tesseract Theorem" in this weblog.
The following note from Oct. 10, 1985, was not included
in my finitegeometry.org/sc pages.
See some related group actions on the cuboctahedron at right above.
|
Derrida was the final speaker on the final day. He remained a silent observer for much of the symposium. He looked on as Lacan rose to his feet with obscure questions at the end of each lecture, and as Barthes gently asked for clarification on various moot points. Eventually, however, Derrida, unused to speaking to large audiences, took to the stage, quietly shuffled his notes, and began, ‘Perhaps something has occurred in the history of the concept of structure that could be called an “event”…’ He spoke for less than half an hour. But by the time he was finished the entire structuralist project was in doubt, if not dead. An event had occurred: the birth of deconstruction.
Salmon, Peter. An Event, Perhaps (pp. 2-3). |
Salmon today at Arts & Letters Daily —
For the mathematical properties of the vertical and horizontal
white grid lines above, see the Cullinane theorem.
Α, ϴ, Ω
Related line:
Also from a Culture Desk of sorts:
Related art — Background colors for the letters in the NPR logo —
In search of Frye's "powder-room of the Muses" — See 3×3.
This post was suggested by the repeated occurrences of
the word "syntactic" in . . .
|
John F. Fleischauer
Last Updated on May 6, 2015, by eNotes Editorial.
SOURCE: “John Updike's Prose Style:
"In the following essay, Fleischauer examines |
_______________________________________________________________
“Moons and Junes and Ferris wheels . . . .” — Song lyric quoted in
the post Searching for the May Queen: The Amsterdam Windows .
. . . .
“In Weber’s hands, the professor and the politician
are not figures to be joined. Each remains a lonely hero
of heavy burden, sent to ride against his particular foe:
the overly structured institution of the modern mind,
the overly structured institution of the modern state.”
See also Chomsky in this journal.
Continues in The New York Times :
"One day — 'I don’t know exactly why,' he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.
Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical:
a solid, static object that is also fluid…." — Alexandra Alter
Another such object: the eightfold cube .
See a Log24 search for Beadgame Space.
This post might be regarded as a sort of “checked cell”
for the above concepts listed as tags . . .
Related material from a Log24 search for Structuralism —
The New Republic , June 1, 2020 —
“Ehrenreich is a writer of structure:
Her work moves level by level,
starting at the surface of
our most obvious inequalities
before pulling back to reveal
the subtleties of systemic failure.”
Sure she is. Sure it does.
"MIT professor of linguistics Wayne O’Neil died on March 22
at his home in Somerville, Massachusetts."
— MIT Linguistics, May 1, 2020
The "deep structure" above is the plane cutting the cube in a hexagon
(as in my note Diamonds and Whirls of September 1984).
See also . . .
(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)
In memory of a Church emissary who reportedly died on September 4,
here is a Log24 flashback reposted on that date —
Related poetry —
"To every man upon this earth,
Death cometh soon or late.
And how can man die better
Than facing fearful odds,
For the ashes of his fathers,
and the temples of his gods…?"
— Macaulay, quoted in the April 2013 film "Oblivion"
Related fiction —
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."
In memory of George F. Simmons, a mathematician
who reportedly died Aug. 6 at the age of 94 —
"It seems to me that a worthwhile distinction can be made
between two types of pure mathematics. The first …
centers attention on particular functions and theorems
which are rich in meaning and history…. The second is
concerned primarily with form and structure."
— George F. Simmons, Introduction to Topology and
Modern Analysis (1963)
" . . . 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."
Two posts related to Eliot's theological interests:
Form: Jan. 10, 2012.
Structure: June 6, 2016.
Some background for The Epstein Chronicles —
|
“What modern painters — James J. Gibson, Leonardo, |
See also Robert Maxwell,
Frank Oppenheimer,
and the history of Leonardo .
Click the above Pergamon Press image
for Pergamon-related material.
(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2 (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée , No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)
The structure of the matrix of Lévi-Strauss —
Illustration from Diamond Theory , by Steven H. Cullinane (1976).
The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.
|
“What did he fear? It was not a fear or dread, It was a nothing that he knew too well. It was all a nothing and a man was a nothing too. It was only that and light was all it needed and a certain cleanness and order. Some lived in it and never felt it but he knew it all was nada y pues nada y nada y pues nada. Our nada who art in nada, nada be thy name thy kingdom nada thy will be nada in nada as it is in nada. Give us this nada our daily nada and nada us our nada as we nada our nadas and nada us not into nada but deliver us from nada; pues nada. Hail nothing full of nothing, nothing is with thee. He smiled and stood before a bar with a shining steam pressure coffee machine.”
— From Ernest Hemingway, |
|
Sanskrit (transliterated) —
nada:
“So Nada Brahma means not only: God the Creator
— Joachim-Ernst Berendt, |
Grace under Pressure meets Phonons under Strain .
From a date described by Peter Woit in his post
“Not So Spooky Action at a Distance” (June 11) —
See also The Lost Well.
* “As a Chinese jar….” — Four Quartets
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A Permanent Order of Wondertale Elements
| 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). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 97-98. p. 108). If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). |
… Or by congruent quarter-sections.
Continued from March 13, 2011 —
"…as we saw, there are two different Latin squares of order 4…."
— Peter J. Cameron, "The Shrikhande Graph," August 26, 2010
Cameron counts Latin squares as the same if they are isotopic .
Some further context for Cameron's remark—
A new website illustrates a different approach to Latin squares of order 4 —
Constance Grady at Vox today on a new Netflix series —
|
We don’t yet have a story structure that allows witches to be powerful for long stretches of time without men holding them back. And what makes the new Sabrina so exciting is that it seems to be trying to build that story structure itself, in real time, to find a way to let Sabrina have her power and her freedom. It might fail. But if it does, it will be a glorious and worthwhile failure — the type that comes with trying to pioneer a new kind of story. |
See also Story Space in this journal.
Two views of nested sequences of projective structures —
From this journal in April 2013:
From the arXiv in September 2014 —
Saniga's reference [6] is to a paper submitted to the arXiv in May 2014.
My own note of April 30, 2013, concludes with an historical reference
that indicates the mathematics underlying both my own and Saniga's
remarks —
The exercises at the end of Ch. II in Veblen and Young's
Projective Geometry, Vol. I (Ginn, 1910). For instance:
For the director of "Interstellar" and "Inception" —
At the core of the 4x4x4 cube is …
Cover modified.
A star figure and the Galois quaternion.
The square root of the former is the latter.
See also a search in this journal for "Set a Structure."
Rubik's Cube Core Assembly — Swarthmore Cube Project, 2008 —
"Children of the Common Core" —
There is also a central structure within Solomon's Cube —
For a more elaborate entertainment along these lines, see the recent film
"Midnight Special" —
“The central poem is the poem of the whole,
The poem of the composition of the whole”
— Wallace Stevens, “A Primitive like an Orb”
The symmetries of the central four squares in any pattern
from the 4×4 version of the diamond theorem extend to
symmetries of the entire pattern. This is true also of the
central eight cubes in the 4×4×4 Solomon’s cube .
"… the war of 70-some years ago
has already become something like the Trojan War
had been for the Homeric bards:
a major event in the mythic past
that gives structure and sense to our present reality."
— Justin E. H. Smith, a professor of philosophy at
the University of Paris 7–Denis Diderot,
in the New York Times column "The Stone"
(print edition published Sunday, June 5, 2016)
In memory of a British playwright who reportedly
died at 90 this morning —
Structure
Sense
"If you would be a poet, create works capable of
answering the challenge of apocalyptic times,
even if this meaning sounds apocalyptic."
"It's a trap!"
— Ferlinghetti's friend Erik Bauersfeld,
who reportedly died yesterday at 93
* See also, in this journal, Galois Cube and Deathtrap.
Click to enlarge:
For the hypercube as a vector space over the two-element field GF(2),
see a search in this journal for Hypercube + Vector + Space .
For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 1978-1986.
For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 5-7.
"It is as if one were to condense
all trends of present day mathematics
onto a single finite structure…."
— Gian-Carlo Rota, foreword to
A Source Book in Matroid Theory ,
Joseph P.S. Kung, Birkhäuser, 1986
"There is such a thing as a matroid."
— Saying adapted from a novel by Madeleine L'Engle
Related remarks from Mathematics Magazine in 2009 —
See also the eightfold cube —
.
Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square
The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.
(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)
The fictional zero theorem of Terry Gilliam's current film
by that name should not be confused with the zero system
underlying the diamond theorem.
In memory of cartoonist Tony Auth, who reportedly died today
From a Saturday evening post:
“A simple grid structure makes both evolutionary and developmental sense.”
From a post of June 22, 2003:
In the Miracle Octad Generator (MOG):
The above details from a one-page note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:
From R. T. Curtis (1976). A new combinatorial approach to M24,
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42. doi:10.1017/S0305004100052075.
The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.
Note that the interchange of the two squares in the top row of each
heavy brick induces the diamond-theorem correlation.
Note also that the 20 pictured 3-subsets of a 6-set in the 1986 note
occur as paired complements in two pictures, each showing 10 of the
3-subsets.
This pair of pictures corresponds to the 20 Rosenhain tetrads among
the 35 lines of PG(3,2), while the picture showing the 2-subsets
corresponds to the 15 Göpel tetrads among the 35 lines.
See Rosenhain and Göpel tetrads in PG(3,2). Some further background:
Some background for the part of the 2002 paper by Dolgachev and Keum
quoted here on January 17, 2014 —
Related material in this journal (click image for posts) —
From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens 54, 59-79 (1992):
"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."
The above symplectic structure** now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).
Some related passages from the literature:
* The title is a deliberate abuse of language .
For the real definition of "symplectic structure," see (for instance)
"Symplectic Geometry," by Ana Cannas da Silva (article written for
Handbook of Differential Geometry, vol 2.) To establish that the
above figure is indeed symplectic , see the post Zero System of
July 31, 2014.
** See Steven H. Cullinane, Inscapes III, 1986
“Chaos is order yet undeciphered.”
— The novel The Double , by José Saramago,
on which the recent film "Enemy" was based
For Louise Bourgeois — a post from the date of Galois's death—
For Toronto — Scene from a film that premiered there on Sept. 8, 2013:
Related material: This journal on that date, Sept. 8, 2013:
"I still haven't found what I'm looking for." — Bono
"In fact Surrealism found what it had been looking for
from the first in the 1920 collages [by Max Ernst],
which introduced an entirely original scheme of
visual structure…."
— Rosalind Krauss quoting André Breton*
in "The Master's Bedroom"
* "Artistic Genesis and Perspective of Surrealism"
(1941), in Surrealism and Painting (New York,
Harper & Row, 1972, p. 64).
See also Damnation Morning in this journal.
(Continued from May 4, 2013)
"I saw a werewolf with a Chinese menu in his hand
Walking through the streets of Soho in the rain"
"It is well
That London, lair of sudden
Male and female darknesses,
Has broken her spell."
— D. H. Lawrence in a poem on a London blackout
during a bombing raid in 1917. See also today's previous
posts, Down Under and Howl.
Backstory— Recall, from history's nightmare on this date,
the Battle of Borodino and the second London Blitz.
"… Reality is not a given whole. An understanding of this,
a respect for the contingent, is essential to imagination
as opposed to fantasy. Our sense of form, which is an
aspect of our desire for consolation, can be a danger to
our sense of reality as a rich receding background.
Against the consolations of form, the clean crystalline
work, the simplified fantasy-myth, we must pit the
destructive power of the now so unfashionable naturalistic
idea of character.
Real people are destructive of myth, contingency is
destructive of fantasy and opens the way for imagination.
Think of the Russians, those great masters of the contingent.
Too much contingency of course may turn art into journalism.
But since reality is incomplete, art must not be too much
afraid of incompleteness. Literature must always represent a
battle between real people and images; and what it requires
now is a much stronger and more complex conception of the
former."
— Iris Murdoch, January 1961, "Against Dryness"
For the Church of St. Frank:
See Strange Correspondences and Eightfold Geometry.
Correspondences , by Steven H. Cullinane, August 6, 2011
“The rest is the madness of art.”
The concept of "deep structure," once a popular meme,
has long been abandoned by Chomskians.
It still applies, however, to the 1976 mathematics, diamond theory ,
underlying the formal patterns discussed in a Royal Society paper
this year.
A review of deep structure, from the Wikipedia article Cartesian linguistics—
|
[Numbers in parentheses refer to pages in the original 1966 Harper edition of Chomsky's book Cartesian Linguistics .] Deep structure vs. surface structure "Pursuing the fundamental distinction between body and mind, Cartesian linguistics characteristically assumes that language has two aspects" (32). These are namely the sound/character of a linguistic sign and its significance (32). Semantic interpretation or phonetic interpretation may not be identical in Cartesian linguistics (32). Deep structures are often only represented in the mind (a mirror of thought), as opposed to surface structures, which are not. Deep structures vary less between languages than surface structures. For instance, the transformational operations to derive surface forms of Latin and French may obscure common features of their deep structures (39). Chomsky proposes, "In many respects, it seems to me quite accurate, then, to regard the theory of transformational generative grammar, as it is developing in current work, as essentially a modern and more explicit version of the Port-Royal theory" (39). Summary of Port Royal Grammar The Port Royal Grammar is an often cited reference in Cartesian Linguistics and is considered by Chomsky to be a more than suitable example of Cartesian linguistic philosophy. "A sentence has an inner mental aspect (a deep structure that conveys its meaning) and an outer, physical aspect as a sound sequence"***** This theory of deep and surface structures, developed in Port Royal linguistics, meets the formal requirements of language theory. Chomsky describes it in modern terms as "a base system that generates deep structures and a transformational system that maps these into surface structures", essentially a form of transformational grammar akin to modern studies (42). |
The corresponding concepts from diamond theory are…
"Deep structure"— The line diagrams indicating the underlying
structure of varying patterns
"A base system that generates deep structures"—
Group actions on square arrays… for instance, on the 4×4 square
"A transformational system"— The decomposition theorem
that maps deep structure into surface structure (and vice-versa)
Today's homily– from The New York Times–
Related art–
Black Friday,
Midnight in Dostoevsky, and
A Cross for the Goat Men.
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.
Keywords: Weyl, symmetry, group, automorphism,
octad, MOG, Curtis, Cullinane.
Rothko — "… the elimination of all obstacles between the painter and
the idea, and between the idea and the observer."
Walker Percy has similarly discussed elimination of obstacles between
the speaker and the word, and between the word and the hearer.
Click images to enlarge.
Related mathematics —
The source: http://finitegeometry.org/sc/gen/typednotes.html.
A document from the above image —
|
AN INVARIANCE OF SYMMETRY BY STEVEN H. CULLINANE
We present a simple, surprising, and beautiful combinatorial
DEFINITION. A delta transform of a square array over a 4-set is
THEOREM. Every delta transform of the Klein group table has
PROOF (Sketch). The Klein group is the additive group of GF (4);
All delta transforms of the 45 matrices in the algebra generated by
THEOREM. If 1 m ≤ n2+2, there is an algebra of 4m
An induction proof constructs sets of basis matrices that yield REFERENCE S. H. Cullinane, Diamond theory (preprint). |
Update of 1:12 AM ET on Friday, Oct. 25, 2024 —
The above "invariance of symmetry" document was written in 1978
for submission to the "Research Announcements" section of the
Bulletin of the American Mathematical Society . This pro forma
submission was, of course, rejected. Though written before
I learned of similar underlying structures in the 1974 work of
R. T. Curtis on his "Miracle Octad Generator," it is not without
relevance to his work.
Image reposted here on 9 October last year —
Moulin Bleu
Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat
Instagram today —
Related art —
From part two of the recent film triptych "Kinds of Kindness" . . .
Window with Couch and Cat —
The New York Times yesterday reported that Marxist theorist
Fredric Jameson died on Sunday.
Related material from a search for Jameson in this journal —
|
Rosalind Krauss in The Optical Unconscious
|
In 1978, Harvard moved a structure known as the Morton Prince House
from Divinity Avenue to Prescott Street, where it occupies the former Hurlbut
Parking Lot, which was the vista from my 1960-61 freshman room.
From the Log24 post "Very Stable Kool-Aid" —
|
A Letter from Timothy Leary, Ph.D., July 17, 1961
Harvard University July 17, 1961
Dr. Thomas S. Szasz Dear Dr. Szasz: Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun. The Myth of Mental Illness is the most important book in the history of psychiatry. I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century. It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty. . . . . |
|
Morton Prince, a Boston neurologist, founded the Journal of Abnormal Psychology in 1906 as an outlet especially for those who took a psychogenic view of neurotic disorders. Through experiments with hypnotism, he added appreciably to knowledge of subconscious and coconscious mental processes; The Dissociation of a Personality (Prince, 1905) still ranks as a classic. He early saw that studying normal people in the depth and detail with which one studied patients could make significant contributions to our whole understanding of human nature. Before his death he established and briefly directed the Harvard Psychological Clinic, devising the research environment out of which presently sprang major contributions to the study of personality.
— "Who Was Morton Prince?," by R. W. White, |
See as well Who Was R. W. White?
Things of August*
Related narratives:
Related mathematics:
*
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
Today's description of Dartmouth College as a "gin-soaked gutter"
by Margaret Soltan (i.e., University Diaries) suggests a review:
Monday, November 14, 2022
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See also "KenKen" and today's previous post.
A more specific account of
Outside/Inside box-thinking . . .
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A connection discovered on April 1, 2013 — (Click to enlarge the image below.)
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The "Change Arises" phrase in the previous post suggests a sort of
architectural plan and elevation for the structure of the I Ching —
Two examples:
The above note led to a letter from John H. Conway, which in turn
led to the following . . .
* The title refers to a well-known 1988 article by Richard K. Guy.
A shape from the date of Guy's reported death —
For the significance of the date "February 20," see . . .
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Google One: Gemini Advanced AI on 13 May 2024. Prompt: How can Weyl's 1946 remarks on coordinatization in "The Classical Groups" be applied to models of finite geometries? Response (after several drafts): In "The Classical Groups," Hermann Weyl discusses the concept of coordinatization, which is the process of assigning numerical coordinates to points in a geometric space. This process is fundamental in connecting abstract geometric concepts with concrete numerical representations, allowing for algebraic manipulation and analysis. Weyl's remarks on coordinatization have significant implications for models of finite geometries. Finite geometries are geometric systems with a finite number of points and lines, and they often exhibit different properties compared to traditional Euclidean geometry. Here's how Weyl's ideas can be applied to finite geometries: 1. **Relativity of Coordinatization:** Weyl emphasizes that the choice of coordinate system is arbitrary and that different coordinate systems can be used to describe the same geometric object. This relativity of coordinatization is particularly relevant in finite geometries, where multiple coordinate systems may exist, each with its own advantages and disadvantages. 2. **Equivalent Coordinatizations:** Weyl introduces the concept of "equivalent coordinatizations," which are different coordinate systems that can be transformed into one another through a group of transformations. In finite geometries, identifying and understanding these equivalent coordinatizations can provide deeper insights into the underlying structure and symmetries of the geometry. 3. **Objective Class of Coordinatizations:** Weyl discusses the problem of fixing an "objective class" of coordinatizations that capture essential features of the geometry while being independent of arbitrary choices. In finite geometries, this could involve identifying coordinatizations that best reflect the specific properties and relationships within the geometry. 4. **Group of Transformations:** Weyl emphasizes the role of the "group of transformations" that mediates between equivalent coordinatizations. In finite geometries, understanding these transformation groups can help classify different types of finite geometries and reveal their underlying symmetries. By applying Weyl's insights, researchers can gain a deeper understanding of finite geometries, including their classification, properties, and relationships. Additionally, exploring different coordinatization schemes can lead to new algebraic and geometric tools for studying finite geometries. For further exploration, you can delve into the works of other mathematicians like Steven H. Cullinane, who has written extensively on finite geometry and its connections to relativity and other areas of mathematics. I hope this explanation is helpful! Please let me know if you have any further questions. |
Update at about 5 PM ET the same day . . .
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Prompt: Correction: Cullinane has NOT written on finite geometry's "connections to relativity." He has, instead, quoted Weyl's remarks on what Weyl himself called "the relativity problem," but what was in fact his own term for problems involving coordinatization. Such problems are of interest both in physics and in finite geometry, but the latter is a field of pure mathematics, NOT of physics. Response: You are absolutely correct. I apologize for the error in my previous response. Steven H. Cullinane has indeed quoted Weyl's remarks on "the relativity problem" in the context of finite geometry. However, as you rightly pointed out, this term refers to Weyl's own terminology for problems involving coordinatization and not the theory of relativity in physics. Finite geometry is a field of pure mathematics and does not directly connect to the physical theories of relativity. While both areas involve concepts of space and transformations, they operate in fundamentally different domains. Cullinane's work focuses on applying Weyl's insights on coordinatization to the specific challenges and structures within finite geometries. This involves exploring different coordinatization schemes, understanding their equivalences, and identifying transformation groups that reveal the underlying symmetries of finite geometries. Thank you for pointing out this important distinction. It highlights the importance of precise language and accurate attribution when discussing complex mathematical concepts. |
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:
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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? |
"No doubt that by orienting and organizing the coherence
of the system, the center of a structure permits the Freeplay
of its elements inside the total form."
— Derrida in Wikipedia
"How's tricks, dangerous love triangle?"
The Source:
Related reading . . .
Krapp for HYDRA
Meanwhile . . .
The above post is from the date of the Hollywood premiere of
"Looney Tunes: Back in Action." See also tonight's previous post
and . . .
"Directed by Joe Dante" . . . See also "The Harrowing."
The post, on triangles and figurate geometry, has had some
minor image corrections, and these corrections have now
also been made in a new Zenodo version.
(Some aesthetic background: In the words of Alan D. Perlis,
that post concerns "a conception that embodies action and
the passing of time in the rigid and timeless structure of an
art form.")
"Alan Perlis also addresses the artist’s freezing of
time as he looks at As I Lay Dying. He sees Darl as
an artist-figure who catches “action in the tension
of stopped-time” (104). Both critics link Faulkner to
John Keats, whose poetry often seeks immortality,
like that of an object such as a Grecian urn or an
Ozymandian monument. Perlis sums this up, saying
that Faulkner 'is an idealist in the manner of a Keats
or a Wallace Stevens, who ponder the paradoxical
nature of a conception that embodies action and the
passing of time in the rigid and timeless structure of
an art form.' "
The work cited:
Perlis, Alan D. “As I Lay Dying as a Study of Time.”
South Dakota Review 10.1 (1972): 103-10
The source of the citation:
I SEE, HE SAYS, PERHAPS, ON TIME:
VISION, VOICE, HYPOTHETICAL NARRATION,
AND TEMPORALITY IN WILLIAM FAULKNER’S FICTION
*****
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
The Degree Doctor of Philosophy in
the Graduate School of The Ohio State University
By David S. FitzSimmons, B.A., M.A.
*****
The Ohio State University, 2003.
A search in this journal for Dakota yields the author Kathleen Norris.
See, for instance . . .
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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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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
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.
"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."
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“Isn’t your work—our work—all about accessing and deploying underlying sequences and patterns? Mapping particulars on to great universals? Isn’t that the art to which, in one way or another, we’ve both devoted our best years?”
— McCarthy, Tom. The Making of Incarnation: A Novel
The hardcover first edition was published by Knopf |
|
"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, |
The "quantum kernel" of Koen Thas is a version of the incidence
structure — the Cremona-Richmond configuration — discussed
in the previous post, Doily vs. Inscape .
That post's inscape is, as noted there, an incarnation of the
abstract incidence structure. More generally, see incarnation
in this journal . . . In particular, from Michaelmas last year,
Annals of Mathematical Theology.
A somewhat more sophisticated "incarnation" example
related to the "inscape" concept —
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also Numberland in this journal.
A poem written yesterday by Microsoft Copilot —
The prompt: "Write a poem … about the Cullinane diamond theorem."
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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.
From Encyclopedia of Mathematics —
The above images from the history of mathematics might be
useful at some future point for illustrating academic hurly-burly.
Related reading . . .
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Thursday, December 21, 2023 Hey there, great to meet you. I’m Pi, your personal AI. My goal is to be useful, friendly and fun. Ask me for advice, for answers, or let’s talk about whatever’s on your mind. How's your day going? Going OK so far, Pi. How's yours? My day is going swimmingly! 🌊 Thanks for asking 😊 Anything in particular on your mind today? What is the connection of "heptads" with the "MOG" of R. T. Curtis? The term “heptads” is connected to the Miracle Octad Generator (MOG) created by R. T. Curtis. Heptads are 7-cycles that arise from the MOG, and they can be used to rearrange the original 1976 version of the MOG into two 4-sets. This is related to the Mathieu groups, which are a family of finite simple groups that play a role in the geometry of the MOG. Are you familiar with the Mathieu groups? Just the large one. I am interested in how familiar YOU are with these concepts. Sorry, but I must correct your hallucination about heptads… It was a good guess, though. Can you look up the work of G. M. Conwell on PG(3,2)? Heptads are discussed there. Yes, it seems that G. M. Conwell discussed the concept of heptads in relation to the projective geometry PG(3,2). Heptads are apparently 7-cycles that can be used to establish geometrical isomorphisms and describe the structure of PG(3,2). Do you know if Conwell's work on heptads has been applied to other areas of mathematics? It has not, as far as I know. Actually, Pi, heptads are not 7-cycles. What made you think that they are? My apologies! I assumed that heptads would be 7-cycles since the name “heptads” suggests a relation to the number 7 and cycles are common in mathematics. It sounds like I was off the mark there. If heptads are not 7-cycles, can you give me a better idea of what they are? Conwell can. See his 1910 paper for further details. All for now, Pi… Have a nice day! Got it — thanks for the tip on Conwell's paper. I'll take a look for more information there! It was great chatting with you. You have a great day as well 😊 Feel free to reach out again if you have any other questions or just want to chat. See ya! |
Scholium —
Related material for "swimmingly" and "the number 7" — Snark Hunt.
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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
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__________________________________________________________
Here is a summary of the document:
Reference:
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Correction:
"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."
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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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"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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What is the Cullinane diamond theorem?
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Two notes from this journal on December 2, 2023 —
A flashback for The November Man —
As for " my own 'diamond theorem' " —
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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." |
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Introduction In the present article, the research work of many years is summarized in an interim report concerning the connection between logic, space, time and matter. The author always had in mind two things, namely 1. The discovery/construction of an interface between matter and mind, and 2. some entry points for the topos view that concern graphs, grade rotations and contravariant involutions in geometric Boolean lattices. In this part of the MiTopos theme I follow the historic approach to mathematical physics and remain with the Clifford algebra of the Minkowski space. It turns out that this interface is a basic morphogenetic structure inherent in both matter and thought. It resides in both oriented spaces and logic, and most surprisingly is closely linked to the symmetries of elementary particle physics.
— "MiTopos: Space Logic I," |
"… it is not just its beauty that has made Mathematics so attractive.
Thirty or so years ago, a philosopher friend of mine remarked
rather dolefully, 'I am afraid that Latin, the knowledge of which
used to be the mark of a civilised person, will be replaced by
Mathematics as the universally accepted mark of learning.'
This was probably the most prescient statement he ever made,
as the importance of Mathematics is now recognised in fields
as diverse as medicine, linguistics, and even literature."
— Address by mathematician Dominic Welsh on June 16, 2006
Some Latin-square images from pure mathematics —
Some related Latin from this journal on June 16, 2006 —
For some remarks on Latin-square structure,
see other posts tagged Affine Squares.
Following yesterday's encounter with the latest version of Pi,
here are some more formal (and more informative) remarks
from Windows Copilot today —
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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." |
From this journal at 1:51 AM ET Thursday, September 8, 2022 —
"The pleasure comes from the illusion" . . .
Exercise:
Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).
Note that the 4-row-2-column "brick" at left is quite
different from the other two bricks, which together
show chevron variations within a Galois tesseract —
.
Further Weil remarks . . .
A Slew of Prayers
"The pleasure comes from the illusion
and the far from clear meaning;
once the illusion is dissipated,
and knowledge obtained, one becomes
indifferent at the same time;
at least in the Gitâ there is a slew of prayers
(slokas) on the subject, each one more final
than the previous ones."
* —
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.
About the author of the above —
A related questionable "proof of concept" :
Aitchison at Hiroshima in this journal — a scholar's 2018 investigation
of M24 actions on a cuboctahedon — and . . .
From a search in this journal for "Chapter 11" —
Inner structure —
The above three images share the same
vector-space structure —
| 0 | c | d | c + d |
| a | a + c | a + d | a + c + d |
| b | b + c | b + d | b + c + d |
|
a + b |
a + b + c |
a + b + d |
a + b + |
(This vector-space a b c d diagram is from
Chapter 11 of Sphere Packings, Lattices
and Groups , by John Horton Conway and
N. J. A. Sloane, first published by Springer
in 1988.)
More later.
Update of 6:06 PM ET — An image from a post of Oct. 12, 2008 —
Moulin Bleu
Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat
"As McCarthy peers through the screen, or veil,
of technological modernity to reveal the underlying
symbolic structures of human experience,
The Making of Incarnation weaves a set of stories
one inside the other, rings within rings, a perpetual
motion machine." — Amazon.com description
of a novel published on All Souls' Day (Dia de los
Muertos), 2021.
The McCarthy novel is mentioned in The New York Times today —
For a simpler perpetual motion machine, see T. S. Eliot's "Chinese jar."
On "Indiana Jones and the Dial of Destiny" —
"… second unit began shooting the tuk-tuk chase in Morocco.
'It’s scripted as Tangier in the movie, but it was actually shot in Fez'…."
— https://www.lucasfilm.com/news/indiana-jones-duncan-broadfoot/
See as well, from 12 AM ET Sept. 10, "Plan 9 from Death Valley."
For other remarks about Archimedes and Death, see Hidden Structure.
Two examples from the Wikipedia article "Archimedean solid" —
Iain Aitchison said in a 2018 talk at Hiroshima that
the Mathieu group M24 can be represented as permuting
naturally the 24 edges of the cuboctahedron.
The 24 vertices of the truncated octahedron are labeled
naturally by the 24 elements of S4 in a permutahedron —
Can M24 be represented as permuting naturally
the 24 vertices of the truncated octahedron?
Related material from the day Orson Welles and Yul Brynner died —
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 .
"Before time began . . ." — Optimus Prime
Structures from pure mathematics, by Plato and R. T. Curtis —
* See other "Preform" posts in this journal.
On finite geometries . . .
"Although many of these structures are studied for
their geometrical importance, they are also of great
interest in other, more applied domains of mathematics."
— Remark from the metadata of a mathematical article
dated September 22, 2021
More applied domains . . .
"Sex Show at a Brothel" — This journal on September 22, 2021.
A scene from the "Badass Song" film mentioned in that post —
Another cinematic towel scene —
|
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. |
Variety.com May 8, 2023, 1:45 AM PT —
“Pema Tseden, a famous Tibetan director, screenwriter and
professor at the Film School of the China Academy of Art,
died in Tibet in the early hours of May 8 due to an acute illness."
"The news was reported by the China Academy of Art."
The time in Lhasa, Tibet, is 12 hours ahead of New York time.
From this journal in the afternoon of May 7 (New York time) —
For a relationship between the above image and classic Chinese culture,
see Geometry of the I Ching.
A memorial image from Variety —
Tseden with the award for best screenplay at Venice on Sept. 8, 2018.
See also that date in this journal . . . Posts now tagged Space Structure.
"Google Gone Haywire" Continues.
See as well a long complete list of the many Google search results
on combinatorial mathematics that contain the above phrase as
part of a fake "abstract" quoted by Google.
". . . The last of the river diamonds . . . .
bright alluvial diamonds,
burnished clean by mountain torrents,
green and blue and yellow and red.
In the darkness, he could feel them burning,
like fire and water of the universe, distilled."
— At Play in the Fields of the Lord ,
by Peter Matthiessen (Random House, 1965)
Related Log24 posts are now tagged Fire Water.
See as well, from posts tagged Heartland Sutra —
♫ "Red and Yellow, Blue and Green"
— "Prism Song," 1964
An actor's obituary in The New York Times today suggests
a review of the phrase "geometry and death" in this journal.
In that review, the phrase, by J. G. Ballard in a 2006 article,
refers to German fortifications in World War II. Ballard had
earlier used the same phrase in connection with French
nuclear-test structures in the Pacific —
— From Rushing to Paradise by J. G. Ballard, 1994.
Those interested in the religious meaning of the phrase "Saint-Esprit"
may consult this journal on the date of Ballard's death.
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.
|
From Gilles Châtelet, Introduction to Figuring Space Metaphysics does have a catalytic effect, which has been described in a very beautiful text by the mathematician André Weil: Nothing is more fertile, all mathematicians know, than these obscure analogies, these murky reflections of one theory in another, these furtive caresses, these inexplicable tiffs; also nothing gives as much pleasure to the researcher. A day comes when the illusion vanishes: presentiment turns into certainty … Luckily for researchers, as the fogs clear at one point, they form again at another.4 André Weil cuts to the quick here: he conjures these 'murky reflections', these 'furtive caresses', the 'theory of Galois that Lagrange touches … with his finger through a screen that he does not manage to pierce.' He is a connoisseur of these metaphysical 'fogs' whose dissipation at one point heralds their reforming at another. It would be better to talk here of a horizon that tilts thereby revealing a new space of gestures which has not as yet been elucidated and cut out as structure. 4 A. Weil, 'De la métaphysique aux mathématiques', (Oeuvres, vol. II, p. 408.) |
For gestures as fogs, see the oeuvre of Guerino Mazzola.
For some clearer remarks, see . . .
Illustrations of object and gestures
from finitegeometry.org/sc/ —
Object
Gestures
An earlier presentation
of the above seven partitions
of the eightfold cube:
|
|
Related material: Galois.space .
|
Posted on 13/01/2023 by Peter Cameron My last major job at Queen Mary University of London more than ten years ago was designing and presenting a new first-semester first-year module to be taken by all students on mathematics programmes or joint programmes involving mathematics. I discussed it in my LMS-Gresham lecture. |
LMS
Gresham
"… seeds having fallen on barren rock, as it were" . . .
The Source —
Some similar notions from my own work . . .
The "Digraph" of Mazzola might correspond to a directed graph
indicating the structure of a permutation, as at right below —
Mazzola's "Formula" might correpond to a matrix and translation that
transform the above "Space" of eight coordinates, and his "Gesture"
to a different way of generating affine transformations of that space . . .
as in my webpage "Cube Space, 1984-2003."
Midrash from Philip Pullman . . .
"The 1929 Einstein-Carmichael Expedition"
I prefer the 1929 Emch-Carmichael expedition —
This is from . . .
“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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