Organic Symmetry
"Benzene is a natural constituent of petroleum
and is one of the elementary petrochemicals.
Due to the cyclic continuous pi bonds between
the carbon atoms, benzene is classed as an
aromatic hydrocarbon. Benzene is a colorless
and highly flammable liquid with a sweet smell,
and is partially responsible for the aroma of gasoline."
Anti-Organic Symmetry —
The exercise posted here on Sept. 11, 2022, suggests a
more precisely stated problem . . .
The 24 coordinate-positions of the 4096 length-24 words of the
extended binary Golay code G24 can be arranged in a 4×6 array
in, of course, 24! ways.
Some of these ways are more geometrically natural than others.
See, for instance, the Miracle Octad Generator of R. T. Curtis.
What is the size of the largest subcode C of G24 that can be
arranged in a 4×6 array in such a way that the set of words of C
is invariant under the symmetry group of the rectangle itself, i.e. the
four-group of the identity along with horizontal and vertical reflections
and 180-degree rotation.
Recent Log24 posts tagged Bitspace describe the structure of
an 8-dimensional (256-word) code in a 4×6 array that has such
symmetry, but it is not yet clear whether that "cube-motif" code
is a Golay subcode. (Its octads are Golay, but possibly not all its
dodecads; the octads do not quite generate the entire code.)
Magma may have an answer, but I have had little experience in
its use.
* Footnote of 30 September 2022. The 4×6 problem is a
special case of a more general symmetric embedding problem.
Given a linear code C and a mapping of C to parts of a geometric
object A with symmetry group G, what is the largest subcode of C
invariant under G? What is the largest such subcode under all
such mappings from C to A?
From "Raiders of the Lost Space," Sept. 11, 2022 —
A related technique appears in a 1989 paper by Cheng and Sloane
that I saw for the first time today:
From 1981 —
From today —
Update —
A Magma check of the motif-generated space shows that
its dimension is only 8, not 12 as with the MOG space.
Four more basis vectors can be added to the 24 motifs to
bring the generated space up to 12 dimensions: the left
brick, the middle brick, the top half (2×6), the left half (4×3).
I have not yet checked the minimum weight in the resulting
12-dimensional 4×6 bit-space.
— SHC 4 PM ET, Sept. 12, 2022.
Gell-Mann Meets Bosch . . .
At Hiroshima . . .
* The Bosch cuboctahedron is from an exhibition at Napoli in 2021.
See also, from that exhibition's starting date,
the Log24 post Desperately Seeking Symmetry.
At Hiroshima on March 9, 2018, Aitchison discussed another
"hexagonal array" with two added points… not at the center, as
in the Gell-Mann picture above, but rather at the ends of one of
a cube's four diagonal axes of symmetry.
See some related illustrations below.
Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.
The exceptional role of 0 and ∞ in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a
Miracle Octad Generator octad —
Transposition of 0 and ∞ in the knight coordinatization
induces the symplectic polarity of PG(3,2) discussed by
(for instance) Anne Duncan in 1968.
A Companion Volume —
See as well this journal on the above publication date —
Continued from April 12, 2022.
"It’s important, as art historian Reinhard Spieler has noted,
that after a brief, unproductive stay in Paris, circa 1907,
Kandinsky chose to paint in Munich. That’s where he formed
the Expressionist art group Der Blaue Reiter (The Blue Rider) —
and where he avoided having to deal with cubism."
— David Carrier,
Remarks by Louis Menand in The New Yorker today —
"The art world isn’t a fixed entity.
It’s continually being reconstituted
as new artistic styles emerge."
(Adapted from Encyclopaedia Britannica,
Eleventh Edition (1911), Crystallography .)
"Before time began, there was the Cube."
— Optimus Prime
See as well Verbum (February 18, 2017).
Related dramatic music —
"Westworld Season 4 begins at Hoover Dam,
with William looking to buy the famous landmark.
What does he consider to be 'stolen' data that is inside?"
Related material — Posts tagged Interality and Seven Seals.
From Hermann Weyl's 1952 classic Symmetry —
"Galois' ideas, which for several decades remained
a book with seven seals but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."
WIkipedia on the URL suffix ".io" —
"In computer science, "IO" or "I/O" is commonly used
as an abbreviation for input/output, which makes the
.io domain desirable for services that want to be
associated with technology. .io domains are often used
for open source projects, application programming
interfaces ("APIs"), startup companies, browser games,
and other online services."
An association with the Bead Game from a post of April 7, 2018 —
|
Glasperlenspiel passage quoted here in Summa Mythologica —
“"I suddenly realized that in the language, or at any rate A less poetic meditation on the above 4x4x4 design cube —
"I saw that in the alternation between front and back, See also a related remark by Lévi-Strauss in 1955:
"…three different readings become possible: |
The recent use by a startup company of the URL "interality.io" suggests
a fourth reading for the 1955 list of Lévi-Strauss — in and out —
i.e., inner and outer group automorphisms — from a 2011 post
on the birthday of T. S. Eliot :
A transformation:
Click on the picture for details.
| Name Tag | .Space | .Group | .Art |
|---|---|---|---|
| Box4 |
2×2 square representing the four-point finite affine geometry AG(2,2). (Box4.space) |
S4 = AGL(2,2) (Box4.group) |
(Box4.art) |
| Box6 |
3×2 (3-row, 2-column) rectangular array representing the elements of an arbitrary 6-set. |
S6 | |
| Box8 | 2x2x2 cube or 4×2 (4-row, 2-column) array. | S8 or A8 or AGL(3,2) of order 1344, or GL(3,2) of order 168 | |
| Box9 | The 3×3 square. | AGL(2,3) or GL(2,3) | |
| Box12 | The 12 edges of a cube, or a 4×3 array for picturing the actions of the Mathieu group M12. | Symmetries of the cube or elements of the group M12 | |
| Box13 | The 13 symmetry axes of the cube. | Symmetries of the cube. | |
| Box15 |
The 15 points of PG(3,2), the projective geometry of 3 dimensions over the 2-element Galois field. |
Collineations of PG(3,2) | |
| Box16 |
The 16 points of AG(4,2), the affine geometry of 4 dimensions over the 2-element Galois field. |
AGL(4,2), the affine group of |
|
| Box20 | The configuration representing Desargues's theorem. | ||
| Box21 | The 21 points and 21 lines of PG(2,4). | ||
| Box24 | The 24 points of the Steiner system S(5, 8, 24). | ||
| Box25 | A 5×5 array representing PG(2,5). | ||
| Box27 |
The 3-dimensional Galois affine space over the 3-element Galois field GF(3). |
||
| Box28 | The 28 bitangents of a plane quartic curve. | ||
| Box32 |
Pair of 4×4 arrays representing orthogonal Latin squares. |
Used to represent elements of AGL(4,2) |
|
| Box35 |
A 5-row-by-7-column array representing the 35 lines in the finite projective space PG(3,2) |
PGL(3,2), order 20,160 | |
| Box36 | Eurler's 36-officer problem. | ||
| Box45 | The 45 Pascal points of the Pascal configuration. | ||
| Box48 | The 48 elements of the group AGL(2,3). | AGL(2,3). | |
| Box56 |
The 56 three-sets within an 8-set or |
||
| Box60 | The Klein configuration. | ||
| Box64 | Solomon's cube. |
— Steven H. Cullinane, March 26-27, 2022
See as well the life of a real astrophysicist.
Update of 12:26 PM ET March 15:
Vide other posts now tagged The Rosenhain Symmetry.
By the Daniel J. Peterson whose Swarthmore honors thesis was quoted
here last night —
"What, then, is the relationship between theory-relative symmetries
(physical symmetries) and theory-independent symmetries
(overarching symmetries)? My statement of this problem is
a bit abstract, so let’s look at an example: classical Newtonian gravity
and classical electromagnetism . . . ."
— Prospects for a New Account of Time Reversal
by Daniel J. Peterson, Ph.D. dissertation, U. Mich., 2013, p. 16.
Another 2013 approach to the word "overarching" and sytmmetries —
Other terms of interest: Tenet , Nolanism , and Magic for Liars .
"There is such a thing as a tesseract."
— Mrs. Whatsit in A Wrinkle in Time (1962)
"Simplify, simplify." — Henry David Thoreau in Walden (1854)
|
A Jungian on this six-line figure:
“They are the same six lines that exist in the I Ching…. Now observe the square more closely: four of the lines are of equal length, the other two are longer…. For this reason symmetry cannot be statically produced and a dance results.” |
Related art — The non-Rubik 3x3x3 cube —
The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.
I prefer the radiance (in the sense of Aquinas) of the central, structuring
eightfold cube at the center of the affine space of six dimensions over
the two-element field GF(2).
Some formal symmetry —
|
"… each 2×4 "brick" in the 1974 Miracle Octad Generator
Folding a 2×4 Curtis array yet again yields — Steven H. Cullinane on April 19, 2016 — The Folding. |
Related art-historical remarks:
The Shape of Time (Kubler, Yale U.P., 1962).
See yesterday's post The Thing .
A Log24 search for "Watercourse" leads to . . .
("Watercourse" is in the Customer review link.)
The "five years ago" link leads to . . .
|
"What modern painters are trying to do,
— James J. Gibson in Leonardo An example of invariant structure:
The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn. Taken as a set, these three line diagrams describe the structure of the bottom colored figure. After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 binary translations acting on the colored figure. A more remarkable invariance — that of symmetry itself — is observed if we arbitrarily and repeatedly permute rows and/or columns and/or 2×2 quadrants of the colored figure above. Each resulting figure has some ordinary or color-interchange symmetry. This sort of mathematics illustrates the invisible "form" or "idea" behind the visible two-color pattern. Hence it exemplifies, in a way, the conflict described by Plato between those who say that "real existence belongs only to that which can be handled" and those who say that "true reality consists in certain intelligible and bodiless forms." |
* See that title in this journal.
"Consider the six-dimensional vector space ( 𝔽2 )6
over the two-element field 𝔽2 ."
— Page 23 of "The Universal Kummer Threefold,"
arXiv:1208.1229v3, 12 June 2013, by Qingchun Ren,
Steven V. Sam, Gus Schrader, and Bernd Sturmfels.
An illustration of that space from 1981 —
The above recollection of the Kummer Threefold remark was suggested by
recent posts now tagged Smallfield . . .
|
"Third Man – an elderly American railway bum, "Art to which I fix my celebrated signature." — "Third Man" in Victor Snaith's play "Changing Stations"
|
If we read the above "art" as a scythe blade to which the "signature" —
Snaith ("the crooked handle or shaft of a scythe") — is attached,
an image of the late art critic Robert Hughes comes to mind:
That image of Hughes appeared here in a post of June 17, 2015 —
"Slow Art, Continued" — that also referenced the Kummer Threefold
paper above.
(A sequel to this morning's post A Subtle Knife for Sean.)
Exhibit A —
Einstein in The Saturday Review, 1949 —
"In any case it was quite sufficient for me
if I could peg proofs upon propositions
the validity of which did not seem to me to be dubious.
For example, I remember that an uncle told me
the Pythagorean theorem before the holy geometry booklet
had come into my hands. After much effort I succeeded
in 'proving' this theorem on the basis of the similarity
of triangles; in doing so it seemed to me 'evident' that
the relations of the sides of the right-angled triangles
would have to be completely determined by one of the
acute angles. Only something which did not in similar fashion
seem to be 'evident' appeared to me to be in need of any proof
at all. Also, the objects with which geometry deals seemed to
be of no different type than the objects of sensory perception,
'which can be seen and touched.' This primitive idea, which
probably also lies at the bottom of the well-known Kantian
problematic concerning the possibility of 'synthetic judgments
a priori' rests obviously upon the fact that the relation of
geometrical concepts to objects of direct experience
(rigid rod, finite interval, etc.) was unconsciously present."
Exhibit B —
Strogatz in The New Yorker, 2015 —
"Einstein, unfortunately, left no … record of his childhood proof.
In his Saturday Review essay, he described it in general terms,
mentioning only that it relied on 'the similarity of triangles.'
The consensus among Einstein’s biographers is that he probably
discovered, on his own, a standard textbook proof in which similar
triangles (meaning triangles that are like photographic reductions
or enlargements of one another) do indeed play a starring role.
Walter Isaacson, Jeremy Bernstein, and Banesh Hoffman all come
to this deflating conclusion, and each of them describes the steps
that Einstein would have followed as he unwittingly reinvented
a well-known proof."
Exhibit C —
Schroeder presents an elegant and memorable proof. He attributes
the proof to Einstein, citing purely hearsay evidence in a footnote.
The only other evidence for Einstein's connection with the proof
is his 1949 Saturday Review remarks. If Einstein did come up with
the proof at age 11 and discuss it with others later, as Schroeder
claims, it seems he might have felt a certain pride and been more
specific in 1949, instead of merely mentioning the theorem in passing
before he discussed Kantian philosophy relating concepts to objects.
Strogatz says that . . .
"What we’re seeing here is a quintessential use of
a symmetry argument… scaling….
Throughout his career, Einstein would continue to
deploy symmetry arguments like a scalpel, getting to
the hidden heart of things."
Connoisseurs of bullshit may prefer a faux-Chinese approach to
"the hidden heart of things." See Log24 on August 16, 2021 —
http://m759.net/wordpress/?p=96023 —
In a Nutshell: The Core of Everything .
The title refers to a recent Ariana Grande video.
Academics may prefer . . .
For a purely mathematical approach to switching the positions see . . .
.
For related philosophical remarks, see "Arrowy, Still Strings."
Update on April 20, 2021 —
The following was added today to the above summary:
“It describes a group of 322,560 permutations, later known as
‘the octad group,’ that now plays a role in speculative high-energy physics.
See Moonshine, Superconformal Symmetry, and Quantum Error Correction .”
Twelves (in memory of Robert de Marrais) —
Receipt date for the above article —
Synchronicity check —
Related reading —
http://www.universityreaders.com/pdf/
Incarnations-of-the-Blaring-Bluesblinger_sneak_preview.pdf
“None of the memories and identities of the people in the city are solid.”
— Grimstrup, Jesper Møller. SHELL BEACH:
The search for the final theory . Kindle Edition.
See also Peter Woit’s “Various Links” post today and Shell Beach in this journal.
For another approach to solidity (from pure mathematics, not physics), see
a Log24 search for Solid Symmetry.
(Adapted from Encyclopaedia Britannica,
Eleventh Edition (1911), Crystallography .)
From today’s post “Logo Animation” —
Related material from the art world —
Related entertainment —
“V. is whatever lights you to the end of the street: she is also the dark annihilation waiting at the end of the street.” (Tony Tanner, page 36, "V. and V-2," in Pynchon: A Collection of Critical Essays, ed. Edward Mendelson. Prentice-Hall, 1978. 16-55). |
Midrash — Other posts tagged Annihilation.
The title is a phrase by Robert Hughes from the previous post.
The previous post contained a passage from Iris Murdoch’s
1961 essay “Against Dryness.” Some related philosophy —
For those who prefer pure mathematics to philosophical ruminations
there are some relevant remarks in my webpage of August 27, 2003.
“For the first thirty years of its history, Columbia was known as King’s College.”
— History of the University Identity
Hence the crown favicon—
“When people talk about the importance of the study of ‘symmetry’
in mathematics, physics, and elsewhere, they often make the mistake
of only paying attention to the symmetry groups. The structure you
actually have is not just a group (the abstract ‘symmetries’), but an
action of that group on some other object, the thing that has symmetries.”
— Peter Woit of Columbia on June 9, reviewing a Quanta Magazine article
* From earlier posts in this journal containing the title phrase.
"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 . . .
|
Saturday, September 17, 2016
|
For those who prefer comedy —
Other toys: Archimedes at Hiroshima and related posts.
The following passage is from Amanda Gefter’s Trespassing
on Einstein’s Lawn (Bantam Books, 2014).
| “You know the story of Plato’s cave?” my father asked. “All the prisoners are chained up in the cave and they can’t see the real world outside, only the shadows on the wall? That’s supposed to be a negative thing, like they’ll never know reality. But the truth is, you have to be stuck inside a limited reference frame for there to be any reality at all! If you weren’t chained to your light cone, you’d see nothing. The H-state.”
I nodded. “You’d have no information. You need the broken symmetry, the shadow, to have information and information gives rise to the world. It from bit.” I couldn’t help but grin with excitement. The message was clear: having a finite frame of reference creates the illusion of a world, but even the reference frame itself is an illusion. Observers create reality, but observers aren’t real. There is nothing ontologically distinct about an observer, because you can always find a frame in which that observer disappears: the frame of the frame itself, the boundary of the boundary. “If physicists discover an invariant someday, the game will be up,” my father mused. “That would rule out the hypothesis that the universe is really nothing.” That was true. But so far, at least, every last invariant had gone the way of space and time, rendered relative and observer-dependent. Spacetime, gravity, electromagnetism, the nuclear forces, mass, energy, momentum, angular momentum, charge, dimensions, particles, fields, the vacuum, strings, the universe, the multiverse, the speed of light— one by one they had been downgraded to illusion. As the surface appearance of reality fell away, only one thing remained. Nothing. |
My path to Gefter’s father’s musing led from a quotation attributed,
probably falsely, to John Archibald Wheeler on page 52 of Octavio
Paz’s Claude Lévi-Strauss: An Introduction (Cornell, 1970) —
“There is a point at which
‘something is nothing and nothing is something.’ “
The quote may actually be by AP writer John Barbour reporting
on a 1967 American Physical Society talk by Wheeler, “The End
of Time.”
Gefter mentions Wheeler 369 times:
See as well Introduction to Quantum Woo.
For an account by R. T. Curtis of how he discovered the Miracle Octad Generator,
see slides by Curtis, “Graphs and Groups,” from his talk on July 5, 2018, at the
Pilsen conference on algebraic graph theory, “Symmetry vs. Regularity: The first
50 years since Weisfeiler-Leman stabilization” (WL2018).
See also “Notes to Robert Curtis’s presentation at WL2018,” by R. T. Curtis.
Meanwhile, here on July 5, 2018 —
| Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oft-repeated and ever-different metaphor of identity.
— Paz, Octavio. The Monkey Grammarian |
The 2018 Log24 post containing the above Paz quote goes on to quote
remarks by Lévi-Strauss. Paz’s phrase “series of transparent strata”
suggests a review of other remarks by Lévi-Strauss in the 2016 post
“Key to All Mythologies.“
Click the ring for Pierre Cartier on the barber of Seville
and “The evolution of concepts of space and symmetry.”
From "Mathieu Moonshine and Symmetry Surfing" —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
"This presentation of the symmetry groups Gi is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group
G = (Z2)4 ⋊ A8 .
It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O9 = {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose."
This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring," by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A-193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the "octad group" in 1993 —
… An abstract artist who reportedly died at 93 yesterday.
A search in this journal for Shubnikov yields…
"Raiders of the Lost Stone" (December 26, 2017).
"… And the song of love's recision . . . ." — E. L. Doctorow
Two of the thumbnail previews
from yesterday's 1 AM post …
Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to …
|
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. . . . . |
The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.
Related "jewel box" material for those who
prefer narrative to mathematics —
"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."
— Tim Doody in The Morning News web 'zine on July 26, 2012**
Oracular advice related to yesterday evening's
"jewel box" post …
A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces. In its incarnation as a Galois tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of 2,304 affine transformations.
Similar remarks apply to the I Ching In its incarnation as
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.
* The volume Wolfe mentions was, according to Fadiman, the I Ching.
** See also this journal on that date — July 26, 2012.
From Wallace Stevens, "The Man with the Blue Guitar":
IX
And the color, the overcast blue
Of the air, in which the blue guitar
Is a form, described but difficult,
And I am merely a shadow hunched
Above the arrowy, still strings,
The maker of a thing yet to be made . . . .
"Arrowy, still strings" from the diamond theorem
(Adapted from Encyclopaedia Britannica,
Eleventh Edition (1911), Crystallography .)
The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel
Many Dimensions published in 1993 by Wm. B. Eerdmans.
Compare and constrast —
Cover of a book by Douglas Hofstadter
The production-company logos for Carpenter B and Bad Robot
in end credits for a 2016 TV mini-series based on the Stephen King
novel 11/22/63 suggest a look at . . .
For the Church of Synchronology —
This weblog on Aug. 11, 2017:
John Horgan in Scientific American magazine on October 8, 2019 —
"In the early 1990s, I came to suspect that the quest
for a unified theory is religious rather than scientific.
Physicists want to show that all things came from
one thing: a force, or essence, or membrane
wriggling in eleven dimensions, or something that
manifests perfect mathematical symmetry. In their
search for this primordial symmetry, however,
physicists have gone off the deep end . . . ."
Other approaches —
See "Story Theory of Truth" in this journal and, from the November 2019
Notices of the American Mathematical Society . . .
|
More fundamental than the label of mathematician is that of human. And as humans, we’re hardwired to use stories to make sense of our world (story-receivers) and to share that understanding with others (storytellers) [2]. Thus, the framing of any communication answers the key question, what is the story we wish to share? Mathematics papers are not just collections of truths but narratives woven together, each participating in and adding to the great story of mathematics itself. The first endeavor for constructing a good talk is recognizing and choosing just one storyline, tailoring it to the audience at hand. Should the focus be on a result about the underlying structures of group actions? . . . .
[2] Gottschall, J. , The Storytelling Animal , — "Giving Good Talks," by Satyan L. Devadoss |
"Before time began, there was the Cube." — Optimus Prime
See as well a webpage from 2000,
"Symmetry from Plato to the
Four-Color Conjecture."
At present, the above Ajna page is not functioning in my Chrome browser
and is only partly functioning in Edge, but seems OK in Firefox.
Click for the pages below at Internet Archive.
Enveloping algebras also appeared later in the work on "crystal bases"
of Masaki Kashiwara. It seems highly unlikely that his work on enveloping
algebras, or indeed any part of his work on crystal bases, has any relation
to my own earlier notes.
A 1995 page by Kashiwara —
Kashiwara was honored with a Kyoto prize in 2018:
Kashiwara's 2018 Kyoto Prize diploma —
"The stars and galaxies seem static, eternal, or moving slowly
in deterministic patterns, becoming the background stage
on which we move. But if we could speed up the sequence,
we would see how dramatic and unpredictable this background
really is — an actor, director, script and stage all at once.
Moreover, it is a unified universe, a single unfolding event
of which we are an embedded part, a narrative of highly
dangerous and fine-tuned events, something more like
a detective thriller with many crimes and last-minute escapes
than the impersonal account of astronomy textbooks.
We are only just beginning to decipher the plot and figure out
the Cosmic Code, as Heinz Pagels puts it."
— Charles Jencks, The Architecture of the Jumping Universe :
A Polemic (How Complexity Science is Changing Architecture
and Culture), Academy Editions, 1995, rev. ed. 1997
"A Grand Unified Theory (GUT) is a model in particle physics…."
— Wikipedia
"Under the GUT symmetry operation these field components
transform into one another. The reason quantum particles
appear to have different properties in nature is that the unifying
symmetry is broken. The various gluons, quarks and leptons
are analogous to the facets of a cut diamond, which appear
differently according to the way the diamond is held but in
fact are all manifestations of the same underlying object."
— Heinz Pagels, Perfect Symmetry , Bantam paperback, 1986, p. 284
See also the recent post Multifaceted Narrative.
This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .
Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.
Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —
Revision history accounting for the above change from yesterday —
The jargon "rm OR" means "remove original research."
The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.
* The 35 squares, each consisting of four 4-element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).
The title abbreviates* that of a collection of Wittgenstein's remarks:
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Ludwig Wittgenstein — Culture and Value Showing 20 results for spirit — page 18, rubble & finally a heap of ashes; but spirits will hover over the ashes. MS 107 229: page 18, Page 5 Only something supernatural can expre page 20, contemplating it from above in its†c flight.† page 21, spirit in which it is written.†f This spirit is, I believe, different from that of t page 21, and American civilization. The spirit of this civilization the expression of page 21, day†h fascism & socialism, is a spirit that is alien & uncongenial†i to the au page 21, he Page Break 9 can work in the spirit of the whole, and his strength can with page 21, straight for what is concrete. Which is chara page 22, danger in a long foreword is that the spirit of a book has to be evident in the book page 22, It is all one to me whether the typical weste page 23, a great temptation to want to make the spirit explicit. MS 109 204: 6-7.11.1930 Page page 23, readers that will be clear just from the fact page 28, Foggy day. Grey autumn haunts us. Laughter se page 42, If one wanted to characterize the essence of page 51, attention from what matters.) The Spirit puts what is essential, essential for y page 51, how far all this is exactly in the spirit of Kierkegaard.) MS 119 151: 22.10.1937 page 51, something feminine about this outlook?) MS 11 page 100, comfortable, clearer expression, but cannot b page 106, act otherwise."–Perhaps, though, one might s page 210, Page 7 †b function Page 7 †c from its Page **************************************************************** |
The above "spirit guide" was suggested by yesterday's post
on Knuth as Yoda and by the paper in today's previous post,
"Shadowhunter Tales."
This post's title, "CV," is from . . .
The recent post "Tales from Story Space," about the 18th birthday
of the protagonist in the TV series "Shadowhunters" (2016-),
suggests a review of the actual 18th birthday of actress Lily Collins.
Collins is shown below warding off evil with a magical rune as
a shadowhunter in the 2013 film "City of Bones" —
She turned 18 on March 18, 2007. A paper on symmetry and logic
referenced here on that date displays the following "runes" of
philosopher Charles Sanders Peirce —
See also Adamantine Meditation (Log24, Oct. 3, 2018)
and the webpage Geometry of the I Ching.
From the former date above —
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Saturday, September 17, 2016 |
From the latter date above —
|
Tuesday, October 18, 2016
Parametrization
|
From March 2018 —
Click image to enlarge —
A portrait from the home page of David Eppstein,
a professor at the University of California, Irvine
“… how can an image with 8 points and 8 lines
possibly represent a space with 7 points and 7 lines???“
— David Eppstein, 21 December 2015
See ” Projective spaces as ‘collapsed vector spaces,’ ”
page 203 in Geometry and Symmetry by Paul B. Yale,
published by Holden-Day in 1968.
The previous post was suggested by some April 17, 2016, remarks
by James Propp on the eightfold cube.
Propp's remarks included the following:
"Here’s a caveat about my glib earlier remark that
'There are only finitely many numbers ' in a finite field.
It’s a bit of a stretch to call the elements of finite fields
'numbers'. Elements of GF(q ) can be thought of as
the integers mod q when q is prime, and they can be
represented by 0, 1, 2, …, q–1; but when q is a prime
raised to the 2nd power or higher, describing the
elements of GF(q ) is more complicated, and the word
'number' isn’t apt."
Related material —
See also this journal on the date of Propp's remarks — April 17, 2016.
A star figure and the Galois quaternion.
The square root of the former is the latter.
See also a passage quoted here a year ago today
(May the Fourth, "Star Wars Day") —
Tom Wolfe in The Painted Word (1975) —
“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 1945-75,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johns-but Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”
The above group of 322,560 permutations appears also in a 2011 book —
— and in 2013-2015 papers by Anne Taormina and Katrin Wendland:
See also the Feb. 17, 2017, post on Bertram Kostant
as well as "Mathieu Moonshine and Symmetry Surfing."
The FBI holding cube in "The Blacklist" —
" 'The Front' is not the whole story . . . ."
— Vincent Canby, New York Times film review, 1976,
as quoted in Wikipedia.
See also Solomon's Cube in this journal.
Some may view the above web page as illustrating the
Glasperlenspiel passage quoted here in Summa Mythologica —
“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”
A less poetic meditation on the above 4x4x4 design cube —
"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."
See also a related remark by Lévi-Strauss in 1955:
"…three different readings become possible:
left to right, top to bottom, front to back."
See as well a search for "Gold Bug" in this journal.
From that search —
Richard Powers, The Gold Bug Variations , first published in 1991—
Botkin, whatever her gifts as a conversationist, is almost as old
as the rediscovery of Mendel. The other extreme in age,
Joe Lovering, beat a time-honored path out of pure math
into muddy population statistics. Ressler has seen the guy
potting about in the lab, although exactly what the excitable kid
does is anybody's guess. He looks decidedly gumfooted holding
any equipment more corporeal than a chi-square. Stuart takes
him to the Y for lunch, part of a court-your-resources campaign.
He has the sub, Lovering the congealed mac and cheese.
Hardly are they seated when Joe whips out a napkin and begins
sketching proofs. He argues that the genetic code, as an
algorithmic formal system, is subject to Gödel's Incompleteness
Theorem. "That would mean the symbolic language of the code
can't be both consistent and complete. Wouldn't that be a kick
in the head?"
Kid talk, competitive showing off, intellectual fantasy.
But Ressler knows what Joe is driving at. He's toyed with similar
ideas, cast in less abstruse terms. We are the by-product of the
mechanism in there. So it must be more ingenious than us.
Anything complex enough to create consciousness may be too
complex for consciousness to understand. Yet the ultimate paradox
is Lovering, crouched over his table napkin, using proofs to
demonstrate proof's limits. Lovering laughs off recursion and takes
up another tack: the key is to find some formal symmetry folded
in this four-base chaos. Stuart distrusts this approach even more.
He picks up the tab for their two untouched lunches, thanking
Lovering politely for the insight.
"The key is to find some formal symmetry…."
"On Tralfamadore, Billy is put in a transparent geodesic dome
exhibit in a zoo; the dome represents a house on Earth.
The Tralfamadorians later abduct a movie star named
Montana Wildhack, who had disappeared and was believed to
have drowned herself in the Pacific Ocean. They intend to
have her mate with Billy." — Wikipedia on Kurt Vonnegut's
Slaughterhouse-Five .
See also the previous post and (from Log24 on Jan. 22) "Hollywood Moment" …
Related material on automorphism groups —
The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.
Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure above
than they are an eight-cube mechanical puzzle like the one below.
Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."
Related material —
The seven points of the Fano plane within
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"Before time began . . . ."
— Optimus Prime
Michael Atiyah on the late Ron Shaw —
Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —
These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .
I prefer other, purely geometric, reasons for the importance of GF(2) —
See Finite Geometry of the Square and Cube.
See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
On a Trinity classmate of Ian Macdonald (see previous post)—
Atiyah's eulogy of Shaw in Trinity Annual Record 2017
is on pages 137 through 146. The conclusion —
A death on the date of the above symmetry chat,
Wednesday, August 17, 2016 —
An Hispanic Hollywood moment:
Ojo de Dios —
Click for related material.
For further Hispanic entertainment,
see Ben Affleck sing
"Aquellos Ojos Verdes "
in "Hollywoodland."
From "The Principle of Sufficient Reason," by George David Birkhoff,
in "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940 —
From the same lecture —
|
Up to the present point my aim has been to consider a variety of applications of the Principle of Sufficient Reason, without attempting any precise formulation of the Principle itself. With these applications in mind I will venture to formulate the Principle and a related Heuristic Conjecture in quasi-mathematical form as follows: PRINCIPLE OF SUFFICIENT REASON. If there appears in any theory T a set of ambiguously determined ( i e . symmetrically entering) variables, then these variables can themselves be determined only to the extent allowed by the corresponding group G. Consequently any problem concerning these variables which has a uniquely determined solution, must itself be formulated so as to be unchanged by the operations of the group G ( i e . must involve the variables symmetrically). HEURISTIC CONJECTURE. The final form of any scientific theory T is: (1) based on a few simple postulates; and (2) contains an extensive ambiguity, associated symmetry, and underlying group G, in such wise that, if the language and laws of the theory of groups be taken for granted, the whole theory T appears as nearly self-evident in virtue of the above Principle. The Principle of Sufficient Reason and the Heuristic Conjecture, as just formulated, have the advantage of not involving excessively subjective ideas, while at the same time retaining the essential kernel of the matter. In my opinion it is essentially this principle and this conjecture which are destined always to operate as the basic criteria for the scientist in extending our knowledge and understanding of the world. It is also my belief that, in so far as there is anything definite in the realm of Metaphysics, it will consist in further applications of the same general type. This general conclusion may be given the following suggestive symbolic form:
While the skillful metaphysical use of the Principle must always be regarded as of dubious logical status, nevertheless I believe it will remain the most important weapon of the philosopher. |
Related remarks by a founding member of the Metaphysical Club:
See also the previous post, "Seven Types of Interality."
"There is always an awareness in her fiction
of the subjectivity of perception, and
the kaleidoscopic permutations
that memory can work on reality."
This is from a New York Times article subtitled
"Alice Munro, Nobel Winner, Mines the Inner Lives
of Girls and Women" …
The New York Times article was linked to by Marjorie Senechal
in a Huffington Post article of All Saints' Day 2013.
Further material on kaleidoscopic permutations —
See the Log24 post Symmetry of May 3, 2016.
For further material on mining, see Diamond-Mine:
"SEE HEAR READ" — Walt Disney Productions
Two Students of Structure
A comment on Sean Kelly's Christmas Morning column on "aliveness"
in the New York Times philosophy series The Stone —
Diana Senechal's 1999 doctoral thesis at Yale was titled
"Diabolical Structures in the Poetics of Nikolai Gogol."
Her mother, Marjorie Senechal, has written extensively on symmetry
and served as editor-in-chief of The Mathematical Intelligencer .
From a 2013 memoir by Marjorie Senechal —
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"While I was in Holland my enterprising student assistant at Smith had found, in Soviet Physics – Crystallography, an article by N. N. Sheftal' on tetrahedral penetration twins. She gave it to me on my return. It was just what I was looking for. The twins Sheftal' described had evidently begun as (111) contact twins, with the two crystallites rotated 60o with respect to one another. As they grew, he suggested, each crystal overgrew the edges of the other and proceeded to spread across the adjacent facet. When all was said and done, they looked like they'd grown through each other, but the reality was over-and-around. Brilliant! I thought. Could I apply this to cubes? No, evidently not. Cube facets are all (100) planes. But . . . these crystals might not have been cubes in their earliest stages, when twinning occurred! I wrote a paper on "The mechanism of certain growth twins of the penetration type" and sent it to Martin Buerger, editor of Neues Jarbuch für Mineralogie. This was before the Wrinch symposium; I had never met him. Buerger rejected it by return mail, mostly on the grounds that I hadn't quoted any of Buerger's many papers on twinning. And so I learned about turf wars in twin domains. In fact I hadn't read his papers but I quickly did. I added a reference to one of them, the paper was published, and we became friends.[5]
After reading Professor Sheftal's paper I wrote to him in Moscow; a warm and encouraging correspondence ensued, and we wrote a paper together long distance.[6] Then I heard about the scientific exchanges between the Academies of Science of the USSR and USA. I applied to spend a year at the Shubnikov Institute for Crystallography, where Sheftal' worked. I would, I proposed, study crystal growth with him, and color symmetry with Koptsik. To my delight, I was accepted for an 11-month stay. Of course the children, now 11 and 14, would come too and attend Russian schools and learn Russian; they'd managed in Holland, hadn't they? Diana, my older daughter, was as delighted as I was. We had gone to Holland on a Russian boat, and she had fallen in love with the language. (Today she holds a Ph.D. in Slavic Languages and Literature from Yale.) . . . . |
Philosophy professors and those whose only interest in mathematics
is as a path to the occult may consult the Log24 posts tagged Tsimtsum.
David E. Wellbery on Goethe
From an interview published on 2 November 2017 at
http://literaturwissenschaft-berlin.de/interview-with-david-wellbery/
as later republished in
The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .
James Propp in the current Math Horizons on the eightfold cube —
For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).
See the previous post and College of the Desert in this journal.
From the latter, see particularly Slide 69 in Geoff Hagopian's Symmetry.
A sequel to the post CP is for Consolation Prize (Sept. 3, 2016)
An image from Log24 on this date last year:
A recent comment on a discussion of CP symmetry —
The elementary shapes at the top of the figure below mirror
the looking-glass property of the classical Lo Shu square.
The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008.
** A well-known construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
Text —
"A field is perhaps the simplest algebraic structure we can invent."
— Hermann Weyl, 1952
Context —
See also yesterday's Personalized Book Search.
|
Full text of Symmetry – Internet Archive — https://archive.org/details/Symmetry_482
A field is perhaps the simplest algebraic 143 structure |
From a Log24 search for Mathematics+Nutshell —
The New York Times online this evening —
"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."
Review —
|
Thursday, September 1, 2011
How It Works
|
See also 1984 Bricks in this journal.
Del Toro and the History of Mathematics ,
Or: Applied Bullshit Continues
For del Toro —
For the history of mathematics —
|
Thursday, September 1, 2011
How It Works
|
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —
A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.
See also Magic Square and Diamond Theorem in this journal.
From a trailer for the 2013 Dutch film "App" —
From the online New York Times yesterday —
See also Overarching + Symmetry in this journal.
In memory of a Disney "imagineer" who reportedly died yesterday.
From the opening scene of a 2017 film, "Gifted":
Frank calls his niece Mary to breakfast on the morning she is
to enter first grade. She is dressed, for the first time, for school —
- Hey! Come on. Let's move! - No! - Let me see. - No. - Come on, I made you special breakfast. - You can't cook. - Hey, Mary, open up. (She opens her door and walks out.) - You look beautiful. - I look like a Disney character. Where's the special? - What? - You said you made me special breakfast. Read more: http://www.springfieldspringfield.co.uk/ movie_script.php?movie=gifted |
Remark on conceptual art quoted in the previous post —
"…he’s giving the concept but not the realization."
A concept — See a note from this date in 1983:
A realization —
Not the best possible realization, but enough for proof of concept .
"For years, the AllSpark rested, sitting dormant
like a giant, useless art installation."
— Vinnie Mancuso at Collider.com yesterday
Related material —
Giant, useless art installation —
Sol LeWitt at MASS MoCA. See also LeWitt in this journal.
In memory of John Severson, the founder of Surfer magazine —
"Freeze-frame surfer, and as a live Hendrix 'E Z Rider' blares
over the soundtrack, the surfer lifts his arms and rises like Christ
into the sky."
— Rolling Stone , August 5, 1971, on the film Rainbow Bridge
Severson reportedly died on Friday, May 26, 2017.
For a rather different sort of surfing, see this journal on that date.
Mark Zuckerberg in a commencement speech
at Harvard yesterday —
"Movies and pop culture get this all wrong.
The idea of a single eureka moment
is a dangerous lie. It makes us feel inadequate
since we haven’t had ours. It prevents people
with seeds of good ideas from getting started.
Oh, you know what else movies get wrong about
innovation? No one writes math formulas on glass.
That’s not a thing."
The Thing from Taormina —
See also Chandrasekharan in a Log24 search for Weyl+Schema.
Update of 6:16 AM Friday, May 12, 2017 —
The phrase "smallest perfect universe" is from Burkard Polster (2001).
For the music box of the title, see the previous post.
See also Mazzola on the Glass Bead Game
(Facebook date June 7, 2016)
and the Log24 post Symmetry (May 3, 2016).
Prequel —
Note that Yale's die design and use of the phrase "rigid motions"
differ from those in the webpage "Solomon's Cube."
From a search in this journal for Seagram + Tradition —
Related art: Saturday afternoon's Twin Pillars of Symmetry.
An arXiv article from Good Friday, 2003, by Igor Dolgachev,
a student of the late Igor Shafarevich (see previous post) —
See also my Dec. 29, 1986, query on Duality and Symmetry.
"Bertram Kostant, professor emeritus of mathematics at MIT,
died at the Hebrew Senior Rehabilitation Center in Roslindale,
Massachusetts, on Thursday, Feb. 2, at the age of 88."
— MIT News, story dated Feb. 16, 2017
See also a search for Kostant in this journal.
Regarding the discussions of symmetries and "facets" found in
that search —
Kostant:
“A word about E(8). In my opinion, and shared by others,
E(8) is the most magnificent ‘object’ in all of mathematics.
It is like a diamond with thousands of facets. Each facet
offering a different view of its unbelievable intricate internal
structure.”
Cullinane:
In the Steiner system S(5, 8, 24) each octad might be
regarded as a "facet," with the order of the system's
automorphism group, the Mathieu group M24 , obtained
by multiplying the number of such facets, 759, by the
order of the octad stabilizer group, 322,560.
Analogously …
From a Dec. 21 obituary posted by the
University of Tennessee at Knoxville —
"Wade was ordained as a pastor and served
at Oakwood Baptist Church in Knoxville."
Other information —
In a Log24 post, "Seeing the Finite Structure,"
of August 16, 2008, Wade appeared as a co-author
of the Walsh series book mentioned above —
Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp et al.,
Taylor & Francis, 1990
From the 2008 post —
The patterns on the faces of the cube on the cover
of Walsh Series above illustrate both the
Walsh functions of order 3 and the same structure
in a different guise, subspaces of the affine 3-space
over the binary field. For a note on the relationship
of Walsh functions to finite geometry, see
Symmetry of Walsh Functions.
Images from Burkard Polster's Geometrical Picture Book —
See as well in this journal the large Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.
Exercise: Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the small-configuration model's striking symmetry?
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model, which requires a free player download.)
Labeling the Tetrahedral Model (Click to enlarge) —
Related folk etymology (see point a above) —
Related literature —
The concept of "fire in the center" at The New Yorker ,
issue dated December 12, 2016, on pages 38-39 in the
poem by Marsha de la O titled "A Natural History of Light."
Cézanne's Greetings.
Before the monograph "Diamond Theory" was distributed in 1976,
two (at least) notable figures were published that illustrate
symmetry properties of the 4×4 square:
Hudson in 1905 —
Golomb in 1967 —
It is also likely that some figures illustrating Walsh functions as
two-color square arrays were published prior to 1976.
Update of Dec. 7, 2016 —
The earlier 1950's diagrams of Veitch and Karnaugh used the
1's and 0's of Boole, not those of Galois.
According to Octavio Paz and Claude Lévi-Strauss
"Poetry…. conceives of the text as a series of transparent strata
within which the various parts—the different verbal and semantic
currents— produce momentary configurations as they intertwine
or break apart, as they reflect each other or efface each other.
Poetry contemplates itself, fuses with itself, and obliterates itself
in the crystallizations of language. Apparitions, metamorphoses,
volatilizations, precipitations of presences. These configurations
are crystallized time…."
— Octavio Paz in The Monkey Grammarian (written in 1970)
"Strata" also seem to underlie the Lévi-Strauss "canonic formula" of myth
in its original 1955 context, described as that of permutation groups —
I do not recommend trying to make sense of the above "formula."
Related material —
From Hermann Weyl's 1952 classic Symmetry —
"Galois' ideas, which for several decades remained
a book with seven seals but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."
Some Galois geometry —
See the previous post for more narrative.
As the Key to All Mythologies
For the theorem of the title, see "Diamond Theorem" in this journal.
"These were heavy impressions to struggle against,
and brought that melancholy embitterment which
is the consequence of all excessive claim: even his
religious faith wavered with his wavering trust in his
own authorship, and the consolations of the Christian
hope in immortality seemed to lean on the immortality
of the still unwritten Key to all Mythologies."
— Middlemarch , by George Eliot, Ch. XXIX
Related material from Sunday's print New York Times —
Sunday's Log24 sermon —
See also the Lévi-Strauss "Key to all Mythologies" in this journal,
as well as the previous post.
The previous post quoted Tom Wolfe on Chomsky's use of
the word "array."
An example of particular interest is the 4×4 array
(whether of dots or of unit squares) —
.
Some context for the 4×4 array —
The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .
Further background on the Kummer lattice:
Alice Garbagnati and Alessandra Sarti,
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action."
To appear in Rocky Mountain J. Math. —
The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite geometry, see the website
Finite Geometry of the Square and Cube.
Some further context …
"To our knowledge, the relation of the Golay code
to the Kummer lattice … is a new observation."
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 "
As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface. The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.
* Update of Sept. 14: "Uncoordinatized," but parametrized by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.
The Lévi-Strauss “canonic formula” of myth in its original 1955 context,
described as that of permutation groups —
Related material in this journal —
Dueling Formulas and Symmetry.
CERN COURIER May 28, 1999:
In hot pursuit of CP violation
"CP was a consolation prize for physicists.
At least it seemed so until 1964."
"James W. Cronin, who shared the Nobel Prize in physics
for discovering a startling breakdown in what was assumed
to be the immutable symmetry of physical law, thereby
helping to explain the behavior and evolution of the universe
as a whole, died Aug. 25 in St. Paul, Minn. He was 84.
Dr. Cronin’s death was announced by the University of Chicago,
where he was a professor emeritus of physics as well as of
astronomy and astrophysics. No cause was reported."
— Martin Weil in The Washington Post , August 28, 2016
CERN COURIER May 28, 1999:
In hot pursuit of CP violation
"CP was a consolation prize for physicists.
At least it seemed so until 1964."
"James W. Cronin, who shared the Nobel Prize in physics
for discovering a startling breakdown in what was assumed
to be the immutable symmetry of physical law, thereby
helping to explain the behavior and evolution of the universe
as a whole, died Aug. 25 in St. Paul, Minn. He was 84.
Dr. Cronin’s death was announced by the University of Chicago,
where he was a professor emeritus of physics as well as of
astronomy and astrophysics. No cause was reported."
— Martin Weil in The Washington Post , August 28, 2016
From an earlier Log24 post —
Friday, July 11, 2014
|
From a post of the next day, July 12, 2014 —
"So there are several different genres and tones
jostling for prominence within Lexicon :
a conspiracy thriller, an almost abstract debate
about what language can do, and an ironic
questioning of some of the things it’s currently used for."
— Graham Sleight in The Washington Post
a year earlier, on July 15, 2013
For the Church of Synchronology, from Log24 on the next day —
From a post titled Circles on the date of Marc Simont's death —
See as well Verhexung in this journal.
See also, from that same day, "24-Part Invention."
* The title is a reference to a 2001 article by Cartier on
"the evolution of concepts of space and symmetry" —
"The editors are also grateful to
T. Kibble and Imperial College Press
for permission to reprint B. Zumino's paper
'Supersymmetry: A Personal View' . . . ."
— Preface to Symmetry in Mathematics and Physics
(AMS, 2009), a book based on talks at
a UCLA conference of Jan. 18-20, 2008
(For the book's title page, see yesterday morning's post Symmetry.)
This suggests a search in this journal for the term "supersymmetry."
That search yields some links that may be of further interest to
devotees of the Church of Synchronology.
According to McLuhan
Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—
"The American mind is not even close to being amenable
to the ideogram principle as yet. The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D. AB:CD.
It can see AB relations. But relations in four terms are still
verboten. This amounts to deep occultation of nearly all
human thought for the U.S.A.
I am trying to devise a way of stating this difficulty as it exists.
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."
For context, see Cameron McEwen,
"Marshall McLuhan, John Pick, and Gerard Manley Hopkins."
(Renascence , Fall 2011, Vol. 64 Issue 1, 55-76)
A relation in four terms —
A : B :: C : D as Model : Crutch :: Metaphor : Ornament —
See also Dueling Formulas and Symmetry.
"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng and H. van Dam,
February 20, 2009
For one such framework,* see posts from that same date
four years earlier — February 20, 2005.
* A 4×4 array. See the 1977, 1978, and 1986 versions by
Steven H. Cullinane, the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —
Cullinane, 1977
Cullinane, 1978
Cullinane, 1986
Curtis, 1987
Update of 10:42 PM ET on Sunday, June 19, 2016 —
The above images are precursors to …
Conway and Sloane, 1988
Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.
A less metaphysical approach to a "pre-form" —
From Wallace Stevens, "The Man with the Blue Guitar":
IX
And the color, the overcast blue
Of the air, in which the blue guitar
Is a form, described but difficult,
And I am merely a shadow hunched
Above the arrowy, still strings,
The maker of a thing yet to be made . . . .
"Arrowy, still strings" from the diamond theorem
See also "preforming" and the blue guitar
in a post of May 19, 2010.
Update of 7:11 PM ET:
More generally, see posts tagged May 19 Gestalt.
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