For another variation on this hallucination, see the previous post.
Wednesday, October 30, 2024
Tuesday, October 29, 2024
Hallucinated Geometry
For fans of the "story theory of truth" —
An example of artificial stupidity:
The phrases "midpoints of opposite faces" and "essentially
creating a smaller cube" are hallucinated bullshit.
The above AI description was created by inanely parroting
verbiage from the Wikipedia article "Diamond cubic" —
which it credits as a source. (See wider view of search.)
That article contains neither the word "theorem" nor the
phrase "unit cube " from the search-request prompt.
AI, like humans, is likely to fall victim to the notorious
"story theory of truth" purveyed by Richard J. Trudeau.
A real "diamond shape formed within a unit cube" is the
octahedron, one of the five classical Platonic solids.
Fans of the opposing "diamond theory of truth" rejected by
Trudeau may prefer . . .
(Log24, July 1, 2019).
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Sunday, October 27, 2024
For the “David Brooks” of “Canary Black” —
Grandiose Eschatological Visions!
Comments Off on For the “David Brooks” of “Canary Black” —
Grandiose Eschatological Visions!
Grandiose Eschatological Visions!
Saturday, October 19, 2024
A Seven-Eleven for Mystics: October 7 . . . 11 Years Ago
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Thursday, October 10, 2024
From Time to Eternity
"Time is the moving image of eternity." — Plato (paraphrased)
Summary, as an illustration of a title by George Mackey —
A more recent famous saying . . .
"Before time began, there was the Cube." — Optimus Prime
Since it is part of the cube, the square figure above
may be seen as a representation of eternity. (The circle,
familiar to us as a clock face, of course represents time.)
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Wednesday, October 2, 2024
Activity Log: “Time Machine as Ball Machine”
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Saturday, September 28, 2024
Architectural Singularity
Embedded in the Sept. 26 New Yorker review of Coppola's
Megalopolis is a ghostly transparent pyramidal figure . . .
The pyramidal figure is not unrelated to Scandia.tech —
|
American Mathematical Monthly, Vol. 92, No. 6 LETTERS TO THE EDITOR Material for this department should be prepared exactly the same way as submitted manuscripts (see the inside front cover) and sent to Professor P. R. Halmos, Department of Mathematics, University of Santa Clara, Santa Clara, CA 95053 Editor: Miscellaneum 129 ("Triangles are square," June-July 1984 Monthly ) may have misled many readers. Here is some background on the item. That n2 points fall naturally into a triangular array is a not-quite-obvious fact which may have applications (e.g., to symmetries of Latin-square "k-nets") and seems worth stating more formally. To this end, call a convex polytope P an n-replica if P consists of n mutually congruent polytopes similar to P packed together. Thus, for n ∈ ℕ, (A) An equilateral triangle is an n-replica if and only if n is a square. Does this generalize to tetrahedra, or to other triangles? A regular tetrahedron is not a (23)-replica, but a tetrahedron ABCD with edges AB, BC, and CD equal and mutually orthogonal is an n-replica if and only if n is a cube. Every triangle satisfies the "if" in (A), so, letting T be the set of triangles, one might surmise that (B) ∀ t ∈ T (t is an n-replica if and only if n is a square). This, however, is false. A. J. Schwenk has pointed out that for any m ∈ ℕ, the 30°-60°-90° triangle is a (3m2)-replica, and that a right triangle with legs of integer lengths a and b is an ((a2 + b2)m2)-replica. As Schwenk notes, it does not seem obvious which other values of n can occur in counterexamples to (B). Shifting parentheses to fix (B), we get a "square-triangle" lemma:
(C) (∀ t ∈ T, t is an n-replica) if and only if n is a square.
Steven H. Cullinane
501 Follett Run Road Warren, PA 16365 |
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Tuesday, September 24, 2024
Software Hardware
The "Cara.app" name in the previous post suggests . . .
Other "techniques d'avant garde" in 1985 —
85-03-26… Visualizing GL(2, p)
85-04-05… Group actions on partitions
85-04-05… GL(2, 3) actions on a cube
85-04-28… Generating the octad generator
85-08-22… Symmetry invariance under M12
85-11-17… Groups related by a nontrivial identity
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Monday, September 23, 2024
Design History for a Guy Fawkes Day
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Friday, September 20, 2024
Adapter Becomes Transformer
"When things go bonkers, you have to adapt."
— Chris Hemsworth as Dementus in "Furiosa" (2024)
"Before time began, there was the Cube."
— Optimus Prime in "Transformers" (2007)
Today, an animated Transformers opens, with
Chris Hemsworth as the voice of Optimus Prime.
Also today: The new tag "Cubehenge" in this journal.
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Wednesday, September 18, 2024
Flatiron Building Timelessness . . .
Directed by Quine, not Hitchcock
Click the "timelessness" quote below for the "Bell, Book and Candle" scene
with Kim Novak and James Stewart atop the Flatiron Building.
"Before time began . . . ." — Optimus Prime
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Directed by Quine, not Hitchcock
Directed by Quine, not Hitchcock
Triskelion
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Saturday, September 14, 2024
Notes on a Friday the 13th Death
A passage accessed via the new URL Starbrick.art* —
Thursday, February 25, 2021
|
A related cultural note suggested by the New York Times obituary today
of fashion designer Mary McFadden, who reportedly died yesterday
(a Friday the Thirteenth) and is described by the Times as a late-life
partner of "eightfold-way" physicist Murray Gell-Mann —
* A reference to the 2-column 4-row matrix (a "brick") that underlies
the patterns in the Miracle Octad Generator of R. T. Curtis. The only
connection of this eight-part matrix to Gell-Mann's "Eightfold Way"
that I know of is simply the number 8 itself.
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Thursday, September 12, 2024
Structures
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.
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Thursday, September 5, 2024
Wednesday, July 10, 2024
Wednesday, July 3, 2024
Vedic Carnival: “Hey Rubik!” *
The Moolakaprithi Cube (as opposed to Rubik's Moola Cube ) —
"The key to these connections lies in a 3 x 3 x 3 cube, which
in Vedic Physics, forms the Moolaprakriti, a key component of
the Substratum, the invisible black hole form of matter."
— viXra.org, "Clifford Clock and the Moolakaprithi Cube"
* See Wikipedia.
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Tuesday, July 2, 2024
Chinatown
|
Updated 8:18 PM EDT, Tue July 2, 2024 "Robert Towne, the Oscar-winning screenwriter of a number of acclaimed movies, including the classic 1974 noir thriller 'Chinatown' starring Jack Nicholson and Faye Dunaway, has died. He was 89 years old. The news was confirmed by Towne’s publicist Carri McClure, who said he died on Monday 'peacefully at home surrounded by his loving family.' No cause of death was provided. Towne won the Academy Award for best original screenplay for 'Chinatown,' which last month celebrated 50 years since being released." |
Related imagery . . .
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Sunday, June 30, 2024
Minecraft as a design tool: The Trial
(Kafkaesque? Maybe, maybe not.)
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(Kafkaesque? Maybe, maybe not.)
(Kafkaesque? Maybe, maybe not.)
Friday, June 28, 2024
Thursday, June 27, 2024
Die Berliner Mitschrift
https://page.math.tu-berlin.de/~felsner/Lehre/DSI11/Mitschrift-EH.pdf
The above S (3,4,8) is the foundation of the "happy family" of
subgroups of the Monster Group. See Griess and . . .
Related narrative and art —
"Battles argues that 'the experience of the physicality
of the book is strongest in large libraries,' and stand
among the glass cube at the center of the British Library,
the stacks upon stacks in Harvard’s Widener Library, or
the domed portico of the Library of Congress and tell me
any differently."
— Ed Simon, Binding the Ghost: Theology, Mystery, and
the Transcendence of Literature. Hardcover – April 19, 2022.
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Friday, June 21, 2024
Thursday, June 13, 2024
Of London Bondage … continues.
From a bondage search . . .
“Loitering in Lara’s dressing room, she tries on
the faux-bondage harness she picked up in London….”
From Geometry for Belgium —
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Monday, June 10, 2024
Art Logos
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Monday, June 3, 2024
Symmetry Plane
"For ten years… " — Song lyric
The previous post, together with the above song lyric, suggests a review
of the date May 19 ten years ago. The result of the review is the new tag
"Symmetry Plane."
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Friday, May 24, 2024
One Lesson
"At the present time there is no direct experimental evidence
that supersymmetry is a fundamental symmetry of nature . . . ."
— Introduction to the 1983 book
Superspace or One Thousand and One Lessons in Supersymmetry
Also from 1983 . . .
For direct experimental evidence of this symmetry, see . . .
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Tuesday, May 21, 2024
For Optimus Prime
"Before time began, there was the Cube." — Optimus Prime in "Transformers"
This journal at 9 PM ET March 17, 2023 —
The use of binary coordinate systems as a conceptual tool
Natural physical transformations of square or cubical arrays
of actual physical cubes (i.e., building blocks) correspond to
natural algebraic transformations of vector spaces over GF(2).
This was apparently not previously known.
See "The Thing and I."
See as well today's post Geometry for Belgium.
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Geometry for Belgium
Other matching patterns . . .
Tuesday Weld in the 1972 film of Didion's Play It As It Lays :
Note the making of a matching pattern.
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Wednesday, May 15, 2024
Transformer Problems (Before the Pretrained Ones)
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Saturday, May 11, 2024
Saturday, May 4, 2024
What Lies Between
"I perceived . . . cinema is that which is between things,
not things [themselves] but between one and another."
— Jean-Luc Godard, "Introduction à une véritable histoire
du cinéma," Albatros , Paris, 1980, p. 145
|
Log24 on 10 Dec. 2008 —
|
Log24 on 12 Dec. 2008 —
|
Between the two image-dates above . . .
" 'The jury is still out on how long – and whether – people are actually
going to understand this.' It took the world 150 years to realize
the true power of the printing press . . . ." — Cade Metz
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Friday, May 3, 2024
Miller Class: The Screws
Related entertainment . . .
A Warren Area High School Key Club IG post of Nov. 10, 2023, and,
on that same date . . .
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Thursday, May 2, 2024
Hitchcock Studios
The New York Times today reports the death at 90 of
Peggy Mellon Hitchcock, who arranged for Timothy Leary's
accomodation at the Hitchcock Estate, on April 9, 2024 . . .
Also on April 9 —
A rather different Hitchcock image —
This is from a Log24 search for Hitchcock Cube.
"Before time began . . ." — Optimus Prime.
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Friday, April 26, 2024
Thursday, April 18, 2024
Summer Solstice Entertainment, 2019
Related Helen Mirren image . . .
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Wednesday, March 27, 2024
Thursday, March 21, 2024
Library Note: Chicago Exposition
|
"Chang noted that 'the story starts slowly, for
1. Chang, Margaret A. "The King in the Window". |
Some will prefer exposition more closely related to Chicago.
From a Log24 search for that word . . .
The above phrase "the intersection of storytelling and visual arts"
suggests a review . . .
Some exposition that does not go back thousands of years —
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The Cross Section
Addendum for Christopher Nolan — Dice and the Eightfold Cube.
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Wednesday, March 20, 2024
Wednesday, March 6, 2024
In Memory of an AutoCAD Cofounder
One of a Kind Malfunction
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Sunday, March 3, 2024
Deep Blue Research: A Report by You.com AI
|
Cullinane Diamond Theorem Research Report by https://you.com/?chatMode=research on March 3, 2024 Overview of the Cullinane Diamond Theorem The Cullinane Diamond Theorem is a mathematical concept developed by Steven H. Cullinane that explores the symmetrical properties of certain geometric patterns. It is particularly concerned with the structure of finite projective geometry as it relates to the 35 square patterns found in R. T. Curtis's Miracle Octad Generator (MOG). The theorem asserts that every G-image of a diamond figure D possesses some form of ordinary or color-interchange symmetry. Symmetry and Group Theory The theorem is deeply rooted in group theory, with G being isomorphic to the affine group A on the linear 4-space over the finite field GF(2). This group has a significant order of 322,560 and underlies the symmetry of both the diamond-theorem figures and the square patterns of the MOG . The 35 structures of the 840 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), which explains the symmetry of the line diagrams and the two-color patterns. The Diamond Theorem in Various Dimensions The diamond theorem can be applied to different cases, such as the 4×4, 2×2, and even 3-dimensional cubes like the 2x2x2 and 4x4x4 cubes. The generalized statement of the theorem involves inscribing a white diamond in a black square and dividing it into four quadrants, each with a diagonal split into black and white halves. Algebraic Implications The theorem has algebraic implications, such as the ability to express every 4-coloring as a sum of three 2-colorings. This algebraic formulation is useful for demonstrating that certain four-colored graphic images can form a ring under multiplication. There is also an infinite family of such 'diamond' rings, isomorphic to rings of matrices over GF(4). Philosophical and Cultural Context Steven H. Cullinane has also explored the philosophical implications of the diamond theorem, referring to it in the context of Plato's Meno dialogue and opposing it to the 'Story Theory of truth' as discussed by Richard J. Trudeau. The theorem is seen as a correct statement of the relation between geometry and the physical universe. Clarification on Cullinan Diamonds It is important to note that the Cullinane Diamond Theorem is unrelated to the Cullinan diamond, which is a famous large gemstone discovered in South Africa and now part of the British Crown Jewels. The Cullinan diamond has its own historical and cultural significance, with its various cut shapes and ownership history. In summary, the Cullinane Diamond Theorem is a mathematical concept that reveals the symmetrical properties of certain geometric patterns, with applications in group theory and algebra. It also has philosophical implications, drawing connections to classical ideas about truth and geometry. |
Instagram ad for You.com AI in research mode
"Show me ALL your sources, babe."
— Line adapted from Leonardo DiCaprio
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Friday, February 23, 2024
Turing-Award Winner Memorial
In memory of the inventor of the Pascal programming language,
who reportedly died on New Year's Day, 2024, an image from
this journal on that date —
"Before time began, there was the Cube." — Optimus Prime
See as well a Pascal that some will prefer, and my own ACM card.
That card gives Warren, PA, as my location… which is no longer
always the case… but Warren is where I learned, and worked, in
computing, after a somewhat belated education in pure mathematics
in New York State.
Some personal background is suggested by…
- Yesterday's post on the words "sexadecimal" and "hexadecimal"
- The related word "sexagesimal," referring to base-60 numbering
-
My high school class of 1960 —
quod vide.
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Tuesday, February 20, 2024
Backlight
The epigraph of the previous post —
"To Phaedrus, this backlight from the conflict between
the Sophists and the Cosmologists adds an entirely
new dimension to the Dialogues of Plato." — Robert M. Pirsig
Related reading and art for academic nihilists — See . . .
Reading and art I prefer —
Love in the Ruins , by Walker Percy, and . . .
Van Gogh (by Ed Arno) and an image and
a passage from The Paradise of Childhood
(by Edward Wiebé):
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Friday, February 2, 2024
“Platonic Good Mathematics”
"Before time began, there was the Cube."
— Optimus Prime
"Spiel ist nicht Spielerei."
— Friedrich Froebel
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Monday, January 15, 2024
Storylines Colliding: The Cornfield Paradox
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Monday, January 1, 2024
Molly’s Date
The Des Moines Register 2018 story pictured at lower right in
the previous post featured one Molly Vincent. A holiday greeting
on her LinkedIn page leads to . . .
Also on the above YouTube date —
Related reading — "Weird Pharaonic Monument" —
"Before time began, there was the Cube." — Optimus Prime
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Monday, December 25, 2023
“Weird Pharaonic Monument”
Epigraph for Cormac McCarthy —
|
"When I got to high school the first place I went was to the library. It was just a small room with a desk and maybe a thousand books. Maybe not that. But among them was a volume of Berkeley. I dont know what it was doing there. Probably because Berkeley was a bishop. Well. Almost certainly because Berkeley was a bishop. But I sat in the floor and I read A New Theory of Vision. And it changed my life. I understood for the first time that the visual world was inside your head. All the world, in fact. I didnt buy into his theological speculations but the physiology was beyond argument. I sat there for a long time. Just letting it soak in."
— McCarthy, Cormac. Stella Maris (p. 39). |
From this journal on April 18, 2023 —
" NY Times columnist's advice to the recent Harvard donor of $300 million —
'At least make them build you some weird pharaonic monument.' "
Illustration suggested by my own high-school library reading many years ago —
Click to enlarge:
"Before time began, there was the Cube." — Optimus Prime
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Wednesday, December 13, 2023
Director’s Exit
For St. Lucy's Day . . . Vide another post now tagged "Cube School."
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Sunday, December 10, 2023
Erin’s Barron’s: Skye High
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Tuesday, December 5, 2023
Wednesday, November 22, 2023
For E. Lily Yu* — Devs Setting
Friday, July 11, 2014
|
* Author of Jewel Box: Stories ( Erewhon Books, Oct. 24, 2023).
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Sunday, November 19, 2023
Six Dimensions
"Sharpie, we have condensed six dimensions into four,
then we either work by analogy into six, or we have to use math
that apparently nobody but Jake and my cousin Ed understands.
Unless you can think of some way to project six dimensions into three–
you seem to be smart at such projections."
I closed my eyes and thought hard. "Zebbie, I don't think it can be done.
Maybe Escher could have done it."
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Friday, November 17, 2023
Classicism Continued: An Apotheosis of Modernity
|
From Chapter 23, "Poetry," by Adam Parkes, in Writing in 1910–11, the English poet and critic T. E. Hulme claimed that the two major traditions in poetry, romanticism and classicism, were as different as a well and a bucket. According to the romantic party, Hulme explained, humankind is “intrinsically good, spoilt by circumstance”; that is, our nature is “a well, a reservoir full of possibilities.” For the classical party, however, human nature is “like a bucket”; it is “intrinsically limited, but disciplined by order and tradition to something fairly decent” (Hulme 1987: 117). But it was not only that romanticism and classicism were as dissimilar as a well and a bucket; their contents were different, too. To draw water from the well of romanticism was, in effect, to pour a “pot of treacle over the dinner table,” while the classical bucket was more likely to be full of little stones – or jewels, perhaps. Romanticism, in Hulme’s view, was the result of displaced religious fervor; it represented the return of religious instincts that the “perverted rhetoric of Rationalism” had suppressed, so that “concepts that are right and proper in their own sphere are spread over, and so mess up, falsify and blur the clear outlines of human experience” (Hulme 1987: 118). Classicism, by contrast, traded in dry goods – dry, hard goods, to be precise. Hulme left little doubt as to which side he was on. “It is essential to prove,” he argued, “that beauty may be in small, dry things. The great aim is accurate, precise and definite description. . . . I prophesy that a period of dry, hard, classical verse is coming” (Hulme 1987: 131–3). If by “dry, hard, classical verse” Hulme meant poems looking like the fragments of Sappho, he didn’t have to wait long to see his prophecy fulfilled.
The hard sand breaks,
Far off over the leagues of it, 228
playing on the wide shore, So wrote Hilda Doolittle in “Hermes of the Ways,” the first poem that she signed “H. D., Imagiste” at the behest of her fellow American expatriate Ezra Pound. From Pound’s perspective, the Imagist movement that he co-founded in 1912 with H. D. and the English poet Richard Aldington was finished well before the First World War began in August 1914; throughout this war-torn decade, however, Imagism continued to spawn the poetry of “small, dry things” whose coming Hulme had predicted a few years before. Indeed, modernist poets weren’t content merely to break down the extended heroic narratives – the “spilt religion,” as Hulme put it – of their treacly nineteenthcentury predecessors; they insisted on breaking down small things into ever-smaller particles and subparticles. This logic of disintegration is clearly at work in poems like “Hermes of the Ways,” where each line is metrically unique, creating a sense of perpetual freshness – an apotheosis of modernity, as it were. REFERENCE Hulme, T. E. (1987). Speculations: Essays on Humanism and the Philosophy of Art, ed. Herbert Read. London and New York: Routledge and Kegan Paul. First published 1924. |
Compare and contrast:
Jeremy Gray,
Plato's Ghost: The Modernist Transformation of Mathematics,
Princeton University Press, first edition Sept. 22, 2008 —
"Here, modernism is defined as an autonomous body of ideas,
having little or no outward reference, placing considerable emphasis
on formal aspects of the work and maintaining a complicated—
indeed, anxious— rather than a naïve relationship with the
day-to-day world, which is the de facto view of a coherent group
of people, such as a professional or discipline-based group
that has a high sense of the seriousness and value of what it is
trying to achieve. This brisk definition…."
(Quoted at the webpage Solomon's Cube.)
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Sunday, November 12, 2023
“Cum grano salis”
The above title is from the Black Mass performed by Boris Karloff
in a classic 1934 horror film. An illustration —
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Friday, November 10, 2023
Altman’s Monster* Version of Musk’s Grok Logo
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Tuesday, November 7, 2023
If It’s Tuesday . . .
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Tuesday, October 24, 2023
A Bond with Reality: The Geometry of Cuts
Illustrations of object and gestures
from finitegeometry.org/sc/ —
Object
Gestures
An earlier presentation of the above
seven partitions of the eightfold cube:
|
|
Related mathematics:
|
The use of binary coordinate systems
Natural physical transformations of square or cubical arrays See "The Thing and I." |
and . . .
Related entertainment:
Or Matt Helm by way of a Jedi cube.
Comments Off on A Bond with Reality: The Geometry of Cuts
Monday, October 23, 2023
Friday, October 13, 2023
Hungarian Puzzle
See "Cube Space" + Lovasz.
This search was suggested by . . .
The conclusion of Solomon Golomb's
"Rubik's Cube and Quarks,"
American Scientist , May-June 1982 —
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Artbox.group
This new URL will forward to http://m759.net/wordpress/?s=Solomon+Cube.
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Lightbox for a Friday the 13th
For a different sort of Lightbox, more closely associated with
the number 13, see instances in this journal of . . .
(Adapted from Encyclopaedia Britannica,
Eleventh Edition (1911), Crystallography .)
"Before time began . . . ." — Optimus Prime
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Monday, September 25, 2023
Cool Kids’ Vocabulary: Anthropic
https://www.merriam-webster.com/dictionary/anthropic —
"Questions abound about how the various proposals intersect with
anthropic reasoning and the infamous multiverse idea."
— Natalie Wolchover, WIRED, 16 June 2019
A more recent, and notable, use of "anthropic" :
https://techcrunch.com/2023/09/25/
amazon-to-invest-up-to-4-billion-in-ai-startup-anthropic/ —
"As part of the investment agreement, Anthropic will use
Amazon’s cloud giant AWS as a primary cloud provider for
mission-critical workloads . . . ."
The cloud giant appeared here recently :
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Friday, September 22, 2023
Figurate Space
For the purpose of defining figurate geometry , a figurate space might be
loosely described as any space consisting of finitely many congruent figures —
subsets of Euclidean space such as points, line segments, squares,
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them.
Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate space, considered as a collection of figures — vertices, edges,
faces — seen in the nineteenth century as acted upon by a group of symmetries .
More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M24 . That group may be viewed as acting
on various subsets of a 24-set… for instance, the 759 octads that are analogous
to the faces of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.
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Tuesday, September 19, 2023
Mystery Box
Sarah Larson in The New Yorker yesterday —
"Having revealed itself, the Perelman Performing Arts Center (PAC NYC),
designed by Joshua Ramus and his firm, REX, retains an air of mystery:
it’s a giant marble-sheathed cube, beige and opaque by day and warmly
aglow by night, fronted by a two-story staircase that evokes the approach
to a Mayan temple or the gangway to an alien spacecraft. What’s inside?"
Always an interesting question . . .
From "Made for Love" (2021) — Lyle Herringbone:
See as well yesterday's post
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Thursday, September 14, 2023
Wednesday, September 13, 2023
Edgelord School
|
Monday, May 8, 2017
New Pinterest Board
|
The face at lower left above is that of an early Design edgelord.
A product of that edgelord's school —
See a design by Prince-Ramus in today's New York Times —
Remarks quoted here on the above San Diego date —
A related void —
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Friday, September 1, 2023
Stella!
An animated GIF that shows the basic unit for
the "design cube" pages at finitegeometry.org —
From a post of Dec. 8, 2010, the (somewhat) related Stella Octangula —
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Friday, August 25, 2023
On the Night Road from Marfa
"I’m really interested in exploring space."
— New Yorker cover artist for the Aug. 28, 2023, issue.
Related cinematic art . . .
From a search in this journal for Nocturnal —
For some Bright Art Blocks Moments , see Cube Epiphany .
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Thursday, August 10, 2023
Lines and Shadows (title taken from Wambaugh)
Lines
"Listen to the wind blow, down comes the night
Running in the shadows, damn your love, damn your lies"
— Fleetwood Mac, "The Chain"
Shadows
Related YouTube and Log24 date: Sept. 27, 2018 —
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Monday, July 24, 2023
Topdot Art
The new URL topdot.art refers to depictions of
the top dot (or point, or vertex) in a vertex-edge diagram
of a square, cube, or hypercube that has been rotated
so that the bottom dot (or point, or vertex), represented by
all-zero coordinates in a labeling, is at the bottom …
and the top dot (or point, or vertex), represented by
all-one coordinates in a labeling, is at the top.
See (for instance) the Log24 post Physicality (Oct. 5, 2022).
Related philosophical remarks: Einheit .
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Friday, July 21, 2023
Sunday, July 2, 2023
Badlands Philosophy
This afternoon's Windows lockscreen is Badlands National Park —
From this morning's post, a phrase from Schopenhauer —
"Apparent Design in the Fate of the Individual."
An apparent design in the philosophy of Optimus Prime —
"Before time began, there was the Cube" —
Click the image for further remarks.
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Saturday, July 1, 2023
Mechanical Plaything (Hinged) vs. Conceptual Art (Unhinged)
"Infinity Cube" … hinged plaything, for sale —
"Eightfold Cube" … un hinged concept, not for sale—
See as well yesterday's Trickster Fuge ,
and a 1906 discussion of the eightfold cube:
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Wednesday, June 21, 2023
Annals of Magical Thinking: The Rushmore Embedding
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Saturday, June 10, 2023
Green, Orange, Black
The colors surrounding Watson's body in the above
"bandeau" photo suggest a review. A search in this journal
for Green+Orange+Black yields . . .
In the above image, the "hard core of objectivity" is represented
by the green-and-white eightfold cube. The orange and black are,
of course, the Princeton colors.
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Sunday, June 4, 2023
The Galois Core
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Friday, June 2, 2023
Geometric Entertainment
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Reichenbach’s Fell Swoop
See The Eightfold Cube and . . .
Art is magic delivered from
the lie of being truth.
— Theodor Adorno, Minima moralia,
London, New Left Books, 1974, p. 222
(First published in German in 1951.)
The director, Carol Reed, makes…
impeccable use of the beauty of black….
— V. B. Daniel on The Third Man
I see your ironical smile.
— Hans Reichenbach
Adorno, The Third Man, and Reichenbach
are illustrated below (l. to r.) above the names of
cities with which they are associated.
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Thursday, June 1, 2023
Antwerp Revisited: Diamond Space
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Tuesday, May 30, 2023
The Tour
In memory of a co-founder of Hollywood's "Magic Castle"
who reportedly died at 92 on Sunday . . .
From posts that were tagged "Blake Tour" on Sunday —
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Sunday, May 14, 2023
“The Structure of Space”
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 —
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Wednesday, May 10, 2023
Saving the Appearances
“… I realized that to me,
Gödel and Escher and Bach
were only shadows
cast in different directions by
some central solid essence.
I tried to reconstruct
the central object, and
came up with this book.”
Related images —
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Friday, May 5, 2023
For St. Corky Lee
Today's Google Doodle honors a Chinese-American photographer
who reportedly died on January 27, 2021.
From his dies natalis (birth into heaven, in the Catholic tradition) —
See as well some background on a Chinese-related cube.
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Thursday, April 20, 2023
Alphabet Meets Gestalt . . .
Continued from April 18 .
"Working with words to create art
and working with your hands to create art
seem like two separate activities to me."
— Cover artist, The New Yorker , on April 17
See also Alphabet Blocks in this journal
as well as Escher's Verbum.
|
|
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Saturday, April 8, 2023
Annals of Journalism
Update of 12:31 PM ET —
The time of this post, 12:27 PM ET,
suggests a 12/27 flashback:
Click the above image for a related Log24 post of 15 years ago today.
A related literary remark —
"Imagine Raiders of the Lost Ark set in 20th-century London, and then
imagine it written by a man steeped not in Hollywood movies but in Dante
and the things of the spirit, and you might begin to get a picture…."
— Doug Thorpe in an Amazon.com book review, not of Dark Materials.
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Thursday, April 6, 2023
A Literary Supplement
Religious remarks in the Times Literary Supplement
issue dated April 7, 2023 (Good Friday) suggest a
review of other remarks — from July 1, 2019 —now
tagged The Exploded Cube. Some will prefer more useful
types of explosions.
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Sunday, March 19, 2023
For Your Consideration
Mank, Baez, Collins — A trip back to Christmas Eve, 2021.
Related art (via Baez) for Josefine Lyche —
See also Lyche in Log24 posts tagged Star Cube.
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Saturday, March 18, 2023
Blocking Groups*
Kitty in Uncanny X-Men #168 (April 1983)
"Try Bing Chat, Kitty."
* A Harvard phrase for a process analogous to that of the Hogwarts Sorting Hat.
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Zu diesem Themenkreis
From last night's update to the previous post —
|
The use of binary coordinate systems
Natural physical transformations of square or cubical arrays See "The Thing and I." |
From a post of May 1, 2016 —
|
Mathematische Appetithäppchen: Autor: Erickson, Martin —
"Weitere Informationen zu diesem Themenkreis finden sich |
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Friday, March 17, 2023
Bing Chat Continues
Update at 9 PM ET March 17: A related observation by SHC —
The use of binary coordinate systems as a conceptual tool
Natural physical transformations of square or cubical arrays
of actual physical cubes (i.e., building blocks) correspond to
natural algebraic transformations of vector spaces over GF(2).
This was apparently not previously known.
See "The Thing and I."
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Sunday, March 12, 2023
Monday, February 13, 2023
The Adelson Shadow
Related search results —
From a different Adelson, in a Log24 post from 2003 —
Related geometric entertainment —
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Friday, February 10, 2023
Interstices and Symmetry
Call a 4×4 array labeled with 4 copies each
of 4 different symbols a foursquare.
The symmetries of foursquares are governed
by the symmetries of their 24 interstices —
(Cullinane, Diamond Theory, 1976.)
From Log24 posts tagged Mathieu Cube —
A similar exercise might involve the above 24 interstices of a 4×4 array.
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Monday, February 6, 2023
Interality Studies
|
You, Xi-lin; Zhang, Peter. "Interality in Heidegger."
The term "interology" is meant as an interventional alternative to traditional Western ontology. The idea is to help shift people's attention and preoccupation from subjects, objects, and entities to the interzones, intervals, voids, constitutive grounds, relational fields, interpellative assemblages, rhizomes, and nothingness that lie between, outside, or beyond the so-called subjects, objects, and entities; from being to nothing, interbeing, and becoming; from self-identicalness to relationality, chance encounters, and new possibilities of life; from "to be" to "and … and … and …" (to borrow Deleuze's language); from the actual to the virtual; and so on. As such, the term wills nothing short of a paradigm shift. Unlike other "logoi," which have their "objects of study," interology studies interality, which is a non-object, a no-thing that in-forms and constitutes the objects and things studied by other logoi. |
Some remarks from this journal on April 1, 2015 —
Manifest O
|
| 83-06-21 | An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. |
| 83-10-01 | Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. |
| 83-10-16 | Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. |
| 84-09-15 | Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. |
The above site, finitegeometry.org/sc, illustrates how the symmetry
of various visual patterns is explained by what Zhang calls "interality."
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Friday, January 27, 2023
The Stone
Here stands the mean, uncomely stone,
’Tis very cheap in price!
The more it is despised by fools,
The more loved by the wise.
— https://jungcurrents.com/
the-story-of-the-stone-at-bollingen
Not so cheap:
Identical copies of the above image are being offered for sale
on three websites as representing a Masonic "cubic stone."
None of the three sites say where, exactly, the image originated.
Image searches for "Masonic stone," "Masonic cube," etc.,
fail to yield any other pictures that look like the above image —
that of a 2x2x2 array of eight identical subcubes.
For purely mathematical — not Masonic — properties of such
an array, see "eightfold cube" in this journal.
The websites offering to sell the questionable image —
|
Getty —
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|
Alamy —
https://www.alamy.com/
|
|
Photo12 —
https://www.photo12.com/en/image/
No price quoted on public page:
|
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Sunday, January 22, 2023
Preform
Sometimes the word "preform" is not a misspelling.
"… there are present in every psyche forms which are unconscious
but nonetheless active — living dispositions, ideas in the Platonic sense,
that preform and continually influence our thoughts and feelings and actions."
The Source: Jung on a facultas praeformandi . . .
Illustration —
"A primordial image . . . .
the axial system of a crystal"
For those who prefer a Jewish approach to these matters —
(Post last updated at about 2:10 PM ET on Jan. 23, 2023.)
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“Preform an affine transformation”
"Before time began, there was the Cube." — Transformers (2007)
Related literature:
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Saturday, January 21, 2023
Monday, January 16, 2023
Saturday, January 14, 2023
Châtelet on Weil — A “Space of Gestures”
|
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 .
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Friday, January 13, 2023
The “Diamond Space” of Mazzola
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."
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Wednesday, January 11, 2023
Raiders of the Lost Dark
A sequel to the previous post, "How the Darkness Gets In" —
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Tuesday, December 27, 2022
The Forms of Being
"If the window is this matrix of ambi- or multivalence,
and the bars of the windows-the grid-are what help us
to see, to focus on, this matrix, they are themselves
the symbol of the symbolist work of art. They function as
the multilevel representation through which the work of art
can allude, and even reconstitute, the forms of Being."
— Page 59, Rosalind Krauss, "Grids," MIT Press,
October , Vol. 9 (Summer, 1979), pp. 50-64
Related material —
Click the above image for a related Log24 post of 15 years ago today.
A related literary remark —
"Imagine Raiders of the Lost Ark set in 20th-century London, and then
imagine it written by a man steeped not in Hollywood movies but in Dante
and the things of the spirit, and you might begin to get a picture…."
— Doug Thorpe in an Amazon.com book review, not of Dark Materials.
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Monday, December 26, 2022
Super-8 Box
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Friday, December 23, 2022
“Was ist Raum?” — Bauhaus Founder Walter Gropius
"Was ist Raum, wie können wir ihn
erfassen und gestalten?"
The Theory and
Organization of the
Bauhaus (1923)
A relevant illustration:
At math.stackexchange.com on March 1-12, 2013:
“Is there a geometric realization of the Quaternion group?” —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
These references will not appeal to those who enjoy modernism as a religion.
(For such a view, see Rosalind Krauss on grids and another writer's remarks
on the religion's 100th anniversary this year.)
Some related nihilist philosophy from Cormac McCarthy —
"Forms turning in a nameless void."
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Wednesday, November 30, 2022
Three Representations
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Tuesday, November 29, 2022
Mere Synchronology
The date — January 9, 2010 — of the Guardian book review
in the previous post was noted here by a top 40 music list
from that same date in an earlier year.
Update of 4:07 AM ET the same morning:
Fans of Cormac McCarthy's recent adventures in unreality
might enjoy interpreting the time — 3:25 AM ET — of this post
as the date 3/25, and comparing the logos, both revisited
and new, in a Log24 post from 3/25 . . .
Helen Mirren with plastic Gankyil .
. . . with the logo of a venue whose motto is
"Reality is not enough."
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Monday, November 28, 2022
Groups, Spaces, and Ripoffs
"Rubik's Cube, and the simpler [2x2x2] Super Cube, represent
one form of mathematical and physical reality."
— Solomon W. Golomb, "Rubik's Cube and Quarks:
Twists on the eight corner cells of Rubik's Cube
provide a model for many aspects of quark behavior,"
American Scientist , Vol. 70, No. 3 (May-June 1982), pp. 257-259
From the last (Nov. 14, 2022) of the Log24 posts now tagged Groups and Spaces —
From the first (June 21, 2010) of the Log24 posts now tagged Groups and Spaces —
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Thursday, November 10, 2022
For Students of the Forked Tongue
The above 1975 book by Robert Greer Cohn, Modes of Art, is
Volume I of a planned three-volume work.
The passage below is from a review of Cohn's Vol. II, Ways of Art —
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Franklin, Ursula (1987) "Book Review: A Critical Work II. . . . . Those not familiar with the author's epistemology should begin with Appendix A of Ways of Art , a schematic demonstration of his tetrapolar-polypolar-dialectic, especially as it concerns the development of the French novel within the European tradition. But this dialectic, which has antecedents in Kierkegaard, Mallarme and Joyce, underlies all art, because: "this dimensional pulsation, or tetrapolar (and polypolar) higher vibrancy is, in short, the stuff of life: life is vibrant in this more complex way as well as in the more bipolar sense" (7). Cohn shows that "far out enough" the male or linear and the female or circular, the male vertical and the female horizontal dimensions "tend to merge as in relativity theory" (19). Ways of Art shows us the way through a historical becoming of art in its complex dialectic in which the metonymic (horizontal) axis constantly interrelates with the metaphoric (vertical). "Life is the mother, art the father" (vii); hence Cohn's quarrel with most contemporary Feminism, which is pronounced throughout the volume. Firmly grounded in its author's tetra-polypolar epistemology, this beautiful book becomes, however, at no point dryly abstract; it is the mature work of a true humanist who stands in clear and open opposition to the dehumanizing trend of "the quasi-scientific reductionism and abstract gimmickry of a great deal of current academic literary study, bellwethered by the structuralists, post-structuralists, and deconstructionists" (vi). Abundant footnotes constitute a substantial part of Ways of Art , on occasion developing insights almost into essays demonstrating crucial points along the general flow of the tradition from "Obscure Beginnings;' the opening chapter, to our "Contemporaries;' the last. Cohn reminds us that "In the Beginning was the Word;' for the Judaeo-Christian tradition at least, which his study fervently embraces; thus, for example, in Appendix 0 on "The Dance of the Sexes;' he censures "those who live by slogans, camps, and peer-opinion, the countless little bastard cults which characterize an era which has massively veered away from our free and beautiful Greco-Judaeo-Christian tradition" (332). Cohn traces man's way and that of his myths and rituals culminating in his art from that beginning along the lines of Freud, Neumann and Cassirer, and many others, always demonstrating the underlying polypolar dialectical rhythm. Thus in "From Barbarism to Young Culture;' we follow the Celts to Druidic ritual, Hebrew beginnings to the Psalms, Dionysian ritual to Greek tragedy, and thence to the beginnings of French dramatic literature originating in the Quem quaeritis sequence of the medieval Mass. Along the way arises artistic symbolism, for Cohn synonymous with "effective poetry;' to finally "ripen in France as never before" (99). Table I (134) graphs this development from the twelfth to the late nineteenth and early twentieth centuries. The author traces the rise of the artistic vocation from its antecedents in the double function of bard and priest, with the figure of Ronsard at the crossroads of that dying institution and the nascent concept of personal glory. "The Enlightenment Vocation" is exemplified in Montaigne, who humanizes the French cultural elite and points the way to French classicism and, farther down the road, after the moral collapse with the outgoing reign of Louis XIV, toward the Age of Reason. Clearly the most significant figure of the French Enlightenment for all of Western civilization is Rousseau, and Cohn beautifully shows us why this is so. Subsequently, "the nineteenth-century stage of the writer's journey will lead, starting from the crossroads of Rousseau, primarily in these two directions: the imperialistic and visionary prose of Balzac, the equally ambitious poetry of Mallarme", brothers under the skin" (199). And these two paths will then be reconciled in Proust's monumental A la recherche du temps perdu . . . . . |
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Tuesday, November 1, 2022
From “Goethe on All Souls’ Day”
The above title is that of a Log24 post on St. Cecilia's Day in 2017
that quoted some earlier All Souls' Day remarks from Berlin.
From that post —
Exercise: Explain why the lead article in the November issue of
Notices of the American Mathematical Society misquotes Weyl
and gives the misleading impression that the example above,
the eightfold cube , might be part of the mathematical pursuit
known as geometric group theory .
Background: Earlier instances here of the phrase "geometric group theory."
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Monday, October 31, 2022
Folklore vs. Mathematics
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Wednesday, October 12, 2022
Wednesday, October 5, 2022
Physicality
"Battles argues that 'the experience of the physicality
of the book is strongest in large libraries,' and stand
among the glass cube at the center of the British Library,
the stacks upon stacks in Harvard’s Widener Library, or
the domed portico of the Library of Congress and tell me
any differently."
— Ed Simon, Binding the Ghost: Theology, Mystery, and
the Transcendence of Literature. Hardcover – April 19, 2022.
… And back to cube:
Related meditation: Beer Summit.
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Thursday, September 29, 2022
The 4×6 Problem*
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?
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Wednesday, September 28, 2022
Bitspace Note
Update of 5:20 AM ET on Sept. 29. 2022 —
The octads of the [24, 8, 8] cube-motif code
can be transformed by the permutation below
into octads recognizable, thanks to the Miracle
Octad Generator (MOG) of R. T. Curtis, as
belonging to the Golay code.
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The Madness of Art
The title is by Henry James.*
For examples, see the Sept. 19 webpage below . . .
… and, in this journal, posts from that same date now tagged Cube Codes.
*
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Friday, September 16, 2022
Symmetric Generation
The above is about a subspace of the
24-dimensional vector space over GF(2)
. . . "An entire world of just 24 squares,"
to adapt a phrase from other Log24
posts tagged "Promises."
Update of 1:45 AM ET Sept. 18, 2022 —
It seems* from a Magma calculation that
the resemblance of the above extended
cube-motif code to the Golay code is only
superficial.
Without the highly symmetric generating codewords that were added
to extend its dimension from 8 to 12, the cube-motifs code apparently
does , like the Golay code, have nonzero weights of only 8, 12, 16, and 24 —
Perhaps someone can prove there is no way that adding more generating
codewords can turn the cube-motif code into the Golay code.
* The "seems" is because I have not yet encountered any of these
relatively rare (42 out of 4096) purported weight-4 codewords. Their
apparent existence may be due to an error in my typing of 0's and 1's.
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Wednesday, September 14, 2022
Tuesday, September 13, 2022
A Helpful Survey of the Literature
Some background for the exercise of 9/11 —
Vera Pless, "More on the uniqueness of the Golay codes,"
Discrete Mathematics 106/107 (1992) 391-398 —
"Several people [1-2,6] have shown that
any set of 212 binary vectors of length 24,
distance ≥ 8, containing 0, must be the
unique (up to equivalence) [24,12,8] Golay code."
[1] P. Delsarte and J.M. Goethals, "Unrestricted codes
with the Golay parameters are unique,"
Discrete Math. 12 (1975) 211-224.
[2] A. Neumeier, private communication, 1990.
[6] S.L. Snover, "The uniqueness of the
Nordstrom-Robinson and the Golay binary codes,"
Ph.D. Thesis, Dept. of Mathematics,
Michigan State Univ., 1973.
Related images —
"Before time began, there was the Cube."
"Remember, remember the fifth of November"
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“We Got This Covered”
The previous post's quotation of the word "leitmotif" suggests a review:
See as well Sunday's post "Raiders of the Lost Space."
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