Saturday, October 24, 2009
A search for “Chinese Cube” (based on the the previous entry’s title) reveals the existence of a most interesting character, who…
“… has attempted in his books to produce a Science and Art of Reasoning using the simplest of the Platonic solids, the Cube. [His] model also parallels, in some ways, the Cube of Space constructed from the Sepher Yetzirah’s attributions for the Hebrew letters and their direction. [He] elucidated his theories at great length….”
— More…
For related remarks, see the link to Solomon’s Cube from the previous entry.
Then of course there is…
Click on figure for details.
Comments Off on Chinese Cubes Continued
Thursday, October 22, 2009
From the Bulletin of the American Mathematical Society, Jan. 26, 2005:
What is known about unit cubes
by Chuanming Zong, Peking University
Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in n-dimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all….
From Log24, now:
What is known about the 4×4×4 cube
by Steven H. Cullinane, unaffiliated
Abstract: The 4×4×4 cube, from one point of view, is among the simplest and the most important objects in n-dimensional binary space. In fact, as one will see from the links below, it is not simple at all.
Solomon's Cube
The Klein Correspondence, Penrose Space-Time, and a Finite Model
Non-Euclidean Blocks
Geometry of the I Ching
Related material:
Monday's entry Just Say NO and a poem by Stevens,
"The Well Dressed Man with a Beard."
Comments Off on Chinese Cubes
Thursday, March 27, 2025
The "Loeb Fellowship in advanced environmental studies"
is not named for Arthur L. Loeb . . .
Related reading from this journal —
Posts now tagged Arthur Lee Loeb.
Related art from this journal —
The setting for the Sidney Lumet film "Deathtrap" (1982)
Monday, March 24, 2025
Related art —
From "Self-Dual Configurations and Regular Graphs" by H. S. M. Coxeter,
Bulletin of the American Mathematical Society, Vol. 56 (1950), pp. 413-455
For a related combinatorial configuration, take Oxbury's "16 lines"
to be the the 16 dots above and take the "8 points of intersection"
to be the four squares
234, 1234, 124, 24
23, 123, 12, 2
3, 13, 1, 0
34, 134, 14, 4
along with the four diamonds
234, 23, 3, 34
1234, 123, 13, 134
124, 12, 1, 14
24, 2, 0, 4.
Then each "line" is on two "points" and each "point" on
four "lines."
Note that these eight "points" — the four squares and the four diamonds
of Coxeter's figure — form the rows and columns of the following matrix:
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234
|
1234
|
124
|
24
|
|
23
|
123
|
12
|
2
|
|
3
|
13
|
1
|
0
|
|
34
|
134
|
14
|
4
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Related reading: Points with Parts .
Sunday, March 23, 2025
The recent URL cubebrick.space forwards to . . .
http://m759.net/wordpress/?tag=brick-space.
The web posts so tagged are, as one would expect,
NOT in the Harvard Library system. I was therefore
somewhat surprised to see the following popup today —
Clicking on the "Get article" link yields . . .
This metadata is actually quite helpful, as the cited article
does, in fact, give good references for what I have called,
using a term from the "Miracle Octad Generator" of R. T. Curtis,
"brick space" — the finite projective space PG(5,2).
Saturday, March 22, 2025
Note that in the above illustration, there are four sets of points
that, with the exception of the top front corner point h0, form
four equilateral triangles . . .
z1, z2, z3
y1– , y2–, y3–
y1+, y2+, y3+
h0, h1, h2. h3 .
Enthusiasts of sacred geometry may investigate the mystical properties
of this four-triangle (plus h0) labeling.
For a less mystical approach to the 2011 Yu-Oh illustration, see . . .
Friday, March 21, 2025
Two posts from the date below in an image from today's
previous post have now also been tagged Congregated Light.
See as well Conway and Congregated Light . . .
and, for Hotel New Hampshire fans,
Conway Scenic Railroad . . .
Sunday, March 16, 2025
Note that the triangles 5-9-12 and 7-8-11 in figure B above correspond
in cube A to vertices ∞ and 0 in the Aitchison Hiroshima cube below.
Comments Off on Compare and Contrast
Monday, March 10, 2025
A rather different planes solution from Log24 on March 7 —
See the illustrations in this journal on that date and . . .
"Note that in the Design Cube image above, the six faces are viewed
as suspended in space on three pairs of parallel planes, with the observer
viewing the parallel images only from the three directions front to back,
right to left, and top to bottom. In the WebSim row of six faces, the images
are pictured as they might be seen by an observer whose viewpoints vary
as he himself floats in space around the cube."
Related chrome art —
Comments Off on Face Planes Mindset Shift
Friday, March 7, 2025
From a much more abstract space, an image from the
Log24 post Symmetry of May 3, 2016 . . .
For a rotating 3D view from 2025, made with the help of AI, see . . .
Note that in the Design Cube image above, the six faces are viewed
as suspended in space on three pairs of parallel planes, with the observer
viewing the parallel images only from the three directions front to back,
right to left, and top to bottom. In the WebSim row of six faces, the images
are pictured as they might be seen by an observer whose viewpoints vary
as he himself floats in space around the cube.
Comments Off on 2025: Space Odysseys
Wednesday, March 5, 2025
Comments Off on “Old men ought to be explorers” — T. S. Eliot
Saturday, February 15, 2025
Comments Off on No Ordinary Venue
Thursday, February 13, 2025
Illustration of a July 1980 title by George Mackey —
Exploitation of Symmetry in 1981 . . .
See also the tetrahedra* in my "square triangles" letter
(1985), as well as "Senechal" in this journal.
"And we both know what memories can bring…" Do we?
* "Schläfli orthoschemes"
Comments Off on The Exploitation of Symmetry . . . Continues.
Friday, January 31, 2025
A date from the above Google search for Whanganui meaning —
September 22, 2019.
See that date in other posts now tagged Simplex.
Comments Off on Sigillum Veri: Maori Date
Wednesday, January 29, 2025
Comments Off on “A Local Habitation and a Name”
Thursday, January 23, 2025
This post was suggested by Dudeney's "Stonemason's Problem," which
in turn was suggested by the number "204" in the opening episode of
the new Apple TV Plus thriller "Prime Target."
Comments Off on The Gotham/Getty problem: “Da hats ein Eck”
Sunday, January 12, 2025
See as well the Diamond Cube at Pinterest —
See also news from the day I turned 11 …
"How many mathematicians …?"
Comments Off on Sontag Variations: I Sing the Body Electromagnetic
Tuesday, January 7, 2025
Comments Off on Parallelisms
Monday, January 6, 2025
More recently in this journal . . .
Comments Off on Ninefold
Sunday, January 5, 2025
Comments Off on Precursor to Dürer’s 1514 Square
Monday, December 16, 2024
"Each of the 64 subcubes is supposed to be marked identically,
with white caps on two opposite vertices and a black band
around the subcube that separates the two white caps."
The Source —
Related whitecap reading . . .
"Dice and the Eightfold Cube."
Comments Off on Whitecaps for Beethoven
Thursday, December 5, 2024
"Where past and future are gathered" — T. S. Eliot
* The UI/UX meaning of "chrome." See the previous post, "Chrome Cube."
Comments Off on Chrome* Square
Tuesday, December 3, 2024
Current events suggest revisiting Koreatown —
Koreatown (Los Angeles) photo by Marcela Nowak.
A different, digital, rewind … back 10 years in this journal … yields —
Click the above geometry image to enlarge it.
Related narratives . . .
Vide the Disney tale of Sith tetrahedron and Jedi cube, and, in this journal,
posts tagged "Congregated Light."
Comments Off on Back to Koreatown
Wednesday, November 27, 2024
An instance of T. S. Eliot's poetic "still point" is the
center of a 3x3x3 Galois cube made up of 27 subcubes …
Not Rubik's puzzle, whose center is a mere mechanical contrivance.
Associated with that Galois cube is the set of
13 symmetry axes of its central subcube.
The figure above is not unrelated to the so-called "free will theorem."
Mathematician Peter J. Cameron's recent quotation of St. Bernard*
on free will and grace, while not impressive as a philosophical
statement, is at least preferable to the TV sitcom "Will and Grace."
See also the notion of free will in other posts tagged "Congregated Light."
Some context: Tom Wolfe, below, on the word "clerisy." It seems that the
word applies to many academics besides those in areas named by Wolfe.
* Vide http://www.catholictradition.org/Tradition/efficacious-grace3.htm#67 —
"De gratia et Libero arbitrio, chaps. 1 and 14."
Comments Off on For the Still Point: “Congregated Light”
The domain bitcube.space has now been renewed for another year.
It leads to — among other things — the following remarks . . .
Towards a Philosophy of Real Mathematics, by David Corfield, Cambridge U. Press, 2003, p. 206:
“Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in the form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey (1992) to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries.”
For a simpler example of this idea, see the entities in The Diamond Theorem, the decomposition in A Four-Color Theorem, and the space in Geometry of the 4×4 Square. The decomposition differs from that of harmonic analysis, although the subspaces involved in the diamond theorem are isomorphic to Walsh functions– well-known as discrete analogues of the trigonometric functions of traditional harmonic analysis.
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* See that phrase in this journal.
Comments Off on Hoarding Space*
Wednesday, November 13, 2024
Alias:
Victor at 194 Tower Avenue in "The Penguin"
Alibi:
Marcela_234 at Likewise.com
Romance in Numberland:
* Technical terms from pure mathematics —
For scholia on "the cube is being moved around," vide . . .
http://m759.net/wordpress/?s="Stack+Exchange" .
Comments Off on Group Theory: The Romance of Alias and Alibi*
Thursday, October 31, 2024
Comments Off on Text and Context
Wednesday, October 30, 2024
For another variation on this hallucination, see the previous post.
Comments Off on Variation on an AI Hallucination
Tuesday, October 29, 2024
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 . . .
Inside the Exploded Cube
(Log24, July 1, 2019).
Comments Off on Hallucinated Geometry
Sunday, October 27, 2024
Comments Off on For the “David Brooks” of “Canary Black” —
Grandiose Eschatological Visions!
Saturday, October 19, 2024
(Reposted from http://m759.net/wordpress/?p=74943.)
See as well Du Sucre (Log24, July 18, 2010).
Comments Off on A Seven-Eleven for Mystics: October 7 . . . 11 Years Ago
Thursday, October 10, 2024
"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.)
Comments Off on From Time to Eternity
Wednesday, October 2, 2024
https://www.behance.net/gallery/19964281/BALL-MACHINE
from . . .
See also "Cuber."
Comments Off on Activity Log: “Time Machine as Ball Machine”
Saturday, September 28, 2024
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 —
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American Mathematical Monthly, Vol. 92, No. 6
(June-July 1985), p. 443
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.
Miscellaneum 129 was a less formal statement of (C), with quotation marks instead of parentheses; this may have led many readers to think (B) was intended. To these readers, my apology.
Steven H. Cullinane
501 Follett Run Road
Warren, PA 16365
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Comments Off on Architectural Singularity
Tuesday, September 24, 2024
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Monday, September 23, 2024
Publisher (click to enlarge) —
See also a Google machine translation of the article to English.
Comments Off on Design History for a Guy Fawkes Day
Friday, September 20, 2024
"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.
Comments Off on Adapter Becomes Transformer
Wednesday, September 18, 2024
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
Comments Off on Flatiron Building Timelessness . . .
Directed by Quine, not Hitchcock
See also Weyl + Palermo in this journal —
Helen Mirren with plastic Gankyil .
Comments Off on Triskelion
Saturday, September 14, 2024
A passage accessed via the new URL Starbrick.art* —
Thursday, February 25, 2021
Filed under: General — Tags: Cube School, Octad,
Octad Generator, The Guy Embedding
— m759 @ 12:31 pm
“… What is your dream—your ideal?
What is your News from Nowhere, or, rather,
What is the result of the little shake your hand has given
to the old pasteboard toy with a dozen bits of colored glass
for contents? And, most important of all, can you present it
in a narrative or romance which will enable me to pass an
idle hour not disagreeably? How, for instance, does it compare
in this respect with other prophetic books on the shelf?”
— Hudson, W. H.. A Crystal Age , 1887.
Open Road Media, Kindle Edition, page 2.
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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.
Comments Off on Notes on a Friday the 13th Death
Thursday, September 12, 2024
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 —
From September 18, 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.
Comments Off on Structures
Thursday, September 5, 2024
I prefer K .
"Before time began . . . ." — Optimus Prime
Comments Off on Cereal Tales
Wednesday, July 10, 2024
Wednesday, July 3, 2024
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.
Comments Off on Vedic Carnival: “Hey Rubik!” *
Tuesday, July 2, 2024
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CNN — By Dan Heching
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."
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Related imagery . . .
Comments Off on Chinatown
Sunday, June 30, 2024
Comments Off on Minecraft as a design tool: The Trial
(Kafkaesque? Maybe, maybe not.)
Friday, June 28, 2024
Comments Off on The Adapter
Thursday, June 27, 2024
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.
Comments Off on Die Berliner Mitschrift
Friday, June 21, 2024
Related material — Solomon's Cube and . . .
Comments Off on OSF Project
Thursday, June 13, 2024
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 —
Comments Off on Of London Bondage … continues.
Monday, June 10, 2024
For fans of the NFT (Web 3) approach to art marketing —
My own approach is somewhat different . . .
Comments Off on Art Logos
Monday, June 3, 2024
"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."
Comments Off on Symmetry Plane
Friday, May 24, 2024
"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 . . .
Comments Off on One Lesson
Tuesday, May 21, 2024
"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.
Comments Off on For Optimus Prime
Other matching patterns . . .
Tuesday Weld in the 1972 film of Didion's Play It As It Lays :
Note the making of a matching pattern.
Comments Off on Geometry for Belgium
Wednesday, May 15, 2024
Related marketing:
Disney Easter eggs —
Comments Off on Transformer Problems (Before the Pretrained Ones)
Saturday, May 11, 2024
Comments Off on Arabesque for Cairo Sweet
Saturday, May 4, 2024
"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
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
Comments Off on What Lies Between
Friday, May 3, 2024
Related entertainment . . .
A Warren Area High School Key Club IG post of Nov. 10, 2023, and,
on that same date . . .
Comments Off on Miller Class: The Screws
Thursday, May 2, 2024
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.
Comments Off on Hitchcock Studios
Friday, April 26, 2024
Thursday, April 18, 2024
Related Helen Mirren image . . .
Comments Off on Summer Solstice Entertainment, 2019
Wednesday, March 27, 2024
Comments Off on The Triple-Threat Problem
Thursday, March 21, 2024
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Wikipedia —
"Chang noted that 'the story starts slowly, for
its complicated and rather far-fetched premises
require quite a bit of exposition, but rises to
an action-packed climax'.[1]"
1. Chang, Margaret A. "The King in the Window".
School Library Journal . Retrieved February 26, 2024 –
via Chicago Public Library.
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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 —
Comments Off on Library Note: Chicago Exposition
Addendum for Christopher Nolan — Dice and the Eightfold Cube.
Comments Off on The Cross Section
Wednesday, March 20, 2024
Comments Off on Annals of Geometric Theology
Wednesday, March 6, 2024
One of a Kind Malfunction
Comments Off on In Memory of an AutoCAD Cofounder
Sunday, March 3, 2024
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Cullinane Diamond Theorem Research Report
by https://you.com/?chatMode=research on March 3, 2024
Overview of the Cullinane Diamond Theorem
The Cullinane Diamond Theorem is a mathematical concept developed by Steven H. Cullinane that explores the symmetrical properties of certain geometric patterns. It is particularly concerned with the structure of finite projective geometry as it relates to the 35 square patterns found in R. T. Curtis's Miracle Octad Generator (MOG).
The theorem asserts that every G-image of a diamond figure D possesses some form of ordinary or color-interchange symmetry.
Symmetry and Group Theory
The theorem is deeply rooted in group theory, with G being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).
This group has a significant order of 322,560 and underlies the symmetry of both the diamond-theorem figures and the square patterns of the MOG . The 35 structures of the 840 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), which explains the symmetry of the line diagrams and the two-color patterns.
The Diamond Theorem in Various Dimensions
The diamond theorem can be applied to different cases, such as the 4×4, 2×2, and even 3-dimensional cubes like the 2x2x2 and 4x4x4 cubes.
The generalized statement of the theorem involves inscribing a white diamond in a black square and dividing it into four quadrants, each with a diagonal split into black and white halves.
Algebraic Implications
The theorem has algebraic implications, such as the ability to express every 4-coloring as a sum of three 2-colorings.
This algebraic formulation is useful for demonstrating that certain four-colored graphic images can form a ring under multiplication. There is also an infinite family of such 'diamond' rings, isomorphic to rings of matrices over GF(4).
Philosophical and Cultural Context
Steven H. Cullinane has also explored the philosophical implications of the diamond theorem, referring to it in the context of Plato's Meno dialogue and opposing it to the 'Story Theory of truth' as discussed by Richard J. Trudeau.
The theorem is seen as a correct statement of the relation between geometry and the physical universe.
Clarification on Cullinan Diamonds
It is important to note that the Cullinane Diamond Theorem is unrelated to the Cullinan diamond, which is a famous large gemstone discovered in South Africa and now part of the British Crown Jewels.
The Cullinan diamond has its own historical and cultural significance, with its various cut shapes and ownership history.
In summary, the Cullinane Diamond Theorem is a mathematical concept that reveals the symmetrical properties of certain geometric patterns, with applications in group theory and algebra. It also has philosophical implications, drawing connections to classical ideas about truth and geometry.
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Instagram ad for You.com AI in research mode
"Show me ALL your sources, babe."
— Line adapted from Leonardo DiCaprio
Comments Off on Deep Blue Research: A Report by You.com AI
Friday, February 23, 2024
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.
Comments Off on Turing-Award Winner Memorial
Tuesday, February 20, 2024
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):
Comments Off on Backlight
Friday, February 2, 2024
"Before time began, there was the Cube."
— Optimus Prime
"Spiel ist nicht Spielerei."
— Friedrich Froebel
Comments Off on “Platonic Good Mathematics”
Monday, January 15, 2024
Also from the above
"Cloverfield Paradox" review date —
From a Log24 search for "Cornfield" —
Comments Off on Storylines Colliding: The Cornfield Paradox
Monday, January 1, 2024
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
Epigraph for Cormac McCarthy —
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"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).
Knopf Doubleday Publishing Group. Kindle Edition.
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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
Comments Off on “Weird Pharaonic Monument”
Wednesday, December 13, 2023
For St. Lucy's Day . . . Vide another post now tagged "Cube School."
Comments Off on Director’s Exit
Sunday, December 10, 2023
More realistic views, near Erin Burnett's ancestral Isle of Skye —
"Lustworthy!" — Instagram
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Tuesday, December 5, 2023
Continued.
Antwerp Chevrons
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Wednesday, November 22, 2023
* Author of Jewel Box: Stories ( Erewhon Books, Oct. 24, 2023).
Comments Off on For E. Lily Yu* — Devs Setting
Sunday, November 19, 2023
Heinlein:
"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."
Escher’s Verbum
Solomon’s Cube
Comments Off on Six Dimensions
Friday, November 17, 2023
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From Chapter 23, "Poetry," by Adam Parkes, in
A Companion to Modernist Literature and Culture,
edited by David Bradshaw and Kevin J. H. Dettmar,
Blackwell Companions to Literature and Culture,
© 2006 by Blackwell Publishing Ltd.
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,
and the grains of it
are clear as wine.
Far off over the leagues of it,
the wind,
228
playing on the wide shore,
piles little ridges,
and the great waves
break over it.
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.
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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.)
Comments Off on Classicism Continued: An Apotheosis of Modernity
Sunday, November 12, 2023
The above title is from the Black Mass performed by Boris Karloff
in a classic 1934 horror film. An illustration —
Related dialogue from Log24 — "Cube mine! "
Comments Off on “Cum grano salis”
Friday, November 10, 2023
* —
A line for the Monster — "Cube mine !"
Comments Off on Altman’s Monster* Version of Musk’s Grok Logo
Tuesday, November 7, 2023
See Antwerp in this journal…
Art related to a different location in Belgium —
Comments Off on If It’s Tuesday . . .
Tuesday, October 24, 2023
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:
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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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and . . .
Galois.space .
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
Comments Off on Little Bitty Pretty Gestalt
Friday, October 13, 2023
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 —
Comments Off on Hungarian Puzzle
Comments Off on Artbox.group
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
Comments Off on Lightbox for a Friday the 13th
Saturday, September 30, 2023
Some less esoteric café alternatives … the ".cafe" domains
cubespace, foursquare, metamorph, and namespace.
And then there is a Morocco café domain for Marcela —
Comments Off on Café Society
Monday, September 25, 2023
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 :
Comments Off on Cool Kids’ Vocabulary: Anthropic
Friday, September 22, 2023
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.
Comments Off on Figurate Space
Tuesday, September 19, 2023
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
"The Mystery Box of Jena Malone."
Comments Off on Mystery Box
Thursday, September 14, 2023
Wednesday, September 13, 2023
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 —
Comments Off on Edgelord School
Friday, September 1, 2023
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 —
Comments Off on Stella!
Friday, August 25, 2023
"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 .
Comments Off on On the Night Road from Marfa
Thursday, August 10, 2023
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 —
Comments Off on Lines and Shadows (title taken from Wambaugh)
Monday, July 24, 2023
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 .
Comments Off on Topdot Art
Friday, July 21, 2023
From https://clever-chat.ai/demo/32f6d9c9107/complete
at about 3 PM ET on Friday, July 21, 2023 —
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Hi there 👋 I'm a self-service chatbot and I'm here to help you with anything related to Finite Geometry of the Square and Cube. Just ask away! 🙂
What is the diamond theorem?
The Diamond Theorem is a mathematical concept that was introduced by Steven H. Cullinane. It is closely related to finite geometry and explores the symmetries and patterns found in diamond-shaped arrays. The theorem has applications in various areas, including combinatorial designs, group theory, and quantum information theory. If you'd like more information or have any specific questions about the Diamond Theorem, feel free to ask! 😊
Is this website related to the work of Peter J. Cameron?
No, the website "Finite Geometry of the Square and Cube" is not directly related to the work of Peter J. Cameron. It is primarily focused on the study of finite geometry, particularly the symmetries and patterns found in square and cube arrays. The website is authored by Steven H. Cullinane, who has made significant contributions to the field. If you have any more questions or need further information, feel free to ask! 😊
What is the theorem of Desargues?
The theorem of Desargues, named after the French mathematician Gérard Desargues, is a fundamental result in projective geometry. It states that if two triangles are perspective from a point, then they are perspective from a line as well. In other words, if the three pairs of corresponding sides of two triangles intersect at three points that are collinear, then the three pairs of corresponding vertices of the triangles are collinear as well. This theorem has important implications in the study of projective geometry and has applications in various fields, including computer graphics and architecture. If you have any more questions or need further information, feel free to ask! 😊
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Comments Off on “Terminator Zero: Rise of the Chatbots” … Continues.
Sunday, July 2, 2023
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.
Comments Off on Badlands Philosophy
Saturday, July 1, 2023
"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:
Comments Off on Mechanical Plaything (Hinged) vs. Conceptual Art (Unhinged)
Wednesday, June 21, 2023
Comments Off on Annals of Magical Thinking: The Rushmore Embedding
Saturday, June 10, 2023
https://www.dailymail.co.uk/tvshowbiz/article-12179599/
Emma-Watson-stuns-revealing-black-bandeau-Prada.html
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.
Comments Off on Green, Orange, Black
Sunday, June 4, 2023
Rubik core:
Swarthmore Cube Project, 2008
Non- Rubik core:
Central structure from a Galois plane
(See image below.)
Comments Off on The Galois Core
Friday, June 2, 2023
From Log24 on the reported date of Singmaster's death —
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Related geometric entertainment —
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Comments Off on Geometric Entertainment
See The Eightfold Cube and . . .
Truth, Beauty, and The Good
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.
Comments Off on Reichenbach’s Fell Swoop
Thursday, June 1, 2023
The opening of the new Netflix film FUBAR suggests a review . . .
Comments Off on Antwerp Revisited: Diamond Space
Tuesday, May 30, 2023
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 —
Comments Off on The Tour
Sunday, May 14, 2023
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 —
Comments Off on “The Structure of Space”
Wednesday, May 10, 2023
Douglas Hofstadter —
“… 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 —
Escher’s Verbum
Solomon’s Cube
Comments Off on Saving the Appearances
Friday, May 5, 2023
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.
Comments Off on For St. Corky Lee
Thursday, April 20, 2023
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.
Comments Off on Alphabet Meets Gestalt . . .
Saturday, April 8, 2023
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.
Comments Off on Annals of Journalism
Thursday, April 6, 2023
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.
Comments Off on A Literary Supplement
Sunday, March 19, 2023
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.
Comments Off on For Your Consideration
Saturday, March 18, 2023
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.
Comments Off on Blocking Groups*
From last night's update to the previous post —
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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."
|
From a post of May 1, 2016 —
|
Mathematische Appetithäppchen:
Faszinierende Bilder. Packende Formeln. Reizvolle Sätze
Autor: Erickson, Martin —
"Weitere Informationen zu diesem Themenkreis finden sich
unter http://www.encyclopediaofmath.org/index.php/
Cullinane_diamond_theorem
und http://finitegeometry.org/sc/gen/coord.html ."
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Comments Off on Zu diesem Themenkreis
Friday, March 17, 2023
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."
Comments Off on Bing Chat Continues
Sunday, March 12, 2023
Comments Off on Las Mañanitas de Nuestro Jardin
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