The dimensions of the "bricks" in the R. T. Curtis
"Miracle Octad Generator": 2×4.
For those who prefer narrative to mathematics . . .
Monday, April 22, 2024
Dimensions
Tuesday, March 5, 2024
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, August 11, 2023
Dimensions for a Coachella Valley Girl
…230811-Palm_Springs-IMDb-soundtrack-song-500wide-373deep.jpg —
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Sunday, February 5, 2023
Dimensions
A Logo for Riri —
The above Nick Romano passage is from Knock on Any Door,
a 1947 novel by Willard Motley. Another Motley novel about
Chicago, from 1958 . . .
|
Page 41 The city was a blue-black panther that slunk along beside them. The tall, skyscraper night-grass hemmed them in. The thousand neon animal eyes watched their going. Page 67 The blue-black panther of a city watched their going. The un- blinking neon animal eyes watched their going. Thousands of neon signs lit their way. In an alley behind West Madison Street half an Page 68 hour before, a bum, drunk, had frozen to death lying in the back doorway of a pawnshop. The blue-black panther crouched over him. Page 70 First the creak of ice as an automobile goes by. Then the frown into your room of the red brick building across the street, its windows frosted over like cold, unfriendly eyes. Then a bum stumbling along trying to keep warm. Now a drunk, unevenly. And the wind like the howling voice of the blue-black panther, hunting, finding. And the clanging of impersonal streetcars. And each bar of neon, cold, dead. No message. The clown takes his bow and it is Christmas Day. Page 79 The blue-black panther followed them, sniffing at their heels. Page 106 Above them the blue-black panther lay on the roof of a tenement house, its feline chin on the cornice, its yellow-green eyes staring down onto the black night street of Maxwell. Its tail, wagging slowly back and forth, was like a lasso, a noose, sending little shivers of pebbles rolling loosely across the roof. Page 154 Then he went down to the Shillelagh Club. Through the pane, in the crowded, noisy place, he saw her. She was sitting at a table near the back, alone. Her cigarette had fallen from her lips and rolled away from her on the table top. It had burned itself to a long gray ash. Her head hung loosely on her neck as if she was asleep. A half-empty glass of beer was in front of her. Please, Mother, please come out, he prayed to her. And he stood next door to the tavern, waiting, his small shoulders drawn in, his head down in shame. And often he walked to the window and stood on tiptoe. She was still there. In the same position. He waited. He would be late to school tomorrow. He waited, keeping the long vigil. He waited. Twelve years old. And the thousand neon-animal eyes stared at him savagely. He waited. The blue-black panther lashed out its tail, flicking its furry tip against his ankles. He waited. Page 250 Alongside the blue-black patrol wagon the blue-black panther walks majestically. Page 262 Outside the door the blue-black panther rubs its back like a house cat. Page 409 Nick held the cigarette listlessly. The smoke curled up his wrist and arm like a snake. The blue-black panther licked his hand. |
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Tuesday, November 8, 2022
Saturday, July 21, 2018
Raiders of the Hidden Dimensions
See as well this journal on the above YouTube date: May 17, 2010.
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Thursday, March 24, 2016
Sunday, February 15, 2015
Dimensions
"Hello darkness, my old friend.
I’ve come to talk with you again."
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Wednesday, November 18, 2009
A Wrinkle in Dimensions…
An educated consumer
is our best customer!
“Acme Klein Bottles — where
yesterday’s future is here today!”
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Monday, November 16, 2009
A Wrinkle in Dimensions
Clifford Pickover now seems to be trying to catch up with Christian fantasists Madeleine L’Engle and Charles Williams. Click on the images below for further details.
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Tuesday, March 5, 2024
Metadata for Sam Levinson: Turning 16
In the recent film "Stars at Noon," the protagonist said . . .
Perhaps she should also have investigated Hell's coordinates . . .
The above XYZ record label metadata refers to
August 23, 1958 … The day I turned sixteen.
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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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Monday, January 15, 2024
Storylines Colliding: The Cornfield Paradox
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Wednesday, January 3, 2024
Flippant
From the Log24 search in the previous post for "Dimensions" —
"Hello darkness, my old friend.
I’ve come to talk with you again."
The above image was flipped to reverse left and right.
Related reading: Other posts tagged Darkness and …
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Text and Context: “A Multidimensional Crisis”
"Academia seems to be in the grip of a multidimensional crisis
that goes beyond ideology, and also beyond Harvard."
— A. O. Scott in The New York Times today
See Dimensions and Multidimensional in this journal.
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Sunday, November 19, 2023
For the First Church of Aquaman . . .
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Friday, November 10, 2023
Logos
Related art —
(For some backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)
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Tuesday, May 2, 2023
Dust in the Wind
See posts tagged "The Next Level."
Perhaps Isadore Singer now has a clue . . .
See his phrases "manic as hell" and "pregnant as hell."
See also Illinois Beltane.
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Saturday, April 29, 2023
The Long Movie
"The history of the length of movies takes place in two dimensions—
on the axis of the ordinary and the axis of the extraordinary, or,
of the rule and the exception."
— Richard Brody, The New Yorker , April 24, "In Praise of the Long Movie."
The Ordinary —
The Extraordinary —
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Monday, February 27, 2023
Saturday, February 11, 2023
Annals of Entertainment
The New York Times on a set designer who
reportedly died at 83 on Monday (Feb. 6, 2023) —
"Adrian Hall, the founding artistic director,
brought him in as resident designer.
(Mr. Hall died on Feb. 4 in Van, Texas.)"
Hall was the founding artistic director of
Trinity Repertory Company, Providence, R.I.
Not-so-holy writ ….
Panthers — "Dimensions," Log24, Feb. 5, 2023.
Beast Belly — Tonight's previous post, "Gutter Mathematics."
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Friday, December 30, 2022
Bullshit Studies: The View from East Lansing
Detail of the above screen (click to enlarge) —
See also this journal on the above date — June 10, 2021.
From this journal on May 6, 2009 —
A related picture of images that "reappear metamorphosed
in the coordinate system of the high region" —
(For the backstory, see Geometry of the I Ching
and the history of Chinese philosophy.)
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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 —
|
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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Friday, October 21, 2022
Dimensional Positioning*
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Wednesday, September 14, 2022
A Linear Code with 4×6 Symmetry
The exercise of 9/11 continues . . .
As noted in an update at the end of the 9/11 post,
these 24 motifs, along with 3 bricks and 4 half-arrays,
generate a linear code of 12 dimensions. I have not
yet checked the code's minimum weight.
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Sunday, September 11, 2022
Raiders of the Lost Space
From 1981 —
From today —
Update —
A Magma check of the motif-generated space shows that
its dimension is only 8, not 12 as with the MOG space.
Four more basis vectors can be added to the 24 motifs to
bring the generated space up to 12 dimensions: the left
brick, the middle brick, the top half (2×6), the left half (4×3).
I have not yet checked the minimum weight in the resulting
12-dimensional 4×6 bit-space.
— SHC 4 PM ET, Sept. 12, 2022.
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Saturday, July 30, 2022
Literary Figures
"… the tesseract, identified with a figure too inclusive,
contradictory, and all-pervasive to be seen as a character,
connects multiple dimensions in a manner counter to
ordinary thought…."
— Catherine Flynn, "From Dowel to Tesseract" (2016)
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Monday, July 11, 2022
Friday, May 27, 2022
Great Escapes
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Saturday, March 26, 2022
Box Geometry: Space, Group, Art (Work in Progress)
| Name Tag | .Space | .Group | .Art |
|---|---|---|---|
| Box4 |
2×2 square representing the four-point finite affine geometry AG(2,2). (Box4.space) |
S4 = AGL(2,2) (Box4.group) |
(Box4.art) |
| Box6 |
3×2 (3-row, 2-column) rectangular array representing the elements of an arbitrary 6-set. |
S6 | |
| Box8 | 2x2x2 cube or 4×2 (4-row, 2-column) array. | S8 or A8 or AGL(3,2) of order 1344, or GL(3,2) of order 168 | |
| Box9 | The 3×3 square. | AGL(2,3) or GL(2,3) | |
| Box12 | The 12 edges of a cube, or a 4×3 array for picturing the actions of the Mathieu group M12. | Symmetries of the cube or elements of the group M12 | |
| Box13 | The 13 symmetry axes of the cube. | Symmetries of the cube. | |
| Box15 |
The 15 points of PG(3,2), the projective geometry of 3 dimensions over the 2-element Galois field. |
Collineations of PG(3,2) | |
| Box16 |
The 16 points of AG(4,2), the affine geometry of 4 dimensions over the 2-element Galois field. |
AGL(4,2), the affine group of |
|
| Box20 | The configuration representing Desargues's theorem. | ||
| Box21 | The 21 points and 21 lines of PG(2,4). | ||
| Box24 | The 24 points of the Steiner system S(5, 8, 24). | ||
| Box25 | A 5×5 array representing PG(2,5). | ||
| Box27 |
The 3-dimensional Galois affine space over the 3-element Galois field GF(3). |
||
| Box28 | The 28 bitangents of a plane quartic curve. | ||
| Box32 |
Pair of 4×4 arrays representing orthogonal Latin squares. |
Used to represent elements of AGL(4,2) |
|
| Box35 |
A 5-row-by-7-column array representing the 35 lines in the finite projective space PG(3,2) |
PGL(3,2), order 20,160 | |
| Box36 | Eurler's 36-officer problem. | ||
| Box45 | The 45 Pascal points of the Pascal configuration. | ||
| Box48 | The 48 elements of the group AGL(2,3). | AGL(2,3). | |
| Box56 |
The 56 three-sets within an 8-set or |
||
| Box60 | The Klein configuration. | ||
| Box64 | Solomon's cube. |
— Steven H. Cullinane, March 26-27, 2022
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Wednesday, February 23, 2022
Friday, February 11, 2022
De Beer’s Consolidated Mine
The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel
Many Dimensions published in 1993 by Wm. B. Eerdmans.
But seriously . . .
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Monday, January 24, 2022
Saturday, January 1, 2022
Clean Lines
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Friday, December 31, 2021
Aesthetics in Academia
Related art — The non-Rubik 3x3x3 cube —
The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.
I prefer the radiance (in the sense of Aquinas) of the central, structuring
eightfold cube at the center of the affine space of six dimensions over
the two-element field GF(2).
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Monday, September 13, 2021
Cube Space Revisited
The above Quanta article mentions
"Maryna Viazovska’s 2016 discovery of the most efficient
ways of packing spheres in dimensions eight and 24."
From a course to be taught by Viazovska next spring:
The Lovasz reference suggests a review of my own webpage
Cube Space, 1984-2003.
See as well a review of Log24 posts on Packing.
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Tuesday, February 16, 2021
Buffalo Logic
From a post of January 8, 2021 —
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
A Midrash for Emma —
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Friday, January 8, 2021
Welcome to Zombieland
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
“Don’t give a damn ’cause I done that already.” — Zombie Jamboree
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Sunday, December 6, 2020
The Undoing
Today’s earlier post “Binary Coordinates” discussed a Dec. 6
revision to the Wikipedia article on PG(3,2), the projective
geometry of 3 dimensions over the 2-element field GF(2).
The revision, which improved the article, was undone later today
by a clueless retired academic, one William “Bill” Cherowitzo,
a professor emeritus of mathematics at U. of Colorado at Denver.
(See his article “Adventures of a Mathematician in Wikipedia-land,”
MAA Focus , December 2014/January 2015.)
See my earlier remarks on this topic . . . specifically, on this passage —
“A 3-(16,4,1) block design has 140 blocks
of size 4 on 16 points, such that each triplet
of points is covered exactly once. Pick any
single point, take only the 35 blocks
containing that point, and delete that point.
The 35 blocks of size 3 that remain comprise
a PG(3,2) on the 15 remaining points.”
As I noted on November 17, this is bullshit. Apparently Cherowitzo
never bothered to find out that an arbitrary “3-(16,4,1) block design”
(an example of a Steiner quadruple system ) does not yield a PG(3,2).
PG(3,2) is derived from the classical 3-(16,4,1) block design formed by the affine
space of 4 dimensions over GF(2). That design has 322,560 automorphisms.
In contrast, see a 3-(16,4,1) block design that is automorphism-free.
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Wednesday, May 20, 2020
Friday, May 1, 2020
Bullshit Studies
The following passage is from Amanda Gefter’s Trespassing
on Einstein’s Lawn (Bantam Books, 2014).
| “You know the story of Plato’s cave?” my father asked. “All the prisoners are chained up in the cave and they can’t see the real world outside, only the shadows on the wall? That’s supposed to be a negative thing, like they’ll never know reality. But the truth is, you have to be stuck inside a limited reference frame for there to be any reality at all! If you weren’t chained to your light cone, you’d see nothing. The H-state.”
I nodded. “You’d have no information. You need the broken symmetry, the shadow, to have information and information gives rise to the world. It from bit.” I couldn’t help but grin with excitement. The message was clear: having a finite frame of reference creates the illusion of a world, but even the reference frame itself is an illusion. Observers create reality, but observers aren’t real. There is nothing ontologically distinct about an observer, because you can always find a frame in which that observer disappears: the frame of the frame itself, the boundary of the boundary. “If physicists discover an invariant someday, the game will be up,” my father mused. “That would rule out the hypothesis that the universe is really nothing.” That was true. But so far, at least, every last invariant had gone the way of space and time, rendered relative and observer-dependent. Spacetime, gravity, electromagnetism, the nuclear forces, mass, energy, momentum, angular momentum, charge, dimensions, particles, fields, the vacuum, strings, the universe, the multiverse, the speed of light— one by one they had been downgraded to illusion. As the surface appearance of reality fell away, only one thing remained. Nothing. |
My path to Gefter’s father’s musing led from a quotation attributed,
probably falsely, to John Archibald Wheeler on page 52 of Octavio
Paz’s Claude Lévi-Strauss: An Introduction (Cornell, 1970) —
“There is a point at which
‘something is nothing and nothing is something.’ “
The quote may actually be by AP writer John Barbour reporting
on a 1967 American Physical Society talk by Wheeler, “The End
of Time.”
Gefter mentions Wheeler 369 times:
See as well Introduction to Quantum Woo.
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Wednesday, April 22, 2020
One with Everything
The title is of course from an old joke about mystic philosophies.
Related remarks by John Archibald Wheeler —
“Remarkable issues connected with the puzzle of existence
confront us today in Hermann Weyl’s domain of thought.
Four among them I bring before you here as especially interesting:
(1) What is the machinery of existence?
(2) What is deeper foundation of the quantum principle?
(3) What is the proper position to take about the existence of
the “continuum” of the natural numbers? And
(4) What can we do to understand time as an entity, not precise and
supplied free of charge from outside physics, but approximate and
yet to be derived from within a new and deeper time-free physics?
In brief, how come time?
What about the continuum?
How come the quantum?
What is existence?”
— John A. Wheeler, “Hermann Weyl and the Unity of Knowledge,”
American Scientist 74, no. 4 (1986): 366–75. Reprinted in
John A. Wheeler, At Home in the Universe
(American Institute of Physics, December 1, 1995).
The above bibliographic data is from . . .
Schrank, Jeffrey. Inventing Reality: Stories We Create To Explain Everything .
Gatekeeper Press. Kindle Edition, March 14, 2020.
For further scholarly details, see a version at JSTOR:
“…the article is adapted from the concluding address given at
the Hermann Weyl Centenary Congress, University of Kiel, 3 July 1985.”
— https://www.jstor.org/stable/27854250.
I prefer Charles Williams on “the Unity of Knowledge.”
See the 15 instances of the phrase “the Unity” in his 1931 novel Many Dimensions .
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Wednesday, March 25, 2020
Sunday in the Park
Some notes suggested by recent posts now also tagged Three Days —
Sporkin in 1975, according to his obituary in this morning’s print edition
of The New York Times —
He reportedly died at 88 of natural causes on Monday, March 23.
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Monday, February 3, 2020
A Kuhnian Register
Ereignis in the Stanford Encyclopedia of Philosophy —
|
Further aspects of the essential unfolding of Being are revealed by what is perhaps the key move in the Contributions—a rethinking of Being in terms of the notion of Ereignis, a term translated variously as ‘event’ (most closely reflecting its ordinary German usage), ‘appropriation’, ‘appropriating event’, ‘event of appropriation’ or ‘enowning’. (For an analysis which tracks Heidegger's use of the term Ereignis at various stages of his thought, see Vallega-Neu 2010). The history of Being is now conceived as a series of appropriating events in which the different dimensions of human sense-making—the religious, political, philosophical (and so on) dimensions that define the culturally conditioned epochs of human history—are transformed. Each such transformation is a revolution in human patterns of intelligibility, so what is appropriated in the event is Dasein and thus the human capacity for taking-as (see e.g., Contributions 271: 343). Once appropriated in this way, Dasein operates according to a specific set of established sense-making practices and structures. In a Kuhnian register, one might think of this as the normal sense-making that follows a paradigm-shift. — Michael Wheeler, 2011 |
See as well "reordering" in Sunday evening's post Tetrads for McLuhan
and in a Log24 search for Reordering + Steiner.
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Wednesday, January 29, 2020
On the Road
From Mosaic Logic, a post of September 3, 2017 —
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
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Saturday, January 4, 2020
Welcome to the Uncanny Valley Country Club
The previous post suggests a look at some Robot Apocalypse remarks:
Related material —
This journal on the above Stranger Dimensions date —
"Thus the theory of description matters most.
It is the theory of the word for those
For whom the word is the making of the world…."
— Wallace Stevens, "Description Without Place, VII"
See also Finite Relativity (St. Cecilia's Day, 2012).
Some other lines from "Description Without Place" —
"An age is green or red. An age believes
Or it denies. An age is solitude
Or a barricade against the singular man
By the incalculably plural."
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Sunday, November 17, 2019
E-Elements Revisited
The German mathematician Wolf Barth in the above post is not the
same person as the Swiss artist Wolf Barth in today's previous post.
An untitled, undated, picture by the latter —
Compare and contrast with an "elements" picture of my own —
— and with . . .
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
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Monday, November 11, 2019
Time and Chance
The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel
Many Dimensions published in 1993 by Wm. B. Eerdmans.
Compare and constrast —
Cover of a book by Douglas Hofstadter
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Friday, October 11, 2019
Quest
John Horgan in Scientific American magazine on October 8, 2019 —
"In the early 1990s, I came to suspect that the quest
for a unified theory is religious rather than scientific.
Physicists want to show that all things came from
one thing: a force, or essence, or membrane
wriggling in eleven dimensions, or something that
manifests perfect mathematical symmetry. In their
search for this primordial symmetry, however,
physicists have gone off the deep end . . . ."
Other approaches —
See "Story Theory of Truth" in this journal and, from the November 2019
Notices of the American Mathematical Society . . .
|
More fundamental than the label of mathematician is that of human. And as humans, we’re hardwired to use stories to make sense of our world (story-receivers) and to share that understanding with others (storytellers) [2]. Thus, the framing of any communication answers the key question, what is the story we wish to share? Mathematics papers are not just collections of truths but narratives woven together, each participating in and adding to the great story of mathematics itself. The first endeavor for constructing a good talk is recognizing and choosing just one storyline, tailoring it to the audience at hand. Should the focus be on a result about the underlying structures of group actions? . . . .
[2] Gottschall, J. , The Storytelling Animal , — "Giving Good Talks," by Satyan L. Devadoss |
"Before time began, there was the Cube." — Optimus Prime
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Wednesday, September 18, 2019
Unity
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Wednesday, November 28, 2018
Geometry and Experience
Einstein, "Geometry and Experience," lecture before the
Prussian Academy of Sciences, January 27, 1921–
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… This view of axioms, advocated by modern axiomatics, purges mathematics of all extraneous elements, and thus dispels the mystic obscurity, which formerly surrounded the basis of mathematics. But such an expurgated exposition of mathematics makes it also evident that mathematics as such cannot predicate anything about objects of our intuition or real objects. In axiomatic geometry the words "point," "straight line," etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics. Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the behavior of real objects. The very word geometry, which, of course, means earth-measuring, proves this. For earth-measuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuring-lines, measuring-wands, etc. It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of real objects of this kind, which we will call practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it in what follows from "purely axiomatic geometry." The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience. …. |
Later in the same lecture, Einstein discusses "the theory of a finite
universe." Of course he is not using "finite" in the sense of the field
of mathematics known as "finite geometry " — geometry with only finitely
many points.
Nevertheless, his remarks seem relevant to the Fano plane , an
axiomatically defined entity from finite geometry, and the eightfold cube ,
a physical object embodying the properties of the Fano plane.
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I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental picture to which, with some practice, we shall soon grow accustomed. First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To "visualize" a theory therefore means to bring to mind that abundance of sensible experiences for which the theory supplies the schematic arrangement. In the present case we have to ask ourselves how we can represent that behavior of solid bodies with respect to their mutual disposition (contact) that corresponds to the theory of a finite universe. |
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Friday, September 14, 2018
Denkraum
Underlying the I Ching structure is the finite affine space
of six dimensions over the Galois field with two elements.
In this field, "1 + 1 = 0," as noted here Wednesday.
See also other posts now tagged Interstice.
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Monday, August 27, 2018
Geometry and Simplicity
From …
Thinking in Four Dimensions
By Dusa McDuff
"I’ve got the rather foolhardy idea of trying to explain
to you the kind of mathematics I do, and the kind of
ideas that seem simple to me. For me, the search
for simplicity is almost synonymous with the search
for structure.
I’m a geometer and topologist, which means that
I study the structure of space …
. . . .
In each dimension there is a simplest space
called Euclidean space … "
— In Roman Kossak, ed.,
Simplicity: Ideals of Practice in Mathematics and the Arts
(Kindle Locations 705-710, 735). Kindle Edition.
For some much simpler spaces of various
dimensions, see Galois Space in this journal.
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Saturday, July 21, 2018
Columbia University Press: The Knight of CUP
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The Unification of Physics and Consciousness (Columbia Series in Science and Religion)
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A look at this publication was suggested by the previous post,
Raiders of the Hidden Dimensions.
From the Columbia University Press description of Hidden Dimensions —
— https://cup.columbia.edu/book/hidden-dimensions/9780231141505
For variations on these themes, see Batman Begins (2005)
and the trailer for Knight of Cups (2015) —
.
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Sunday, July 15, 2018
Jewish Oases
"… Lincoln Plaza Cinemas, the Juilliard String Quartet,
and the Strand Book Store remained oases
for cultural and intellectual stimulation."
— John S. Friedman in The Forward , Jan. 21, 2018
Read more:
https://forward.com/culture/392483/
how-fred-bass-dan-talbot-robert-mann
-shaped-new-york-culture/
From the Oasis in Steven Spielberg's "Ready Player One" (2018) —
I prefer, from a Log24 search for Flux Capacitor …
From "Raiders of the Lost Images" —
"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."
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Wednesday, June 27, 2018
Taken In
A passage that may or may not have influenced Madeleine L'Engle's
writings about the tesseract :
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From Mere Christianity , by C. S. Lewis (1952) —
"Book IV – Beyond Personality: I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully. You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares. Do you see the point? A world of one dimension would be a straight line. In a two-dimensional world, you still get straight lines, but many lines make one figure. In a three-dimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels. Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine. In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something super-personal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already. You may ask, "If we cannot imagine a three-personal Being, what is the good of talking about Him?" Well, there isn't any good talking about Him. The thing that matters is being actually drawn into that three-personal life, and that may begin any time —tonight, if you like. . . . . |
But beware of being drawn into the personal life of the Happy Family .
https://www.jstor.org/stable/24966339 —
"The colorful story of this undertaking begins with a bang."
And ends with …
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
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Saturday, May 19, 2018
Flux Capacitor
For Tom Hanks and Dan Brown —
From "Raiders of the Lost Images" —
"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."
See as well a Google search for flux philosophy —
https://www.google.com/search?q=flux+philosophy.
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Tuesday, February 27, 2018
Raiders of the Lost Images
On the recent film "Justice League" —
From DC Extended Universe Wiki, "Mother Box" —
"However, during World War I, the British rediscovered
mankind's lost Mother Box. They conducted numerous studies
but were unable to date it due to its age. The Box was then
shelved in an archive, up until the night Superman died,
where it was then sent to Doctor Silas Stone, who
recognized it as a perpetual energy matrix. . . ." [Link added.]
The cube shape of the lost Mother Box, also known as the
Change Engine, is shared by the Stone in a novel by Charles Williams,
Many Dimensions . See the Solomon's Cube webpage.
See too the matrix of Claude Lévi-Strauss in posts tagged
Verwandlungslehre .
Some literary background:
Who speaks in primordial images speaks to us
as with a thousand trumpets, he grips and overpowers,
and at the same time he elevates that which he treats
out of the individual and transitory into the sphere of
the eternal. — C. G. JUNG
"In the conscious use of primordial images—
the archetypes of thought—
one modern novelist stands out as adept and
grand master: Charles Williams.
In The Place of the Lion he incarnates Plato’s
celestial archetypes with hair-raising plausibility.
In Many Dimensions he brings a flock of ordinary
mortals face to face with the stone bearing
the Tetragrammaton, the Divine Name, the sign of Four.
Whether we understand every line of a Williams novel
or not, we feel something deep inside us quicken
as Williams tells the tale.
Here, in The Greater Trumps , he has turned to
one of the prime mysteries of earth . . . ."
— William Lindsay Gresham, Preface (1950) to
Charles Williams's The Greater Trumps (1932)
For fans of what the recent series Westworld called "bulk apperception" —
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Tuesday, November 28, 2017
For Quantum Mystics
"An awful lot of important dualities in four and fewer dimensions
follow from this six-dimensional theory and its properties."
— Edward Witten, interviewed by Natalie Wolchover,
in Quanta Magazine on November 28, 2017
See also Six Dimensions in this journal.
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Sunday, September 3, 2017
Mosaic Logic
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
While you're waiting …
Click the above illustration for
some remarks on mosaics.
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Sunday, August 13, 2017
Logos
In memoriam —
Zadeh is known for the unfortunate phrase "fuzzy logic."
Not-so-fuzzy related material —
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
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Monday, June 12, 2017
Bubble
The "bubble" passage in the previous post suggests a review of
a post from December 21, 2006, with the following images —
Update of 11:01 PM ET the same day, June 12, 2017 —
Related material for the Church of Synchronology —
From a tech-article series that began on Halloween 2006 and
ended on the date of the above Geometry's Tombstones post —
Compare and contrast (from a post of Feb. 27, 2017) —
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
See also "The Geometry of Logic:
Finite Geometry and the 16 Boolean Connectives"
by Steven H. Cullinane in 2007.
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Sunday, March 19, 2017
Gravedigger’s Handbook
In memory of Jimmy Breslin, who reportedly died today at 88 —
From "Dimensions," (Log24, Feb. 15, 2015) —
"Hello darkness, my old friend.
I’ve come to talk with you again."
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Monday, February 27, 2017
Logic for Jews
Adam Gopnik in The New Yorker today reacts to the startling
outcomes of three recent contests: the presidential election,
the Super Bowl, and the Oscar for Best Picture —
"The implicit dread logic is plain."
Related material —
Transformers in this journal and …
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
See also …
The above figure is from Ian Stewart's 1996 revision of a 1941 classic,
What Is Mathematics? , by Richard Courant and Herbert Robbins.
One wonders how the confused slave boy of Plato's Meno would react
to Stewart's remark that
"The number of copies required to double an
object's size depends on its dimension."
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Saturday, January 28, 2017
Cranking It Up
From "Core," a post of St. Lucia's Day, Dec. 13, 2016 —
In related news yesterday —
California yoga mogul’s mysterious death:
Trevor Tice’s drunken last hours detailed
"Police found Tice dead on the floor in his home office,
blood puddled around his head. They also found blood
on walls, furniture, on a sofa and on sheets in a nearby
bedroom, where there was a large bottle of Grey Goose
vodka under several blood-stained pillows on the floor."
See as well an image from "The Stone," a post of March 18, 2016 —
Some backstory —
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
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Wednesday, August 10, 2016
Narratives
The novel Blood on Snow , set in Oslo, was published
by Knopf on April 7, 2015. This journal on that date —
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Log24 on Tuesday, April 7, 2015 Filed under: Uncategorized — m759 @ 7:00 PM Seven years ago in this journal —
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A related image —
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Sunday, April 3, 2016
Logic
See also Stone Logical Dimensions …
“Lord Arglay had a suspicion that the Stone
would be purely logical. Yes, he thought,
but what, in that sense, were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
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Friday, March 11, 2016
Spacey
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Monday, December 14, 2015
Dirac and Geometry
See a post by Peter Woit from Sept. 24, 2005 — Dirac's Hidden Geometry.
The connection, if any, with recent Log24 posts on Dirac and Geometry
is not immediately apparent. Some related remarks from a novel —
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From Broken Symmetries by Paul Preuss "He pondered the source of her fascination with the occult, which sooner or later seemed to entangle a lot of thoughtful people who were not already mired in establishmentarian science or religion. It was the religious impulse, at base. Even reason itself could function as a religion, he supposed— but only for those of severely limited imagination. He’d toyed with 'psi' himself, written a couple of papers now much quoted by crackpots, to his chagrin. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand— for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox. Quantum theory was inextricable from the uncertainty relations. The classical fox knows many things, but the quantum-mechanical hedgehog knows only one big thing— at a time. 'Complementarity,' Bohr had called it, a rubbery notion the great professor had stretched to include numerous pairs of opposites. Peter Slater was willing to call it absurdity, and unlike some of his older colleagues who, following in Einstein’s footsteps, demanded causal explanations for everything (at least in principle), Peter had never thirsted after 'hidden variables' to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods. The psychic investigators, on the other hand, demanded to know how the mind and the psychical world were related. Through ectoplasm, perhaps? Some fifth force of nature? Extra dimensions of spacetime? All these naive explanations were on a par with the assumption that psi is propagated by a species of nonlocal hidden variables, the favored explanation of sophisticates; ignotum per ignotius . 'In this connection one should particularly remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all…' The words were Heisenberg’s, lecturing in 1929 on the irreducible ambiguity of the uncertainty relations. They reminded Peter of Evan Harris Walker’s ingenious theory of the psi force, a theory that assigned psi both positive and negative values in such a way that the mere presence of a skeptic in the near vicinity of a sensitive psychic investigation could force null results. Neat, Dr. Walker, thought Peter Slater— neat, and totally without content. One had to be willing to tolerate ambiguity; one had to be willing to be crazy. Heisenberg himself was only human— he’d persuasively woven ambiguity into the fabric of the universe itself, but in that same set of 1929 lectures he’d rejected Dirac’s then-new wave equations with the remark, 'Here spontaneous transitions may occur to the states of negative energy; as these have never been observed, the theory is certainly wrong.' It was a reasonable conclusion, and that was its fault, for Dirac’s equations suggested the existence of antimatter: the first antiparticles, whose existence might never have been suspected without Dirac’s crazy results, were found less than three years later. Those so-called crazy psychics were too sane, that was their problem— they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry." |
Particularly relevant …
"Mathematical relationships were enough to satisfy him,
mere formal relationships which existed at all times,
everywhere, at once."
Some related pure mathematics —
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Tuesday, December 1, 2015
Pascal’s Finite Geometry
See a search for "large Desargues configuration" in this journal.
The 6 Jan. 2015 preprint "Danzer's Configuration Revisited,"
by Boben, Gévay, and Pisanski, places this configuration,
which they call the Cayley-Salmon configuration , in the
interesting context of Pascal's Hexagrammum Mysticum .
They show how the Cayley-Salmon configuration is, in a sense,
dual to something they call the Steiner-Plücker configuration .
This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3-space." The six-sets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.
The duality was, as they note, previously described in 1898.
Related material on six-set geometry from the classical literature—
Baker, H. F., "Note II: On the Hexagrammum Mysticum of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236
Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen (1900), Volume 53, Issue 1-2, pp 161-176
Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160
Related material on six-set geometry from a more recent source —
Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage
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Wednesday, September 16, 2015
The World as Myth
Three approaches to The World as Myth…
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From Heinlein's 1985 The Cat Who Walks Through Walls … The World as Myth is a subtle concept. It has sometimes been called multiperson solipsism, despite the internal illogic of that phrase. Yet illogic may be necessary, as the concept denies logic. For many centuries religion held sway as the explanation of the universe- or multiverse. The details of revealed religions differed wildly but were essentially the same: Somewhere up in the sky-or down in the earth-or in a volcano-any inaccessible place- there was an old man in a nightshirt who knew everything and was all powerful and created everything and rewarded and punished… and could be bribed. "Sometimes this Almighty was female but not often because human males are usually bigger, stronger, and more belligerent; God was created in Pop's image. "The Almighty-God idea came under attack because it explained nothing; it simply pushed all explanations one stage farther away. In the nineteenth century atheistic positivism started displacing the Almighty-God notion in that minority of the population that bathed regularly. "Atheism had a limited run, as it, too, explains nothing, being merely Godism turned upside down. Logical positivism was based on the physical science of the nineteenth century which, physicists of that century honestly believed, fully explained the universe as a piece of clockwork. "The physicists of the twentieth century made short work of that idea. Quantum mechanics and Schrodringer's cat tossed out the clockwork world of 1890 and replaced it with a fog of probability in which anything could happen. Of course the intellectual class did not notice this for many decades, as an intellectual is a highly educated man who can't do arithmetic with his shoes on, and is proud of his lack. Nevertheless, with the death of positivism, Godism and Creationism came back stronger than ever. "In the late twentieth century -correct me when I' m wrong, Hilda-Hilda and her family were driven off Earth by a devil, one they dubbed 'the Beast.' They fled in a vehicle you have met, Gay Deceiver, and in their search for safety they visited many dimensions, many universes… and Hilda made the greatest philosophical discovery of all time." "I'll bet you say that to all the girls!" "Quiet, dear. They visited, among more mundane places, the Land of Oz-" I sat up with a jerk. Not too much sleep last night and Dr. Harshaw's lecture was sleep-inducing. "Did you say 'Oz'?" "I tell you three times. Oz, Oz, Oz. They did indeed visit the fairyland dreamed up by L. Frank Baum. And the Wonderland invented by the Reverend Mr. Dodgson to please Alice. And other places known only to fiction. Hilda discovered what none of us had noticed before because we were inside it: The World is Myth. We create it ourselves-and we change it ourselves. A truly strong myth maker, such as Homer, such as Baum, such as the creator of Tarzan, creates substantial and lasting worlds … whereas the fiddlin', unimaginative liars and fabulists shape nothing new and their tedious dreams are forgotten. …. |
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Friday, November 6, 2009
Where Entertainment is God (continued)
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Wednesday, May 6, 2015
Narrative Metaphysics
(Continued from Dec. 13, 2014.)
David Lavery's enthusiasm today for the Marvel Comics
"Infinity Stones" suggests a review of The Foundation Stone
mentioned in the post Narrative Metaphysics of 12/13/2014.
See as well "Many Dimensions" in this journal.
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Saturday, April 25, 2015
Faustian Merry-Go-Round
Thanks to David Lavery for the following:
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"Voilà! Stevens has managed to create out of nothing a palpable imaginative space, an interiority without material dimensions, replete with its own achieved and accomplished music. And in truth, in a world of Heisenbergian uncertainties and shifting star masses, it may be enough for the dizzying, ever-shifting merry-go-round of the Faustian mind simply to slow down and let itself come to rest, at least for the moment." — Paul Mariani, "God and the Imagination," Aug. 10, 1996 |
http://imagejournal.org/page/journal/articles/issue-18/mariani-essays
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Tuesday, April 7, 2015
Logic
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Thursday, February 26, 2015
Brit Award
"The Brit Awards are… the British equivalent
of the American Grammy Awards." — Wikipedia
Detail of an image from yesterday's 5:30 PM ET post:
Related material:
From a review: "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 of Charles Williams's
novel Many Dimensions ."
See also Solomon's Seal (July 26, 2012).
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Monday, January 19, 2015
Shema
In memory of a mathematician who
reportedly died on Dec. 16, 2014:
"Four dimensions is where things change a lot."
Backstory: Or Only Die.
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Thursday, January 8, 2015
Gitterkrieg
From the abstract of a talk, "Extremal Lattices," at TU Graz
on Friday, Jan. 11, 2013, by Prof. Dr. Gabriele Nebe
(RWTH Aachen) —
"I will give a construction of the extremal even
unimodular lattice Γ of dimension 72 I discovered
in summer 2010. The existence of such a lattice
was a longstanding open problem. The
construction that allows to obtain the
minimum by computer is similar to the one of the
Leech lattice from E8 and of the Golay code from
the Hamming code (Turyn 1967)."
On an earlier talk by Nebe at Oberwolfach in 2011 —
"Exciting new developments were presented by
Gabriele Nebe (Extremal lattices and codes ) who
sketched the construction of her recently found
extremal lattice in 72 dimensions…."
Nebe's Oberwolfach slides include one on
"The history of Turyn's construction" —
Nebe's list omits the year 1976. This was the year of
publication for "A New Combinatorial Approach to M24"
by R. T. Curtis, the paper that defined Curtis's
"Miracle Octad Generator."
Turyn's 1967 construction, uncredited by Curtis, may have
been the basis for Curtis's octad-generator construction.
See Turyn in this journal.
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Saturday, December 27, 2014
Hopefully
The previous post, More To Be Done, quotes an
opera lyric by physicist Lisa Randall :
The opera, about a physics of hidden dimensions,
may of course be given a theological spin —
See Hope of Heaven in this journal.
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Friday, December 5, 2014
Wittgenstein’s Picture
From Zettel (repunctuated for clarity):
249. « Nichts leichter, als sich einen 4-dimensionalen Würfel
vorstellen! Er schaut so aus… »
"Nothing easier than to imagine a 4-dimensional cube!
It looks like this…
[Here the editor supplied a picture of a 4-dimensional cube
that was omitted by Wittgenstein in the original.]
« Aber das meine ich nicht, ich meine etwas wie…
"But I don't mean that, I mean something like…
…nur mit 4 Ausdehnungen! »
but with four dimensions!
« Aber das ist nicht, was ich dir gezeigt habe,
eben etwas wie…
"But isn't what I showed you like…
…nur mit 4 Ausdehnungen? »
…only with four dimensions?"
« Nein; das meine ich nicht! »
"No, I don't mean that!"
« Was aber meine ich? Was ist mein Bild?
Nun der 4-dimensionale Würfel, wie du ihn gezeichnet hast,
ist es nicht ! Ich habe jetzt als Bild nur die Worte und
die Ablehnung alles dessen, was du mir zeigen kanst. »
"But what do I mean? What is my picture?
Well, it is not the four-dimensional cube
as you drew it. I have now for a picture only
the words and my rejection of anything
you can show me."
"Here's your damn Bild , Ludwig —"
Context: The Galois Tesseract.
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Monday, September 22, 2014
Wednesday, August 27, 2014
Schau der Gestalt
(Continued from Aug. 19, 2014)
“Christian contemplation is the opposite
of distanced consideration of an image:
as Paul says, it is the metamorphosis of
the beholder into the image he beholds
(2 Cor 3.18), the ‘realisation’ of what the
image expresses (Newman). This is
possible only by giving up one’s own
standards and being assimilated to the
dimensions of the image.”
— Hans Urs von Balthasar,
The Glory of the Lord:
A Theological Aesthetics,
Vol. I: Seeing the Form
[ Schau der Gestalt ],
Ignatius Press, 1982, p. 485
A Bauhaus approach to Schau der Gestalt :
I prefer the I Ching ‘s approach to the laws of cubical space.
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Thursday, August 14, 2014
Full Frontal Aesthetics
A prequel to today's noon post
Rosalind Krauss in "Grids" —
"The physical qualities of the surface, we could say,
are mapped onto the aesthetic dimensions of
the same surface. And those two planes—
the physical and the aesthetic— are demonstrated
to be the same plane…. the bottom line of the grid is
a naked and determined materialism.”
A writer I prefer:
Barsotti's classic illustrated:
The naked and determined Annette Bening in "The Grifters" —
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Thursday, July 17, 2014
Paradigm Shift:
Continuous Euclidean space to discrete Galois space*
Euclidean space:
Counting symmetries in Euclidean space:
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
* For related remarks, see posts of May 26-28, 2012.
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Thursday, June 26, 2014
The McLuhan Dimension
"History is a deep and complicated puzzle—
especially when it involves more dimensions than time."
— Introduction to a novella in Analog Science Fiction
"Annenberg Hall" at Harvard was originally part of a memorial for
Civil War dead. Formerly "Alumni Hall," it was renamed in 1996.
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Sunday, June 1, 2014
Black Key
The title was suggested by a post on The Piano
and by the dimensions of an image in this morning’s
previous post: 404 x 211 pixels, suggesting
4/04, a date significant to author Katherine Neville,
and 2/11, the date of a Log24 post from 2014.
These dates are both related to the post…
Everybody Comes to Rick’s
(original title of Casablanca ).
Whimsical, yes, but see Iris Murdoch
on the contingent in literature and the word
“whimsical” in a post of January 26, 2004
(in a series of posts involving Michael Sprinker).
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Tuesday, February 18, 2014
Chamber Music
The New York Times online this evening
has two passages of interest.
From an obituary by Helen T. Verongos of
fiction writer Mavis Gallant—
"Ms. Gallant also endowed children with
special powers that vanish as they grow up.
In 'The Doctor,' she wrote: 'Unconsciously,
everyone under the age of 10 knows
everything. Under-ten can come into a room
and sense at once everything felt, kept
silent, held back in the way of love, hate and
desire, though he may not have the right
words for such sentiments. It is part of the
clairvoyant immunity to hypocrisy we are born
with and that vanishes just before puberty.' "
From a review by William Grimes of a memoir
by non-fiction writer Joachim Fest—
"Not I shrinks the Wagnerian scale of
German history in the 1930s and 1940s to
chamber music dimensions. It is intensely
personal, cleareyed and absolutely riveting,
partly because the author, thrust into an
outsider’s position, developed a keen
appreciation of Germany’s contradictions
and paradoxes."
Related material in this journal—
Octobers for Fest (Sept. 13, 2006).
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Monday, November 11, 2013
The Mystic Hexastigm…
Or: The Nutshell
What about Pascal?
For some background on Pascal's mathematics,
not his wager, see…
Richmond, H. W.,
"On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics ,
Volume 31 (1900), pp. 125-160,
dated by Richmond March 30,1899
Richmond, H. W.,
"The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen ,
Volume 53 (1900), Issue 1-2, pp 161-176,
dated by Richmond February 1, 1899
See also Nocciolo in this journal.
Recall as well that six points in space may,
if constrained to lie on a circle, be given
a religious interpretation. Richmond's
six points are secular and more general.
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Saturday, September 21, 2013
Mathematics and Narrative (continued)
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013
|
Narrative:
Aooo.
Happy birthday to Stephen King.
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Wednesday, September 4, 2013
Friday, August 16, 2013
Six-Set Geometry
From April 23, 2013, in
"Classical Geometry in Light of Galois Geometry"—
Click above image for some background from 1986.
Related material on six-set geometry from the classical literature—
Baker, H. F., "Note II: On the Hexagrammum Mysticum of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236
Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen (1900), Volume 53, Issue 1-2, pp 161-176
Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions,"
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160
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Monday, July 29, 2013
St. Walter’s Day
Today is the dies natalis of group theorist Walter Feit.
"The Steiner systems S (5,6,12) and S (5,8,24) are remarkable combinatorial
configurations unlike any others. Their automorphism groups are the Mathieu
groups M12 and M24. These are the only 5-transitive permutation groups other
than symmetric and alternating groups: (a fact long conjectured but only
proved as a consequence of the classification). The Leech lattice is a blown up
version of S (5,8,24). It is the unique even unimodular lattice in 24 dimensions
with no vectors of weight 2. This uniqueness is an essential reason why it is a
geometric object of fundamental importance. The automorphism group Co.O
of the Leech lattice involves about half of the sporadic groups and generally it
is felt that these are well understood."
— Walter Feit, book review, Bulletin of the American Mathematical Society ,
Vol. 8 (1983), 120-124, page 123
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Sunday, April 21, 2013
Sermon
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Sunday, March 31, 2013
For Baker
Baker, Principles of Geometry, Vol. IV (1925), Title:
Baker, Principles of Geometry, Vol. IV (1925), Frontispiece:
Baker's Vol. IV frontispiece shows "The Figure of fifteen lines
and fifteen points, in space of four dimensions."
Another such figure in a vector space of four dimensions
over the two-element Galois field GF(2):
(Some background grid parts were blanked by an image resizing process.)
Here the "lines" are actually planes in the vector 4-space over GF(2),
but as planes through the origin in that space, they are projective lines .
For some background, see today's previous post and Inscapes.
Update of 9:15 PM March 31—
The following figure relates the above finite-geometry
inscape incidences to those in Baker's frontispiece. Both the inscape
version and that of Baker depict a Cremona-Richmond configuration.
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Saturday, January 5, 2013
Vector Addition in a Finite Field
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—
The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“Self-Dual Configurations and Regular Graphs.”
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Tuesday, December 4, 2012
December Days
For the Dec. 3rd-4th graduate conference
at the University of Cambridge on
"Occultism, Magic, and the History of Art"—
Four novels by Charles Williams—
See also the life, and Dec. 1st death, of a former Chief Justice of South Africa.
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Friday, November 2, 2012
Time and Chance (continued)
For clergymen who embrace Trudeau's
"Story Theory of Truth" (see last evening's
7:20 PM post on geometry and A Wrinkle in Time )…
Here are two meditations suggested by
yesterday evening's New York Lottery :
1. Page 141 in Philosophies of India—
2. Post 4658 in this journal— A Wrinkle in Dimensions.
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Friday, October 19, 2012
Midnight Politics
For Mitt …
See "A Deskful of Girls" in Fritz Leiber's Selected Stories .
See also the Feast of St. Mary Magdalene in 2009.
… And for Clint—
From "Deskful":
I quickly settled myself in the chair, not to be gingerly
about it. It was rather incredibly comfortable, almost
as if it had adjusted its dimensions a bit at the last
instant to conform to mine. The back was narrow at
the base but widened and then curled in and over to
almost a canopy around my head and shoulders.
The seat too widened a lot toward the front, where
the stubby legs were far apart. The bulky arms
sprang unsupported from the back and took my own
just right, though curving inwards with the barest
suggestion of a hug. The leather or unfamiliar plastic
was as firm and cool as young flesh and its texture
as mat under my fingertips.
"An historic chair," the Doctor observed, "designed
and built for me by von Helmholtz of the Bauhaus…."
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Thursday, September 13, 2012
Backstory
Yesterday's online Los Angeles Times
on a film that inspired recent protests in Cairo—
The film… was shown on June 23
to an audience of less than 10
at a theater on Hollywood Boulevard,
a source familiar with the screening said….
The screening was at The Vine Theater,
which rents itself out for private screenings,
said one person involved in the theater.
An image from this journal on that same day, June 23—
Source: Rudolf Koch, The Book of Signs
For some background on the symbol, see Damnation Morning.
See also Don Henley's Hollywood hymn "Garden of Allah."
Update of 8 PM Sept. 13, 2012—
Other sources give the film's screening date not as June 23,
2012, but rather as June 30, 2012. (BBC News, LAWEEKLY blogs)
The following post from this journal on that date may or
may not have some religious relevance.
| Saturday, June 30, 2012
Filed under: Uncategorized — m759 @ 7:20 PM "… to snare the spirits of mankind in nets of magic" — The aim of the artist, according to Thomas Wolfe Related entertainment— High-minded— Many Dimensions . Not so high-minded— The Cosmic Cube . |
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Friday, July 27, 2012
Raiders of the Lost Ring
Wikipedia on a magical ring—
Background— The Ring and the Stone, a story linked to here Wednesday.
"By then he was familiar with the work of the Vienna Actionists….
He once said that he had his first taste of the movement
when he heard the screams of his mother’s dental patients
from her office next door to the family’s apartment."
— Obituary of a Viennese artist who reportedly died Wednesday
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Thursday, July 26, 2012
Solomon’s Seal
(Mathematics and Narrative, continued)
Narrative—
The Ring and The Stone from yesterday’s post, and…
“In Medieval Jewish, Christian and Islamic legends,
the Seal of Solomon was a magical signet ring
said to have been possessed by King Solomon….”
— Wikipedia article, Seal of Solomon
Mathematics—
A fact related to the mathematical
“Solomon’s seal” described above by Bell:
The reference to Edge is as follows—
[3] Edge, W. L., Quadrics over GF(2) and
their relevance for the cubic surface group,
Canadian J. Maths. 11 (1959) ….
(This reference relates Hirschfeld’s remarks
quoted above to the 64-point affine space
illustrated below (via the associated
63-point projective space PG (5, 2)).
See Solomon’s Cube.
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Wednesday, July 25, 2012
Thursday, July 19, 2012
But Seriously…
For those who prefer fiction:
"Many Dimensions (1931) — An evil antiquarian illegally purchases
the fabled Stone of Suleiman (Williams uses this Muslim form
rather than the more familiar King Solomon) from its Islamic guardian
in Baghdad and returns to England to discover not only that the Stone
can multiply itself infinitely without diminishing the original, but that it
also allows its possessor to transcend the barriers of space and time."
— Wikipedia article on the author Charles Williams
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Saturday, June 30, 2012
Snares
"… to snare the spirits of mankind in nets of magic"
— The aim of the artist, according to Thomas Wolfe
Related entertainment—
High-minded— Many Dimensions .
Not so high-minded— The Cosmic Cube .
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Friday, June 8, 2012
Cartoon Graveyard
For some background, see "Cartoon Graveyard" and "Many Dimensions."
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Sunday, April 1, 2012
The Palpatine Dimension
A physics quote relayed at Peter Woit's weblog today—
"The relation between 4D N=4 SYM and the 6D (2, 0) theory
is just like that between Darth Vader and the Emperor.
You see Darth Vader and you think 'Isn’t he just great?
How can anyone be greater than that? No way.'
Then you meet the Emperor."
Some related material from this weblog—
(See Big Apple and Columbia Film Theory)
The Meno Embedding:
Some related material from the Web—
See also uses of the word triality in mathematics. For instance…
A discussion of triality by Edward Witten—
Triality is in some sense the last of the exceptional isomorphisms,
and the role of triality for n = 6 thus makes it plausible that n = 6
is the maximum dimension for superconformal symmetry,
though I will not give a proof here.
— "Conformal Field Theory in Four and Six Dimensions"
and a discussion by Peter J. Cameron—
There are exactly two non-isomorphic ways
to partition the 4-subsets of a 9-set
into nine copies of AG( 3,2).
Both admit 2-transitive groups.
— "The Klein Quadric and Triality"
Exercise: Is Witten's triality related to Cameron's?
(For some historical background, see the triality link from above
and Cameron's Klein Correspondence and Triality.)
Cameron applies his triality to the pure geometry of a 9-set.
For a 9-set viewed in the context of physics, see A Beginning—
|
From MIT Commencement Day, 2011— A symbol related to Apollo, to nine, and to "nothing"— A minimalist favicon—
This miniature 3×3 square— |
Happy April 1.
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Monday, January 23, 2012
How It Works
J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M24. His approach at that time was
purely algebraic, not geometric—
For earlier (and later) discussions of the geometry (not the algebra )
of that order-16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2-element field), see The Galois Tesseract.
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Wednesday, January 11, 2012
Cuber
“Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing. And once you have made or acquired a new ‘cube’… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube. What is the essence of each operator? One senses a deep invariant lying somehow ‘down underneath’ it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment. In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….
… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube. It is the answer; it simply has the right spirit .”
— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern (Kindle edition, locations 11557-11572)
See also Many Dimensions in this journal and Solomon’s Cube.
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Thursday, December 1, 2011
Paranoia Strikes Deep
Tens of Millions of Smartphones Come With Spyware
Preinstalled, Security Analyst Says
Published December 01, 2011 – FoxNews.com
For details, see comments at YouTube.
Related entertainment—
1. Tara Fitzgerald in "New World Disorder" (1999)—
We skipped the light fandango
turned cartwheels 'cross the floor
I was feeling kinda seasick
but the crowd called out for more
2. Tara Fitzgerald in "Broken Glass" (2011)—
And so it was that later
as the miller told his tale
that her face, at first just ghostly,
turned a whiter shade of pale
— Procol Harum song at beginning and end of "The Net" (1995)
“Lord Arglay had a suspicion that the Stone
would be purely logical. Yes, he thought,
but what, in that sense, were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams,
quoted here on Kristallnacht 2011
See also, from "The Net"—
Decompiling Wolfenstein
"In Wolfenstein 3D , the player assumes the role of an American soldier
of Polish descent… attempting to escape from the Nazi stronghold of
Castle Wolfenstein." — Wikipedia
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Thursday, November 17, 2011
The Rolling Donut
See notes related to the discussion of the torus within the hypercube
in Thomas F. Banchoff 's 1996 text Beyond the Third Dimension .
The hypercube torus is more intelligible in the light of an
animation at the weblog post "Gleaming the Hypercube"—
(Animation source: MIQEL.com)
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Wednesday, November 9, 2011
Polish Logic–
The Big Lukasiewicz
“Lord Arglay had a suspicion that the Stone
would be purely logical. Yes, he thought,
but what, in that sense, were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
See also Łukasiewicz in Wikipedia and Lukasiewicz in this journal.
The latter's Christian references seem preferable to yesterday's
link to a scene from the Coen brothers' film "The Big Lebowski."
For those who prefer a Christ-for-Jews there is
also Harvard's version. See The Crimson Passion.
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Monday, August 29, 2011
Many = Six.
A comment today on yesterday's New York Times philosophy column "The Stone"
notes that "Augustine… incorporated Greek ideas of perfection into Christianity."
Yesterday's post here for the Feast of St. Augustine discussed the 2×2×2 cube.
Today's Augustine comment in the Times reflects (through a glass darkly)
a Log24 post from Augustine's Day, 2006, that discusses the larger 4×4×4 cube.
For related material, those who prefer narrative to philosophy may consult
Charles Williams's 1931 novel Many Dimensions . Those who prefer mathematics
to either may consult an interpretation in which Many = Six.
Click image for some background.
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Sunday, August 28, 2011
The Cosmic Part
Yesterday’s midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik’s mechanical contrivance as a rather absurd “Cosmic Cube.”
A simpler candidate for the “Cube” part of that phrase:
The Eightfold Cube
As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.
“Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions.”
— Alexandre V. Borovik in “Coxeter Theory: The Cognitive Aspects“
Borovik has a such a diagram—
The planes in Borovik’s figure are those separating the parts of the eightfold cube above.
In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.
In light of Borovik’s remarks, the eightfold cube might serve to illustrate the “Cosmic” part of the Marvel Comics phrase.
For some related theological remarks, see Cube Trinity in this journal.
Happy St. Augustine’s Day.
* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.
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Sunday, June 19, 2011
Metalibrarianship
This screenshot was suggested by the word "metalibrarianship" in a Bloomsday 2011 post by artist Steve Richards.
Note the words in the search field.
The phrase "Dublin Core metadata" refers to a city in Ohio, not in Ireland.
Related material— The highlighted phrase below is in the epigraph to Borges's "The Library of Babel"—
"I would, for these causes, wish him that is melancholy, to use both humane and divine authors, voluntarily to impose some taske upon himself, to divert his melancholy thoughts; to study the art of memory, Cosmus Rosselius, Pet. Ravennas, Scenkelius Detectus, or practise brachygraphy, &c. that will aske a great deal of attention: or let him demonstrate a proposition in Euclide in his five last books, extract a square root, or studie Algebra: than which, as gClavius holds, in all humane disciplines nothing can be more excellent and pleasant, so abstruse and recondite, so bewitching, so miraculous, so ravishing, so easie withal and full of delight, omnem humanum captum superare videtur . By this means you may define ex ungue leonem , as the diverbe is, by his thumb alone the bigness of Hercules, or the true dimensions of the great hColossus, Solomons temple, and Domitians amphitheater, out of a little part. By this art you may contemplate the variation of the 23 letters…."
g Ad. 2. definit.2. elem. In disciplinis humanis nihil praestantius reperitur: quippe miracula quaedam numerorum eruit tam abstrusa et recondita, tanta nihilo minus facilitate et voluptate, ut, &c.
h Which contained 1080000 weight of brass.
— The Anatomy of Melancholy, Part. 2, Sec. 2, Mem. 4
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Tuesday, May 10, 2011
Groups Acting
The LA Times on last weekend's film "Thor"—
"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."
Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).
A high-minded— if not Shakespearean— non-Nordic approach to groups acting—
"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394
"Groups act "… For some examples, see
- The 2×2×2 Cube,
- The Diamond 16 Puzzle,
- The Diamond Theorem, and
- Finite Geometry of the Square and Cube.
Related entertainment—
High-minded— Many Dimensions—
Not so high-minded— The Cosmic Cube—
One way of blending high and low—
The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.
The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—
The above illustration is from undated software by Ed Pegg Jr.
For mathematical background, see a 1985 note, "Visualizing GL(2,p)."
For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).
For the Nordic-averse, some background on the Hebrew connection—
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Wednesday, April 20, 2011
Romancing the Cube
It was a dark and stormy night…
— Page 180, Logicomix
“… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field.”
– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups”
For some context, see the small cube in “A Simple Reflection Group of Order 168.”
See also the larger cube in “Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).
That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix page above—
A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night—
“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”
— Gian-Carlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.
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Thursday, December 2, 2010
Caesarian
The Dreidel Is Cast
The Nietzschean phrase "ruling and Caesarian spirits" occurred in yesterday morning's post "Novel Ending."
That post was followed yesterday morning by a post marking, instead, a beginning— that of Hanukkah 2010. That Jewish holiday, whose name means "dedication," commemorates the (re)dedication of the Temple in Jerusalem in 165 BC.
The holiday is celebrated with, among other things, the Jewish version of a die— the dreidel . Note the similarity of the dreidel to an illustration of The Stone* on the cover of the 2001 Eerdmans edition of Charles Williams's 1931 novel Many Dimensions—
For mathematics related to the dreidel , see Ivars Peterson's column on this date fourteen years ago.
For mathematics related (if only poetically) to The Stone , see "Solomon's Cube" in this journal.
Here is the opening of Many Dimensions—
For a fanciful linkage of the dreidel 's concept of chance to The Stone 's concept of invariant law, note that the New York Lottery yesterday evening (the beginning of Hanukkah) was 840. See also the number 840 in the final post (July 20, 2002) of the "Solomon's Cube" search.
Some further holiday meditations on a beginning—
Today, on the first full day of Hanukkah, we may or may not choose to mark another beginning— that of George Frederick James Temple, who was born in London on this date in 1901. Temple, a mathematician, was President of the London Mathematical Society in 1951-1953. From his MacTutor biography—
"In 1981 (at the age of 80) he published a book on the history of mathematics. This book 100 years of mathematics (1981) took him ten years to write and deals with, in his own words:-
those branches of mathematics in which I had been personally involved.
He declared that it was his last mathematics book, and entered the Benedictine Order as a monk. He was ordained in 1983 and entered Quarr Abbey on the Isle of Wight. However he could not stop doing mathematics and when he died he left a manuscript on the foundations of mathematics. He claims:-
The purpose of this investigation is to carry out the primary part of Hilbert's programme, i.e. to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced."
For a brief review of Temple's last work, see the note by Martin Hyland in "Fundamental Mathematical Theories," by George Temple, Philosophical Transactions of the Royal Society, A, Vol. 354, No. 1714 (Aug. 15, 1996), pp. 1941-1967.
The following remarks by Hyland are of more general interest—
"… one might crudely distinguish between philosophical and mathematical motivation. In the first case one tries to convince with a telling conceptual story; in the second one relies more on the elegance of some emergent mathematical structure. If there is a tradition in logic it favours the former, but I have a sneaking affection for the latter. Of course the distinction is not so clear cut. Elegant mathematics will of itself tell a tale, and one with the merit of simplicity. This may carry philosophical weight. But that cannot be guaranteed: in the end one cannot escape the need to form a judgement of significance."
— J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.
Here Hyland appears to be discussing semantic ("philosophical," or conceptual) and syntactic ("mathematical," or structural) approaches to proof theory. Some other remarks along these lines, from the late Gian-Carlo Rota—
See also "Galois Connections" at alpheccar.org and "The Galois Connection Between Syntax and Semantics" at logicmatters.net.
* Williams's novel says the letters of The Stone are those of the Tetragrammaton— i.e., Yod, He, Vau, He (cf. p. 26 of the 2001 Eerdmans edition). But the letters on the 2001 edition's cover Stone include the three-pronged letter Shin , also found on the dreidel . What esoteric religious meaning is implied by this, I do not know.
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Tuesday, November 23, 2010
Tale
A reviewer says Steve Martin finds in his new novel An Object of Beauty "a sardonic morality tale."
From this journal on the day The Cube was published (see today's Art Object ) —
Monday February 20, 2006
|
See also a post on Mathematics and Narrative from Nov. 14, 2009.
That post compares characters in Many Dimensions to those in Logicomix—
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Friday, October 15, 2010
Mathematics and Narrative, continued
The Story of N
Roberta Smith in the New York Times of July 7, 2006—
Art Review
Endgame Art? It's Borrow, Sample and Multiply in an Exhibition at Bard College
"… The show has an endgame, end-time mood, as if we are looking at the end of the end of the end of Pop, hyperrealism and appropriation art. The techniques of replication and copying have become so meticulous that they are beside the point. This is truly magic realism: the kind you can't see, that has to be explained. It is also a time when artists cultivate hybridism and multiplicity and disdain stylistic coherence, in keeping with the fashionable interest in collectivity, lack of ego, the fluidity of individual identity. But too often these avoidance tactics eliminate the thread of a personal sensibility or focus.
I would call all these strategies fear of form, which can be parsed as fear of materials, of working with the hands in an overt way and of originality. Most of all originality. Can we just say it? This far from Andy Warhol and Duchamp, the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.
Fear of form above all means fear of compression— of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible. With a few exceptions, forms of collage and assemblage dominate this show: the putting together (or simply putting side by side) of existing images and objects prevails. The consistency of this technique in two and three dimensions should have been a red flag for the curators. Collage has driven much art since the late 1970's. Lately, and especially in this exhibition, it often seems to have become so distended and pulled apart that its components have become virtually autonomous and unrelated, which brings us back to square one. This is most obvious in the large installations of graphic works whose individual parts gain impact and meaning from juxtaposition but are in fact considered distinct artworks."
Margaret Atwood on art and the trickster—
"The pleasures of fabulation, the charming and playful lie— this line of thought leads Hyde* to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning 'to join,' 'to fit,' and 'to make.' If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart."
* Lewis Hyde, Trickster Makes This World: Mischief, Myth, and Art, Farrar Straus & Giroux, January 1998
Smith mentions "an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."
Atwood mentions "a seamless whole."
For some related remarks, see "A Study in Art Education" and the central figure pictured above. (There "N" can stand for "number," "nine," or "narrative.")
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Friday, August 20, 2010
The Moore Correspondence
There is a remarkable correspondence between the 35 partitions of an eight-element set H into two four-element sets and the 35 partitions of the affine 4-space L over GF(2) into four parallel four-point planes. Under this correspondence, two of the H-partitions have a common refinement into 2-sets if and only if the same is true of the corresponding L-partitions (Peter J. Cameron, Parallelisms of Complete Designs, Cambridge U. Press, 1976, p. 60). The correspondence underlies the isomorphism* of the group A8 with the projective general linear group PGL(4,2) and plays an important role in the structure of the large Mathieu group M24.
A 1954 paper by W.L. Edge suggests the correspondence should be named after E.H. Moore. Hence the title of this note.
Edge says that
It is natural to ask what, if any, are the 8 objects which undergo
permutation. This question was discussed at length by Moore…**.
But, while there is no thought either of controverting Moore's claim to
have answered it or of disputing his priority, the question is primarily
a geometrical one….
Excerpts from the Edge paper—
Excerpts from the Moore paper—
Pages 432, 433, 434, and 435, as well as the section mentioned above by Edge— pp. 438 and 439
* J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford U. Press, 1985, p. 72
** Edge cited "E.H. Moore, Math. Annalen, 51 (1899), 417-44." A more complete citation from "The Scientific Work of Eliakim Hastings Moore," by G.A. Bliss, Bull. Amer. Math. Soc. Volume 40, Number 7 (1934), 501-514— E.H. Moore, "Concerning the General Equations of the Seventh and Eighth Degrees," Annalen, vol. 51 (1899), pp. 417-444.
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