Click the "timelessness" quote below for the "Bell, Book and Candle" scene
with Kim Novak and James Stewart atop the Flatiron Building.
"Before time began . . . ." — Optimus Prime
Wednesday, September 18, 2024
Flatiron Building Timelessness . . .
Directed by Quine, not Hitchcock
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Directed by Quine, not Hitchcock
Directed by Quine, not Hitchcock
Sunday, September 8, 2024
Sunday Morning Koppel
Today's host for a special political edition of CBS Sunday Morning
is Ted Koppel. Vocabulary review:
Koppel's appearance today was backed by the usual CBS Sunday Morning
sun-disk Apollo symbol. An Apollo symbol that some may prefer —
The Ninefold Square
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, the field of Reason…." |
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Sunday, July 28, 2024
For Gigi — With Chill and Irony
Today's Google Doodle, honoring soccer at the Paris games,
suggests a review of Hillman's "acorn theory" of the soul in
a Log24 post of September 10, 2022 . . .
"… I loved Gigi. It fed directly into my Francophilia.
I was convinced that at some future date, I, like
Gigi, would be trained as a courtesan. I, too, would
cause some hard case, experienced roué to abandon
his chill and irony." — Jessica Kardon
Related reading:
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Sunday, June 9, 2024
Friday, April 5, 2024
Thursday, February 15, 2024
For Harlan Kane: The Aggies Prayer
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Wednesday, October 11, 2023
Void Game
Fans of the phrase "God-shaped hole" may have some opinions
about what should fill the inner 3×3 void of the above 5×5 array.
Update of 3:53 pm ET — The White Paper —
The Source —
The Atlantic . . . Technology:
Washington and Beijing have been locked in a conflict
over AI development. Now a new battle line is being drawn.
By Karen Hao. October 11, 2023, 9:13 AM ET
Comments Off on Void Game
Saturday, August 19, 2023
Landings
One of the scenes from "Spencer" shows a Christmas weigh-in
at Sandringham with a staircase, and landing, in the background.
Another landing — On the staircase between the first and second
floors at Skillmans in Bemus Point, NY, where in a summer not too
many years ago I saw displayed a copy of Dorm Room Feng Shui .
I ordered this book online and enjoyed it when it arrived.
On the cover is a 3×3 array of images, with the caption "You are here"
in the center square.
Interpret this as you will.
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Wednesday, August 9, 2023
Waiting for the Low-Hanging Fruit
How many miles to Babylon?*
Three score miles and ten.
Can I get there by candle-light?**
Yes, and back again.
Mary Gaitskill's latest substack meditation —
"I am thinking of Susan Sontag, writer, philosopher,
political activist and some-time pain in the ass;
she went to Sarajevo during the siege in order to
put on a theatrical production of Waiting for Godot.
She didn’t get paid and none of the actors did either.
They rehearsed in the dark and performed by sparse
candlelight . . . ."
"How many bananas ?"
"Drei . . . or else Vier ."
See also the comedy writers of Elsevier —
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“The Seed Crystal” — Plan 9 from Science Direct
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Monday, August 22, 2022
Tokens/Totems
See Ballet Blanc and Black Art in this journal.
From the former:
"A blank underlies the trials of device."
— Wallace Stevens
From the latter:
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Monday, August 15, 2022
Holding the Center: A Study in Composition
Earlier . . .
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Thursday, August 11, 2022
Monday, July 25, 2022
Tuesday, May 24, 2022
Playing the Palace
From a Jamestown (NY) Post-Journal article yesterday on
"the sold-out 10,000 Maniacs 40th anniversary concert at
The Reg Lenna Center Saturday" —
" 'The theater has a special place in our hearts. It’s played
a big part in my life,' Gustafson said.
Before being known as The Reg Lenna Center for The Arts,
it was formerly known as The Palace Theater. He recalled
watching movies there as a child…."
This, and the band's name, suggest some memories perhaps
better suited to the cinematic philosophy behind "Plan 9 from
Outer Space."
"With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series
The above 3×3 Tablet of Ahkmenrah image comes from
a Log24 search for the finite (i.e., Galois) field GF(3) that
was, in turn, suggested by last night's post "Making Space."
See as well a mysterious document from a website in Slovenia
that mentions a 3×3 array "relating to nine halls of a mythical
palace where rites were performed in the 1st century AD" —
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Wednesday, April 27, 2022
Ennead (Pace Moon Knight)
Putting the graphic in lexicographic —
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Sunday, April 10, 2022
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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Tuesday, March 22, 2022
Esprit for Pascal and Galois: Finesse vs. Geometrie
Finesse —
|
Sunday December 10, 2006 m759 @ 9:00 PM
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Geometrie —
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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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Friday, December 10, 2021
Unhinged Melody
The time of the previous post was 4:46 AM ET today.
Fourteen minutes later —
"I'm a groupie, really." — Murray Bartlett in today's online NY Times
The previous post discussed group actions on a 3×3 square array. A tune
about related group actions on a 4×4 square array (a Galois tesseract ) . . .
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Friday, October 22, 2021
Frye on Structure
In search of Frye's "powder-room of the Muses" — See 3×3.
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Tuesday, August 10, 2021
Ex Fano Apollinis
|
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. 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." (254) 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. |
"As a Chinese jar . . . ."
— Four Quartets
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, |
The "Katz" of the August 7 post Art Angles
is a product of Princeton's
Department of Art and Archaeology.
ART —
.jpg)
ARCHAEOLOGY —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
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Thursday, July 29, 2021
Catchphrases from the City of Angels
"Go away — I'm asleep."
— Epitaph of the late Joan Hackett.
Hackett is at top center
in the poster below.
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Saturday, July 3, 2021
Here, There, and Chicago
The above phrase "the intersection of storytelling and visual arts"
suggests a review . . .
Storytelling —
Visual arts —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
A Midrash for Michener —
For a connection of the above "Holy Field"
with pure mathematics, see Coxeter's Aleph.
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Tuesday, March 9, 2021
Alternate Past: LA/91506
(Title suggested by the beanie label "Alternate Future: NYC/10001")
A version of the Salinger story title "Pretty Mouth and Green My Eyes" —
"… her mouth is red and large, with Disney overtones. But it is her eyes,
a pale green of surprising intensity, that hold me."
— Violet Henderson in Vogue , 30 August 2017
See also that date in this journal.
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Friday, January 1, 2021
Geometry for Child Buyers
Yesterday’s flashback to the “Square Ice” post of
St. Francis’s Day, 2016 —
This suggests a review of the July 16, 2013, post “Child Buyers.”
Related images from “Tomorrowland” (2015) —
An ignorant, but hopeful, space fan —
The space fan knocks on one door-panel of a 3×3 array —
Related image from “Hereafter” (2010) —
Matt Damon with his 3×3 door-panel array.
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Saturday, December 5, 2020
Thursday, May 14, 2020
Tournamonde
In memory of Wallace Stevens, a not-so-gay tournamonde —
Beware, beware, her flashing eyes, her floating hair:
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Monday, February 17, 2020
RIP Charles Portis
See also "True Grid " in this journal.
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, the field of Reason…." |
See as well . . .
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Friday, August 30, 2019
The Coxeter Aleph
The previous post displayed part of a page from
a newspaper published the day Olivia Newton-John
turned 21 — Friday, September 26, 1969.
A meditation, with apologies to Coleridge:
In Xanadu did Newton-John
A stately pleasure-square decree
Where Aleph the sacred symbol ran
Through subsquares measureless to man.
A related video —
Beware, beware, her flashing eyes, her floating hair:
Set design —
As opposed to block design —
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Thursday, August 29, 2019
As Well
For some backstory, see
http://m759.net/wordpress/?s=”I+Ching”+48+well .
See as well “elegantly packaged” in this journal.
“Well” in written Chinese is the hashtag symbol,
i.e., the framework of a 3×3 array.
My own favorite 3×3 array is the ABC subsquare
at lower right in the figure below —
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Monday, June 3, 2019
Jar Story
“. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”
— T. S. Eliot, Four Quartets
From Writing Chinese Characters:
“It is practical to think of a character centered
within an imaginary square grid . . . .
The grid can… be… subdivided, usually to
9 or 16 squares. . . .“
These “Chinese jars” (as opposed to their contents)
are as follows:
.
See as well Eliot’s 1922 remarks on “extinction of personality”
and the phrase “ego-extinction” in Weyl’s Philosophy of Mathematics —
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Tuesday, May 28, 2019
Quaternion at Candlebrow
From a Groundhog Day post in 2009 —
|
The Candlebrow Conference The conferees had gathered here from all around the world…. Their spirits all one way or another invested in, invested by, the siegecraft of Time and its mysteries. "Fact is, our system of so-called linear time is based on a circular or, if you like, periodic phenomenon– the earth's own spin. Everything spins, up to and including, probably, the whole universe. So we can look to the prairie, the darkening sky, the birthing of a funnel-cloud to see in its vortex the fundamental structure of everything–"
"Um, Professor–"…. … Those in attendance, some at quite high speed, had begun to disperse, the briefest of glances at the sky sufficing to explain why. As if the professor had lectured it into being, there now swung from the swollen and light-pulsing clouds to the west a classic prairie "twister"…. … In the storm cellar, over semiliquid coffee and farmhouse crullers left from the last twister, they got back to the topic of periodic functions…. "Eternal Return, just to begin with. If we may construct such functions in the abstract, then so must it be possible to construct more secular, more physical expressions." "Build a time machine." "Not the way I would have put it, but if you like, fine." Vectorists and Quaternionists in attendance reminded everybody of the function they had recently worked up…. "We thus enter the whirlwind. It becomes the very essence of a refashioned life, providing the axes to which everything will be referred. Time no long 'passes,' with a linear velocity, but 'returns,' with an angular one…. We are returned to ourselves eternally, or, if you like, timelessly." "Born again!" exclaimed a Christer in the gathering, as if suddenly enlightened. Above, the devastation had begun. |
"As if the professor had lectured it into being . . . ."
See other posts now tagged McLuhan Time.
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Sunday, December 9, 2018
Quaternions in a Small Space
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —
-
Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square. -
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube. - Quaternions in an Affine Galois Plane — A webpage from 2010.
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
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Sunday, September 9, 2018
Plan 9 Continues.
"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.
Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."
— From p. 192 of "The Phenomenology of Mathematical Proof,"
by Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183-196. Published by: Springer.
Stable URL: https://www.jstor.org/stable/20117627.
Related figures —
Note the 3×3 subsquare containing the triangles ABC, etc.
"That in which space itself is contained" — Wallace Stevens
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Thursday, August 9, 2018
True Grids
From a search in this journal for "True Grid,"
a fanciful description of the 3×3 grid —
"This is the garden of Apollo,
the field of Reason…."
– John Outram, architect
A fanciful instance of the 4×2 grid in
a scene from the film "The Master" —
A fanciful novel referring to the number 8,
and a not -so-fanciful reference:
-actions-500w.jpg)
Illustrated above are Katherine Neville's novel The Eight and the
"knight" coordinatization of the 4×2 grid from a page on the exceptional
isomorphism between PSL(3,2) (alias GL(3,2)) and PSL(2,7) — groups
of, respectively, degree 7 and degree 8.
Literature related to the above remarks on grids:
Ross Douthat's New York Times column yesterday purported, following
a 1946 poem by Auden, to contrast students of the humanities with
technocrats by saying that the former follow Hermes, the latter Apollo.
I doubt that Apollo would agree.
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Wednesday, August 1, 2018
Publish or …
|
From The New York Times online on July 29 — " Ms. Appelbaum’s favorite authors, she said in an interview with The Internet Writing Journal in 1998, were too many to count, but they included George Eliot, Anthony Trollope, Anne Tyler and Julian Barnes. 'I love to see writers expand our range of understanding, experience, knowledge, even happiness,' she said in that interview. 'Publishing has always struck me as a way to change the world.' " A version of this article appears in print on , on Page B6 of the New York edition with the headline: Judith Appelbaum, Guru On Publishing, Dies at 78. |
See a review of the new Anne Tyler novel Clock Dance
in today's online New York Times .
For a more abstract dance, see Ballet Blanc .
"A blank underlies the trials of device." — Wallace Stevens
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Tuesday, July 31, 2018
Wednesday, June 27, 2018
Friday, May 11, 2018
A Pure Geometry
From posts tagged Modernism —
|
m759 @ 9:00 PM
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
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Friday, April 6, 2018
Tuesday, February 13, 2018
A Titan of the Field
|
On the late Cambridge astronomer Donald Lynden-Bell — "As an academic at a time when students listened and lecturers lectured, he had the disconcerting habit of instead picking on a random undergraduate and testing them on the topic. One former student, now a professor, remembered how he would 'ask on-the-spot questions while announcing that his daughter would solve these problems at the breakfast table'. He got away with it because he was genuinely interested in the work of his colleagues and students, and came to be viewed with great affection by them. He also got away with it because he was well established as a titan of the field." — The London Times on Feb. 8, 2018, at 5 PM (British time) |
Related material —
Two Log24 posts from yesteday, Art Wars and The Void.
See as well the field GF(9) …
… and the 3×3 grid as a symbol of Apollo
(an Olympian rather than a Titan) —
.
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Wednesday, December 27, 2017
For Day 27 of December 2017
See the 27-part structure of
the 3x3x3 Galois cube
as well as Autism Sunday 2015.
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Friday, December 1, 2017
The Architect and the Matrix
In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —
Update from the morning of December 2 —
The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:
.
See as well Tsimtsum in this journal.
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Wednesday, October 18, 2017
Dürer for St. Luke’s Day
Structure of the Dürer magic square
16 3 2 13
5 10 11 8 decreased by 1 is …
9 6 7 12
4 15 14 1
15 2 1 12
4 9 10 7
8 5 6 11
3 14 13 0 .
Base 4 —
33 02 01 30
10 21 22 13
20 11 12 23
03 32 31 00 .
Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays —
3 0 0 3 3 2 1 0
1 2 2 1 0 1 2 3
2 1 1 2 0 1 2 3
0 3 3 0 3 2 1 0 .
Base 2 –
1111 0010 0001 1100
0100 1001 1010 0111
1000 0101 0110 1011
0011 1110 1101 0000 .
Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —
1001 1001 1100 1010
0110 1001 0011 0101
1001 0110 0011 0101
0110 0110 1100 1010 .
— Steven H. Cullinane,
October 18, 2017
See also recent related analyses of
noted 3×3 and 5×5 magic squares.
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Tuesday, October 17, 2017
Plan 9 Continues
See also Holy Field in this journal.
Some related mathematics —
Analysis of the Lo Shu structure —
Structure of the 3×3 magic square:
4 9 2
3 5 7 decreased by 1 is …
8 1 6
3 8 1
2 4 6
7 0 5
In base 3 —
10 22 01
02 11 20
21 00 12
As orthogonal Latin squares
(a well-known construction) —
1 2 0 0 2 1
0 1 2 2 1 0
2 0 1 1 0 2 .
— Steven H. Cullinane,
October 17, 2017
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Friday, October 13, 2017
Sicut Erat
Smith College in 2011 on some music by Dan Brown's brother —
"Using the conventions of a traditional five-movement
Roman Catholic Mass to revere Darwin’s body of work,
Gregory Brown, Smith’s assistant director of choral
activities and a composer of choral music, is
collaborating with Craig Phillips, an early music specialist
and member of the classical a cappella male quartet
New York Polyphony, to create the piece Missa Charles Darwin .
Brown is building the work in three large-scale sections and
scoring it for a male vocal quartet, which will be performed by
New York Polyphony."
— https://www.smith.edu/insight/stories/darwin.php
Dan Brown has said his brother's Missa helped suggest his new novel Origin .
Material from Smith College related to a performance of
Missa Charles Darwin at the college on Feb. 4, 2011 —
Dan Brown, in the following passage, claims that an eight-ray star with arrowheads
at the rays' ends is "the mathematical symbol for entropy." Brown may have first
encountered this symbol at a questionable "Sacred Science" website. Wikipedia
discusses some even less respectable uses of the symbol.
My own version of the above symbol (from the pure mathematics of group actions
on a 3×3 square) appeared here the day before the Friday, Feb. 4, 2011,
Smith College Darwin Mass . . .
See posts now tagged The Next Thing.
Comments Off on Sicut Erat
Monday, October 9, 2017
Still Point for a Dance
"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."
— T. S. Eliot, Four Quartets
See also a recurrent image
from this journal —
.
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Monday, August 14, 2017
Sound Track
Two songs by Chuck Berry on Chess Records in 1958 —
Sweet Little Sixteen and Sweet Little Rock and Roller .
She's 9 years old and sweet as she can be
All dressed up like a downtown Christmas tree
Dancin' and hummin' a rock-roll melody
For meditations on Sixteen , see Berry + Sixteen in this journal.
A meditation on Rock and Roller —
Related material — From the above post's date,
March 21, 2017, a memoir by one Siva Vaidhyanathan,
"Robertson Professor of Media Studies and Director of
the Center for Media and Citizenship at the University of Virginia."
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Thursday, July 27, 2017
Keeping It Simple
Michiko Kakutani in The New York Times —
"The detective story genre concerns the finding of clues
and the search for hidden designs, and its very form
underscores Mr. Pynchon’s obsession with conspiracies
and the existence of systems too complicated to understand."
— Review of Pynchon's Bleeding Edge , Sept. 10, 2013
Background: "Moss on the Wall," this journal on that date.
A less complicated system —
"Plan 9 deals with the resurrection of the dead."
— Bill Murray in "Ed Wood"
(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)
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Sunday, April 16, 2017
Tuesday, April 4, 2017
Plan 9 Continues
"Plan 9 deals with the resurrection of the dead."
— Bill Murray in "Ed Wood"
(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)
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Monday, April 3, 2017
Tuesday, March 21, 2017
Res Ipsa
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Sunday, March 19, 2017
Delos Incorporated* Sunday School
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Wednesday, January 4, 2017
A Drama of Many Forms
According to art historian Rosalind Krauss in 1979,
the grid's earliest employers
"can be seen to be participating in a drama
that extended well beyond the domain of art.
That drama, which took many forms, was staged
in many places. One of them was a courtroom,
where early in this century, science did battle with God,
and, reversing all earlier precedents, won."
The previous post discussed the 3×3 grid in the context of
Krauss's drama. In memory of T. S. Eliot, who died on this date
in 1965, an image of the next-largest square grid, the 4×4 array:
See instances of the above image.
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Tuesday, January 3, 2017
Cultist Space
The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —
The previous post illustrated a 3×3 grid. That cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.
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Friday, December 9, 2016
Saturday, September 17, 2016
Monday, June 20, 2016
Sacred Space
For further background to this morning's post Plan 9 Continues,
see posts tagged Sacred Space —
.
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Sunday, March 13, 2016
Space Sermon
In memory of the late architect Patrick Hodgkinson
Harvey Court at Gonville & Caius College, Cambridge
For the architect, see yesterday's post "Brick-Perfect."
See as well a meditation on the numbers 9 and 13
in the post "Space" on day 13 of May, 2015.
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Sunday, November 29, 2015
There the Dance Is
In memory of ballet designer
Yolanda Sonnabend, who
reportedly died at 80 on Nov. 9,
see posts on Apollo, Ballet Blanc,
maps of New Haven, etc., etc., etc.
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Sunday, October 18, 2015
Coordinatization Problem
There are various ways to coordinatize a 3×3 array
(the Chinese "Holy Field'). Here are some —
See Cullinane, Coxeter, and Knight tour.
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Wednesday, September 9, 2015
Saturday, August 8, 2015
Raiders of the Lost Windows
"Why is it called Windows 10 and not Windows 9?"
Good question.
See Sunday School (Log24 on June 13, 2010) —
.
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Thursday, July 9, 2015
Man and His Symbols
A post of July 7, Haiku for DeLillo, had a link to posts tagged "Holy Field GF(3)."
As the smallest Galois field based on an odd prime, this structure
clearly is of fundamental importance.
It is, however, perhaps too small to be visually impressive.
A larger, closely related, field, GF(9), may be pictured as a 3×3 array…
|
… hence as the traditional Chinese Holy Field.
Marketing the Holy Field
The above illustration of China's Holy Field occurred in the context of
Log24 posts on Child Buyers. For more on child buyers, see an excellent
condemnation today by Diane Ravitch of the U. S. Secretary of Education.
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Saturday, June 20, 2015
Conceptual Art for Basel
The previous post's link to The Lindbergh Manifesto
and Thursday's post on Basel-born artist Wolf Barth
suggest the following —
See as well a June 14 New York Times
piece on Art Basel.
The logo of the University of Basel …
… suggests a review of The Holy Field —
.
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Saturday, June 13, 2015
Plan 9 Continues…
A version of the song in the previous post that I prefer:
A related meditation —
For a more abstract version of the
"matrix of the cosmic process,"
see "3×3 Grid" in this journal.
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Tuesday, May 12, 2015
Writing Well*
See Stevens + New Haven.
* The above figure may be viewed as
the Chinese “Holy Field” or as the
Chinese character for “Well”
inscribed in a square.
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Sunday, May 10, 2015
Language Animal Farm
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Tuesday, December 30, 2014
An Education
AP Today in History
Thought for the Day:
“I respect faith, but doubt is what
gives you an education.”
— Wilson Mizner,
American playwright (1876-1933)*
From this journal on the (wide) release date
of "X-Men: First Class" —
A minimalist 3×3 matrix favicon—
This may, if one likes, be viewed as the "nothing"
present at the Creation. See Jim Holt on physics.
* A source —
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Thursday, December 18, 2014
Platonic Analogy
(Five by Five continued)
As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.
See posts tagged Galois-Plane Models.
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Tuesday, November 25, 2014
Euclidean-Galois Interplay
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3-space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
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Tuesday, November 18, 2014
The Abacus Conundrum…
Prequel from 1961 (click image for context):
Detail that may be interpreted as the Chinese
3×3 "Holy Field" and a Chinese temple bell—
"Ting-a-ling." — Kurt Vonnegut.
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Thursday, November 13, 2014
Progressive Matrix
Yesterday's post and recent Hollywood news suggest
a meditation on a Progressive Matrix —
Click to enlarge.
"My card."
Structurally related images —
A sample Raven's Progressive Matrices test item
(such items share the 3×3 structure of the hash symbol above):
Structural background —
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Saturday, October 25, 2014
Foundation Square
In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean geometry or of Galois geometry.
In Euclidean geometry, these grids illustrate a property of
the inner triangle.
In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids. This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).
The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:
See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.
For 5×5 geometry that is not so elementary, see…
-
"The Hoffman-Singleton Graph and its Automorphisms," by
Paul R. Hafner, Journal of Algebraic Combinatorics , 18 (2003), 7–12, and -
the Web pages "Hoffman-Singleton Graph" and "Higman-Sims Graph"
of A. E. Brouwer.
Hafner's abstract:
We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.
The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)
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Tuesday, October 21, 2014
Art as a Tool
Two news items on art as a tool:
Two Log24 posts related to the 3×3 grid, the underlying structure for China’s
ancient Lo Shu “magic” square:
Finally, leftist art theorist Rosalind Krauss in this journal
on AntiChristmas, 2010:
Which is the tool here, the grid or Krauss?
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Wednesday, October 15, 2014
Diabolically Complex
The title is from a Log24 post, "Diabolically Complex Riddle," of Sept. 27, 2014.
(See also a search for "Diabolic" in this journal, which yields an application to
"magic" squares.)
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Monday, October 13, 2014
Sallows on “The Lost Theorem”
Parallelograms and the structure of the 3×3 array —
Click to enlarge:
A different approach to parallelograms and arrays —
Click for original post:
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Raiders of the Lost Theorem
(Continued from Nov. 16, 2013.)
The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.
One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:
In context (click to enlarge):
Update of later the same day —
Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —
“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”
* Defined by Sallows later in the book:
“Geometric or, less formally, geomagic is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles or pieces ) may appear in the cells instead of numbers.”
** See some geometric matrices by Cullinane in a March 2001 webpage.
Earlier actual specimens — see Diamond Theory excerpts published in
February 1977 and a brief description of the original 1976 monograph:
“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”
— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society
The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.
Comments Off on Raiders of the Lost Theorem
Saturday, May 17, 2014
It’s Time for You to…
See the Field
“There have long been rumors of a mythical Ninth Element
that grants ultimate power to the Wizard who masters it.
The Order of Magick says there is no such thing. But….”
— Website of Magicka: The Ninth Element Novel
William Worthy in Beijing —
This journal on the date of Worthy’s death,
May 4, 2014, had a link to…
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Sunday, May 11, 2014
For the Perplexed
From a New York Times obituary by Bruce Weber tonight—
Charles Marowitz, Director and Playwright, Dies at 82
“There are two kinds of bafflement in the theater: the kind that fascinates as it perplexes, and the kind that just perplexes,” he wrote in The Times in 1969 in an essay about Mr. Shepard’s play “La Turista,” which had recently opened in London. “If a play doesn’t make quick sense, but enters into some kind of dialogue with our subconscious, we tend to admit it to that lounge where we entertain interesting-albeit-unfamiliar strangers.
“If it only baffles, there are several courses open to us: we can assume it is ‘above our heads’ or directed ‘to some other kind of person,’ or regretfully conclude that it confuses us because it is itself confused. However, the fear of being proved wrong is so great today that almost every new work which isn’t patently drivel gets the benefit of the doubt.”
Another play by Sam Shepard mentioned in the obituary suggests a review of…
- “From the Witch Ball,” a post from May 2,
the reported date of Marowitz’s death - “Two Satanic Majesties Request All-Devouring Fame,”
a NY Times review from October 9, 2006 - This journal on that date— October 9, 2006—
“Art Wars: To Apollo.”
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Sunday, April 27, 2014
Sunday School
Galois and Abel vs. Rubik
"Abel was done to death by poverty, Galois by stupidity.
In all the history of science there is no completer example
of the triumph of crass stupidity…."
— Eric Temple Bell, Men of Mathematics
Gray Space (Continued)
… For The Church of Plan 9.
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Wednesday, March 26, 2014
Wednesday, March 12, 2014
Obiter Dictum
The title is both a legal phrase and a phrase
used by Tom Wolfe in his writings on art.
See, too, the pattern of nine triangular half-squares
arranged in a 3×3 square used in the logo of the
Jean Stephen art galleries in Minneapolis…
… and in a print at the Tate in London (click to enlarge)—
See as well an obit of the print’s artist, Justin Knowles, who reportedly died
on Feb. 24, 2004.
Some instances of that date in this journal are related to Knowles’s aesthetics.
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Friday, February 14, 2014
Epiphany
“… the object sets up a kind of
frame or space or field
within which there can be epiphany.”
Related material —
Star Wars (January 11, 2014),
The Lyche Gate Asterisk , from 10:31 AM ET on May 22, 2010,
the date of Martin Gardner's death —
— and the March 2014 issue of the
Notices of the American Mathematical Society —
See as well Epiphany 2014 (Jan. 6) in this journal and the
March Notices on the Shaw prize —
"Established under the auspices of Run Run Shaw
in November 2002, the prize is managed and
administered by the Shaw Prize Foundation
based in Hong Kong."
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Saturday, November 16, 2013
Raiders of the Lost Theorem
Yes. See …
The 48 actions of GL(2,3) on a 3×3 coordinate-array A,
when matrices of that group right-multiply the elements of A,
with A =
| (1,1) (1,0) (1,2) (0,1) (0,0) (0,2) (2,1) (2,0) (2,2) |
Actions of GL(2,p) on a pxp coordinate-array have the
same sorts of symmetries, where p is any odd prime.
Note that A, regarded in the Sallows manner as a magic square,
has the constant sum (0,0) in rows, columns, both diagonals, and
all four broken diagonals (with arithmetic modulo 3).
For a more sophisticated approach to the structure of the
ninefold square, see Coxeter + Aleph.
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Saturday, October 5, 2013
For Baron Samedi
Click on the image for a video.
See also Josefine Lyche's "Grids, you say?"
I prefer Lyche's versions of the diagonal
3×3 grid. Her versions have no lettering.
(This post was suggested by a photo of magical sigils
that Lyche posted a few hours ago at Facebook.
The above seems to be another such sigil that may
or may not be intended to function like those posted
today by Lyche.)
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Monday, August 26, 2013
Dark Side Tales
"Got to keep the loonies on the path."
— Lyrics to Dark Side of the Moon
For those who, like Tom Stoppard, prefer the dark side—
|
NEW ANGLE:
INT. OFFICE BUILDING – NIGHT
NIGHT WATCHMAN
Bateman wheels around and shoots him.
NEW ANGLE:
INT. PIERCE & PIERCE LOBBY – NIGHT
— AMERICAN PSYCHO |
Not quite so dark—
"And then one day you find ten years have got behind you."
— Lyrics to Dark Side of the Moon
This journal ten years ago, on August 25, 2003—
|
… We seek
The poem of pure reality, untouched
At the exactest point at which it is itself,
The eye made clear of uncertainty, with the sight
Everything, the spirit's alchemicana
The solid, but the movable, the moment,
— Wallace Stevens, "An Ordinary Evening |
"A view of New Haven, say…." —
"This is the garden of Apollo,
the field of Reason…."
John Outram, architect
A similar version of this Apollonian image —
Detail:
Related material for the loonies:
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Tuesday, July 16, 2013
Child Buyers
The title refers to a classic 1960 novel by John Hersey.
“How do you get young people excited about space?”
— Megan Garber in The Atlantic , Aug. 16, 2012
(Italics added.) (See previous four posts.)
Allyn Jackson on “Simplicity, in Mathematics and in Art,”
in the new August 2013 issue of Notices of the American
Mathematical Society—
“As conventions evolve, so do notions of simplicity.
Franks mentioned Gauss’s 1831 paper that
established the respectability of complex numbers.”
This suggests a related image by Gauss, with a
remark on simplicity—
Here Gauss’s diagram is not, as may appear at first glance,
a 3×3 array of squares, but is rather a 4×4 array of discrete
points (part of an infinite plane array).
Related material that does feature the somewhat simpler 3×3 array
of squares, not seen as part of an infinite array—
Marketing the Holy Field
Click image for the original post.
For a purely mathematical view of the holy field, see Visualizing GL(2,p).
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Monday, May 13, 2013
Sunday, May 5, 2013
Night of Lunacy*
Structure vs. Character continued…
|
Structure |
|
Related vocabulary:
Nick Tosches on the German word “Quell “
* The title is from Heidegger.
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Wednesday, April 10, 2013
Caution: Slow Art
"Of course, DeLillo being DeLillo,
it’s the deeper implications of the piece —
what it reveals about the nature of
film, perception and time — that detain him."
— Geoff Dyer, review of Point Omega
Related material:
A phrase of critic Robert Hughes,
"slow art," in this journal.
A search for that phrase yields the following
figure from a post on DeLillo of Oct. 12, 2011:
The above 3×3 grid is embedded in a
somewhat more sophisticated example
of conceptual art from April 1, 2013:
Update of April 12, 2013
The above key uses labels from the frontispiece
to Baker's 1922 Principles of Geometry, Vol. I ,
that shows a three-triangle version of Desargues's theorem.
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
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Sunday, March 31, 2013
Tuesday, February 19, 2013
Configurations
Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.
My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010.
For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books and Amazon.com):
For a similar 1998 treatment of the topic, see Burkard Polster's
A Geometrical Picture Book (Springer, 1998), pp. 103-104.
The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's
symmetry planes , contradicting the usual use of of that term.
That argument concerns the interplay between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)
Related material: Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.
* Earlier guides: the diamond theorem (1978), similar theorems for
2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
(1985). See also Spaces as Hypercubes (2012).
Comments Off on Configurations
Thursday, November 29, 2012
Lines of Symbols
C. P. Snow on G. H. Hardy, in Snow's foreword to A Mathematician's Apology—
"One morning early in 1913, he found, among the letters on his breakfast table, a large untidy envelope decorated with Indian stamps. When he opened it, he found sheets of paper by no means fresh, on which, in a non-English holograph, were line after line of symbols. Hardy glanced at them without enthusiasm. He was by this time, at the age of thirty-six, a world famous mathematician: and world famous mathematicians, he had already discovered, are unusually exposed to cranks. He was accustomed to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon has inserted in the plays of the so-called Shakespeare."
Some related material (click to enlarge)—
The author links to, but does not name, the source of the above
"line after line of symbols." It is "Visualizing GL(2,p)." See that webpage
for some less esoteric background.
See also the two Wikipedia articles Finite geometry and Hesse configuration
and an image they share—
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Monday, November 5, 2012
Sitting Specially
Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.
Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.
A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.
Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine article from April 2009—
Click for a pdf of the article.
Related material:
“The Religion of Cubism” (May 9, 2003) and “Art and Lies”
(Nov. 16, 2008).
This post was suggested by following the link in yesterday’s
Sunday School post to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date—
The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.
Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).
Comments Off on Sitting Specially
Friday, October 26, 2012
High White Noon
Today's 11:01 AM post discussed time concepts
in Eliot's Four Quartets.
For the temporally challenged, here is
a somewhat simpler conceptual framework—
|
Three Trios From a post of Columbus Day
|
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Sunday, October 21, 2012
Saturday, August 18, 2012
Summer Reading
On the author of a novel published August 14th,
"Where'd You Go, Bernadette"—
"Semple moved to the Pacific Northwest several years ago
seeking refuge from Los Angeles, but that doesn't mean
that the Emerald City gets a free pass from Semple's
sharp, satirical eye."
— Stewart Oksenhorn yesterday in The Aspen Times
See also a detail from Thursday's 1.3 MB image
"Search for the Lost Tesseract"—
Update of 9 PM EDT (6 PM LA time) the same day, Saturday, Aug. 18—
Comments Off on Summer Reading
Tuesday, August 7, 2012
The Space of Horizons
"In the space of horizons that neither love nor hate"
— Wallace Stevens, "Things of August"
Seven years ago yesterday—
For some context, see Rosetta Stone as a Metaphor.
Related material from the University of Western Australia—
Projective plane of order 3
(The four points on the curve
at the right of the image are
the points on the line at infinity.)
Art critic Robert Hughes, who nearly died in Western
Australia in a 1999 car crash, actually met his death
yesterday at Calvary Hospital in the Bronx.
See also Hughes on "slow art" in this journal.
Comments Off on The Space of Horizons
Saturday, June 2, 2012
Matrix Problem Revolutions
(The sequel to yesterday's Matrix Problem Reloaded)
Wikipedia on the sci-fi weblog io9.com—
Newitz explained the significance of the name "io9":
"Well, io9s are input-output devices that let you see into the future.
They're brain implants that were outlawed because they drove
anyone who used one insane. We totally made that (device) up
to name the blog."
— Jenna Wortham at wired.com, Jan. 2, 2008
From io9.com itself—
"Science fiction writer Ken MacLeod has another term for io9ers.
He calls them rapture fuckers.*"
For the relevance of the term "revolutions" in this post's title, see
Wikipedia on Ken MacLeod.
I prefer to associate the number 9 with The Holy Field.
* MacLeod used this phrase in one of his novels, Newton's Wake.
Comments Off on Matrix Problem Revolutions
Friday, April 27, 2012
An April 27–
The 3×3×3 Galois Cube
Backstory— The Talented, from April 26 last year,
and Atlas Shrugged, from April 27 last year.
Comments Off on An April 27–
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.
Comments Off on The Palpatine Dimension
Saturday, March 24, 2012
The David Waltz…
"At the still point…" — T. S. Eliot
In memory of David L. Waltz, artificial-intelligence pioneer,
who died Thursday, March 22, 2012—
- The Log24 post of March 22 on the square-triangle theorem
- The March 18 post, Square-Triangle Diamond
- Remarks from the BBC on linguistic embedding
that begin as follows—
"If we draw a large triangle and embed smaller triangles in it,
how does it look?"—
and include discussion of a South American "tribe called Piranha" [sic ] - The result of a Cartoon Bank search suggested by no. 3 above—
(Click image for some related material.)
- A suggestion from the Cartoon Bank—
-
The following from the First of May, 2010—
Some context–
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71 - The phrase "embedding the stone" —
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Monday, February 20, 2012
Coxeter and the Relativity Problem
In the Beginning…
"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)
From some 1949 remarks of Weyl—
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541
Weyl in 1946—:
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16
Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—
"… the successive powers of the primitive root λ or 10 are
λ = 10, λ2 = 21, λ3 = 22, λ4 = 02,
λ5 = 20, λ6 = 12, λ7 = 11, λ8 = 01.
These are the proper coordinate symbols….
(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"
Coxeter's Figure 10 yields...
The Aleph
The details:
Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).
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Saturday, January 28, 2012
The Sweet Smell of Avon
The twin topics of autism and of narrowing definitions
suggested the following remarks.
The mystical number "318" in the pilot episode
of Kiefer Sutherland's new series about autism, "Touch,"
is so small that it can easily apply (as the pilot
illustrated) to many different things: a date, a
time, a bus number, an address, etc.
The last 3/18 Log24 post— Defining Configurations—
led, after a false start and some further research,
to the writing of the webpage Configurations and Squares.
An image from that page—
Interpreting this, in an autistic manner, as the number
287501346 lets us search for more specific items
than those labeled simply 318.
The search yields, among other things, an offer of
Night Magic Cologne (unsold)—
For further mystery and magic, see, from the date
the Night Magic offer closed— May 8, 2010— "A Better Story."
See also the next day's followup, "The Ninth Gate."
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Saturday, January 14, 2012
Defining Form (continued)
Detail of Sylvie Donmoyer picture discussed
here on January 10—
The "13" tile may refer to the 13 symmetry axes
in the 3x3x3 Galois cube, or the corresponding
13 planes through the center in that cube. (See
this morning's post and Cubist Geometries.)
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Sunday, January 1, 2012
Sunday Shul
"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."
— Rosalind Krauss in "Grids,"
October (Summer 1979), 9: 50-64.
Counterexample—
The Ninefold Square
See Coxeter and the Aleph and Ayn Sof—
| Mathematics and Narrative, Illustrated |
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Mathematics |
 Narrative |
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Friday, December 30, 2011
Quaternions on a Cube
The following picture provides a new visual approach to
the order-8 quaternion group's automorphisms.
Click the above image for some context.
Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.
See also…
- The 1985 note from which the above figures were drawn
- Visualizing GL(2,p)
- Quaternions in an Affine Galois Plane
Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.
* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
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Thursday, November 3, 2011
The China Conundrum
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"Buckle up!" — Harlan Kane, in the spirit of strategic stupidity.
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Wednesday, October 12, 2011
High White Noon
Grid from a post linked to in yesterday's 24 Hour DeLillo—
For an example of this grid as slow art , consider the following—
"One can show that the binary tetrahedral group
is isomorphic to the special linear group SL(2,3)—
the group of all 2×2 matrices over the finite field F3
with unit determinant." —Wikipedia
As John Baez has noted, these two groups have the same structure as the geometric 24-cell.
For the connection of the grid to the groups and the 24-cell, see Visualizing GL(2,p).
Related material—
The 3×3 grid has been called a symbol of Apollo (Greek god of reason and of the sun).
"This is where we sat through his hushed hour,
a torchlit sky, the closeness of hills barely visible
at high white noon." — Don DeLillo, Point Omega
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Friday, September 9, 2011
A Beginning
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From MIT Commencement Day, 2011— A symbol related to Apollo, to nine, and to "nothing"— A minimalist favicon—
This miniature 3×3 square— |
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Thursday, September 8, 2011
Starring the Diamond
"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 83
are joined by four concurrent lines."
— H. S. M. Coxeter (see below)
Continued from Tuesday, Sept. 6—
The Diamond Star
The above is a version of a figure from Configurations and Squares.
Yesterday's post related the the Pappus configuration to this figure.
Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to the figure.
Some excerpts from Coxeter—
The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8-cycle in the Galois field GF(9).
The relabeled configuration is used in a discussion of Pappus—
(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)
Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues configuration—
The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts—
"The value of Desargues' theorem and the reason why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities that relate geometry and algebra in unexpected ways."
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Monday, June 6, 2011
Tree of Life — Jewish Version
Today's midday NY Lottery number was 753, the number of a significant page in Gravity's Rainbow .
An excerpt from that page ((Penguin Classics paperback, June 1, 1995)—
"… the Abyss had crept intolerably close, only an accident away…."
Midrash— See Ben Stein in this journal.
But seriously… See "Geometry and Death" in this journal.
See also PlanetMath.org on the Hesse configuration—
A picture of the Hesse configuration—
.
Some context— A Study in Art Education.
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Friday, June 3, 2011
MIT Day
Today is Commencement Day at MIT.
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“To measure the changes — Shing-Tung Yau, To measure the changes:
The smartest are nothing:
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Well, perhaps not quite nothing.
The above pictures were posted here on the day the following book was published—
The lives of the nine Jews in the above book amount to more than Yau's "nothing."
Note, however, that claims by Jews (see Jill Abramson yesterday)
that their secular publications constitute a substitute for religion
and contain only "absolute truth" should be viewed with at least one
raised eyebrow.
Abramson's remark yesterday that her promotion to New York Times executive editor
was like "ascending to Valhalla" had a religious flavor worthy of yesterday's
Feast of the Ascension.
In related news from yesterday's Times—
See also a symbol related to Apollo, to nine, and to "nothing"—
A minimalist 3×3 matrix favicon—
This may, if one likes, be viewed as the "nothing"
present at the Creation. See Jim Holt on physics.
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Monday, May 23, 2011
The Stoner Series
A reader comments on yesterday afternoon's New York Times
"The Stone" column by Justin E.H. Smith—
"I did indeed appreciate Mr. Smith’s essay.
And I’m curious as to what future contributions of his,
to the Stoner series, that we can look forward to."
From August 24, 2010—
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Happy day 23 of Mental Health Month.
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Wednesday, May 18, 2011
Minimalist Icon
The source of the mysterious generic
3×3 favicon with one green cell —
— has been identified.
For minimalists, here is a purer 3×3 matrix favicon—
This may, if one likes, be viewed as the "nothing"
present at the Creation. See Jim Holt on physics.
See also Visualizing GL(2,p), Coxeter and the Aleph, and Ayn Sof.
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