The source —
Monday, October 15, 2018
Tesserae for a Tesseract
Monday, March 12, 2018
“Quantum Tesseract Theorem?”
Remarks related to a recent film and a not-so-recent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The Saniga-Planat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Saturday, May 20, 2017
van Lint and Wilson Meet the Galois Tesseract*
Click image to enlarge.
The above 35 projective lines, within a 4×4 array —
The above 15 projective planes, within a 4×4 array (in white) —
* See Galois Tesseract in this journal.
Thursday, May 11, 2017
Reopening the Tesseract
Dialogue from the film "Interstellar" —
Cooper: Did it work?
TARS: I think it might have.
Cooper: How do you know?
TARS: Because the bulk beings
are closing the tesseract.
Related material — "Bulk apperception"
in this journal, and …
Wednesday, December 28, 2016
Rosetta Tesseracts
Tuesday, March 24, 2015
Brouwer on the Galois Tesseract
Yesterday's post suggests a review of the following —
Andries Brouwer, preprint, 1982:
"The Witt designs, Golay codes and Mathieu groups" Pages 8-9: Substructures of S(5, 8, 24) An octad is a block of S(5, 8, 24). Theorem 5.1
Let B_{0} be a fixed octad. The 30 octads disjoint from B_{0}
the design of the points and affine hyperplanes in AG(4, 2), Proof…. … (iv) We have AG(4, 2).
(Proof: invoke your favorite characterization of AG(4, 2) An explicit construction of the vector space is also easy….) |
Related material: Posts tagged Priority.
Tuesday, December 10, 2013
Wittgenstein’s Tesseract
See also last night's "Pink Champagne on Ice" post.
The "ice" in that post's title refers to the white lines
forming a tesseract in the book cover's background—
"icy white and crystalline," as Johnny Mercer put it.
(A Tune for Josefine, Nov. 25.)
See also the tag Diamond Theory tesseract in this journal.
Saturday, July 6, 2013
The People’s Tesseract*
From Andries Brouwer —
* Related material: Yesterday's evening post and The People's Cube.
(By the way, any 4×4 array is a tesseract .)
Thursday, August 16, 2012
Raiders of the Lost Tesseract
(Continued from August 13. See also Coxeter Graveyard.)
Here the tombstone says
"GEOMETRY… 600 BC — 1900 AD… R.I.P."
In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.
An 1892 figure by Jowett illustrating Plato's Meno—
Jowett's picture is nonetheless of interest for
its resemblance to a figure drawn some decades later
by the Toronto geometer H. S. M. Coxeter.
A similar 1950 figure by Coxeter illustrating a tesseract—
For a less scholarly, but equally confusing, view of the number 8,
see The Eight , a novel by Katherine Neville.
Monday, August 13, 2012
Raiders of the Lost Tesseract
(An episode of Mathematics and Narrative )
A report on the August 9th opening of Sondheim's Into the Woods—
Amy Adams… explained why she decided to take on the role of the Baker’s Wife.
“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com
Related material—
Amy Adams in Sunshine Cleaning "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro
Compare and contrast…
1. The following item from Walpurgisnacht 2012—
2. The six partitions of a tesseract's 16 vertices
into four parallel faces in Diamond Theory in 1937—
Sunday, July 29, 2012
The Galois Tesseract
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vector-space structure:
0 | c | d | c + d |
a | a + c | a + d | a + c + d |
b | b + c | b + d | b + c + d |
a + b | a + b + c | a + b + d | a + b + c + d |
(This vector-space a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
Source:
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.
Monday, June 4, 2012
Cube to Tesseract
Yesterday's post Child's Play displayed a cube formed
by a Hasse diagram of the 8 subsets of a 3-set.*
This suggests a review of a post from last January—
* See a comment on yesterday's post relating it to earlier,
very similar, remarks by Margaret Masterman.
I was unaware yesterday that those remarks exist.
Tuesday, January 31, 2012
Tesseract
“… a finite set with n elements Tesseract formed from a 4-set— The same 16 subsets or points can “There is such a thing as a 4-set.” |
Update of August 12, 2012:
Figures like the above, with adjacent vertices differing in only one coordinate,
appear in a 1950 paper of H. S. M. Coxeter—
Saturday, September 3, 2011
The Galois Tesseract (continued)
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M_{24}, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
Thursday, September 1, 2011
Tuesday, August 6, 2019
Mathematics and Narrative: The Crosswicks Curse Continues.
"There is such a thing as a desktop."
— Saying adapted from a 1962 young-adult novel.
Sunday, July 14, 2019
Old Pathways in Science:
The Quantum Tesseract Theorem Revisited
"The secret is that the super-mathematician expresses by the anticommutation
of his operators the property which the geometer conceives as perpendicularity
of displacements. That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.
Thus far we have touched only the fringe of the structure of our set of sixteen E-operators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."
A related illustration, from posts tagged Dirac and Geometry —
Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.
Tuesday, July 9, 2019
Perception of Space
The three previous posts have now been tagged . . .
Tetrahedron vs. Square and Triangle vs. Cube.
Related material —
Tetrahedron vs. Square:
Labeling the Tetrahedral Model (Click to enlarge) —
Triangle vs. Cube:
… and, from the date of the above John Baez remark —
Monday, July 8, 2019
Exploring Schoolgirl Space
See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."
Thursday, July 4, 2019
From Devil’s Night 2014
And now, General, time presses; and America is in a hurry.
Have you realized that though you may occupy towns and win battles,
you cannot conquer a nation? — The Devil's Disciple
A figure related to Dürer's "magic" square posted during Devil's Night —
Tuesday, April 9, 2019
Zero Dark Nine:
The Crosswicks Curse Continues . . .
"There is such a thing as geometry."
— Saying adapted from a 1962 young-adult novel.
Monday, March 11, 2019
Ant-Man Meets Doctor Strange
The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .
Overarching Metanarratives
See also "Overarching + Tesseract" in this journal. From the results
of that search, some context for the "inscape" of the previous post —
Wednesday, March 6, 2019
The Relativity Problem and Burkard Polster
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 |
For some context, see Relativity Problem in this journal.
In the case of PG(3,2), there is a choice of geometric models
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.
The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.
For comparison of the two models, see a figure posted here on
May 21, 2014 —
Labeling the Tetrahedral Model (Click to enlarge) —
"Citation needed" —
The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)," a "reliable source."
Thursday, February 28, 2019
Wikipedia Scholarship
Besides omitting the name Cullinane, the anonymous Wikipedia author
also omitted the step of representing the hypercube by a 4×4 array —
an array called in this journal a Galois tesseract.
Sunday, February 17, 2019
See Also …
"And the Führer digs for trinkets in the desert."
"See also Acht "
— Cambridge German-English Dictionary, article on "Elf "
Wednesday, December 12, 2018
An Inscape for Douthat
Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —
In finite geometry and combinatorics,
an inscape is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:
Related material — the phrase
"Quantum Tesseract Theorem" and …
A. An image from the recent
film "A Wrinkle in Time" —
B. A quote from the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
Tuesday, November 13, 2018
Blackboard Jungle Continues.
From the 1955 film "Blackboard Jungle" —
From a trailer for the recent film version of A Wrinkle in Time —
Detail of the phrase "quantum tesseract theorem":
From the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
Related mathematics from Koen Thas that some might call a
"quantum tesseract theorem" —
Some background —
See also posts tagged Dirac and Geometry. For more
background on finite geometry, see a web page
at Thas's institution, Ghent University.
Monday, October 15, 2018
History at Bellevue
The previous post, "Tesserae for a Tesseract," contains the following
passage from a 1987 review of a book about Finnegans Wake —
"Basically, Mr. Bishop sees the text from above
and as a whole — less as a sequential story than
as a box of pied type or tesserae for a mosaic,
materials for a pattern to be made."
A set of 16 of the Wechsler cubes below are tesserae that
may be used to make patterns in the Galois tesseract.
Another Bellevue story —
“History, Stephen said, is a nightmare
from which I am trying to awake.”
— James Joyce, Ulysses
Tuesday, September 4, 2018
MBTI at the Church of St. Frank*
Wednesday, June 27, 2018
Taken In
A passage that may or may not have influenced Madeleine L'Engle's
writings about the tesseract :
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."
Thursday, June 21, 2018
Models of Being
A Buddhist view —
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
A Christian view —
A formal view —
From a Log24 search for High Concept:
See also Galois Tesseract.
Monday, June 11, 2018
Arty Fact
The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.
The Eye of ARTI —
See also a post of May 19, "Uh-Oh" —
— and a post of June 6, "Geometry for Goyim" —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
An arty fact I prefer, suggested by the triangular computer-eye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
Sunday, June 10, 2018
Pieces of April
This journal on April 16, 2018 —
Happy birthday to Pope Emeritus Benedict XVI.
Related material from another weblog in a post also dated April 16, 2018 —
"As I write this, it’s April 5, midway through the eight-day
festival of Passover. During this holiday, we Jews air our
grievances against the ancient Pharaoh who enslaved
and oppressed us, and celebrate the feats of strength
with which the Almighty delivered us from bondage —
wait a minute, I think I’m mixing up Passover with Festivus."
. . . .
"Next month: Time and Tesseracts."
From that next post, dated May 16, 2018 —
"The tesseract entered popular culture through
Madeleine L’Engle’s 'A Wrinkle in Time' . . . ."
The post's author, James Propp, notes that
" L’Engle caused some of her readers confusion
when one of the characters … the prodigy
Charles Wallace Murray [sic ] , declared 'Well, the fifth
dimension’s a tesseract.' "
Propp is not unfamiliar with prodigies:
"When I was a kid living in the Long Island suburbs,
I sometimes got called a math genius. I didn’t think
the label was apt, but I didn’t mind it; being put in
the genius box came with some pretty good perks."
— "The Genius Box," a post dated March 16, 2018
To me, Propp seems less like Charles Wallace
and more like the Prime Coordinator —
For further details, see the following synchronicity checks:
Tuesday, May 1, 2018
Wake
Remarks on space from 1998 by sci-fi author Robert J. Sawyer quoted
here on Sunday (see the tag "Sawyer's Space") suggest a review of
rather similar remarks on space from 1977 by sci-fi author M. A. Foster
(see the tag "Foster's Space"):
Quoted here on September 26, 2012 —
"All she had to do was kick off and flow."
"I'se so silly to be flowing but I no canna stay."
Another work by Sawyer —
Sunday, April 29, 2018
Wednesday, April 25, 2018
Thursday, March 29, 2018
“Before Creation Itself . . .”
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
"Is this Time Cube?"
Notes toward an answer —
And from Six-Set Geometry in this journal . . .
Sunday, March 11, 2018
Blackboard Jungle Continues . . .
. . . With intolerable disrespect for the word …
In particular, the word "theorem."
See also "Quantum Tesseract Theorem" in this journal.
Thursday, March 8, 2018
Thursday, January 25, 2018
Beware of Analogical Extension
"By an archetype I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."
— Max Black in Models and Metaphors
(Cornell, 1962, p. 241)
"Others … spoke of 'ultimate frames of reference' …."
— Ibid.
A "frame of reference" for the concept four quartets —
A less reputable analogical extension of the same
frame of reference —
Madeleine L'Engle in A Swiftly Tilting Planet :
"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."
See also the phrase Galois tesseract .
Wednesday, January 24, 2018
The Pentagram Papers
From a Log24 post of March 4, 2008 —
SINGER, ISAAC:
"Sets forth his own aims in writing for children and laments
— An Annotated Listing of Criticism
"She returned the smile, then looked across the room to
— A Swiftly Tilting Planet,
For "the dimension of time," see A Fold in Time, Time Fold,
A Swiftly Tilting Planet is a fantasy for children |
Ibid. —
The pen's point:
John Trever, Albuquerque Journal, 2/29/08
Note the figure on the cover of National Review above —
A related figure from Pentagram Design —
See, more generally, Isaac Singer in this journal.
Tuesday, January 9, 2018
Koen Thas and Quantum Theory
This post supplies some background for earlier posts tagged
Quantum Tesseract Theorem.
Monday, January 8, 2018
Raiders of the Lost Theorem
The Quantum Tesseract Theorem —
Raiders —
A Wrinkle in Time
starring Storm Reid,
Reese Witherspoon,
Oprah Winfrey &
Mindy Kaling
Time Magazine December 25, 2017 – January 1, 2018
Thursday, December 28, 2017
Rocky Start
The above prose suggests a musical alternative to the Dec. 21
Camazotz song in the posts tagged Quantum Tesseract Theorem . . .
Saturday, December 23, 2017
The Right Stuff
A figure related to the general connecting theorem of Koen Thas —
See also posts tagged Dirac and Geometry in this journal.
Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."
The Patterning
Friday, December 22, 2017
Thursday, December 21, 2017
Wrinkles
TIME magazine, issue of December 25th, 2017 —
" In 2003, Hand worked with Disney to produce a made-for-TV movie.
Thanks to budget constraints, among other issues, the adaptation
turned out bland and uninspiring. It disappointed audiences,
L’Engle and Hand. 'This is not the dream,' Hand recalls telling herself.
'I’m sure there were people at Disney that wished I would go away.' "
Not the dream? It was, however, the nightmare, presenting very well
the encounter in Camazotz of Charles Wallace with the Tempter.
From a trailer for the latest version —
Detail:
From the 1962 book —
"There's something phoney in the whole setup, Meg thought.
There is definitely something rotten in the state of Camazotz."
Song adapted from a 1960 musical —
"In short, there's simply not
A more congenial spot
For happy-ever-aftering
Than here in Camazotz!"
Thursday, October 19, 2017
Design Grammar***
The elementary shapes at the top of the figure below mirror
the looking-glass property of the classical Lo Shu square.
The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008.
** A well-known construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
Saturday, September 23, 2017
The Turn of the Frame
"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."
— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.
See also the previous post and The Galois Tesseract.
Sunday, August 27, 2017
Black Well
The "Black" of the title refers to the previous post.
For the "Well," see Hexagram 48.
Related material —
The Galois Tesseract and, more generally, Binary Coordinate Systems.
Saturday, August 26, 2017
Aesthetic Distance
Naive readers may suppose that this sort of thing is
related to what has been dubbed "geometric group theory."
It is not. See posts now tagged Aesthetic Distance.
Sunday, July 23, 2017
The Partitioned Self
Tuesday, July 11, 2017
Dialogue from Plato’s Cave
Why was the Cosmic Cube named the Tesseract
in the Marvel movie series? Is there any specific reason
for the name change? According to me, Cosmic Cube
seems a nice and cooler name.
— Asked March 14, 2013, by Dhwaneet Bhatt
At least it wasn't called 'The AllSpark.'
It's not out of the realm of possibility.
— Solemnity, March 14, 2013
Saturday, June 3, 2017
Expanding the Spielraum (Continued*)
Or: The Square
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy
* See Expanding the Spielraum in this journal.
Tuesday, May 23, 2017
Saturday, May 20, 2017
The Ludicrous Extreme
From a review of the 2016 film "Arrival" —
"A seemingly off-hand reference to Abbott and Costello
is our gateway. In a movie as generally humorless as Arrival,
the jokes mean something. Ironically, it is Donnelly, not Banks,
who initiates the joke, naming the verbally inexpressive
Heptapod aliens after the loquacious Classical Hollywood
comedians. The squid-like aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
Sapir-Whorf taken to the ludicrous extreme."
— Jordan Brower in the Los Angeles Review of Books
Further on in the review —
"Banks doesn’t fully understand the alien language, but she
knows it well enough to get by. This realization emerges
most evidently when Banks enters the alien ship and, floating
alongside Costello, converses with it in their picture-language.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."
For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —
van Lint and Wilson Meet the Galois Tesseract.
The death process of van Lint occurred on Sept. 28, 2004.
Tuesday, May 2, 2017
Image Albums
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
Saturday, March 25, 2017
Twin Pillars of Symmetry
The phrase "twin pillars" in a New York Times Fashion & Style
article today suggests a look at another pair of pillars —
This pair, from the realm of memory, history, and geometry disparaged
by the late painter Mark Rothko, might be viewed by Rothko
as "parodies of ideas (which are ghosts)." (See the previous post.)
For a relationship between a 3-dimensional simplex and the {4, 3, 3},
see my note from May 21, 2014, on the tetrahedron and the tesseract.
Sunday, March 19, 2017
Norwegian Sermon
Saturday, December 10, 2016
Folk Etymology
Images from Burkard Polster's Geometrical Picture Book —
See as well in this journal the large Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.
Exercise: Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the small-configuration model's striking symmetry?
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model, which requires a free player download.)
Labeling the Tetrahedral Model (Click to enlarge) —
Related folk etymology (see point a above) —
Related literature —
The concept of "fire in the center" at The New Yorker ,
issue dated December 12, 2016, on pages 38-39 in the
poem by Marsha de la O titled "A Natural History of Light."
Cézanne's Greetings.
Friday, December 9, 2016
Snow Dance
See Ballet Blanc in this journal.
For a darker perspective, click on the image below.
See also Cartier in The Hexagon of Opposition.
Happy birthday to Kirk Douglas.
Wednesday, October 5, 2016
Sources
From a Google image search yesterday —
Sources (left to right, top to bottom) —
Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)
Tuesday, August 16, 2016
Midnight Narrative
The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —
Cross and Boolean lattice.
The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —
Update of 10 am on August 16, 2016 —
See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
Friday, July 15, 2016
Autistic Enchantment*
Robert Nye, author of the novel Falstaff , reportedly died
at 77 on July 2, 2016.
Harvey D. Heinz, expert on magic squares, cubes,
tesseracts, etc., reportedly died at 82 on July 6, 2013.
In memoriam —
From the date of Nye's death:
From Nye's book:
From the date of Heinz's death:
* See also a search for the title in this journal.
Wednesday, May 25, 2016
Framework
"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng and H. van Dam,
February 20, 2009
For one such framework,* see posts from that same date
four years earlier — February 20, 2005.
* A 4×4 array. See the 1977, 1978, and 1986 versions by
Steven H. Cullinane, the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —
Cullinane, 1977
Cullinane, 1978
Cullinane, 1986
Curtis, 1987
Update of 10:42 PM ET on Sunday, June 19, 2016 —
The above images are precursors to …
Conway and Sloane, 1988
Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.
Wednesday, May 11, 2016
Jewel of Odin
The tesseract in last night's post Game Theory
suggests a search in Log24 for "Jewel of Odin."
See also Trinkets.
Tuesday, May 10, 2016
Game Theory
The following passage appeared in this journal
on the night of May 23-24, 2015.
The afternoon of May 23, 2015, was significant
for devotees of mathematics and narrative.
Monday, April 18, 2016
The Philosopher’s Apprentice…
… is a novel by James Morrow reviewed in The New York Times
on March 23, 2008:
"Morrow’s inventiveness is beguiling, as are his delight
in Western philosophy and his concern for the sorry state
of the world. Yet there’s also something comic-bookish
about his novel…."
"Something comic-bookish"
in memory of Albert Einstein,
who reportedly died on this date
in 1955 —
Saturday, April 16, 2016
Matinee (continued)
Today is Kelli O'Hara's last Saturday matinee in "The King and I."
A show that some may prefer —
Related to the plot of Dante's film —
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , Chapter 5, "The Tesseract" |
Monday, February 22, 2016
Schoolgirl Problems…
and versions of "Both Sides Now"
See a New York Times version of "Both Sides Now."
I prefer a version by Umberto Eco.
Related material for storytellers and the Church of Synchronology —
This journal on the date of the above shooting script, 03/19/15.
Friday, January 29, 2016
Excellent Adventure*
(Continued from Dec. 9, 2013)
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , |
From a media weblog yesterday, a quote from the video below —
"At 12:03 PM Eastern Standard Time, January 12th, 2016…."
This weblog on the previous day (January 11th, 2016) —
"There is such a thing as harmonic analysis of switching functions."
— Saying adapted from a young-adult novel
* For some backstory, see a Caltech page.
Thursday, January 14, 2016
Raiders of the Lost Box
Monday, January 11, 2016
Space Oddity
It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.
A Google search today for "Galois spaces" + "Boolean spaces"
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.
Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …
"Harmonic Analysis of Switching Functions" ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.
"Galois Switching Functions and Their Applications,"
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975
D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978
"Switching functions constructed by Galois extension fields,"
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981
An illustration from the Lechner paper above —
"There is such a thing as harmonic analysis of switching functions."
— Saying adapted from a young-adult novel
Friday, January 8, 2016
Triumph of the Will
Monday, October 12, 2015
Ex Tenebris
“By groping toward the light we are made to realize how deep the darkness is around us.” — Arthur Koestler, The Call Girls: A Tragi-Comedy, Random House, 1973, page 118 |
"The Tesseract is where it belongs: out of our reach."
— Samuel L. Jackson as Nick Fury,
quoted here on Epiphany 2013
Earlier … (See Jan. 27, 2012) …
"And the Führer digs for trinkets in the desert."
Monday, September 28, 2015
Hypercube Structure
Click to enlarge:
For the hypercube as a vector space over the two-element field GF(2),
see a search in this journal for Hypercube + Vector + Space .
For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 1978-1986.
For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 5-7.
Friday, June 19, 2015
Footnote
There is such a thing as geometry.*
* Proposition adapted from A Wrinkle in Time , by Madeleine L'Engle.
Tuesday, June 9, 2015
Colorful Song
For geeks* —
" Domain, Domain on the Range , "
where Domain = the Galois tesseract and
Range = the four-element Galois field.
This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.
* A term from the 1947 film "Nightmare Alley."
Saturday, May 23, 2015
Group
On the artist Hilma af Klint (1862-1944):
"She belonged to a group called 'The Five'…."
Related material — Real Life (Log24, May 20, 2015).
From that post:
Wednesday, May 20, 2015
Real Life
From the Milwaukee Journal Sentinel Tuesday afternoon —
A 46-year-old Jesuit priest who was a Marquette University
assistant professor of theology collapsed on campus
Tuesday morning and died, President Michael Lovell
announced to the campus community in an email….
"Rev. Lúcás (Yiu Sing Luke) Chan, S.J., died after
collapsing this morning in Marquette Hall. Just last Sunday,
Father Chan offered the invocation at the Klingler College
of Arts and Sciences graduation ceremony…."
Synchronicity check…
From this journal on the above publication date of
Chan's book — Sept. 20, 2012 —
From a Log24 post on the preceding day, Sept. 19, 2012 —
“The Game in the Ship cannot be approached as a job,
a vocation, a career, or a recreation. To the contrary,
it is Life and Death itself at work there. In the Inner Game,
we call the Game Dhum Welur , the Mind of God."
— The Gameplayers of Zan
Thursday, March 26, 2015
The Möbius Hypercube
The incidences of points and planes in the
Möbius 8_{4 } configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 8_{4}," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x_{4}, x_{3}, x_{2}, x_{1}) over the two-element
Galois field.^{†} In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "Self-Dual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413-455
^{†} The subscripts' usual 1-2-3-4 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16-element Galois field GF(2^{4}). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
Tuesday, March 24, 2015
Hirzebruch
(Continued from July 16, 2014.)
Some background from Wikipedia:
"Friedrich Ernst Peter Hirzebruch ForMemRS^{[2]}
(17 October 1927 – 27 May 2012)
was a German mathematician, working in the fields of topology,
complex manifolds and algebraic geometry, and a leading figure
in his generation. He has been described as 'the most important
mathematician in Germany of the postwar period.'
^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]"}
A search for citations of the A. E. Brouwer paper in
the previous post yields a quotation from the preface
to the third ("2013") edition of Wolfgang Ebeling's
Lattices and Codes: A Course Partially Based
on Lectures by Friedrich Hirzebruch , a book
reportedly published on September 19, 2012 —
"Sadly, on May 27 this year, Friedrich Hirzebruch, Hannover, July 2012 Wolfgang Ebeling "
(Prof. Dr. Wolfgang Ebeling, Institute of Algebraic Geometry, |
Also sadly …
Monday, March 23, 2015
Gallucci’s Möbius Configuration
From H. S. M. Coxeter's 1950 paper
"Self-Dual Configurations and Regular Graphs,"
a 4×4 array and a more perspicuous rearrangement—
(Click image to enlarge.)
The above rearrangement brings Coxeter's remarks into accord
with the webpage The Galois Tesseract.
Update of Thursday, March 26, 2015 —
For an explanation of Coxeter's Fig. 24, see Thursday's later
post titled "The Möbius Hypercube."
Monday, January 26, 2015
Savior for Atheists…
Continued from June 17, 2013
(John Baez as a savior for atheists):
As an atheists-savior, I prefer Galois…
The geometry underlying a figure that John Baez
posted four days ago, "A Hypercube of Bits," is
Galois geometry —
See The Galois Tesseract and an earlier
figure from Log24 on May 21, 2007:
For the genesis of the figure,
see The Geometry of Logic.
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.
Friday, November 7, 2014
The Crosswicks Curse…
There is such a thing as an MBTI Tesseract.
See a thread at http://www.typologycentral.com/forums/
from August 17 and 18, 2010.
See also this journal on those dates: The Kermode Game.
Friday, October 31, 2014
Structure
Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square
The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.
(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)
Thursday, September 11, 2014
Monday, August 4, 2014
The Omega Portal
Version from “The Avengers” (2012) —
Version from Josefine Lyche (2009) —
See also this journal on the date that the above Avengers video was uploaded.
A Wrinkle in Space
"There is such a thing as a tesseract." — Madeleine L'Engle
An approach via the Omega Matrix:
See, too, Rosenhain and Göpel as The Shadow Guests .
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.
Friday, July 11, 2014
Back to 1955
Nick Fury takes the Tesseract…
… which travels back to 1955
(see The Call Girls, Nov. 3, 2013)…
Above: A 1955 cover design by Robert Flynn.
Images from December 1955…
… and a fictional image imagined in an earlier year:
Wednesday, May 21, 2014
The Tetrahedral Model of PG(3,2)
The page of Whitehead linked to this morning
suggests a review of Polster's tetrahedral model
of the finite projective 3-space PG(3,2) over the
two-element Galois field GF(2).
The above passage from Whitehead's 1906 book suggests
that the tetrahedral model may be older than Polster thinks.
Shown at right below is a correspondence between Whitehead's
version of the tetrahedral model and my own square model,
based on the 4×4 array I call the Galois tesseract (at left below).
(Click to enlarge.)
Thursday, May 8, 2014
Wrinkles in Time
Rivka Galchen, in a piece mentioned here in June 2010—
On Borges: Imagining the Unwritten Book
"Think of it this way: there is a vast unwritten book that the heart reacts to, that it races and skips in response to, that it believes in. But it’s the heart’s belief in that vast unwritten book that brought the book into existence; what appears to be exclusively a response (the heart responding to the book) is, in fact, also a conjuring (the heart inventing the book to which it so desperately wishes to respond)."
Related fictions
Galchen's "The Region of Unlikeness" (New Yorker , March 24, 2008)
Ted Chiang's "Story of Your Life." A film adaptation is to star Amy Adams.
… and non-fiction
"There is such a thing as a 4-set." — January 31, 2012
Friday, March 14, 2014
The Search for Charles Wallace
The search in the previous post for the source of a quotation from Poincaré yielded, as a serendipitous benefit, information on an interesting psychoanalyst named Wilfred Bion (see the Poincaré quotation at a webpage on Bion). This in turn suggested a search for the source of the name of author Madeleine L'Engle's son Bion, who may have partly inspired L'Engle's fictional character Charles Wallace. Cynthia Zarin wrote about Bion in The New Yorker of April 12, 2004 that
"According to the family, he is the person for whom L’Engle’s insistence on blurring fiction and reality had the most disastrous consequences."
Also from that article, material related to the name Bion and to what this journal has called "the Crosswicks Curse"*—
"Madeleine L’Engle Camp was born in 1918 in New York City, the only child of Madeleine Hall Barnett, of Jacksonville, Florida, and Charles Wadsworth Camp, a Princeton man and First World War veteran, whose family had a big country place in New Jersey, called Crosswicks. In Jacksonville society, the Barnett family was legendary: Madeleine’s grandfather, Bion Barnett, the chairman of the board of Jacksonville’s Barnett Bank, had run off with a woman to the South of France, leaving behind a note on the mantel. Her grandmother, Caroline Hallows L’Engle, never recovered from the blow. ….
… The summer after Hugh and Madeleine were married, they bought a dilapidated farmhouse in Goshen, in northwest Connecticut. Josephine, born in 1947, was three years old when they moved permanently to the house, which they called Crosswicks. Bion was born just over a year later."
* "There is such a thing as a tesseract."
Tuesday, March 11, 2014
Depth
"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."
— G. H. Hardy, A Mathematician's Apology
Part I: An Inch Deep
Part II: An Inch Wide
See a search for "square inch space" in this journal.
See also recent posts with the tag depth.
Sunday, March 2, 2014
Sermon
Raiders of the Lost (Continued)
"Socrates: They say that the soul of man is immortal…."
From August 16, 2012—
In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.
An 1892 figure by Jowett illustrating Plato's Meno—
A more correct version, from hermes-press.com —
Socrates: He only guesses that because the square is double, the line is double.Meno: True.
Socrates: Observe him while he recalls the steps in regular order. (To the Boy.) Tell me, boy, do you assert that a double space comes from a double line? Remember that I am not speaking of an oblong, but of a figure equal every way, and twice the size of this-that is to say of eight feet; and I want to know whether you still say that a double square comes from double line? [Boy] Yes. Socrates: But does not this line (AB) become doubled if we add another such line here (BJ is added)? [Boy] Certainly.
Socrates: And four such lines [AJ, JK, KL, LA] will make a space containing eight feet? [Boy] Yes. Socrates: Let us draw such a figure: (adding DL, LK, and JK). Would you not say that this is the figure of eight feet? [Boy] Yes. Socrates: And are there not these four squares in the figure, each of which is equal to the figure of four feet? (Socrates draws in CM and CN) [Boy] True. Socrates: And is not that four times four? [Boy] Certainly. Socrates: And four times is not double? [Boy] No, indeed. Socrates: But how much? [Boy] Four times as much. Socrates: Therefore the double line, boy, has given a space, not twice, but four times as much. [Boy] True. Socrates: Four times four are sixteen— are they not? [Boy] Yes. |
As noted in the 2012 post, the diagram of greater interest is
Jowett's incorrect version rather than the more correct version
shown above. This is because the 1892 version inadvertently
illustrates a tesseract:
A 4×4 square version, by Coxeter in 1950, of a tesseract—
This square version we may call the Galois tesseract.
Thursday, February 27, 2014
The Crosswicks Curse
"There is such a thing as a tesseract."
— Saying from Crosswicks
See also March 5, 2011.
Adapted from the above passage —
"So did L'Engle understand four-dimensional geometry?"
Saturday, January 25, 2014
Rotatable Hypercube
The archived Java rotatable hypercube of
Harry J. Smith is no longer working.
For an excellent JavaScript replacement,
see Pete Michaud's
http://petemichaud.github.io/4dhypercube/.
This JavaScript version can easily be saved.
Friday, January 17, 2014
The 4×4 Relativity Problem
The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.
“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
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)
See also a 2011 publication of the Mathematical Association of America —
Friday, December 20, 2013
For Emil Artin
(On His Dies Natalis )…
This is asserted in an excerpt from…
"The smallest non-rank 3 strongly regular graphs
which satisfy the 4-vertex condition"
by Mikhail Klin, Mariusz Meszka, Sven Reichard, and Alex Rosa,
BAYREUTHER MATHEMATISCHE SCHRIFTEN 73 (2005), 152-212—
(Click for clearer image)
Note that Theorem 46 of Klin et al. describes the role
of the Galois tesseract in the Miracle Octad Generator
of R. T. Curtis (original 1976 version). The tesseract
(a 4×4 array) supplies the geometric part of the above
exceptional geometric-combinatorial isomorphism.
Wednesday, December 18, 2013
A Hand for the Band
"How about another hand for the band?
They work real hard for it.
The Cherokee Cowboys, ladies and gentlemen."
— Ray Price, video, "Danny Boy Mid 80's Live"
Other deathly hallows suggested by today's NY Times—
Click the above image for posts from December 14.
That image mentions a death on August 5, 2005, in
"entertainment Mecca" Branson, Missouri.
Another note from August 5, 2005, reposted here
on Monday—
Happy birthday, Keith Richards.
Monday, December 16, 2013
Quartet
Happy Beethoven's Birthday.
Related material: Abel 2005 and, more generally, Abel.
See also Visible Mathematics.
Sunday, December 15, 2013
Sermon
Odin's Jewel
Jim Holt, the author of remarks in yesterday's
Saturday evening post—
"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."
— Vanity Fair Daily , July 16, 2012
Backstory: Odin + Jewel in this journal.
See also Odin on the Kyoto school —
For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —
Tuesday, December 10, 2013
Pink Champagne on Ice
The title refers to a post of April 26, 2009.
Monday, December 9, 2013
Being There
Or: The Naked Blackboard Jungle
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , |
Related material: Machete Math and…
Starring the late Eleanor Parker as Swiftly Mrs. Who.
Saturday, September 21, 2013
Geometric Incarnation
The Kummer 16_{6} configuration is the configuration of sixteen
6-sets within a 4×4 square array of points in which each 6-set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.
See Configurations and Squares.
The Wikipedia article Kummer surface uses a rather poetic
phrase* to describe the relationship of the 16_{6} to a number
of other mathematical concepts — "geometric incarnation."
Related material from finitegeometry.org —
* Apparently from David Lehavi on March 18, 2007, at Citizendium .
Monday, August 12, 2013
Form
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The Galois tesseract is the basis for a representation of the smallest
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday's post.
The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—
As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator (MOG) of
R. T. Curtis.
Tuesday, July 16, 2013
Space Itself
"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space …
but in space itself?"
— Megan Garber in The Atlantic , Aug. 16, 2012
One approach:
"There is such a thing as a tesseract" and
Diamond Theory in 1937.
See, too, Baez in this journal.
Tuesday, July 9, 2013
Vril Chick
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) |
Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
Friday, July 5, 2013
Mathematics and Narrative (continued)
Short Story — (Click image for some details.)
Parts of a longer story —
Sunday, June 23, 2013
Random Dudes
Here is the link to an MIT Scratch project from the above comment.
See also a comment by a Random Norwegian Dude:
For related art, see
"4D AMBASSADOR (HYPERCUBE)" for Steven H. Cullinane
by the Norwegian artist Josefine Lyche.
Tuesday, June 4, 2013
Cover Acts
The Daily Princetonian today:
A different cover act, discussed here Saturday:
See also, in this journal, the Galois tesseract and the Crosswicks Curse.
"There is such a thing as a tesseract." — Crosswicks saying
Tuesday, May 28, 2013
Codes
The hypercube model of the 4-space over the 2-element Galois field GF(2):
The phrase Galois tesseract may be used to denote a different model
of the above 4-space: the 4×4 square.
MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).
The thirty-five 4×4 structures within the MOG:
Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:
A later book co-authored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.
Between the 1977 and 1988 Sloane books came the diamond theorem.
Update of May 29, 2013:
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):
Sunday, May 19, 2013
Sermon
Best vs. Bester
The previous post ended with a reference mentioning Rosenhain.
For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).
From the next day, April 2, 2013:
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
– Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
See, too, in the Conway-Sloane book, the Galois tesseract …
and, in this journal, Geometry for Jews and The Deceivers , by Bester.
Priority Claim
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F_{2}^{4} built on I \ O_{9}. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O_{9}."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345-353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M _{24 },"
arXiv.org > hep-th > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
The vector space structure as it occurs in a 4×4 array |
See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).