The second Logos figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—
Click for further details.
The second Logos figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—
Click for further details.
… and Schoolgirl Space
"This poem contrasts the prosaic and sensual world of the here and now
with the transcendent and timeless world of beauty in art, and the first line,
'That is no country for old men,' refers to an artless world of impermanence
and sensual pleasure."
— "Yeats' 'Sailing to Byzantium' and McCarthy's No Country for Old Men :
Art and Artifice in the New Novel,"
Steven Frye in The Cormac McCarthy Journal ,
Vol. 5, No. 1 (Spring 2005), pp. 1420.
See also Schoolgirl Space in this journal.
* See, for instance, Lewis Hyde on the word "artifice" and . . .
Metaphysical conceit  literature  Britannica.com

This post's title refers to a metaphysical conceit
in the previous post, Desperately Seeking Clarity.
Related material —
The source of the above mystical octahedron —
See also Jung's Imago Dei in this journal.
See as well posts mentioning "An Object of Beauty."
Update of 12 AM June 11 — A screenshot of this post
is now available at http://dx.doi.org/10.17613/hqk7nx97 .
The opening lines of Eliot's Four Quartets —
"Time present and time past
Are both perhaps present in time future,
And time future contained in time past."
Perhaps.
Those who prefer geometry to rhetoric may also prefer
to Eliot's lines the immortal opening of the Transformers saga —
"Before time began, there was the Cube."
One version of the Cube —
(Continued from this morning)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also other Log24 posts tagged Kummerhenge.
https://newatlas.com/angelparticleownantiparticle/50579/
Scientists discover "angel particle"
Michael Irving . . . . "Our team predicted exactly where to find the Majorana fermion and what to look for as its 'smoking gun' experimental signature," says Shoucheng Zhang, one of the senior authors of the research paper. "This discovery concludes one of the most intensive searches in fundamental physics, which spanned exactly 80 years." . . . . Zhang proposes that the team's discovery be named the "angel particle" after the Dan Brown novel Angels and Demons , which features a bomb powered by the meeting of matter and antimatter. In the long run, Majoranas could find practical application in making quantum computers more secure. The research was published in the journal Science . . . . 
See as well Stanford News yesterday —
Shoucheng Zhang … died on Dec. 1. He was 55.
Zhang’s death was unexpected and followed
a “battle with depression,” according to his family.
Clicking on Zong in the above post leads to a 2005 article
in the Bulletin of the American Mathematical Society .
See also the eightfold cube and interality .
Click to enlarge:
Above are the 7 frames of an animated gif from a Wikipedia article.
* For the Furey of the title, see a July 20 Quanta Magazine piece —
See also the eightfold cube in this journal.
"Before time began . . . ." — Optimus Prime
From a post of July 25, 2008, "56 Triangles," on the Klein quartic
and the eightfold cube —
"Baez's discussion says that the Klein quartic's 56 triangles
can be partitioned into 7 eighttriangle Egan 'cubes' that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eighttriangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane."
Related material from 1975 —
More recently …
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, "UhOh" —
— 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 computereye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
A star figure and the Galois quaternion.
The square root of the former is the latter.
See also a passage quoted here a year ago today
(May the Fourth, "Star Wars Day") —
For Greta Gerwig and Saoirse Ronan —
See also a Log24 post from the above Cube Theory date —
April 12, 2016 — Lyrics for a Cartoon Graveyard — as well as . . .
The Java applets at the webpage "Diamonds and Whirls"
that illustrate Cullinane cubes may be difficult to display.
Here instead is an animated GIF that shows the basic unit
for the "design cube" pages at finitegeometry.org.
James Propp in the current Math Horizons on the eightfold cube —
For another puerile approach to the eightfold cube,
see Cube Space, 19842003 (Oct. 24, 2008).
See also a figure from 2 AM ET April 26 …
" Partner, anchor, decompose. That's not math.
That's the plot to 'Silence of the Lambs.' "
"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20thcentury art."
"Space: what you
damn well have to see."
— James Joyce, Ulysses
Another view of the previous post's art space —
More generally, see Solomon's Cube in Log24.
See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above
cube arrangement from "Affine groups on small binary spaces" (1984).
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —
Two such considerations —
“The man who lives in contact with what he believes to be a living Church
is a man always expecting to meet Plato and Shakespeare tomorrow
at breakfast.”
— G. K. Chesterton
Or Sunday dinner.
Platonic 
Shakespearean 
Not to mention Euclid and Picasso.  


In the above pictures, Euclid is represented by 
Cassirer vs. Heidegger at Harvard —
A remembrance for Michaelmas —
A version of Heidegger's "Sternwürfel " —
From Log24 on the upload date for the above figure —
A KUNSTforum.as article online today (translation by Google) —
Update of Sept. 7, 2016: The corrections have been made,
except for the misspelling "Cullinan," which was caused by
Google translation, not by KUNSTforum.
Foreword by Sir Michael Atiyah —
"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
Judy Bass, Los Angeles Times , March 12, 1989 —
"Like Rubik's Cube, The Eight demands to be pondered."
As does a figure from 1984, Cullinane's Cube —
For natural group actions on the Cullinane cube,
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
See also the recent post Cube Bricks 1984 —
Related remark from the literature —
Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 19842003."
(From Anatomy of a Cube, Sept. 18, 2011.)
The following page quotes "Raiders of the Lost Crucible,"
a Log24 post from Halloween 2015.
From KUNSTforum.as, a Norwegian art quarterly, issue no. 1 of 2016.
Related posts — See Lyche Eightfold.
Related aesthetics —
"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
"… the A B C of being…." — Wallace Stevens
Scholia —
Compare to my own later note, from March 4, 2010 —
"It seems that Guitart discovered these 'A, B, C' generators first,
though he did not display them in their natural setting,
the eightfold cube." — Borromean Generators (Log24, Oct. 19)
See also Raiders of the Lost Crucible (Halloween 2015)
and "Guitar Solo" from the 2015 CMA Awards on ABC.
An eightfold cube appears in this detail
of a photo by Josefine Lyche of her
installation "4D Ambassador" at the
Norwegian Sculpture Biennial 2015 —
(Detail from private Instagram photo.)
Catalog description of installation —
Google Translate version —
In a small bedroom to Foredragssalen populate
Josefine Lyche exhibition with a group sculptures
that are part of the work group 4D Ambassador
(20142015). Together they form an installation
where she uses light to amplify the feeling of
stepping into a new dimension, for which the title
suggests, this "ambassadors" for a dimension we
normally do not have access to. "Ambassadors"
physical forms presents nonphysical phenomena.
Lyches works have in recent years been placed
in something one might call an "esoteric direction"
in contemporary art, and defines itself this
sculpture group humorous as "glamminimalist."
She has in many of his works returned to basic
geometric shapes, with hints to the occult,
"new spaceage", mathematics and where
everything in between.
See also Lyche + "4D Ambassador" in this journal and
her website page with a 2012 version of that title.
For Aaron Sorkin and Walter Isaacson —
Related material —
Bauhaus Cube, Design Cube, and
Nabokov's Transparent Things .
Click to enlarge:
For the hypercube as a vector space over the twoelement 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, 19781986.
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. 57.
Spielerei —
"On the most recent visit, Arthur had given him
a brightly colored cube, with sides you could twist
in all directions, a new toy that had just come onto
the market."
— Daniel Kehlmann, F: A Novel (2014),
translated from the German by
Carol Brown Janeway
Nicht Spielerei —
A figure from this journal at 2 AM ET
on Monday, August 3, 2015
Also on August 3 —
FRANKFURT — "Johanna Quandt, the matriarch of the family
that controls the automaker BMW and one of the wealthiest
people in Germany, died on Monday in Bad Homburg, Germany.
She was 89."
MANHATTAN — "Carol Brown Janeway, a Scottishborn
publishing executive, editor and awardwinning translator who
introduced American readers to dozens of international authors,
died on Monday in Manhattan. She was 71."
Related material — Heisenberg on beauty, Munich, 1970
Omega is a Greek letter, Ω , used in
mathematics to denote a set on which
a group acts.
The previous post mentioned a new mobile, "Triangle Constellation,"
commissioned for the Harvard Art Museums.
Related material (click to enlarge) —
The above review is of an exhibition by the "Constellation" artist,
Carlos Amorales, that opened on Sept. 26, 2008 — "just in time for
Halloween and the Day of the Dead."
See also this journal on that date.
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 faceplanes 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 pointplane
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 twoelement
Galois field.^{†} In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "SelfDual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413455
^{†} The subscripts' usual 1234 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 16element Galois field GF(2^{4}). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
The Blacklist “Pilot” Review
"There is an element of camp to this series though. Spader is
quite gleefully channeling Anthony Hopkins, complete with being
a well educated, elegant man locked away in a supercell.
Speaking of that supercell, it’s kind of ridiculous. They’ve got him
locked up in an abandoned post office warehouse on a little
platform with a chair inside a giant metal cube that looks like
it could have been built by Tony Stark. And as Liz approaches
to talk to him, the entire front of the cube opens and the whole
thing slides back to leave just the platform and chair. Really?
FUCKING REALLY ? "
— Kate Reilly at Geekenstein.com (Sept. 27, 2013)
Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :
An unarticulated (but colored) cube:
A 2x2x2 articulated cube:
A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:
A screenshot of the new page on the eightfold cube at Froebel Decade:
Click screenshot to enlarge.
For Reba McEntire on her birthday:
Complex Reflection and Naturalized Epistemology.
The New Yorker on Cubism:
"The style wasn’t new, exactly— or even really a style,
in its purest instances— though it would spawn no end
of novelties in art and design. Rather, it stripped naked
certain characteristics of all pictures. Looking at a Cubist
work, you are forced to see how you see. This may be
gruelling, a gymnasium workout for eye and mind.
It pays off in sophistication."
— Online "Culture Desk" weblog, posted today by Peter Schjeldahl
Nonstyle from 1911:
See also Cube Symmetry Planes in this journal.
A comment at The New Yorker related to Schjeldahl's phrase "stripped naked"—
"Conceptualism is the least seductive modernart movement."
POSTED 4/11/2013, 3:54:37 PM BY CHRISKELLEY
(The "conceptualism" link was added to the quoted comment.)
Yesterday's 11 AM post Mad Day concluded
with a link to a 2001 American Mathematical Society
article by Pierre Cartier that sums up the religion and
politics of many mathematicians…
"Here ends the infancy narrative of the gospel…."
"… while Simone Weil's Catholicism was violently
antiSemitic (in 1942!), Grothendieck's Buddhism
bears a strong resemblance to the practices of
his Hasidic ancestors."
See also Simone Weil in this journal.
Note esp. a post of April 6, 2004 that provides
a different way of viewing Derrida's notion of
inscription .
From Don DeLillo's novel Point Omega — I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create." "What reality?" "This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreedupon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become threedimensional. The reality stands, it walks, it squats. Except when it doesn't." He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku." I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely. "Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear." 
What's there—
This view of a die's faces 3, 6, and 5, in counter
clockwise order (see previous post) suggests a way
of labeling the eight corners of a die (or cube):
123, 135, 142, 154, 246, 263, 365, 456.
Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counterclockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertexlabeling may be used in describing
the automorphisms of the order8 quaternion group.
For a more literary approach to quaternions, see
Pynchon's novel Against the Day .
* From Peter J. Cameron's weblog:
"The big name associated with this is Major MacMahon,
an associate of Hardy, Littlewood and Ramanujan,
of whom Robert Kanigel said,
His expertise lay in combinatorics, a sort of
glorified dicethrowing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.
Glorified dicethrowing, indeed…"
Continued from April 2, 2012.
Some predecessors of the Cullinane design cubes of 1984
that lack the Cullinane cubes' symmetry properties—
Kohs cubes (see 1920 article)
Wechsler cubes (see Wechsler in this journal), and
Horowitz cubes (see links below).
Yesterday's post Child's Play displayed a cube formed
by a Hasse diagram of the 8 subsets of a 3set.*
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.
A search today (Élie Cartan's birthday) for material related to triality*
yielded references to something that has been called a Bhargava cube .
Two pages from a 2006 paper by Bhargava—
Bhargava's reference [4] above for "the story of the cube" is to…
Higher Composition Laws I:
A New View on Gauss Composition,
and Quadratic Generalizations
Manjul Bhargava
The Annals of Mathematics
Second Series, Vol. 159, No. 1 (Jan., 2004), pp. 217250
Published by: Annals of Mathematics
Article Stable URL: http://www.jstor.org/stable/3597249
A brief account in the context of embedding problems (click to enlarge)—
For more ways of slicing a cube,
see The Eightfold Cube —
* Note (1) some remarks by Tony Smith
related to the above Dynkin diagram
and (2) another colorful variation on the diagram.
"Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing. And once you have made or acquired a new 'cube'… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube. What is the essence of each operator? One senses a deep invariant lying somehow 'down underneath' it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment. In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….
… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube. It is the answer; it simply has the right spirit ."
— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern (Kindle edition, locations 1155711572)
See also Many Dimensions in this journal and Solomon's Cube.
The following picture provides a new visual approach to
the order8 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…
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 cofounded—
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(cofounded by Margaret Wertheim)
R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—
"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955
The figure by Cullinane included above shows a way to visualize Sprott's remarks.
For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
Update of 7:42 PM Sept. 18, 2011—
From a Summer 2011 course on discrete structures at a Berlin website—
A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—
Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 19842003."
Yesterday's midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik's mechanical contrivance as a rather absurd "Cosmic Cube."
A simpler candidate for the "Cube" part of that phrase:
The Eightfold Cube
As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.
"Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions."
— Alexandre V. Borovik in "Coxeter Theory: The Cognitive Aspects"
Borovik has a such a diagram—
The planes in Borovik's figure are those separating the parts of the eightfold cube above.
In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.
In light of Borovik's remarks, the eightfold cube might serve to illustrate the "Cosmic" part of the Marvel Comics phrase.
For some related theological remarks, see Cube Trinity in this journal.
Happy St. Augustine's Day.
* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2element Galois field, these hyperplanes are certain sets of four subcubes.
Prequel — (Click to enlarge)
Background —
See also Rubik in this journal.
* For the title, see Groups Acting.
The New York Times has a skateboarder obit with a URL date of July 9.
Here is an earlier version from the LA Times—
By Keith Thursby, Los Angeles Times
Chris Cahill, one of the original Dogtown ZBoys
who brought seismic changes to skateboarding
with their style and attitude, has died. He was 54.
Cahill was found June 24 at his Los Angeles home,
said Larry Dietz of the Los Angeles County
coroner's office. A cause of death has not been
determined and tests are ongoing, Dietz said.
Related material from Midsummer Day, June 24, the day Cahill was found dead—
The Gleaming and The Cube.
An illustration from the latter—
The above was adapted from a 1996 cover—
Vintage Books, July 1996. Cover: Evan Gaffney.
For the significance of the flames,
see PyrE in the book. For the significance
of the cube in the altered cover, see
The 2×2×2 Cube and The Diamond Archetype.
See the signature link in last night's post for a representation of Madison Avenue.
For a representation by Madison Avenue, see today's New York Times—
"As a movement Pop Art came and went in a flash, but it was the kind of flash that left everything changed. The art public was now a different public— larger, to be sure, but less serious, less introspective, less willing or able to distinguish between achievement and its trashy simulacrum. Moreover, everything connected with the life of art— everything, anyway, that might have been expected to offer some resistance to this wholesale vulgarization and demoralization— was now cheapened and corrupted. The museums began their rapid descent into show biz and the retail trade. Their exhibitions were now mounted like Broadway shows, complete with set designers and lighting consultants, and their directors pressed into service as hucksters, promoting their wares in radio and television spots and selling their facilities for cocktail parties and other entertainments, while their socalled education programs likewise degenerated into sundry forms of entertainment and promotion. The critics were coopted, the art magazines commercialized, and the academy, which had once taken a certain pride in remaining aloof from the blandishments of the cultural marketplace, now proved eager to join the crowd— for there was no longer any standard in the name of which a sellout could be rejected. When the boundary separating art and fashion was breached, so was the dividing line between high art and popular culture, and upon all those institutions and professions which had been painstakingly created to preserve high art from the corruptions of popular culture. The effect was devastating. Some surrendered their standards with greater alacrity than others, but the drift was unmistakable and all in the same direction— and the momentum has only accelerated with the passage of time."
— Hilton Kramer, The Triumph of Modernism: The Art World, 19852005 , publ. by Ivan R. Dee on Oct. 26, 2006, pp. 146147
Related material— Rubik in this journal, Exorcist in this journal, and For the Class of '11.
Click the above image for some background.
Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.
The title refers not to numbers of the form p^{ 3}, p prime, but to geometric cubes with p ^{3} subcubes.
Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation groups of degree (not order ) p^{ 3}.
For the case p =2, see The Eightfold Cube.
For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")
For symmetries of group actions for larger primes, see the related 1985 remark* on two dimensional linear groups—
"Actions of GL(2,p ) on a p ×p coordinatearray
have the same sorts of symmetries,
where p is any odd prime."
It was a dark and stormy night…
— Page 180, Logicomix
“… the class of reﬂections is larger in some sense over an arbitrary ﬁeld than over a characteristic zero ﬁeld.”
– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups”
For some context, see the small cube in “A Simple Reflection Group of Order 168.”
See also the larger cube in “Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).
That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix page above—
A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night—
“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”
— GianCarlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.
"An image comes to mind of a white, ideal space
that, more than any single picture, may be
the archetypal image of 20thcentury art."
"May be" —
Image from this journal
at noon (EST) Tuesday
"The geometry of unit cubes is a meeting point
of several different subjects in mathematics."
— Chuanming Zong
"A meeting point" —
The above death reportedly occurred "early Wednesday in Beijing."
Another meeting point —
(Click on logo and on meeting image for more details.)
See also "no ordinary venue."
There is more than one way
to look at a cube.
From Cambridge U. Press on Feb. 20, 2006 —
and from this journal on June 30, 2010 —
In memory of Wu Guanzhong, Chinese artist
who died in Beijing on June 25, 2010 —
See also this journal on Feb. 20, 2006
(the day The Cube was published).
Keanu vs. the Devil, continued
Al Pacino and Keanu Reeves in Devil's Advocate
For Keanu —
For Keanu's mentor —
… There is a Cave
Within the Mount of God, fast by his Throne,
Where light and darkness in perpetual round
Lodge and dislodge by turns, which makes through Heav'n
Grateful vicissitude, like Day and Night….
— Paradise Lost , by John Milton
Click on figure for details. 
Al Pacino in Devil's Advocate 
A search for “Chinese Cube” (based on the the previous entry’s title) reveals the existence of a most interesting character, who…
“… has attempted in his books to produce a Science and Art of Reasoning using the simplest of the Platonic solids, the Cube. [His] model also parallels, in some ways, the Cube of Space constructed from the Sepher Yetzirah’s attributions for the Hebrew letters and their direction. [He] elucidated his theories at great length….”
— More…
For related remarks, see the link to Solomon’s Cube from the previous entry.
Then of course there is…
Click on figure for details.
From the Bulletin of the American Mathematical Society, Jan. 26, 2005:
What is known about unit cubes
by Chuanming Zong, Peking University
Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in ndimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all….
From Log24, now:
What is known about the 4×4×4 cube
by Steven H. Cullinane, unaffiliated
Abstract: The 4×4×4 cube, from one point of view, is among the simplest and the most important objects in ndimensional binary space. In fact, as one will see from the links below, it is not simple at all.
The Klein Correspondence, Penrose SpaceTime, and a Finite Model
Related material:
Monday’s entry Just Say NO and a poem by Stevens,
"Those early works are succinct and uncompromising
in how they give shape to the philosophical perplexities
of form and idea…."
J. J. Charlesworth, artnet news, Dec. 16, 2014
"Form" and "idea" are somewhat synonymous,
as opposed to "form" and "substance." A reading:
The Dream of the Expanded Field continues…
From Klein's 1893 Lectures on Mathematics —
"The varieties introduced by Wirtinger may be called Kummer varieties…."
— E. Spanier, 1956
From this journal on March 10, 2013 —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —
Two such considerations —
Update of 10 PM ET March 7, 2014 —
The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64point vector space and
to the Weyl group of type E_{7}, W (E_{7}):
The Cayley reference is to "Algorithm for the characteristics of the
triple ϑfunctions," Journal für die Reine und Angewandte
Mathematik 87 (1879): 165169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441445
of Volume 10 of his Collected Mathematical Papers .
A NY Times researcher from this morning’s previous post
tweeted last fall about art forgery and China.
Related material — Art Cube.
Illustration from December 25, 2013.
"She never looked up while her mind rotated the facts,
trying to see them from all sides, trying to piece them
together into theory. All she could think was that she
was flunking an IQ test."
— Steve Martin, An Object of Beauty
"So you should not feel so all alone…"
— Adapted song lyric
Related images —
See also other posts tagged Arti Facts.
This post was suggested by those posts and by the following
attempt at humor —
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 —
“I am always the figure in someone else’s dream. I would really rather
sometimes make my own figures and make my own dreams.”
— John Malkovich at squarespace.com, January 10, 2017
Also on that date . . .
See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."
Anonymous remarks on the schoolgirl problem at Wikipedia —
"This solution has a geometric interpretation in connection with
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing."
See also Polster + Tetrahedron in this journal.
There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square model rather
than a tetrahedral model. The square model of PG(3,2) last
appeared in the schoolgirlproblem article on Feb. 11, 2017, just
before a revision that removed it.
" 'My public image is unshakably that of
America’s wholesome virgin, the girl next door,
carefree and brimming with happiness,'
she said in Doris Day: Her Own Story ,
a 1976 book . . . ."
From "Angels & Demons Meet Hudson Hawk" (March 19, 2013) —
From the March 1 post "Solomon and the Image," a related figure —
The title refers to CalabiYau spaces.
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A less "cosmic" but still noteworthy code — The Golay code.
This resides in a 12dimensional space over GF(2).
Related material from Plato and R. T. Curtis —
A related CalabiYau "Chinese jar" first described in detail in 1905 —
A figure that may or may not be related to the 4x4x4 cube that
holds the classical Chinese "cosmic code" — the I Ching —
ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3pix.pdf
"The purpose of mathematics cannot be derived from an activity
inferior to it but from a higher sphere of human activity, namely,
religion."
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also Ultron Cube.
From a New York Times book review of a new novel about
Timothy Leary that was in the Times online on April 10 —
"Most of the novel resides in the perspective
of Fitzhugh Loney, one of Leary’s graduate students."
"A version of this article appears in print on ,
on Page 10 of the Sunday Book Review with the headline:
Strange Days."
For material about one of Leary's non fictional grad students,
Ralph Metzner, see posts now tagged Metzner's Pi Day.
Related material —
The reported publication date of Searching for the Philosophers' Stone
was January 1, 2019.
A related search published here on that date:
* Title suggested by two of Ralph Metzner's titles,
The Expansion of Consciousness and The Unfolding Self .
(Continued from the previous post.)
InBetween "Spacing" and the "Chôra " (Ch. 2 in Henk Oosterling & Ewa Plonowska Ziarek (Eds.), Intermedialities: Philosophy, Arts, Politics , Lexington Books, October 14, 2010) "The term 'spacing' ('espacement ') is absolutely central to Derrida's entire corpus, where it is indissociable from those of différance (characterized, in the text from 1968 bearing this name, as '[at once] spacing [and] temporizing' ^{1}), writing (of which 'spacing' is said to be 'the fundamental property' ^{2}) and deconstruction (with one of Derrida's last major texts, Le Toucher: JeanLuc Nancy , specifying 'spacing ' to be 'the first word of any deconstruction' ^{3})." 1 Jacques Derrida, “La Différance,” in Marges – de la philosophie (Paris: Minuit, 1972), p. 14. Henceforth cited as D . 2 Jacques Derrida, “Freud and the Scene of Writing,” trans. A. Bass, in Writing and Difference (Chicago: University of Chicago Press, 1978), p. 217. Henceforth cited as FSW . 3 Jacques Derrida, Le Toucher, JeanLuc Nancy (Paris: Galilée, 2000), p. 207. . . . . "… a particularly interesting point is made in this respect by the French philosopher, Michel Haar. After remarking that the force Derrida attributes to différance consists simply of the series of its effects, and is, for this reason, 'an indefinite process of substitutions or permutations,' Haar specifies that, for this process to be something other than a simple 'actualisation' lacking any real power of effectivity, it would need “a soubassement porteur ' – let’s say a 'conducting underlay' or 'conducting medium' which would not, however, be an absolute base, nor an 'origin' or 'cause.' If then, as Haar concludes, différance and spacing show themselves to belong to 'a pure Apollonism' 'haunted by the groundless ground,' which they lack and deprive themselves of,^{16} we can better understand both the threat posed by the 'figures' of space and the mother in the Timaeus and, as a result, Derrida’s insistent attempts to disqualify them. So great, it would seem, is the menace to différance that Derrida must, in a 'properly' apotropaic gesture, ward off these 'figures' of an archaic, chthonic, spatial matrix in any and all ways possible…." 16 Michel Haar, “Le jeu de Nietzsche dans Derrida,” Revue philosophique de la France et de l’Etranger 2 (1990): 207227. . . . . … "The conclusion to be drawn from Democritus' conception of rhuthmos , as well as from Plato's conception of the chôra , is not, therefore, as Derrida would have it, that a differential field understood as an originary site of inscription would 'produce' the spatiality of space but, on the contrary, that 'differentiation in general' depends upon a certain 'spatial milieu' – what Haar would name a 'groundless ground' – revealed as such to be an 'inbetween' more 'originary' than the play of differences it informs. As such, this conclusion obviously extends beyond Derrida's conception of 'spacing,' encompassing contemporary philosophy's continual privileging of temporization in its elaboration of a preontological 'opening' – or, shall we say, 'inbetween.' 
For permutations and a possible "groundless ground," see
the eightfold cube and group actions both on a set of eight
building blocks arranged in a cube (a "conducting base") and
on the set of seven natural interstices (espacements ) between
the blocks. Such group actions provide an elementary picture of
the isomorphism between the groups PSL(2,7) (acting on the
eight blocks) and GL(3,2) (acting on the seven interstices).
Espacements
For the Church of Synchronology —
See also, from the reported publication date of the above book
Intermedialities , the Log24 post Synchronicity.
"Cell 461" quote from Curzio Malaparte superimposed on a scene from
the 1963 Godard film "Le Mépris " ("Contempt") —
"The architecture… beomes closely linked to the script…."
Malaparte's cell number , 461, is somewhat less closely linked
to the phrase "eternal blazon" —
Irving was quoted here on Dec. 22, 2008 —
The Tale of
the Eternal Blazon
by Washington Irving
“Blazon meant originally a shield , and then
the heraldic bearings on a shield .
Later it was applied to the art of describing
or depicting heraldic bearings in the proper
manner; and finally the term came to signify
ostentatious display and also description or
record by words or other means . In Hamlet ,
Act I Sc. 5, the Ghost, while talking with
Prince Hamlet, says:
‘But this eternal blazon must not be
To ears of flesh and blood.’
Eternal blazon signifies revelation or description
of things pertaining to eternity .”
— Irving’s Sketch Book , p. 461
Update of 6:25 PM ET —
"SelfBlazon… of Edenic Plenitude"
(The Issuu text is taken from Speaking about Godard , by Kaja Silverman
and Harun Farocki, New York University Press, 1998, page 34.)
Casa Malaparte, also known as Villa Malaparte —
Related film image with architectural quotation superimposed —
Related art prose —
"Maybe an image is too strong
Or maybe is not strong enough."
— "Solomon and the Witch,"
by William Butler Yeats
Some related material in this journal — See a search for k6.gif.
Some related material from Harvard —
Elkies' "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K_{6} and to the 15 2subsets of a 6set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2subsets of a 6set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6set
in the work of other authors in 1994 and 2002 . . .
This post was suggested by the phrase "Froebel Decade" from
the search results below.
This journal a decade ago had a post on the late Donald Westlake,
an author who reportedly died of a heart attack in Mexico on Dec. 31,
2008, while on his way to a New Year's Eve dinner.
One of Westlake's books —
Related material —
From religionnews.com —
"The word 'Hanukkah' means dedication.
It commemorates the rededicating of the
ancient Temple in Jerusalem in 165 B.C. . . . ."
From The New York Times this morning —
Related material —
From this journal on Wednesday, December 5, 2018 —
Megan Fox in "Transformers" (2007) —
From a Google image search this morning —
The image search was suggested by recent posts tagged Aitchison
and by this morning's previous post.
The Finkelstein Talisman —
"Before time began, there was the Cube."
— Optimus Prime in "Transformers" (Paramount, 2007)
Wikipedia on Hasbro —
Three American Jewish brothers,^{[6]} Herman, Hillel, and Henry Hassenfeld^{[7]}
founded Hassenfeld Brothers in Providence, Rhode Island in 1923 . . . .
The Hassenfeld Auction —
Also on September 16, 2015 —
The Hindman Image —
The Hood Warenkorb —
Under the Hood —
Megan Fox in "Transformers" (2007) —
This Way to the Egress —
Einstein, "Geometry and Experience," lecture before the
Prussian Academy of Sciences, January 27, 1921–
… This view of axioms, advocated by modern axiomatics, purges mathematics of all extraneous elements, and thus dispels the mystic obscurity, which formerly surrounded the basis of mathematics. But such an expurgated exposition of mathematics makes it also evident that mathematics as such cannot predicate anything about objects of our intuition or real objects. In axiomatic geometry the words "point," "straight line," etc., stand only for empty conceptual schemata. That which gives them content is not relevant to mathematics. Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the behavior of real objects. The very word geometry, which, of course, means earthmeasuring, proves this. For earthmeasuring has to do with the possibilities of the disposition of certain natural objects with respect to one another, namely, with parts of the earth, measuringlines, measuringwands, etc. It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of real objects of this kind, which we will call practicallyrigid bodies. To be able to make such assertions, geometry must be stripped of its merely logicalformal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practicallyrigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it in what follows from "purely axiomatic geometry." The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience. …. 
Later in the same lecture, Einstein discusses "the theory of a finite
universe." Of course he is not using "finite" in the sense of the field
of mathematics known as "finite geometry " — geometry with only finitely
many points.
Nevertheless, his remarks seem relevant to the Fano plane , an
axiomatically defined entity from finite geometry, and the eightfold cube ,
a physical object embodying the properties of the Fano plane.
I want to show that without any extraordinary difficulty we can illustrate the theory of a finite universe by means of a mental picture to which, with some practice, we shall soon grow accustomed. First of all, an observation of epistemological nature. A geometricalphysical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To "visualize" a theory therefore means to bring to mind that abundance of sensible experiences for which the theory supplies the schematic arrangement. In the present case we have to ask ourselves how we can represent that behavior of solid bodies with respect to their mutual disposition (contact) that corresponds to the theory of a finite universe. 
The New York Times 's Sunday School today —
I prefer the three bricks of the Miracle Octad Generator —
Venturi reportedly died on Tuesday, September 18.*
See also this journal on that date.
* Fact check:
Norwegian artist Josefine Lyche —
Lyche's shirt honors the late Kurt Cobain.
"Here we are now, entertain us."
"The creation of a new world
starts now.
Once again I am tied
to the logic of this
Hypersymmetricaldimension."
"Husserl is not the greatest philosopher of all times. — Kurt Gödel as quoted by GianCarlo Rota Some results from a Google search — Eidetic reduction  philosophy  Britannica.com Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart … Phenomenology Online » Eidetic Reduction
The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research Eidetic reduction – New World Encyclopedia Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience. 
For example —
The reduction of twocolorings and fourcolorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between* the subsquares or subcubes.
See the diamond theorem and the eightfold cube.
* Cf. posts tagged Interality and Interstice.
Review of yesterday's post Perception of Space —
From Harry Potter and the Philosopher's Stone (1997),
republished as "… and the Sorcerer's Stone ," Kindle edition:
In a print edition from Bloomsbury (2004), and perhaps in the
earliest editions, the above word "movements" is the first word
on page 168:
Click the above ellipse for some Log24 posts on the eightfold cube,
the source of the 168 automorphisms ("movements") of the Fano plane.
"Refined interpretation requires that you know that
someone once said the offspring of reality and illusion
is only a staggering confusion."
— Poem, "The Game of Roles," by Mary Jo Bang
Related material on reality and illusion —
an ad on the back cover of the current New Yorker —
"Hey, the stars might lie, but the numbers never do." — Song lyric
“All right, Jessshica. It’s time to open the boxsssschhh.” “Gahh,” she said. She began to walk toward the box, but her heart failed her and she retreated back to the chair. “Fuck. Fuck.” Something mechanical purred. The seam she had found cracked open and the top of the box began to rise. She squeezed shut her eyes and groped her way into a corner, curling up against the concrete and plugging her ears with her fingers. That song she’d heard the busker playing on the train platform with Eliot, “Lucy in the Sky with Diamonds”; she used to sing that. Back in San Francisco, before she learned card tricks. It was how she’d met Benny: He played guitar. Lucy was the best earner, Benny said, so that was mainly what she sang. She must have sung it five times an hour, day after day. At first she liked it but then it was like an infection, and there was nothing she could do and nowhere she could go without it running across her brain or humming on her lips, and God knew she tried; she was smashing herself with sex and drugs but the song began to find its way even there. One day, Benny played the opening chord and she just couldn’t do it. She could not sing that fucking song. Not again. She broke down, because she was only fifteen, and Benny took her behind the mall and told her it would be okay. But she had to sing. It was the biggest earner. She kind of lost it and then so did Benny and that was the first time he hit her. She ran away for a while. But she came back to him, because she had nothing else, and it seemed okay. It seemed like they had a truce: She would not complain about her bruised face and he would not ask her to sing “Lucy.” She had been all right with this. She had thought that was a pretty good deal. Now there was something coming out of a box, and she reached for the most virulent meme she knew. “Lucy in the sky!” she sang. “With diamonds!” • • •
Barry, Max. Lexicon: A Novel (pp. 247248). 
Related material from Log24 on All Hallows' Eve 2013 —
"Just another shake of the kaleidoscope" —
Related material:
Kaleidoscope Puzzle,
Design Cube 2x2x2, and
Through the Looking Glass: A Sort of Eternity.
McCarthy's "materialization of plot and character" does not,
for me, constitute a proof that "there is being, after all,
beyond the arbitrary flux of existence."
Neither does the above materialization of 281 as the page
number of her philosophical remark.
See also the materialization of 281 as a page number in
the book Witchcraft by Charles Williams —
The materialization of 168 as a page number in a
Stephen King novel is somewhat more convincing,
but less convincing than the materialization of Klein's
simple group of of 168 elements in the eightfold cube.
"… Lincoln Plaza Cinemas, the Juilliard String Quartet,
and the Strand Book Store remained oases
for cultural and intellectual stimulation."
— John S. Friedman in The Forward , Jan. 21, 2018
Read more:
https://forward.com/culture/392483/
howfredbassdantalbotrobertmann
shapednewyorkculture/
From the Oasis in Steven Spielberg's "Ready Player One" (2018) —
I prefer, from a Log24 search for Flux Capacitor …
From "Raiders of the Lost Images" —
"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."
The walkerart.org passage above is from Feb. 17, 2011.
See also this journal on Feb. 17, 2011 —
"… Only by the form, the pattern,
Can words or music reach
The stillness…."
— T. S. Eliot,
Four Quartets
For further details, see Time Fold.
The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film "Triple Cross."
Related structures —
Greg Egan's animated image of the Klein quartic —
For a tetrahedral key to the arrangement of the 56 triangles within the above
structure, see a book chapter by Michael Huber of Tübingen —
For further details, see the June 29 post Triangles in the Eightfold Cube.
See also, from an April 2013 philosophical conference:
Abstract for a talk at the City University of New York:
The Experience of Meaning Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to 'being simple': consider Eliot's Four Quartets or Mozart's late symphonies. Some truths are complex, and they are simplified at the cost of distortion, at the cost of ceasing to be truths. Nonetheless, it's often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we've seen into the heart of things. I'll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them. 
For the talk itself, see a YouTube video.
The conference talks also appear in a book.
The book begins with an epigraph by Hilbert —
The phrase "Blue Dream" in the previous post
suggests a Web search for Traumnovelle .
That search yields an interesting weblog post
from 2014 commemorating the 1999 dies natalis
(birth into heaven) of St. Stanley Kubrick.
Related material from March 7, 2014,
in this journal —
That 2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.
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 twodimensional world, you still get straight lines, but many lines make one figure. In a threedimensional 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 superpersonal—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 threepersonal 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 threepersonal 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."
Backstory for fiction fans, from Log24 on June 11 —
Related non fiction —
See as well the structure discussed in today's previous post.
From Nanavira Thera, "Early Letters," in Seeking the Path —
"nine possibilities arising quite naturally" —
Compare and contrast with Hudson's parametrization of the
4×4 square by means of 0 and the 15 2subsets of a 6set —
The previous post was suggested by some April 17, 2016, remarks
by James Propp on the eightfold cube.
Propp's remarks included the following:
"Here’s a caveat about my glib earlier remark that
'There are only finitely many numbers ' in a finite field.
It’s a bit of a stretch to call the elements of finite fields
'numbers'. Elements of GF(q ) can be thought of as
the integers mod q when q is prime, and they can be
represented by 0, 1, 2, …, q–1; but when q is a prime
raised to the 2nd power or higher, describing the
elements of GF(q ) is more complicated, and the word
'number' isn’t apt."
Related material —
See also this journal on the date of Propp's remarks — April 17, 2016.
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