" Welcher Art ist die ursprüngliche Einheit,
daß sie sich in diese Scheidung auseinanderwirft,
und in welchem Sinn sind die Geschiedenen
hier als Wesung der Ab-gründigkeit gerade einig?
Hier kann es sich nicht um irgend eine »Dialektik«
handeln, sondern nur um die Wesung des Grundes
(der Wahrheit also) selbst."
"Before time began, there was the Cube." — Optimus Prime
|
Thomas Mann on "the mystery of the unity" —
"Denn um zu wiederholen, was ich anfangs sagte:
in dem Geheimnis der Einheit von Ich und Welt,
Sein und Geschehen, in der Durchschauung des
scheinbar Objectiven und Akzidentellen als
Veranstaltung der Seele glaube ich den innersten Kern
der analytischen Lehre zu erkennen." (GW IX 488)
An Einheit-Geheimnis that is perhaps* more closely related
to pure mathematics** —
"What is the nature of the original unity
that throws itself apart in this separation,
and in what sense are the separated ones
here as the essence of the abyss?
Here it cannot be a question of any kind of 'dialectic,'
but only of the essence of the ground
(that is, of truth) itself." [Tr. by Google]
" Welcher Art ist die ursprüngliche Einheit,
daß sie sich in diese Scheidung auseinanderwirft,
und in welchem Sinn sind die Geschiedenen
hier als Wesung der Ab-gründigkeit gerade einig?
Hier kann es sich nicht um irgend eine »Dialektik«
handeln, sondern nur um die Wesung des Grundes
(der Wahrheit also) selbst."
* Or perhaps not .
** For a relevant Scheidung , see Eightfold Cube.
A followup to Wednesday's post Deep Space —
Related material from this journal on July 9, 2019 —
Cube Bricks 1984 —
From "Tomorrowland" (2015) —
From other posts tagged 1984 Cubes —
From "Siri + Wechsler" in this journal —
|
For Little Man Tate —
Related material — Wechsler in this journal and
Mark and Lucille, Bill and Violet, Al and Regina,
|
Related material —
". . . It is as if
We had come to an end of the imagination,
Inanimate in an inert savoir.”
— Wallace Stevens, “The Plain Sense of Things”
"In my end . . . ." — T. S. Eliot
|
Thursday, March 22, 2018
The Diamond Cube
|
See a note from Sept. 15, 1984
(perhaps the last day of life for Richard Brautigan).
A question attributed to John Horton Conway
about configurations in his Game of Life —
"Indeed, is there a Godlike still-life,
one that can only have existed
for all time . . . . ?"
A simple answer … but not from Conway's Game —
"Before time began, there was the Cube." — Optimus Prime
Related remarks: Ogdoad.
"Metaphor in language — the prime mover"
— George Steiner in Real Presences (1989)
Not so prime —
See also the "Transformers" marketing saga.
Related marketing:
Disney Easter eggs —
Continued from this journal's posts of March 1, 2021 —
As If
|
"Time past and time future . . ." — T. S. Eliot
Related art — The non-Rubik 3x3x3 cube —
The above structure illustrates the affine space of three dimensions
over the three-element finite (i.e., Galois) field, GF(3). Enthusiasts
of Judith Brown's nihilistic philosophy may note the "radiance" of the
13 axes of symmetry within the "central, structuring" subcube.
I prefer the radiance (in the sense of Aquinas) of the central, structuring
eightfold cube at the center of the affine space of six dimensions over
the two-element field GF(2).
Some formal symmetry —
|
"… each 2×4 "brick" in the 1974 Miracle Octad Generator
Folding a 2×4 Curtis array yet again yields — Steven H. Cullinane on April 19, 2016 — The Folding. |
Related art-historical remarks:
The Shape of Time (Kubler, Yale U.P., 1962).
See yesterday's post The Thing .
From Log24 on Epiphany 2012 —
A version of the Zemeckis Cube —
* See Turning Nine (Log24, Nov. 8, 2021).
Update of later the same evening —
The subtitle for the hashtag symbol at left is "Explore." Appropriate.
"On September 2, 2020, at the age of 59,
David Graeber died of necrotizing pancreatitis
while on vacation in Venice. The news hit me
like a blow. How many books have we lost,
I thought, that will never get written now?
How many insights, how much wisdom,
will remain forever unexpressed? The appearance of
The Dawn of Everything: A New History of Humanity
is thus bittersweet, at once a final, unexpected gift
and a reminder of what might have been."
— William Deresiewicz
This is from The Atlantic on St. Luke's Day, 2021.
Note the article's illustration, and related material from
this journal on the date of the death described above:
Yesterday morning's post "First Step" quoted an essay by Michael Spitzer
published online in Aeon on October 18 —
https://aeon.co/essays/
can-music-give-you-an-orgasm-the-short-answer-is-yes .
A look at earlier essays in that publication reveals . . .
Related material — From a search for Wertheim in this journal —
“Space: what you
damn well have to see.”
— James Joyce, Ulysses
"Before time began, there was the Cube." — Optimus Prime
See also Design Cube.
* According to The Wall Street Journal … and possibly also
dead on October 11th, according to The New York Times .
Continued from September 8 .
The New York Times yesterday —
“Art is another way to try to exercise your imagination
at connecting incongruous things,” Anthony Doerr said.
“It’s a way to say, hey, reader, let’s work together and
practice and train our imagination to connect things
that you don’t readily think of as connected.”
It is often good to remember that writers of headlines (and subheadlines)
are usually not the same people as the authors of the following texts.
In particular, in the above example, neither the word "touchstone" nor
the use of "enquires" to mean "enquiries" appears in the text proper.
Still, the mixed metaphor of "razor" as "touchstone" is not without interest.
See The Eightfold Cube and Modernist Cuts.
Starring J. J. Abrams as Leonhard Euler?
Related material —
The Cornell cap in the recent HBO "White Lotus" —
"I'm just playing the hand I was dealt."
"Two years ago . . . ." — Synopsis of the August 3 film "Hum"
Two years ago on August 3 . . .
What is going on in this picture?
The above is an image from
the August 3, 2019,
post "Butterfield's Eight."
"Within the week . . . ."
— The above synopsis of "Hum"
This suggests a review of a post
from August 5, 2019, that might
be retitled . . .
"The void she knows,
the tune she hums."
From "A Four-Color Theorem:
Function Decomposition Over a Finite Field" —
Related material —
An image from Monday's post
"Scholastic Observation" —
Miller died on February 7, 2021.
See that date in this journal —
“Before time began, there was the Cube.”
— Hassenfeld Brothers cinematic merchandising slogan.
Update at noon on Wednesday, June 9, 2021 —
Related material on Frye and deconstruction —
|
From "The First Major Theoretician? Northrop Frye Towards the end of his career, when it was clear that literary theory had taken hold in the academy, Frye began to reflect on literary theory. In an interview with Deanne Bogdan, Frye laments, “I am feeling out of the great critical trends today”…. Northrop Frye was right that he was “out of fashion,” both in terms of his own theories and his place in literary theory; however, he did seek to reverse the course. Frye hoped to reclaim literary studies from deconstruction, which had become, in a sense, his chief opponent …. |
In Scientific American today —
For a more sophisticated approach to the phrase
“blocks in a box,” search for “the 759 blocks” and
then see box759.wordpress.com.
The mathematics there is based on an apparently
less sophisticated example of “blocks in a box” —
See also Cube Space in this journal.
RA Wilson —”[Submitted on 20 Apr 2021 (v1),
last revised 23 Apr 2021 (this version, v2)]”
SH Cullinane — See as well
box759.wordpress.com.
References to a 1960 conference paper by Freudenthal in this journal
suggest another paper from the same conference …
See as well other posts now tagged . . .
For my own work on models, see
Finite Geometry of the Square and Cube.
Drilling down . . .
My own, more abstract, academic interests are indicated by
a post from this journal on January 20, 2020 —
Dyadic Harmonic Analysis: The Fourfold Square and Eightfold Cube.
Those poetically inclined may regard that post as an instance of the
“intersection of the timeless with time.”
See Trinity Cube in this journal and . . .
McDonnell’s illustration is from 9 June 1983.
See as well a less official note from later that June.
" . . . It is as if
We had come to an end of the imagination,
Inanimate in an inert savoir."
— Wallace Stevens, "The Plain Sense of Things"
For such a savoir, see Cube School.
See as well the Stevens online concordance.
Also on February Seventh —
Sunday, February 7, 2021
Transformers Continues.“Before time began, there was the Cube.”
Et cetera . |
(Adapted from Encyclopaedia Britannica,
Eleventh Edition (1911), Crystallography .)
“Knight move” remark from The Eiger Sanction —
“I like to put people on myself by skipping logical steps
in the conversation until they’re dizzy.”
The following logical step — a check of the date Nov. 18, 2017 —
was omitted in the post Futon Dream on this year’s St. Stephen’s Day.
For further context, see James Propp in this journal.
“A quick note on terminology. Members of the Circle
were logical empiricists, sometimes called logical positivists.
Positivism is the view that our knowledge derives from
the natural world and includes the idea that we can have
positive knowledge of it. The Circle combined this position
with the use of modern logic; the aim was to build a new
philosophy.”
— Edmonds, David. The Murder of Professor Schlick (p. vii).
Princeton University Press. Kindle Edition.
For aficionados of associative logic —
See Triple Cross in this journal and the Fano-plane circle
in the illustration below.
From old posts tagged Change Arises —
| From Christmas 2005:
For the eightfold cube For an rather more
Click on image for details. |
The phrase “change arises” is from Arkani-Hamed in 2013, describing
calculations in physics related to properties of the positive Grassmannian —
A related recent illustration from Quanta Magazine —
The above illustration of seven cells is not unrelated to
the eightfold-cube model of the seven projective points in
the Fano plane.
“… her art was rarely exhibited until the 1970s,
and then only sporadically and in small venues . . . .”
— New York Times obituary suggested by
today’s review,
https://www.nytimes.com/2020/12/22/
arts/artists-who-died-2020.html
“No ordinary venue.” — Song lyric
Related material now linked to in the previous post —
“… and the song of love’s recision is the music of the spheres.”
— E. L. Doctorow, City of God
Doctorow’s remark was quoted here earlier, on February 5, 2009 —
The central aim of Western religion–
"Each of us has something to offer the Creator... the bridging of masculine and feminine, life and death. It's redemption.... nothing else matters." -- Martha Cooley in The Archivist (1998) The central aim of Western philosophy– Dualities of Pythagoras as reconstructed by Aristotle: Limited Unlimited Odd Even Male Female Light Dark Straight Curved ... and so on .... “Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy.” — Jamie James in The Music of the Spheres (1993) “In the garden of Adding — The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000) A quotation today at art critic Carol Kino’s website, slightly expanded: “Art inherited from the old religion — Octavio Paz,”Seeing and Using: Art and Craftsmanship,” in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52 From Brian O’Doherty’s 1976 Artforum essays– not on museums, but rather on gallery space: “We have now reached
“Space: what you — James Joyce, Ulysses |
Hurt’s dies natalis (date of death, in the saints’ sense) was,
it now seems, 25 January 2017, not 27.
A connection, for fantasy fans, between the Philosopher’s Stone
(represented by the eightfold cube) and the Deathly Hallows
(represented by the usual Fano-plane figure) —
Images from a Log24 search for “Holocron.”
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —
Related material — The Eightfold Cube.
Update at 10:51 PM ET the same day —
Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
The new domain qube.link forwards to . . .
http://finitegeometry.org/sc/64/solcube.html .
More generally, qubes.link forwards to this post,
which defines qubes .
Definition: A qube is a positive integer that is
a prime-power cube , i.e. a cube that is the order
of a Galois field. (Galois-field orders in general are
customarily denoted by the letter q .)
Examples: 8, 27, 64. See qubes.site.
Update on Nov. 18, 2020, at about 9:40 PM ET —
Problem:
For which qubes, visualized as n×n×n arrays,
is it it true that the actions of the two-dimensional
galois-geometry affine group on each n×n face, extended
throughout the whole array, generate the affine group
on the whole array? (For the cases 8 and 64, see Binary
Coordinate Systems and Affine Groups on Small
Binary Spaces.)
"The bond with reality is cut."
— Hans Freudenthal, 1962
Related screenshot of a book review
from the November AMS Notices —
“On their way to obscurity, the Simulmatics people
played minor parts in major events, appearing Zelig-like
at crucial moments of 1960s history.”
— James Gleick reviewing a new book by Jill Lepore
Continues in The New York Times :
“One day — ‘I don’t know exactly why,’ he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.
Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical:
a solid, static object that is also fluid….” — Alexandra Alter
Another such object: the eightfold cube .
Metaphysical ruminations of Coleridge that might be applied to
the eightfold cube —
See also "Sprechen Sie Neutsch?".
Update of December 29, 2022 —
The title is from a Cornell page in the previous post.
Related material (click to enlarge) —
The above remarks on primitive mentality suggest
a review of Snakes on a Plane.
An illustration from the Vox article —
Another approach to Nolan theory —
Or Matt Helm by way of a Jedi cube.
From a web page —
From YouTube, for the Church of Synchronology —
Meanwhile, elsewhere . . .
* See that book title in this journal.
The title is a phrase by Robert Hughes from the previous post.
“He recounted the story of Adam and Eve, who were banished
from paradise because of their curiosity. Their inability to resist
the temptation of the forbidden fruit. Which itself was a metaphorical
stand-in for knowledge and power. He urged us to find the restraint
needed to resist the temptation of the cube—the biblical apple
in modern garb. He urged us to remain in Eden until we were able
to work out the knowledge the apple offered, all by ourselves.”
— Richards, Douglas E.. The Enigma Cube (Alien Artifact Book 1)
(pp. 160-161). Paragon Press, 2020. Kindle Edition.
The biblical apple also appears in the game, and film, Assassin’s Creed .
Related material —
See the cartoon version of Alfred North Whitehead in the previous post,
and some Whitehead-related projective geometry —
The previous post reported, perhaps inaccurately, a publication
date of February 13, 2020, for the novel The Enigma Cube .
A variant publication date, Jan. 21, 2020, is reported below.
This journal on that date —
A film not unrelated to the screen career
of Sophia Lillis: Inside Daisy Clover.
I prefer Inside the White Cube.
"MIT professor of linguistics Wayne O’Neil died on March 22
at his home in Somerville, Massachusetts."
— MIT Linguistics, May 1, 2020
The "deep structure" above is the plane cutting the cube in a hexagon
(as in my note Diamonds and Whirls of September 1984).
See also . . .
The resemblance to the eightfold cube is, of course,
completely coincidental.
Some background from the literature —
From a paper cited in the above story:
“Fig. 4 A lattice geometry for a surface code.” —
The above figure suggests a search for “surface code” cube :
Related poetic remarks — “Illumination of a surface.”
"Let me say this about that." — Richard Nixon
Interpenetration in Weyl's epistemology —
Interpenetration in Mazzola's music theory —
Interpenetration in the eightfold cube — the three midplanes —
A deeper example of interpenetration:
Aitchison has shown that the Mathieu group M24 has a natural
action on the 24 center points of the subsquares on the eightfold
cube's six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.
The Tesseract Timeline:
Where The Cube Has Been In The …
www.cinemablend.com › news › the-tesseract-timeline-…
Mar 13, 2019 – With HYDRA. In 1942, Johann Schmidt, a.k.a.
the Red Skull, arrived in Tønsberg to procure the Tesseract
from an ancient church. While he …
Related material from posts tagged Aqua
(suggested by a name in the previous post) —
See also the previous post.
The above date “4 avril 2016”
suggests a review of that date
in this journal: “Cube for Berlin.”
The previous post displayed a photo from November 2014.
Remarks quoted here in November 2014 —
“Before time began, there was the Cube.”
— Optimus Prime
A brief summary of the eightfold cube is now at octad.us.
Mathematics: See Tetrahedron vs. Square in this journal
(Notes on two different models of schoolgirl space ).
Narrative: Replacing the square from the above posts by
a related cube …
… yields a merchandising inspiration —
Dueling Holocrons:
Jedi Cube vs. Sith Tetrahedron —
* See also earlier posts on Mathematics and Narrative.
The above arrangement of graphic images on cube faces is purely
decorative and static, and of little mathematical interest.
(A less static, but structurally chaotic, artifact might be made by
pasting the above 24 graphic images in the "Cosets in S4" picture
above onto the 24 faces of a 2x2x2 Rubik cube. This suggests the
reflection below on the poet Wallace Stevens, whose "Connoisseur
of Chaos" first appeared on page 90 of Twentieth Century Verse ,
Numbers 12-13, October 1938.)
If mathematically interesting permutations of the graphic images
are to be done, the images should be imagined as situated on
parallel planes, as in the permutahedron below —
Click the above permutahedron for an analysis of its structure.
The title is from a post of January 10, 2019.
A figure from this journal on June 1, 2019 —
The following figure may help relate labelings of the
truncated octahedron ("permutahedron") to labelings
of its fellow Archimedean solid, the cuboctahedron.
See as well other posts tagged Aitchison.
For fans of "The Zero Theorem" —
The 24 permutations of S4 arranged on a cube
by Cristi Stoica of Bucharest at
http://www.unitaryflow.com/2009/06/polyhedra-and-groups.html:
See the title in this journal.
Such generation occurs both in Euclidean space …
… and in some Galois spaces —
.
In Galois spaces, some care must be taken in defining "reflection."
Friday, July 11, 2014Spiegel-Spiel des GeviertsFiled under: Uncategorized — m759 @ 12:00 PM See Cube Symbology.
|
Freeman Dyson on his staircase at Trinity College
(University of Cambridge) and on Ludwig Wittgenstein:
“I held him in the highest respect and was delighted
to find him living in a room above mine on the same
staircase. I frequently met him walking up or down
the stairs, but I was too shy to start a conversation.”
Frank Close on Ron Shaw:
“Shaw arrived there in 1949 and moved into room K9,
overlooking Jesus Lane. There is nothing particularly
special about this room other than the coincidence that
its previous occupant was Freeman Dyson.”
— Close, Frank. The Infinity Puzzle (p. 78).
Basic Books. Kindle Edition.
See also other posts now tagged Trinity Staircase.
Illuminati enthusiasts may enjoy the following image:
|
Roberta Smith on Donald Judd’s BY ALEX GREENBERGER February 28, 2020 1:04pm If Minimalist artist Donald Judd is known as a writer at all, it’s likely for one important text— his 1965 essay “Specific Objects,” in which he observed the rise of a new kind of art that collapsed divisions between painting, sculpture, and other mediums. But Judd was a prolific critic, penning shrewd reviews for various publications throughout his career—including ARTnews . With a Judd retrospective going on view this Sunday at the Museum of Modern Art in New York, ARTnews asked New York Times co-chief art critic Roberta Smith— who, early in her career, worked for Judd as his assistant— to comment on a few of Judd’s ARTnews reviews. How would she describe his critical style? “In a word,” she said, “great.” . . . . |
And then there is Temple Eight, or Ex Fano Apollinis —
Cicero, In Verrem II. 1. 46 —
He reached Delos. There one night he secretly 46 carried off, from the much-revered sanctuary of Apollo, several ancient and beautiful statues, and had them put on board his own transport. Next day, when the inhabitants of Delos saw their sanc- tuary stripped of its treasures, they were much distressed . . . .
Delum venit. Ibi ex fano Apollinis religiosissimo noctu clam sustulit signa pulcherrima atque anti- quissima, eaque in onerariam navem suam conicienda curavit. Postridie cum fanum spoliatum viderent ii qui Delum incolebant, graviter ferebant . . . .
From the author who in 2001 described "God's fingerprint"
(see the previous post) —
From the same publisher —
From other posts tagged Triskele in this journal —
Other geometry for enthusiasts of the esoteric —
|
Monday, November 4, 2019
As Above, So Below*
|
|
|
"Although art is fundamentally everywhere and always the same,
nevertheless two main human inclinations, diametrically opposed
to each other, appear in its many and varied expressions. ….
The first aims at representing reality objectively, the second subjectively."
— Mondrian, 1936 [Links added.]
An image search today (click to enlarge) —
Also on January 27, 2017 . . .
For other appearances of John Hurt here,
see 1984 Cubes.
Update of 12:45 AM Feb. 22 —
A check of later obituaries reveals that Hurt may well
have died on January 25, 2017, not January 27 as above.
Thus the following remarks may be more appropriate:
Not to mention what, why, who, and how.
The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.
A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a cuboctahedron :
The Curtis octads are related to symmetries of the square.
See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —
Aitchison's octads are instead related to symmetries of the cube.
Note that essentially the same model as Aitchison's can be pictured
by using, instead of the 24 edges of a cuboctahedron, the 24 outer
faces of subcubes in the eightfold cube .
Image from Christmas Day 2005.
* http://www.math.sci.hiroshima-u.ac.jp/branched/files/2018/
presentations/Aitchison-Hiroshima-2-2018.pdf.
See also Aitchison in this journal.
The plane at left is modeled naturally by
seven types of “cuts” in the cube at right.
* A weblog motto. See …
http://enowning.blogspot.com/
2007/07/alfred-denkers-dictionary-on-ereignis.html.
That Ereignis post is dated July 3, 2007.
Related material for the Church of Synchronology —
"The deepest strain in a religion is the particular
and particularistic doctrine it asserts at its heart,
in the company of such pronouncements as
‘Thou shalt have no other Gods before me.’
Take the deepest strain of religion away…
and what remains are the surface pieties —
abstractions without substantive bite —
to which everyone will assent
because they are empty, insipid, and safe."
— Stanley Fish, quoted here on July 3, 2007…
The opening date of the film "Transformers."
The opening pronouncement of "Transformers" —
Two of the thumbnail previews
from yesterday's 1 AM post …
Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to …
|
A Letter from Timothy Leary, Ph.D., July 17, 1961
Harvard University July 17, 1961
Dr. Thomas S. Szasz Dear Dr. Szasz: Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun. The Myth of Mental Illness is the most important book in the history of psychiatry. I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century. It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty. . . . . |
The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.
Related "jewel box" material for those who
prefer narrative to mathematics —
"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."
— Tim Doody in The Morning News web 'zine on July 26, 2012**
Oracular advice related to yesterday evening's
"jewel box" post …
A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces. In its incarnation as a Galois tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of 2,304 affine transformations.
Similar remarks apply to the I Ching In its incarnation as
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.
* The volume Wolfe mentions was, according to Fadiman, the I Ching.
** See also this journal on that date — July 26, 2012.
The Fourfold Square and Eightfold Cube
Related material: A Google image search for “field dream” + log24.
The above image is from
"A Four-Color Theorem:
Function Decomposition Over a Finite Field,"
http://finitegeometry.org/sc/gen/mapsys.html.
These partitions of an 8-set into four 2-sets
occur also in Wednesday night's post
Miracle Octad Generator Structure.
This post was suggested by a Daily News
story from August 8, 2011, and by a Log24
post from that same date, "Organizing the
Mine Workers" —
(Adapted from Encyclopaedia Britannica,
Eleventh Edition (1911), Crystallography .)
Shown below are Aitchison's March 2018 M24 permutations
and their relabeling, with digits only, for MAGMA checking.
In the versions below, r g b stand for red, green, blue.
Infinity has been replaced by 7 (because a digit was needed,
and the position of the infinity symbol in the Aitchison cube
was suited to the digit 7).
(r7,r1)(b2,g4)(r3,r5)(r6,g0)
mu0= (g7,g2)(r4,b1)(g6,g3)(g5,b0)
(b7,b4)(g1,r2)(b5,b6)(b3,r0)
mu1 = (r7,r2,)(b3,g5)(r4,r6)(r0,g1)
(g7,g3)(r5,b2)(g0,g4)(g6,b1)
(b7,b5)(g2,r3)(b6,b0)(b4,r1)
mu2 = (r7,r3)(b4,g6)(r5,r0)(r1,g2)
(g7,g4)(r6,b3)(g1,g5)(g0,b2)
(b7,b6)(g3,r4)(b0,b1)(b5,r2)
mu3 = (r7,r4)(b5,g0)(r6,r1)(r2,g3)
(g7,g5)(r0,b4)(g2,g6)(g1,b3)
(b7,b0)(g4,r5)(b1,b2)(b6,r3)
mu4 = (r7,r5)(b6,g1)(r0,r2)(r3,g4)
(g7,g6)(r1,b5)(g3,g0)(g2,b4)
(b7,b1)(g5,r6)(b2,b3)(b0,r4)
mu5 = (r7,r6)(b0,g2)(r1,r3)(r4,g5)
(g7,g0)(r2,b6)(g4,g1)(g3,b5)
(b7,b2)(g6,r0)(b3,b4)(b1,r5)
mu6 = (r7,r0)(b1,g3)(r2,r4)(r5,g6)
(g7,g1)(r3,b0)(g5,g2)(g4,b6)
(b7,b3)(g0,r1)(b4,b5)(b2,r6)
Table 1 —
0 1 2 3 4 5 6 7
r 1 2 3 4 5 6 7 8
g 9 10 11 12 13 14 15 16
b 17 18 19 20 21 22 23 24
The wReplace program was used with Table 1 above
to rewrite mu0-mu6 for MAGMA.
The resulting code for MAGMA —
|
G := sub< Sym(24) |
(8,3)(20,14)(5,7)(1,10)
(8,4)(21,15)(6,1)(2,11)
(8,5)(22,9)(7,2)(3,12)
(8,6)(23,10)(1,3)(4,13)
(8,7)(17,11)(2,4)(5,14)
(8,1)(18,12)(3,5)(6,15)
G; |
The Aitchison generators passed the MAGMA test.
An image from All Souls' Day 2010 —
This is from earlier posts tagged Permutahedron.
See also
Wallace Stevens:
A World of Transforming Shapes.
From that book (click to enlarge) —
"Before time began, there was the Cube."
— Optimus Prime.
Also from earlier posts tagged Permutahedron —
The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel
Many Dimensions published in 1993 by Wm. B. Eerdmans.
Compare and constrast —
Cover of a book by Douglas Hofstadter
This journal on the last full day of Matthiessen's life —
John Horgan in Scientific American magazine on October 8, 2019 —
"In the early 1990s, I came to suspect that the quest
for a unified theory is religious rather than scientific.
Physicists want to show that all things came from
one thing: a force, or essence, or membrane
wriggling in eleven dimensions, or something that
manifests perfect mathematical symmetry. In their
search for this primordial symmetry, however,
physicists have gone off the deep end . . . ."
Other approaches —
See "Story Theory of Truth" in this journal and, from the November 2019
Notices of the American Mathematical Society . . .
|
More fundamental than the label of mathematician is that of human. And as humans, we’re hardwired to use stories to make sense of our world (story-receivers) and to share that understanding with others (storytellers) [2]. Thus, the framing of any communication answers the key question, what is the story we wish to share? Mathematics papers are not just collections of truths but narratives woven together, each participating in and adding to the great story of mathematics itself. The first endeavor for constructing a good talk is recognizing and choosing just one storyline, tailoring it to the audience at hand. Should the focus be on a result about the underlying structures of group actions? . . . .
[2] Gottschall, J. , The Storytelling Animal , — "Giving Good Talks," by Satyan L. Devadoss |
"Before time began, there was the Cube." — Optimus Prime
From Wallace Stevens —
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
Of dense investiture, with luminous vassals."
— “An Ordinary Evening in New Haven” VI
From The Point magazine yesterday, October 8, 2019 —
Parricide: On Irad Kimhi's Thinking and Being .
Book review by Steven Methven.
The conclusion:
"Parricide is nothing that the philosopher need fear . . . .
What sustains can be no threat. Perhaps what the
unique genesis of this extraordinary work suggests is that
the true threat to philosophy is infanticide."
This remark suggests revisiting a post from Monday —
|
Monday, October 7, 2019
Berlekamp Garden vs. Kinder Garten
|
|
American Mathematical Monthly , June-July 1984 — MISCELLANEA, 129 Triangles are square
"Every triangle consists of n congruent copies of itself" |
* See Cube Bricks 1984 in previous post.
Stevens's Omega and Alpha (see previous post) suggest a review.
Omega — The Berlekamp Garden. See Misère Play (April 8, 2019).
Alpha — The Kinder Garten. See Eighfold Cube.
Illustrations —
The sculpture above illustrates Klein's order-168 simple group.
So does the sculpture below.
Cube Bricks 1984 —
"The 15 Puzzle and the Magic Cube
are spiritual kin …."
— "Metamagical Themas" column,
Douglas R. Hofstadter, Scientific American ,
Vol. 244, No. 3 (March 1981), pp. 20-39
As are the 15 Schoolgirls and the Eightfold Cube.
The previous post dealt with “magic” cubes, so called because of the
analogous “magic” squares. Douglas Hofstadter has written about a
different, physical , object, promoted as “the Magic Cube,” that Hofstadter
felt embodied “a deep invariant”:
Related literary remarks — Raiders of the Lost Birthday.
"… objects have a notion of 'this' or 'self.' " — Wikipedia
For related notions, see other posts tagged Quark Rock and …
"Before time began, there was the Cube." — Optimus Prime
See as well an obituary for Mrs. Wertham from 1987.
Related art —
Friday, July 11, 2014
|
For further details, search the Web for "Wertham Professor" + Eck.
Down the Up Ladder
" … this beautiful love story . . . ."
An image from the previous post:
The above line "From the producer of Transformers " suggests
a story from March 18, 2019 . . .
Misreading the words of di Bonaventura
yields a phrase that might be applied to
the Church of Rome . . .
"A franchise based on release dates."
See dies natalis in this journal.
For the Church of Synchronology, see
the above di Bonaventura date, March 18.
Then there is the Church of Cubism . . .
"Before time began, there was the Cube."
— Optimus Prime, Transformers , 2007
"In the fantasy, Owen is still working on his Rubik’s Cube.
Finally, he finishes — he’s put together all 6 sides."
— "Maniac" Season 1, Episode 9 recap: ‘Utangatta’
by Cynthia Vinney at showsnob.com, Oct. 9, 2018
Related material —
See also Exploded in this journal.
See the above title in this journal. See also . . .
Click the image for some remarks on a related novel.
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