“Before time began, there was the Cube.”
— Hassenfeld Brothers merchandising slogan
Sunday, February 21, 2021
Cube Woo
Saturday, September 19, 2020
Cube School
The new domain http://cube.school
points to posts tagged Cube School here.
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Sunday, July 5, 2020
The Enigma Cube
Promotional material —
“Did you buckle up?” — Harlan Kane
The publication date of The Enigma Cube reported above was February 13, 2020.
Related material — Log24 posts around that date now tagged The Reality Bond.
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Monday, February 24, 2020
For “Time Cube” Fans
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Tuesday, May 21, 2019
Monday, May 13, 2019
Thursday, March 22, 2018
The Diamond Cube
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.
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Saturday, November 18, 2017
Cube Space Continued
James Propp in the current Math Horizons on the eightfold cube —
For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).
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Sunday, June 4, 2017
In Memory of the Time Cube Page*
From this journal on August 18, 2015, "A Wrinkle in Terms" —
For two misuses by John Baez of the phrase “permutation group”
at the n-Category Café, see “A Wrinkle in the Mathematical Universe”
and “Re: A Wrinkle…” —
“There is such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle
* See RIP, Time Cube at gizmodo.com (September 1, 2015).
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Tuesday, April 4, 2017
White Cube
“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 20th-century art.”
“Space: what you
damn well have to see.”
— James Joyce, Ulysses
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Tuesday, April 5, 2016
“Puzzle Cube of a Novel”
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Monday, April 4, 2016
Cube for Berlin
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, 1984-2003.”
(From Anatomy of a Cube, Sept. 18, 2011.)
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Sunday, December 28, 2014
Cube of Ultron
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 super-cell.
Speaking of that super-cell, 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)
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Monday, May 19, 2014
Cube Space
A sequel to this afternoon’s Rubik Quote:
“The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos.”
— Professor Ernő Rubik at Chrome Cube Lab
(Click image below to enlarge.)
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Thursday, January 24, 2013
Cube Space
For the late Cardinal Glemp of Poland,
who died yesterday, some links:
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Friday, December 28, 2012
Cube Koan
|
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 agreed-upon 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 three-dimensional. 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 counter-clockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertex-labeling may be used in describing
the automorphisms of the order-8 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 dice-throwing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.
Glorified dice-throwing, indeed…"
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Sunday, August 5, 2012
Cube Partitions
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.
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Sunday, February 5, 2012
Wednesday, January 11, 2012
Cuber
“Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing. And once you have made or acquired a new ‘cube’… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube. What is the essence of each operator? One senses a deep invariant lying somehow ‘down underneath’ it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment. In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….
… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube. It is the answer; it simply has the right spirit .”
— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern (Kindle edition, locations 11557-11572)
See also Many Dimensions in this journal and Solomon’s Cube.
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Friday, December 30, 2011
Quaternions on a Cube
The following picture provides a new visual approach to
the order-8 quaternion group's automorphisms.
Click the above image for some context.
Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.
See also…
- The 1985 note from which the above figures were drawn
- Visualizing GL(2,p)
- Quaternions in an Affine Galois Plane
Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.
* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
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Sunday, September 18, 2011
Anatomy of a Cube
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, 1984-2003.”
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Saturday, August 27, 2011
Cosmic Cube*
Prequel — (Click to enlarge)
Background —
See also Rubik in this journal.
* For the title, see Groups Acting.
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Friday, June 24, 2011
The Cube
Click the above image for some background.
Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.
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Monday, June 21, 2010
Cube Spaces
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Example 1— The 2×2×2 Cube—
also known as the eightfold cube—
Group actions on the eightfold cube, 1984—
Version by Laszlo Lovasz et al., 2003—
Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
Example 2— The 3×3×3 Cube
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Example 3— The 4×4×4 Cube
A note from 27 years ago today—
As far as I know, this version of the
group-actions theorem has not yet been ripped off.
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Saturday, February 27, 2021
The Pencil Case
Clue
Here is a midrash on “desmic,” a term derived from the Greek desmé
( δέσμη: bundle, sheaf , or, in the mathematical sense, pencil —
French faisceau ), which is related to the term desmos , bond …
(The term “desmic,” as noted earlier, is relevant to the structure of
Heidegger’s Sternwürfel .)
“Gadzooks, I’ve done it again!” — Sherlock Hemlock
Thursday, February 25, 2021
Compare and Contrast
“… What is your dream—your ideal? What is your News from Nowhere,
or, rather, What is the result of the little shake your hand has given to
the old pasteboard toy with a dozen bits of colored glass for contents?
And, most important of all, can you present it in a narrative or romance
which will enable me to pass an idle hour not disagreeably? How, for instance,
does it compare in this respect with other prophetic books on the shelf?”
— Hudson, W. H.. A Crystal Age (p. 2). Open Road Media. Kindle Edition.
See as well . . .
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
Wednesday, February 24, 2021
Annals of Dim Antiquity
“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.
This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”
— The Bourne Enigma , published on June 21, 2016
Passing, on June 21, 2016, into a higher dimension —
For those who prefer Borges to Bourne —
Monday, February 15, 2021
Raiders of the Lost Building Blocks
In memory of a Dead Sea Scrolls scholar who
reportedly died on December 29, 2020, here are
links to two Log24 posts from that date:
I Ching Geometry and Raiders of the Lost Coordinates.
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Tuesday, December 29, 2020
I Ching Geometry
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Sunday, December 27, 2020
Knight Move for Trevanian
“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.
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Wednesday, September 23, 2020
Geometry of Even Subsets
Various posts here on the geometry underlying the Mathieu group M24
are now tagged with the phrase “Geometry of Even Subsets.”
For example, a post with this diagram . . .
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Monday, September 21, 2020
Zelig-Like?
“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
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Saturday, September 19, 2020
The Summerfield Prize
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Thursday, September 17, 2020
Structure and Mutability . . .
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 .
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Thursday, September 10, 2020
Wednesday, September 9, 2020
Portrait with Holocron
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Thursday, July 9, 2020
The Enigma Glyphs
For those who prefer fiction —
“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.
This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”
— “Robert Ludlum’s” The Bourne Enigma , published on June 21, 2016
Passing, on June 21, 2016, into a higher dimension —
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Sunday, July 5, 2020
It’s Still the Same Old Story …
“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 —
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Enigma Variations
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 —
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Saturday, May 23, 2020
Eightfold Geometry: A Surface Code “Unit Cell”
The resemblance to the eightfold cube is, of course,
completely coincidental.
Some background from the literature —
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Friday, May 22, 2020
Surface Code News
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.”
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Thursday, March 5, 2020
Pythagorean Letter Meets Box of Chocolates
Friday, July 11, 2014Spiegel-Spiel des GeviertsFiled under: Uncategorized — m759 @ 12:00 PM See Cube Symbology.
|
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Friday, February 21, 2020
To and Fro, Back and …
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.
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Monday, November 11, 2019
Time and Chance
The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel
Many Dimensions published in 1993 by Wm. B. Eerdmans.
Compare and constrast —
Cover of a book by Douglas Hofstadter
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Sunday, September 29, 2019
Stage Direction: “Comments Off.”
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”:
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Wednesday, July 10, 2019
Artifice* of Eternity …
… 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. 14-20.
See also Schoolgirl Space in this journal.
* See, for instance, Lewis Hyde on the word "artifice" and . . .
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Tuesday, July 9, 2019
Schoolgirl Space: 1984 Revisited
Cube Bricks 1984 —
From "Tomorrowland" (2015) —
From John Baez (2018) —
See also this morning's post Perception of Space
and yesterday's Exploring Schoolgirl Space.
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Perception of Space
The three previous posts have now been tagged . . .
Tetrahedron vs. Square and Triangle vs. Cube.
Related material —
Tetrahedron vs. Square:
Labeling the Tetrahedral Model (Click to enlarge) —
Triangle vs. Cube:
… and, from the date of the above John Baez remark —
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Dreamtimes
“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 . . .
.
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Monday, July 8, 2019
Exploring Schoolgirl Space
See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."
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Sunday, July 7, 2019
Schoolgirl Problem
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 schoolgirl-problem article on Feb. 11, 2017, just
before a revision that removed it.
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Saturday, June 8, 2019
Art Object, continued and continued
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/hqk7-nx97 .
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Monday, May 13, 2019
Doris Day at the Hudson Rock
" '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 —
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Tuesday, May 7, 2019
Monday, May 6, 2019
In Memoriam Goro Shimura (d. May 3, 2019)
From Richard Taylor, "Modular arithmetic: driven by inherent beauty
and human curiosity," The Letter of the Institute for Advanced Study [IAS],
Summer 2012, pp. 6– 8 (links added) :
"Stunningly, in 1954, Martin Eichler (former IAS Member)
found a totally new reciprocity law . . . .
Within less than three years, Yutaka Taniyama and Goro Shimura
(former IAS Member) proposed a daring generalization of Eichler’s
reciprocity law to all cubic equations in two variables. A decade later,
André Weil (former IAS Professor) added precision to this conjecture,
and found strong heuristic evidence supporting the Shimura-Taniyama
reciprocity law. This conjecture completely changed the development of
number theory."
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Friday, March 1, 2019
Solomon and the Image
"Maybe an image is too strong
Or maybe is not strong enough."
— "Solomon and the Witch,"
by William Butler Yeats
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Wednesday, October 24, 2018
Shadowlands
The previous post suggests a review.
Following the above reference to March 30, 2016 —
Following the above reference to Lovasz —
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Saturday, August 25, 2018
“Waugh, Orwell. Orwell, Waugh.”
Suggested by a review of Curl on Modernism —
Related material —
Waugh + Orwell in this journal and …
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Wednesday, June 6, 2018
Geometry for Goyim
Mystery box merchandise from the 2011 J. J. Abrams film Super 8 —
A mystery box that I prefer —
Click image for some background.
See also Nicht Spielerei .
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Thursday, March 29, 2018
“Before Creation Itself . . .”
From the Diamond Theorem Facebook page —
A question three hours ago at that page —
“Is this Time Cube?”
Notes toward an answer —
And from Six-Set Geometry in this journal . . .
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Saturday, March 24, 2018
Sure, Whatever.
The search for Langlands in the previous post
yields the following Toronto Star illustration —
From a review of the recent film "Justice League" —
"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."
For other cubic adventures, see yesterday's post on A Piece of Justice
and the block patterns in posts tagged Design Cube.
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Friday, March 23, 2018
Reciprocity
Copy editing — From Wikipedia
"Copy editing (also copy-editing or copyediting, sometimes abbreviated ce)
is the process of reviewing and correcting written material to improve accuracy,
readability, and fitness for its purpose, and to ensure that it is free of error,
omission, inconsistency, and repetition. . . ."
An example of the need for copy editing:
Related material: Langlands and Reciprocity in this journal.
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From the Personal to the Platonic
On the Oslo artist Josefine Lyche —
"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."
— Ann Cathrin Andersen,
http://bryggmagasin.no/2017/behind-the-glitter/
Personal —
The Rushkoff Logo
— From a 2016 graphic novel by Douglas Rushkoff.
See also Rushkoff and Talisman in this journal.
Platonic —
Compare and contrast the shifting hexagon logo in the Rushkoff novel above
with the hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).
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Thursday, March 22, 2018
Wednesday, March 7, 2018
Unite the Seven.
Related material —
The seven points of the Fano plane within
|
"Before time began . . . ."
— Optimus Prime
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Monday, January 22, 2018
Hollywood Moment
A death on the date of the above symmetry chat,
Wednesday, August 17, 2016 —
An Hispanic Hollywood moment:
Ojo de Dios —
Click for related material.
For further Hispanic entertainment,
see Ben Affleck sing
"Aquellos Ojos Verdes "
in "Hollywoodland."
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Wednesday, September 13, 2017
Summer of 1984
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
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Tuesday, September 12, 2017
Think Different
The New York Times online this evening —
"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."
Review —
|
Thursday, September 1, 2011
How It Works
|
See also 1984 Bricks in this journal.
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Chin Music
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Tuesday, June 20, 2017
Epic
Continuing the previous post's theme …
Group actions on partitions —
Cube Bricks 1984 —
Related material — Posts now tagged Device Narratives.
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Wednesday, June 7, 2017
Three Things at Once
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Wednesday, April 12, 2017
Contracting the Spielraum
The contraction of the title is from group actions on
the ninefold square (with the center subsquare fixed)
to group actions on the eightfold cube.
From a post of June 4, 2014 …
At math.stackexchange.com on March 1-12, 2013:
“Is there a geometric realization of the Quaternion group?” —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
Comments Off on Contracting the Spielraum
Thursday, November 3, 2016
Triple Cross
(Continued … See the title in this journal, as well as Cube Bricks.)
Cube Bricks 1984 —
Related material —
Dirac and Geometry in this journal,
Kummer’s Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor’s Edge and Nanavira Thera.
See as well Bill Murray’s 1984 film “The Razor’s Edge” …
Movie poster from 1984 —
“A thin line separates
love from hate,
success from failure,
life from death.”
Three other dualities, from Nanavira Thera in 1959 —
“I find that there are, in every situation,
three independent dualities….”
(Click to enlarge.)
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Tuesday, October 4, 2016
Westworld
The title refers to a Log24 post of 9:45 AM ET Sunday, Oct. 2.
From the "Westworld" post of Sunday, Oct. 2 —
"It was rather like watching a play."
QED.
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Sunday, October 2, 2016
Westworld
On a new HBO series that opens at 9 PM ET tonight —
|
Watching Westworld , you can sense a grand mythology unfolding before your eyes. The show’s biggest strength is its world-building, an aspect of screenwriting that many television series have botched before. Often shows will rush viewers into plot, forgetting to instill a sense of place and of history, that you’re watching something that doesn’t just exist in a vacuum but rather is part of some larger ecosystem. Not since Lost can I remember a TV show so committed to immersing its audience into the physical space it inhabits. (Indeed, Westworld can also be viewed as a meta commentary on the art of screenwriting itself: brainstorming narratives, building characters, all for the amusement of other people.) Westworld is especially impressive because it builds two worlds at once: the Western theme park and the futuristic workplace. The Western half of Westworld might be the more purely entertaining of the two, with its shootouts and heists and chases through sublime desert vistas. Behind the scenes, the theme park’s workers show how the robot sausage is made. And as a dystopian office drama, the show does something truly original. — Adam Epstein at QUARTZ, October 1, 2016 |
"… committed to immersing its audience
into the physical space it inhabits…."
See also, in this journal, the Mimsy Cube —
|
"Mimsy Were the Borogoves," "… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."
|
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Friday, September 30, 2016
Desmic Midrash
The author of the review in the previous post, Dara Horn, supplies
below a midrash on “desmic,” a term derived from the Greek desmé
( δέσμη: bundle, sheaf , or, in the mathematical sense, pencil —
French faisceau ), which is related to the term desmos , bond …
(The term “desmic,” as noted earlier, is relevant to the structure of
Heidegger’s Sternwürfel .)
The Horn midrash —
(The “medieval philosopher” here is not the remembered pre-Christian
Ben Sirah (Ecclesiasticus ) but the philosopher being read — Maimonides:
Guide for the Perplexed , 3:51.)
Here of course “that bond” may be interpreted as corresponding to the
Greek desmos above, thus also to the desmic structure of the
stellated octahedron, a sort of three-dimensional Star of David.
See “desmic” in this journal.
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Thursday, September 29, 2016
Articulation
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 —
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Wednesday, September 28, 2016
Star Wars
See also in this journal “desmic,” a term related
to the structure of Heidegger’s Sternwürfel .
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Wednesday, March 30, 2016
Hungarian Algorithm
“Of all the Hungarian friends I’ve ever had…
I can’t remember one who didn’t want me to think of him…
as a king of con men.”
” ‘The omelet, you know that, don’t you? Sure. It’s a classic.
An omelet, it’s in our Hungarian cookbook.
“To make an omelet,” it says… “first, steal an egg.” ‘ ”
— Orson Welles, in his last completed film.
See also Lovasz in this journal.
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Thursday, December 17, 2015
Hint of Reality
From an article* in Proceedings of Bridges 2014 —
|
As artists, we are particularly interested in the symmetries of real world physical objects. Three natural questions arise: 1. Which groups can be represented as the group of symmetries of some real-world physical object? 2. Which groups have actually been represented as the group of symmetries of some real-world physical object? 3. Are there any glaring gaps – small, beautiful groups that should have a physical representation in a symmetric object but up until now have not? |
The article was cited by Evelyn Lamb in her Scientific American
weblog on May 19, 2014.
The above three questions from the article are relevant to a more
recent (Oct. 24, 2015) remark by Lamb:
"… finite projective planes [in particular, the 7-point Fano plane,
about which Lamb is writing] seem like a triumph of purely
axiomatic thinking over any hint of reality…."
For related hints of reality, see Eightfold Cube in this journal.
* "The Quaternion Group as a Symmetry Group," by Vi Hart and Henry Segerman
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Friday, August 7, 2015
Parts
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 Scottish-born
publishing executive, editor and award-winning 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
Comments Off on Parts
Thursday, May 7, 2015
Paradigm for Pedagogues
Illustrations from a post of Feb. 17, 2011:
Plato’s paradigm in the Meno —
Changed paradigm in the diamond theorem (2×2 case) —
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Ultron: By the Book
If The New York Times interviewed Ultron for its
Sunday Book Review "By the Book" column —
What books are currently on your night stand?
Steve Fuller's Thomas Kuhn: A Philosophical History for Our Times
Gerald Holton's Thematic Origins of Scientific Thought
John Gray's The Soul of the Marionette
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Wednesday, May 6, 2015
Soul
Nonsense…
See Gary Zukav, Harvard ’64, in this journal.
and damned nonsense —
“Every institution has a soul.”
— Gerald Holton in Harvard Gazette today
Commentary —
“The Ferris wheel came into view again….”
— Malcom Lowry, Under the Volcano
See also Holton in a Jan. 1977 interview:
“If people have souls, and I think a few have, it shows….”
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Wednesday, April 1, 2015
Würfel-Märchen
Continued from yesterday, the date of death for German
billionaire philanthropist Klaus Tschira —
For Tschira in this journal, see Stiftung .
For some Würfel illustrations, see this morning's post
Manifest O. A related webpage —
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Manifest O
The title was suggested by
http://benmarcus.com/smallwork/manifesto/.
The "O" of the title stands for the octahedral group.
See the following, from http://finitegeometry.org/sc/map.html —
|
|
An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. |
| 83-10-01 | Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. |
| 83-10-16 | Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. |
| 84-09-15 | Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. |
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Wednesday, September 17, 2014
Raiders of the Lost Articulation
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:
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Friday, August 29, 2014
Raum
A possible answer to the 1923 question of Walter Gropius, “Was ist Raum?“—
See also yesterday’s Source of the Finite and the image search
on the Gropius question in last night’s post.
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Thursday, August 28, 2014
Brutalism Revisited
Yesterday's 11 AM post was a requiem for a brutalist architect.
Today's LA Times has a related obituary:
"Architectural historian Alan Hess, who has written several books on
Mid-Century Modern design, said Meyer didn't have a signature style,
'which is one reason he is not as well-known as some other architects
of the period. But whatever style he was working in, he brought a real
sense of quality to his buildings.'
A notable example is another bank building, at South Beverly Drive
and Pico Boulevard, with massive concrete columns, a hallmark of
the New Brutalism style. 'This is a really good example of it,' Hess said."
— David Colker, 5:43 PM LA time, Aug. 28, 2014
A related search, suggested by this morning's post Source of the Finite:
(Click to enlarge.)
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Source of the Finite
"Die Unendlichkeit ist die uranfängliche Tatsache: es wäre nur
zu erklären, woher das Endliche stamme…."
— Friedrich Nietzsche, Das Philosophenbuch/Le livre du philosophe
(Paris: Aubier-Flammarion, 1969), fragment 120, p. 118
Cited as above, and translated as "Infinity is the original fact;
what has to be explained is the source of the finite…." in
The Production of Space , by Henri Lefebvre. (Oxford: Blackwell,
1991 (1974)), p. 181.
This quotation was suggested by the Bauhaus-related phrase
"the laws of cubical space" (see yesterday's Schau der Gestalt )
and by the laws of cubical space discussed in the webpage
Cube Space, 1984-2003.
For a less rigorous approach to space at the Harvard Graduate
School of Design, see earlier references to Lefebvre in this journal.
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Wednesday, August 27, 2014
Altar
"To every man upon this earth,
Death cometh soon or late.
And how can man die better
Than facing fearful odds,
For the ashes of his fathers,
and the temples of his gods…?"
— Macaulay, quoted in the April 2013 film "Oblivion"
"Leave a space." — Tom Stoppard, "Jumpers"
Related material: The August 16, 2014, sudden death in Scotland
of an architect of the above Cardross seminary, and a Log24 post,
Plato's Logos, from the date of the above photo: June 26, 2010.
See also…
Here “eidolon” should instead be “eidos .”
An example of eidos — Plato's diamond (from the Meno ) —
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Schau der Gestalt
(Continued from Aug. 19, 2014)
“Christian contemplation is the opposite
of distanced consideration of an image:
as Paul says, it is the metamorphosis of
the beholder into the image he beholds
(2 Cor 3.18), the ‘realisation’ of what the
image expresses (Newman). This is
possible only by giving up one’s own
standards and being assimilated to the
dimensions of the image.”
— Hans Urs von Balthasar,
The Glory of the Lord:
A Theological Aesthetics,
Vol. I: Seeing the Form
[ Schau der Gestalt ],
Ignatius Press, 1982, p. 485
A Bauhaus approach to Schau der Gestalt :
I prefer the I Ching ‘s approach to the laws of cubical space.
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Saturday, July 12, 2014
Sequel
A sequel to the 1974 film
Thunderbolt and Lightfoot :
Contingent and Fluky
Some variations on a thunderbolt theme:
These variations also exemplify the larger
Verbum theme:
A search today for Verbum in this journal yielded
a Georgetown University Chomskyite, Professor
David W. Lightfoot.
"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."
Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.
See Tecumseh Fitch in this journal.
For other material related to the large Verbum cube,
see posts for the 18th birthday of Harry Potter.
That birthday was also the upload date for the following:
See esp. the comments section.
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Wednesday, June 4, 2014
Monkey Business
The title refers to a Scientific American weblog item
discussed here on May 31, 2014:
Some closely related material appeared here on
Dec. 30, 2011:
A version of the above quaternion actions appeared
at math.stackexchange.com on March 12, 2013:
"Is there a geometric realization of Quaternion group?" —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note "GL(2,3) actions on a cube" of April 5, 1985).
Comments Off on Monkey Business
Saturday, May 31, 2014
Quaternion Group Models:
The ninefold square, the eightfold cube, and monkeys.
For posts on the models above, see quaternion
in this journal. For the monkeys, see
"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American weblog, May 19, 2014:
The Scientific American item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):
See also Finite Geometry and Physical Space.
Comments Off on Quaternion Group Models:
Sunday, April 27, 2014
Sunday School
Galois and Abel vs. Rubik
“Abel was done to death by poverty, Galois by stupidity.
In all the history of science there is no completer example
of the triumph of crass stupidity….”
— Eric Temple Bell, Men of Mathematics
Gray Space (Continued)
… For The Church of Plan 9.
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Friday, April 4, 2014
Eight Gate
From a Huffington Post discussion of aesthetics:
“The image below on the left… is… overly simplistic, and lacks reality:
It’s all a matter of perspective: the problem here is that opposite sides
of the cube, which are parallel in real life, actually look parallel in the
left image! The image on the right is better….”
A related discussion: Eight is a Gate.
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Thursday, March 27, 2014
Diamond Space
Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.
The reason for the name:
Click to enlarge.
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Thursday, December 26, 2013
How It Works
“Design is how it works.” — Steve Jobs
“By far the most important structure in design theory
is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer (Ch. 14 (pp. 693-746),
Section 16 (p. 716) of Handbook of Combinatorics, Vol. I ,
MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel,
and László Lovász)
For some background on that Steiner system, see the footnote to
yesterday’s Christmas post.
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Wednesday, December 25, 2013
Rotating the Facets
“… her mind rotated the facts….”
Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:
“I’ve heard of affairs that are strictly Platonic”
* Footnote added on Dec. 26, 2013 —
See Arnold Emch, “Triple and Multiple Systems, Their Geometric
Configurations and Groups,” Trans. Amer. Math. Soc. 31 (1929),
No. 1, 25–42.
On page 42, Emch describes the above method of rotating a
hypercube’s 8 facets (i.e., three-dimensional cubes) to count
rotational symmetries —
See also Diamond Theory in 1937.
Also on p. 42, Emch mentions work of Carmichael on a
Steiner system with the Mathieu group M11 as automorphism
group, and poses the problem of finding such systems and
groups that are larger. This may have inspired the 1931
discovery by Carmichael of the Steiner system S(5, 8, 24),
which has as automorphisms the Mathieu group M24 .
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Monday, December 9, 2013
Heaven Descending
An I Ching study quoted in Waiting for Ogdoad (St. Andrew’s Day, 2013)—
(Click for clearer image.)
The author of the above I Ching study calls his lattice “Arising Heaven.”
The following lattice might, therefore, be called “Heaven Descending.”
Click for the source, mentioned in Anatomy of a Cube (Sept. 18, 2011).
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Saturday, November 30, 2013
Waiting for Ogdoad
Continued from October 30 (Devil’s Night), 2013.
“In a sense, we would see that change
arises from the structure of the object.”
— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013
This suggests a review of mathematics and the
“Classic of Change ,” the I Ching .
The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching as the corners of a cube.
See also…
(Click for clearer image.)
The Cullinane image above illustrates the seven points of
the Fano plane as seven of the eight I Ching trigrams and as
seven natural ways of slicing the cube.
For a different approach to the mathematics of cube slices,
related to Gauss’s composition law for binary quadratic forms,
see the Bhargava cube in a post of April 9, 2012.
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Thursday, October 10, 2013
Nobel Jam
From this date five years ago in The Guardian—
Alice Munro: An Appreciation by Margaret Atwood—
"The central Christian tenet is that
two disparate and mutually exclusive elements—
divinity and humanity— got jammed together
in Christ, neither annihilating the other.
The result was not a demi-god, or a God
in disguise: God became totally a human being
while remaining at the same time totally divine.
To believe either that Christ was only a man or
that he was simply God was declared heretical
by the early Christian church. Christianity thus
depends on a denial of either/or classifying logic
and an acceptance of both-at-once mystery.
Logic says that A cannot be both itself and non-A
at the same time; Christianity says it can. The
formulation 'A but also non-A' is indispensable to it."
Related literary material— "Excluded Middle" and "Couple of Tots."
See also "The Divided Cube" and "Mimsy Were the Borogoves."
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Thursday, June 13, 2013
Gate
“Eight is a Gate.” — Mnemonic rhyme
Today’s previous post, Window, showed a version
of the Chinese character for “field”—
This suggests a related image—
The related image in turn suggests…
Unlike linear perspective, axonometry has no vanishing point,
and hence it does not assume a fixed position by the viewer.
This makes axonometry ‘scrollable’. Art historians often speak of
the ‘moving’ or ‘shifting’ perspective in Chinese paintings.
Axonometry was introduced to Europe in the 17th century by
Jesuits returning from China.
As was the I Ching. A related structure:
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Saturday, May 11, 2013
Core
Promotional description of a new book:
“Like Gödel, Escher, Bach before it, Surfaces and Essences will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking.”
“Like Gödel, Escher, Bach before it….”
Or like Metamagical Themas .
Rubik core:
Non- Rubik cores:
| Of the odd nxnxn cube:
|
Of the even nxnxn cube:
|
Related material: The Eightfold Cube and…
“A core component in the construction
is a 3-dimensional vector space V over F2 .”
— Page 29 of “A twist in the M24 moonshine story,”
by Anne Taormina and Katrin Wendland.
(Submitted to the arXiv on 13 Mar 2013.)
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Tuesday, March 19, 2013
Mathematics and Narrative (continued)
Angels & Demons meet Hudson Hawk
Dan Brown's four-elements diamond in Angels & Demons :
The Leonardo Crystal from Hudson Hawk :
Mathematics may be used to relate (very loosely)
Dan Brown's fanciful diamond figure to the fanciful
Leonardo Crystal from Hudson Hawk …
-
Compare Brown's fictional Illuminati Diamond to the
nonfictional figures in The Diamond Theorem and
Theme and Variations. -
Compare the fictional Leonardo Crystal to Hudson's
nonfictional desmic system of tetrahedra (above), and
see, in Rosenhain and Göpel Tetrads in PG(3,2), how
the diamond theorem is related to Hudson's work.
For the tetrads ' relationship to tetrahedra , see
Hudson's own book.
"Giving himself a head rub, Hawk bears down on
the three oddly malleable objects. He TANGLES
and BENDS and with a loud SNAP, puts them together,
forming the Crystal from the opening scene."
— A screenplay of Hudson Hawk
Happy birthday to Bruce Willis.
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Thursday, December 27, 2012
Object Lesson
Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 1910-1925" suggests a renewed look at
abstraction and a fundamental building block: the cube.
From a recent Harvard University Press philosophical treatise on symmetry—
The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.
For greater depth see Cassirer's 1910 passage on Vorstellung :
This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.
For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.
"The sacramental nature of the space becomes clear…." — Brian O'Doherty
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Monday, December 24, 2012
Eternal Recreation
Memories, Dreams, Reflections
by C. G. Jung
Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989
From pages 195-196:
“Only gradually did I discover what the mandala really is:
‘Formation, Transformation, Eternal Mind’s eternal recreation.’*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions.”
* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:
… Gestaltung, Umgestaltung,
Des ewigen Sinnes ewige Unterhaltung….
Jung’s “Formation, Transformation” quote is from the realm of
the Mothers (Faust , Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.
See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:
“In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust’s next encounter with
the feminine, but it’s obviously of a very different kind than his
relationship with Gretchen.”
The phrase “Gestaltung, Umgestaltung ” suggests a more mathematical
approach to the Unterhaltung . Hence…
Part I: Mothers
“The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time.”
— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity ,
Johns Hopkins Press, 1969, page 37
Part II: Matrices
Part III: Spaces and Hypercubes
Click image for some background.
Part IV: Forms
Forms from the I Ching :
Click image for some background.
Forms from Diamond Theory :
Click image for some background.
Comments Off on Eternal Recreation
Wednesday, November 14, 2012
Group Actions
The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—
(Only the top part of the ad is shown; for further details
see an ICERM page.)
(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)
The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—
The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—
© 2005 The Institute for Figuring
Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)
The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.
These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.
See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.
Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—
.
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Tuesday, June 26, 2012
Bright Black
“‘In the dictionary next to [the] word “bright,” you should see Paula’s picture,’ he said. ‘She was super smart, with a sparkling wit. … She had a beautiful sense of style and color.'”
— Elinor J. Brecher in The Miami Herald on June 8, quoting Palm Beach Post writer John Lantigua on the late art historian Paula Hays Harper
This journal on the date of her death—
For some simpleminded commentary, see László Lovász on the cube space.
Some less simpleminded commentary—
“Was ist Raum, wie können wir ihn
erfassen und gestalten?”
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Sunday, June 17, 2012
Congruent Group Actions
A Google search today yielded no results
for the phrase "congruent group actions."
Places where this phrase might prove useful include—
- Actions of the quaternion group in finite geometry
- Affine group actions in finite geometry
- The "symmetric generation" technique of R. T. Curtis
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Saturday, June 16, 2012
Chiral Problem
In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—
Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post—
"ANDOVER–HARVARD THEOLOGICAL LIBRARY"
Detail from Knowles's obituary in this morning's New York Times—
William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.
Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
From Pilate Goes to Kindergarten—
The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube—
Problem: Is there a different coordinatization
that yields greater symmetry in the pictures of
quaternion group actions?
A paper written in a somewhat similar spirit—
"Chiral Tetrahedrons as Unitary Quaternions"—
ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….
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Sunday, June 3, 2012
Child’s Play
“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”
– Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art , Cornell U. Press, 1975
Related material—
Saturday, May 19, 2012
G8
"The group of 8" is a phrase from politics, not mathematics.
Of the five groups of order 8 (see today's noon post),
the one pictured* in the center, Z2 × Z2 × Z2 , is of particular
interest. See The Eightfold Cube. For a connection of this
group of 8 to the last of the five pictured at noon, the
quaternion group, see Finite Geometry and Physical Space.
* The picture is of the group's cycle graph.
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Monday, May 7, 2012
More on Triality
John Baez wrote in 1996 ("Week 91") that
"I've never quite seen anyone come right out
and admit that triality arises from the
permutations of the unit vectors i, j, and k
in 3d Euclidean space."
Baez seems to come close to doing this with a
somewhat different i , j , and k — Hurwitz
quaternions— in his 2005 book review
quoted here yesterday.
See also the Log24 post of Jan. 4 on quaternions,
and the following figures. The actions on cubes
in the lower figure may be viewed as illustrating
(rather indirectly) the relationship of the quaternion
group's 24 automorphisms to the 24 rotational
symmetries of the cube.
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