Log24

Monday, February 24, 2020

For “Time Cube” Fans

Filed under: General — Tags: — m759 @ 3:53 AM

See also Time Cube elsewhere in this  journal.

Friday, January 17, 2020

Design Theory

Filed under: General — Tags: , — m759 @ 12:57 PM

On a recently deceased professor emeritus of architecture
at Princeton —

"… Maxwell  'established the school as a principal
center of design research, history and theory.' ”

"This is not the Maxwell you're looking for."

Monday, December 16, 2019

Design Notes Dec. 11

Filed under: General — Tags: — m759 @ 3:01 AM

From The New York Times on Dec. 11 —

See also some other posts in this  journal now tagged "Design Notes Dec. 11."

Tuesday, May 21, 2019

Inside the White Cube

Filed under: General — Tags: , , — m759 @ 11:01 AM

(Continued)

Monday, May 13, 2019

Star Cube

Filed under: General — Tags: — m759 @ 1:00 PM

"Before time began . . . ." — Optimus Prime

Sunday, April 8, 2018

Design

Filed under: General,Geometry — m759 @ 11:00 PM

From a Log24 post of Feb. 5, 2009 —

Design Cube 2x2x2 for demonstrating Galois geometry

An online logo today —

See also Harry Potter and the Lightning Bolt.

 

Thursday, March 22, 2018

The Diamond Cube

Filed under: General,Geometry — Tags: — m759 @ 11:32 AM

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.

Sunday, June 4, 2017

In Memory of the Time Cube Page*

Filed under: General,Geometry — Tags: — m759 @ 10:00 PM

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).

Tuesday, April 5, 2016

“Puzzle Cube of a Novel”

Filed under: General,Geometry — Tags: , — m759 @ 2:00 AM

"To know the mind of the creator"

Or that of Orson Welles

Related material — Cube Coloring.

Thursday, July 23, 2015

Design Cube

Filed under: General — m759 @ 8:24 PM

Broken Symmetries  in  Diamond Space —

Sunday, December 28, 2014

Cube of Ultron

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

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)

Friday, December 28, 2012

Cube Koan

Filed under: General,Geometry — Tags: , , , — m759 @ 4:56 AM
 

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…"

Sunday, August 5, 2012

Cube Partitions

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM

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)—

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

Click for further details.

Sunday, February 5, 2012

Cuber

Filed under: General,Geometry — Tags: — m759 @ 5:15 PM

(Continued from January 11, 2012)

Wednesday, January 11, 2012

Cuber

Filed under: General,Geometry — Tags: , , , — m759 @ 12:00 PM

“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.

Friday, December 30, 2011

Quaternions on a Cube

Filed under: General,Geometry — Tags: , , — m759 @ 5:48 AM

The following picture provides a new visual approach to
the order-8 quaternion  group's automorphisms.

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.

See also…

Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.

* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

Saturday, August 27, 2011

Cosmic Cube*

Filed under: General,Geometry — Tags: , , , — m759 @ 12:25 PM

IMAGE- Anthony Hopkins exorcises a Rubik cube

Prequel (Click to enlarge)

IMAGE- Galois vs. Rubik: Posters for Abel Prize, Oslo, 2008

Background —

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument by a cuber for the image's use

See also Rubik in this journal.

* For the title, see Groups Acting.

Friday, June 24, 2011

The Cube

Filed under: General — Tags: , — m759 @ 12:00 PM

IMAGE- 'The Stars My Destination' (with cover slightly changed)

Click the above image for some background.

Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.

Monday, August 6, 2018

The Girl with Kaleidoscope Eyes

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM

http://www.log24.com/log/pix18/180806-Lexicon-image-search.jpg

“All right, Jessshica. It’s time to open the boxsssschhh.”

“Gahh,” she said. She began to walk toward the box, but her heart failed her and she retreated back to the chair. “Fuck. Fuck.” Something mechanical purred. The seam she had found cracked open and the top of the box began to rise. She squeezed shut her eyes and groped her way into a corner, curling up against the concrete and plugging her ears with her fingers. That song she’d heard the busker playing on the train platform with Eliot, “Lucy in the Sky with Diamonds”; she used to sing that. Back in San Francisco, before she learned card tricks. It was how she’d met Benny: He played guitar. Lucy was the best earner, Benny said, so that was mainly what she sang. She must have sung it five times an hour, day after day. At first she liked it but then it was like an infection, and there was nothing she could do and nowhere she could go without it running across her brain or humming on her lips, and God knew she tried; she was smashing herself with sex and drugs but the song began to find its way even there. One day, Benny played the opening chord and she just couldn’t do it. She could not sing that fucking song. Not again. She broke down, because she was only fifteen, and Benny took her behind the mall and told her it would be okay. But she had to sing. It was the biggest earner. She kind of lost it and then so did Benny and that was the first time he hit her. She ran away for a while. But she came back to him, because she had nothing else, and it seemed okay. It seemed like they had a truce: She would not complain about her bruised face and he would not ask her to sing “Lucy.” She had been all right with this. She had thought that was a pretty good deal.

Now there was something coming out of a box, and she reached for the most virulent meme she knew. “Lucy in the sky!” she sang. “With diamonds!”

•   •   •

Barry, Max. Lexicon: A Novel  (pp. 247-248).
Penguin Publishing Group. Kindle Edition.

Related material from Log24 on All Hallows' Eve 2013

"Just another shake of the kaleidoscope" —

Related material:

Kaleidoscope Puzzle,  
Design Cube 2x2x2, and 
Through the Looking Glass: A Sort of Eternity.

Tuesday, March 27, 2018

Compare and Contrast

Filed under: General,Geometry — Tags: — m759 @ 4:28 PM

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Related material on automorphism groups —

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture 
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

Thursday, December 1, 2016

What’s in a Name

Filed under: General,Geometry — m759 @ 11:23 PM

Design Cube 2x2x2 for demonstrating Galois geometry

   Backstory Aug. 21, 2016, and Quora.com.

Sunday, August 21, 2016

Imperium Emporium

Filed under: General,Geometry — m759 @ 11:30 PM

Design Cube 2x2x2 for demonstrating Galois geometry

Harry Potter with lightning-bolt scar

Harry Potter, star of the new film
"Imperium," with lightning-bolt
scar on his forehead

Wednesday, April 1, 2015

Würfel-Märchen

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 PM

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 —

Saturday, July 12, 2014

Sequel

Filed under: General,Geometry — Tags: , — m759 @ 9:00 AM

A sequel to the 1974 film
Thunderbolt and Lightfoot :

Contingent and Fluky

Some variations on a thunderbolt  theme:

Design Cube 2x2x2 for demonstrating Galois geometry

These variations also exemplify the larger
Verbum  theme:

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

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.

Thursday, October 31, 2013

Interpenetrative Ogdoad

Filed under: General,Geometry — m759 @ 2:21 PM

The title is from an essay by James C. Nohrnberg

(Click to enlarge.)

"Just another shake of the kaleidoscope" —

Related material:

Kaleidoscope Puzzle,  
Design Cube 2x2x2, and 
Through the Looking Glass: A Sort of Eternity.

Saturday, November 27, 2010

Simplex Sigillum Veri

Filed under: General,Geometry — m759 @ 7:20 AM

An Adamantine View of "The [Philosophers'] Stone"

The New York Times  column "The Stone" on Sunday, Nov. 21 had this—

"Wittgenstein was formally presenting his Tractatus Logico-Philosophicus , an already well-known work he had written in 1921, as his doctoral thesis. Russell and Moore were respectfully suggesting that they didn’t quite understand proposition 5.4541 when they were abruptly cut off by the irritable Wittgenstein. 'I don’t expect you to understand!' (I am relying on local legend here….)"

Proposition 5.4541*—

http://www.log24.com/log/pix10B/101127-WittgensteinSimplex.jpg

Related material, found during a further search—

A commentary on "simplex sigillum veri" leads to the phrase "adamantine crystalline structure of logic"—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

For related metaphors, see The Diamond Cube, Design Cube 2x2x2, and A Simple Reflection Group of Order 168.

Here Łukasiewicz's phrase "the hardest of materials" apparently suggested the commentators' adjective "adamantine." The word "diamond" in the links above refers of course not to a material, but to a geometric form, the equiangular rhombus. For a connection of this sort of geometry with logic, see The Diamond Theorem and The Geometry of Logic.

For more about God, a Stone, logic, and cubes, see Tale  (Nov. 23).

* 5.4541 in the German original—

  Die Lösungen der logischen Probleme müssen einfach sein,
  denn sie setzen den Standard der Einfachheit.
  Die Menschen haben immer geahnt, dass es
  ein Gebiet von Fragen geben müsse, deren Antworten—
  a priori—symmetrisch, und zu einem abgeschlossenen,
  regelmäßigen Gebilde vereint liegen.
  Ein Gebiet, in dem der Satz gilt: simplex sigillum veri.

  Here "einfach" means "simple," not "neat," and "Gebiet" means
  "area, region, field, realm," not (except metaphorically) "sphere."

Monday, August 23, 2010

Diamond Puzzle Downloads

Filed under: General,Geometry — m759 @ 2:00 AM

The Diamond 16 Puzzle and the Kaleidoscope Puzzle can now be downloaded in the normal way from a browser, with the save-as web-page-complete option, and have their JavaScript still work— if  the files are saved with the name indicated in the instructions on the puzzles' web pages. (There was a problem with file names in the JavaScript that has been fixed.)

The JavaScript pages Design Cube 2x2x2 and Design Cube 4x4x4 have not been changed. To download these, it is necessary to…

  1. Do a web-page-complete save to get an image-files folder, then
  2. do an HTML-only save to the image-files folder  to put an unaltered copy of the the web page there, then
  3. rename the image-files folder to unlink it from the altered HTML page downloaded in step 1, then
  4. delete the altered HTML page downloaded in step 1.

The result is a folder containing both image files and the HTML page, just as it is on the Web.

Thursday, February 5, 2009

Thursday February 5, 2009

Filed under: General,Geometry — Tags: , , , — m759 @ 1:00 PM

Through the
Looking Glass:

A Sort of Eternity

From the new president’s inaugural address:

“… in the words of Scripture, the time has come to set aside childish things.”

The words of Scripture:

9 For we know in part, and we prophesy in part.
10 But when that which is perfect is come, then that which is in part shall be done away.
11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known. 

First Corinthians 13

“through a glass”

[di’ esoptrou].
By means of
a mirror [esoptron]
.

Childish things:

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
 

Not-so-childish:

Three planes through
the center of a cube
that split it into
eight subcubes:
Cube subdivided into 8 subcubes by planes through the center
Through a glass, darkly:

A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180-degree rotation:

Design Cube 2x2x2 for demonstrating Galois geometry

(Click on image
for further details.)

But then face to face:

A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.

Galois age fifteen, drawn by a classmate.

Galois age fifteen,
drawn by a classmate.

These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.

For some generalizations,
see Galois Geometry.

Related material:

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
live Even and Odd…
And the song of love’s recision
is the music of the spheres.”

— 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
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history.”

— 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:

Inside the 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.”

http://www.log24.com/log/pix09/090205-cube2x2x2.gif

“Space: what you
damn well have to see.”

— James Joyce, Ulysses  

Saturday, May 23, 2020

Eightfold Geometry: A Surface Code “Unit Cell”

Filed under: General — Tags: — m759 @ 1:50 AM

A unit cell in 'a lattice geometry for a surface code'

The resemblance to the eightfold cube  is, of course,
completely coincidental.

Some background from the literature —

Friday, May 22, 2020

Surface Code News

Filed under: General — Tags: — m759 @ 5:50 PM

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.”

Friday, February 21, 2020

To and Fro, Back and …

Filed under: General — Tags: , — m759 @ 11:44 PM

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.

Friday, January 17, 2020

September Morn

Filed under: General — Tags: — m759 @ 10:17 PM

Epigraph from Ch. 4 of Design Theory , Vol. I:

"Es is eine alte Geschichte,
 doch bleibt sie immer neu 
"
 —Heine (Lyrisches Intermezzo  XXXIX)

This epigraph was quoted here earlier on
the morning of September 1, 2011.

Wednesday, December 11, 2019

Miracle Octad Generator Structure

Filed under: General — Tags: — m759 @ 11:44 PM

Miracle Octad Generator — Analysis of Structure

(Adapted from Eightfold Geometry, a note of April 28, 2010.
  See also the recent post Geometry of 6 and 8.)

Canvassing

Filed under: General — Tags: — m759 @ 11:26 AM

The Seagram Case

Filed under: General — Tags: — m759 @ 10:48 AM

From a search in this journal for Seagram

A Seagram 'colorful tale'

Klein Quadric

Filed under: General — Tags: , — m759 @ 1:06 AM

The architecture of the recent post 
Geometry of 6 and 8 is in part
a reference to the Klein quadric.

Sunday, September 29, 2019

Stage Direction: “Comments Off.”

Filed under: General — Tags: , — m759 @ 11:29 AM

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”:

Wednesday, July 10, 2019

Artifice* of Eternity …

Filed under: General — Tags: , — m759 @ 10:54 AM

… 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 . . .

Tuesday, July 9, 2019

Schoolgirl Space: 1984 Revisited

Filed under: General — Tags: , — m759 @ 9:24 PM

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From John Baez (2018) —

See also this morning's post Perception of Space 
and yesterday's Exploring Schoolgirl Space.

Perception of Space

Filed under: General — Tags: , , — m759 @ 10:45 AM

(Continued)

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 —

Dreamtimes

Filed under: General — Tags: , — m759 @ 4:27 AM

“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 . . .

.

Monday, July 8, 2019

Exploring Schoolgirl Space

See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."

Sunday, July 7, 2019

Schoolgirl Problem

Filed under: General — Tags: , — m759 @ 11:18 PM

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.

Saturday, June 8, 2019

Art Object, continued and continued

Filed under: General — Tags: , — m759 @ 1:21 PM

Notes on a remark by Chuanming Zong

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 .

Monday, May 13, 2019

Doris Day at the Hudson Rock

Filed under: General — Tags: — m759 @ 12:00 PM

" '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 —

Friday, March 1, 2019

Solomon and the Image

Filed under: General — Tags: , — m759 @ 2:27 AM

"Maybe an image is too strong
Or maybe is not strong enough."

— "Solomon and the Witch,"
      by William Butler Yeats

Saturday, August 25, 2018

“Waugh, Orwell. Orwell, Waugh.”

Filed under: General,Geometry — Tags: , — m759 @ 4:00 PM

Suggested by a review of Curl on Modernism —

http://www.log24.com/log/pix18/180825-Ballard-on-Modernism.gif

Related material —

Waugh + Orwell in this journal and

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Thursday, March 29, 2018

“Before Creation Itself . . .”

Filed under: General,Geometry — Tags: , , — m759 @ 10:13 AM

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 . . .

Saturday, March 24, 2018

Slight?

Filed under: General,Geometry — Tags: — m759 @ 12:30 PM

Sure, Whatever.

Filed under: General,Geometry — Tags: , — m759 @ 11:13 AM

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.

Friday, March 23, 2018

Reciprocity

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM

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.

Piece Prize

Filed under: General,Geometry — Tags: — m759 @ 6:15 PM

The Waymark Prize from 'A Piece of Justice' (1995) by Jill Paton Walsh

The Waymark Prize Mystery - 'A Piece of Justice' (1995) p. 138

From the Personal to the Platonic

Filed under: General,Geometry — Tags: , — m759 @ 11:01 AM

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 —

The Diamond Cube.

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).

Thursday, March 22, 2018

In Memoriam

Filed under: General,Geometry — Tags: — m759 @ 10:10 PM

Also on March 18, 2015 . . .

Wednesday, March 7, 2018

Unite the Seven.

Filed under: General,Geometry — Tags: , — m759 @ 10:31 AM


Related material —

The seven points of the Fano plane within 

The Eightfold Cube.
 

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching


"Before time began . . . ."

  — Optimus Prime

Monday, January 22, 2018

Hollywood Moment

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Matt B. Roscoe and Joe Zephyrs, both of Missoula, Montana, authors of article on quilt block symmetries

A death on the date of the above symmetry chat,
Wednesday, August 17, 2016

'Love Story' director dies

An Hispanic Hollywood moment:

Ojo de Dios —

Click for related material.

For further Hispanic entertainment,
see Ben Affleck sing 
"Aquellos Ojos Verdes "
in "Hollywoodland."

Wednesday, September 13, 2017

Summer of 1984

Filed under: General,Geometry — Tags: , , — m759 @ 9:11 AM

The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —

Group actions on partitions —

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

Tuesday, September 12, 2017

Think Different

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM

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."

James Poniewozik 

Review —

Thursday, September 1, 2011

How It Works

Filed under: Uncategorized — Tags:  — m759 @ 11:00 AM 

"Design is how it works." — Steven Jobs (See Symmetry and Design.)

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
 — "Block Designs," by Andries E. Brouwer

. . . .

See also 1984 Bricks in this journal.

Chin Music

Filed under: General,Geometry — Tags: — m759 @ 9:45 PM

Related image suggested by "A Line for Frank" (Sept. 30, 2013) —

Tuesday, June 20, 2017

Epic

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 PM

Continuing the previous post's theme  

Group actions on partitions

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material — Posts now tagged Device Narratives.

Wednesday, April 12, 2017

Contracting the Spielraum

Filed under: General,Geometry — Tags: , , , — m759 @ 10:00 AM

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).

Tuesday, March 28, 2017

Bit by Bit

Filed under: General,Geometry — Tags: , , — m759 @ 11:45 AM

From Log24, "Cube Bricks 1984" —

An Approach to Symmetric Generation of the Simple Group of Order 168

Also on March 9, 2017 —

For those who prefer graphic  art —

Broken Symmetries  in  Diamond Space  

Tuesday, February 21, 2017

Arrow Economics

Filed under: General — m759 @ 11:29 PM

Broken Symmetries  in  Diamond Space 

Saturday, January 7, 2017

Conceptualist Minimalism

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

"Clearly, there is a spirit of openhandedness in post-conceptual art
uses of the term 'Conceptualism.' We can now endow it with a
capital letter because it has grown in scale from its initial designation
of an avant-garde grouping, or various groups in various places, and
has evolved in two further phases. It became something like a movement,
on par with and evolving at the same time as Minimalism. Thus the sense
it has in a book such as Tony Godfrey’s Conceptual Art.  Beyond that,
it has in recent years spread to become a tendency, a resonance within
art practice that is nearly ubiquitous." — Terry Smith, 2011

See also the eightfold cube

The Eightfold Cube

 

Friday, September 30, 2016

Desmic Midrash

Filed under: General — Tags: — m759 @ 9:19 AM

The author of the review in the previous post, Dara Horn, supplies
below a midrash on "desmic," a term derived from the Greek desme 
δεσμή , bundle, sheaf, or, in the mathematical sense, pencil —
French faisceau ), which is apparently 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.

Thursday, September 29, 2016

Articulation

Filed under: General — Tags: — m759 @ 10:30 PM

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 —

Wednesday, September 28, 2016

Star Wars

Filed under: General — Tags: , — m759 @ 11:00 PM

See also in this journal “desmic,” a term related
to the structure of Heidegger’s Sternwürfel .

Scholia

Filed under: General — Tags: — m759 @ 2:48 PM

Heidegger- 'The world's darkening never reaches to the light of being'

Scholia —

D. H. Lawrence quote from 'Kangaroo'

South Australia goes dark

Tuesday, June 14, 2016

Model Kit

Filed under: General,Geometry — m759 @ 12:14 PM

The title refers to the previous post, which quotes a 
remark by a poetry critic in the current New Yorker .

Scholia —

From the post Structure and Sense of June 6, 2016 —

Structure

Sense

A set of 7 partitions of the 2x2x2 cube that is invariant under PSL(2, 7) acting on the 'knight' coordinatization

From the post Design Cube of July 23, 2015 —

Broken Symmetries  in  Diamond Space 

Thursday, December 17, 2015

Hint of Reality

Filed under: General,Geometry — Tags: , , — m759 @ 12:45 PM

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

Friday, August 7, 2015

Parts

Filed under: General,Geometry — Tags: — m759 @ 2:19 AM

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                       

Thursday, May 7, 2015

Paradigm for Pedagogues

Filed under: General,Geometry — Tags: — m759 @ 7:14 PM

Illustrations from a post of Feb. 17, 2011:

Plato’s paradigm in the Meno —

http://www.log24.com/log/pix11/110217-MenoFigure16bmp.bmp

Changed paradigm in the diamond theorem (2×2 case) —

http://www.log24.com/log/pix11/110217-MenoFigureColored16bmp.bmp

Ultron: By the Book

Filed under: General,Geometry — Tags: — m759 @ 6:45 PM

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

Lede

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

"Who is Ultron? What is he?"

See too the previous post and Cube of Ultron.

Wednesday, May 6, 2015

Soul

Filed under: General,Geometry — Tags: , — m759 @ 4:30 PM

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…."

Wednesday, April 1, 2015

Manifest O

Filed under: General,Geometry — Tags: , — m759 @ 4:44 AM

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 —

83-06-21 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.

Tuesday, February 10, 2015

In Memoriam…

Filed under: General,Geometry — Tags: — m759 @ 12:25 PM

industrial designer Kenji Ekuan —

Eightfold Design.

The adjective "eightfold," intrinsic to Buddhist
thought, was hijacked by Gell-Mann and later 
by the Mathematical Sciences Research Institute
(MSRI, pronounced "misery").  The adjective's
application to a 2x2x2 cube consisting of eight
subcubes, "the eightfold cube," is not intended to
have either Buddhist or Semitic overtones.  
It is pure mathematics.

Tuesday, October 21, 2014

Tools

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

(Night at the Museum continues.)

"Strategies for making or acquiring tools

While the creation of new tools marked the route to developing the social sciences,
the question remained: how best to acquire or produce those tools?"

— Jamie Cohen-Cole, “Instituting the Science of Mind: Intellectual Economies
and Disciplinary Exchange at Harvard’s Center for Cognitive Studies,”
British Journal for the History of Science  vol. 40, no. 4 (2007): 567-597.

Obituary of a co-founder, in 1960, of the Center for Cognitive Studies at Harvard:

"Disciplinary Exchange" —

In exchange for the free Web tools of HTML and JavaScript,
some free tools for illustrating elementary Galois geometry —

The Kaleidoscope Puzzle,  The Diamond 16 Puzzle
The 2x2x2 Cube, and The 4x4x4 Cube

"Intellectual Economies" —

In exchange for a $10 per month subscription, an excellent
"Quilt Design Tool" —

This illustrates not geometry, but rather creative capitalism.

Related material from the date of the above Harvard death:  Art Wars.

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:

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

A 2x2x2 articulated cube:

IMAGE- Eightfold cube with detail of triskelion structure

A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:

Image-- Solomon's Cube

Solomon’s Cube

Friday, August 29, 2014

Raum

Filed under: General,Geometry — Tags: , — m759 @ 8:00 AM

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.

Thursday, August 28, 2014

Brutalism Revisited

Filed under: General,Geometry — Tags: — m759 @ 11:59 PM

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.)

Source of the Finite

Filed under: General,Geometry — Tags: — m759 @ 10:20 AM

"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.

Wednesday, August 27, 2014

Not Quite

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Click image to enlarge.

Altar

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

"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…

IMAGE- T. Lux Feininger on 'Gestaltung'

Here “eidolon” should instead be “eidos .”

An example of eidos — Plato's diamond (from the Meno ) —

http://www.log24.com/log/pix10A/100607-PlatoDiamond.gif

Schau der Gestalt

Filed under: General,Geometry — Tags: , , — m759 @ 5:01 AM

(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.

Wednesday, June 4, 2014

Monkey Business

Filed under: General,Geometry — Tags: — m759 @ 8:48 PM

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:

IMAGE- Quaternion group acting on an eightfold cube

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).

Saturday, May 31, 2014

Quaternion Group Models:

Filed under: General,Geometry — Tags: — m759 @ 10:00 AM

The ninefold square, the eightfold cube, and monkeys.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

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.

Thursday, March 27, 2014

Diamond Space

Filed under: General,Geometry — Tags: — m759 @ 2:28 PM

(Continued)

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:

IMAGE - The Diamond Theorem, including the 4x4x4 'Solomon's Cube' case

Click to enlarge.

Monday, December 9, 2013

Heaven Descending

Filed under: General,Geometry — Tags: , — m759 @ 2:02 PM

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.”

IMAGE- Construction of 'Heaven Descending' lattice

Click for the source, mentioned in Anatomy of a Cube (Sept. 18, 2011).

Saturday, May 11, 2013

Core

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

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:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Of the odd  nxnxn cube:

Of the even  nxnxn cube:

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Related material: The Eightfold Cube and

“A core component in the construction
is a 3-dimensional vector space  over F.”

—  Page 29 of “A twist in the M24 moonshine story,”
by Anne Taormina and Katrin Wendland.
(Submitted to the arXiv on 13 Mar 2013.)

Tuesday, March 19, 2013

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

Angels & Demons meet Hudson Hawk

Dan Brown's four-elements diamond in Angels & Demons :

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

The Leonardo Crystal from Hudson Hawk :

Hudson:

Mathematics may be used to relate (very loosely)
Dan Brown's fanciful diamond figure to the fanciful
Leonardo Crystal from Hudson Hawk 

"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.

Thursday, December 27, 2012

Object Lesson

Filed under: General,Geometry — Tags: , — m759 @ 3:17 AM

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 :

IMAGE- Ernst Cassirer on 'representation' or 'Vorstellung' in 'Substance and Function' as 'the riddle of knowledge'

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

Monday, December 24, 2012

Eternal Recreation

Filed under: General,Geometry — Tags: , , , — m759 @ 3:17 AM

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.

Wednesday, November 14, 2012

Group Actions

Filed under: General,Geometry — Tags: , — m759 @ 4:30 PM

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
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—

 

Froebel's third gift, the eightfold cube

© 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—

.

Sunday, June 17, 2012

Congruent Group Actions

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 PM

A Google search today yielded no results
for the phrase "congruent group actions."

Places where this phrase might prove useful include—

Saturday, June 16, 2012

Chiral Problem

Filed under: General,Geometry — Tags: , — m759 @ 1:06 AM

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

"ANDOVERHARVARD 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….

Sunday, June 3, 2012

Child’s Play

Filed under: General,Geometry — Tags: , — m759 @ 2:56 PM

(Continued)

“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

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

Related material—

The Trinity Cube

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

Saturday, May 19, 2012

G8

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

"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.

Monday, May 7, 2012

More on Triality

Filed under: General,Geometry — Tags: — m759 @ 4:20 PM

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 kHurwitz
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.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

Sunday, January 22, 2012

Souvenir*

Filed under: General,Geometry — Tags: — m759 @ 8:09 PM

From life's box of chocolates

Happy birthday to Piper Laurie.

* Those who prefer their
souvenirs without sentiment
may consult the quaternions.

Wednesday, January 11, 2012

Language Game

Filed under: General,Geometry — Tags: — m759 @ 8:08 AM

Tension in the Common Room

IMAGE- 'Launched from Cuber' scene in 'X-Men: First Class'

In memory of population geneticist James F. Crow,
who died at 95 on January 4th.

Wednesday, January 4, 2012

Revision

Filed under: General,Geometry — Tags: , — m759 @ 8:00 PM

I revised the cubes image and added a new link to
an explanatory image in posts of Dec. 30 and Jan. 3
(and at finitegeometry.org). (The cubes now have
quaternion "i , j , k " labels and the cubes now
labeled "k " and "-k " were switched.)

I found some relevant remarks here and here.

Tuesday, January 3, 2012

Theorum

Filed under: General,Geometry — Tags: — m759 @ 7:48 AM

In memory of artist Ronald Searle

IMAGE- Ronald Searle, 'Pythagoras puzzled by one of my theorums,' from 'Down with Skool'

Searle reportedly died at 91 on December 30th.

From Log24 on that date

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Update of 9:29 PM EST Jan. 3, 2012

Theorum

 

From RationalWiki

Theorum (rhymes with decorum, apparently) is a neologism proposed by Richard Dawkins in The Greatest Show on Earth  to distinguish the scientific meaning of theory from the colloquial meaning. In most of the opening introduction to the show, he substitutes "theorum" for "theory" when referring to the major scientific theories such as evolution.

Problems with "theory"

Dawkins notes two general meanings for theory; the scientific one and the general sense that means a wild conjecture made up by someone as an explanation. The point of Dawkins inventing a new word is to get around the fact that the lay audience may not thoroughly understand what scientists mean when they say "theory of evolution". As many people see the phrase "I have a theory" as practically synonymous with "I have a wild guess I pulled out of my backside", there is often confusion about how thoroughly understood certain scientific ideas are. Hence the well known creationist argument that evolution is "just  a theory" – and the often cited response of "but gravity is also just  a theory".

To convey the special sense of thoroughness implied by the word theory in science, Dawkins borrowed the mathematical word "theorem". This is used to describe a well understood mathematical concept, for instance Pythagoras' Theorem regarding right angled triangles. However, Dawkins also wanted to avoid the absolute meaning of proof associated with that word, as used and understood by mathematicians. So he came up with something that looks like a spelling error. This would remove any person's emotional attachment or preconceptions of what the word "theory" means if it cropped up in the text of The Greatest Show on Earth , and so people would (in "theory ") have no other choice but to associate it with only the definition Dawkins gives.

This phrase has completely failed to catch on, that is, if Dawkins intended it to catch on rather than just be a device for use in The Greatest Show on Earth . When googled, Google will automatically correct the spelling to theorem instead, depriving this very page its rightful spot at the top of the results.

See also

 

Some backgound— In this journal, "Diamond Theory of Truth."

Thursday, September 1, 2011

How It Works

Filed under: General,Geometry — Tags: , , , — m759 @ 11:00 AM

Design is how it works.” — Steven Jobs (See Symmetry and Design.)

“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer

IMAGE- Harvard senior thesis on Mathieu groups, 2010, and supporting material from book 'Design Theory'

The name Carmichael is not to be found in Booher’s thesis.  A book he does  cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937.  It should  be dated 1931, as the following quotation shows—

From Log24 on Feb. 20, 2010

“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”

– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Epigraph from Ch. 4 of Design Theory , Vol. I:

Es is eine alte Geschichte,
doch bleibt sie immer neu

—Heine (Lyrisches Intermezzo  XXXIX)

See also “Do you like apples?

Saturday, July 30, 2011

Groups and Symmetry

Filed under: General,Geometry — m759 @ 1:00 PM

"… the best way to understand a group is to
see it as the group of symmetries of something."

— John Baez, p. 239, Bulletin (New Series) of the
American Mathematical Society
, Vol. 42, No. 2,
April 2005, book review on pp. 229–243
electronically published on January 26, 2005

"Imagine yourself as a gem cutter,
turning around this diamond…."

Ibid ., p. 240

See also related material from Log24.

Friday, June 24, 2011

For Stephen King Fans

Filed under: General,Geometry — Tags: — m759 @ 6:48 AM

The Gleaming

http://www.log24.com/log/pix11A/110624-Cubes-and-NYT.jpg

The column at left is from Galois Geometry.

Thursday, May 5, 2011

Beyond Forgetfulness

Filed under: General,Geometry — Tags: , — m759 @ 10:10 AM

From this journal on July 23, 2007

It is not enough to cover the rock with leaves.
We must be cured of it by a cure of the ground
Or a cure of ourselves, that is equal to a cure

Of the ground, a cure beyond forgetfulness.
And yet the leaves, if they broke into bud,
If they broke into bloom, if they bore fruit
,

And if we ate the incipient colorings
Of their fresh culls might be a cure of the ground.

– Wallace Stevens, "The Rock"

This quotation from Stevens (Harvard class of 1901) was posted here on when Daniel Radcliffe (i.e., Harry Potter) turned 18 in July 2007.

Other material from that post suggests it is time for a review of magic at Harvard.

On September 9, 2007, President Faust of Harvard

"encouraged the incoming class to explore Harvard’s many opportunities.

'Think of it as a treasure room of hidden objects Harry discovers at Hogwarts,' Faust said."

That class is now about to graduate.

It is not clear what "hidden objects" it will take from four years in the Harvard treasure room.

Perhaps the following from a book published in 1985 will help…

http://www.log24.com/log/pix11A/110505-MetamagicalIntro.gif

The March 8, 2011, Harvard Crimson  illustrates a central topic of Metamagical Themas , the Rubik's Cube

http://www.log24.com/log/pix11A/110427-CrimsonAtlas300w.jpg

Hofstadter in 1985 offered a similar picture—

http://www.log24.com/log/pix11A/110505-RubikGlobe.gif

Hofstadter asks in his Metamagical  introduction, "How can both Rubik's Cube and nuclear Armageddon be discussed at equal length in one book by one author?"

For a different approach to such a discussion, see Paradigms Lost, a post made here a few hours before the March 11, 2011, Japanese earthquake, tsunami, and nuclear disaster—

http://www.log24.com/log/pix11A/110427-ParadigmsLost.jpg

Whether Paradigms Lost is beyond forgetfulness is open to question.

Perhaps a later post, in the lighthearted spirit of Faust, will help. See April 20th's "Ready When You Are, C.B."

Monday, February 21, 2011

The Abacus Conundrum*

Filed under: General,Geometry — Tags: , , — m759 @ 2:02 PM

From Das Glasperlenspiel  (Hermann Hesse, 1943) —

“Bastian Perrot… constructed a frame, modeled on a child’s abacus, a frame with several dozen wires on which could be strung glass beads of various sizes, shapes, and colors. The wires corresponded to the lines of the musical staff, the beads to the time values of the notes, and so on. In this way he could represent with beads musical quotations or invented themes, could alter, transpose, and develop them, change them and set them in counterpoint to one another. In technical terms this was a mere plaything, but the pupils liked it.… …what later evolved out of that students’ sport and Perrot’s bead-strung wires bears to this day the name by which it became popularly known, the Glass Bead Game.”

From “Mimsy Were the Borogoves” (Lewis Padgett, 1943)—

…”Paradine looked up. He frowned, staring. What in—
…”Is that an abacus?” he asked. “Let’s see it, please.”
…Somewhat unwillingly Scott brought the gadget across to his father’s chair. Paradine blinked. The “abacus,” unfolded, was more than a foot square, composed of thin,  rigid wires that interlocked here and there. On the wires the colored beads were strung. They could be slid back and forth, and from one support to another, even at the points of jointure. But— a pierced bead couldn’t cross interlocking  wires—
…So, apparently, they weren’t pierced. Paradine looked closer. Each small sphere had a deep groove running around it, so that it could be revolved and slid along the wire at the same time. Paradine tried to pull one free. It clung as though magnetically. Iron? It looked more like plastic.
…The framework itself— Paradine wasn’t a mathematician. But the angles formed by the wires were vaguely shocking, in their ridiculous lack of Euclidean logic. They were a maze. Perhaps that’s what the gadget was— a puzzle.
…”Where’d you get this?”
…”Uncle Harry gave it to me,” Scott said on the spur of the moment. “Last Sunday, when he came over.” Uncle Harry was out of town, a circumstance Scott well knew. At the age of seven, a boy soon learns that the vagaries of adults follow a certain definite pattern, and that they are fussy about the donors of gifts. Moreover, Uncle Harry would not return for several weeks; the expiration of that period was unimaginable to Scott, or, at least, the fact that his lie would ultimately be discovered meant less to him than the advantages of being allowed to keep the toy.
…Paradine found himself growing slightly confused as he attempted to manipulate the beads. The angles were vaguely illogical. It was like a puzzle. This red bead, if slid along this  wire to that  junction, should reach there— but it didn’t. A maze, odd, but no doubt instructive. Paradine had a well-founded feeling that he’d have no patience with the thing himself.
…Scott did, however, retiring to a corner and sliding beads around with much fumbling and grunting. The beads did  sting, when Scott chose the wrong ones or tried to slide them in the wrong direction. At last he crowed exultantly.
…”I did it, dad!”
…””Eh? What? Let’s see.” The device looked exactly the same to Paradine, but Scott pointed and beamed.
…”I made it disappear.”
…”It’s still there.”
…”That blue bead. It’s gone now.”
…Paradine didn’t believe that, so he merely snorted. Scott puzzled over the framework again. He experimented. This time there were no shocks, even slight. The abacus had showed him the correct method. Now it was up to him to do it on his own. The bizarre angles of the wires seemed a little less confusing now, somehow.
…It was a most instructive toy—
…It worked, Scott thought, rather like the crystal cube.

* Title thanks to Saturday Night Live  (Dec. 4-5, 2010).

Thursday, November 25, 2010

Art Object, continued

Filed under: General,Geometry — Tags: — m759 @ 4:00 AM

Inside the White Cube

"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."

"May be" —

http://www.log24.com/log/pix10B/101123-plain_cube_200x227.gif

     Image from this journal
     at noon (EST) Tuesday

"The geometry of unit cubes is a meeting point
 of several different subjects in mathematics."
                                    — Chuanming Zong

http://www.log24.com/log/pix10B/101125-ZongAMS.jpg

    (Click to enlarge.)

"A meeting point" —

http://www.log24.com/log/pix10B/101125-NYTobit-UN.jpg

  The above death reportedly occurred "early Wednesday in Beijing."

Another meeting point —

                            http://www.log24.com/log/pix10B/101125-McDonaldLogoSm.jpg

http://www.log24.com/log/pix10B/101125-DayTheEarth.jpg

(Click on logo and on meeting image for more details.)

See also "no ordinary venue."

Tuesday, February 16, 2010

Mysteries of Faith

Filed under: General,Geometry — m759 @ 9:00 AM

From today's NY Times

http://www.log24.com/log/pix10/100216-NYTobits.jpg

Obituaries for mystery authors
Ralph McInerny and Dick Francis

From the date (Jan. 29) of McInerny's death–

"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.'"

Harvard University Press on Persons and Things (Walpurgisnacht, 2008), by Barbara Johnson

From the date (Feb. 14) of Francis's death–

2x2x2 cube

The EIghtfold Cube

The "something missing" in the above figure is an eighth cube, hidden behind the others pictured.

This eighth cube is not, as Johnson would have it, a void and "vanishing point," but is instead the "still point" of T.S. Eliot. (See the epigraph to the chapter on automorphism groups in Parallelisms of Complete Designs, by Peter J. Cameron. See also related material in this journal.) The automorphism group here is of course the order-168 simple group of Felix Christian Klein.

For a connection to horses, see
a March 31, 2004, post
commemorating the birth of Descartes
  and the death of Coxeter–

Putting Descartes Before Dehors

     Binary coordinates for a 4x2 array  Chess knight formed by a Singer 7-cycle

For a more Protestant meditation,
see The Cross of Descartes

Descartes

Descartes's Cross

"I've been the front end of a horse
and the rear end. The front end is better."
— Old vaudeville joke

For further details, click on
the image below–

Quine and Derrida at Notre Dame Philosophical Reviews

Notre Dame Philosophical Reviews

Friday, February 27, 2009

Friday February 27, 2009

Filed under: General,Geometry — Tags: , — m759 @ 7:35 PM
Time and Chance
continued

Today's Pennsylvania lottery numbers suggest the following meditations…

Midday:  Lot 497, Bloomsbury Auctions May 15, 2008– Raum und Zeit (Space and Time), by Minkowski, 1909. Background: Minkowski Space and "100 Years of Space-Time."*

Evening: 5/07, 2008, in this journal– "Forms of the Rock."

Related material:

A current competition at Harvard Graduate School of Design, "The Space of Representation," has a deadline of 8 PM tonight, February 27, 2009.

The announcement of the competition quotes the Marxist Henri Lefebvre on "the social production of space."

A related quotation by Lefebvre (cf. 2/22 2009):

"… an epoch-making event so generally ignored that we have to be reminded of it at every moment. The fact is that around 1910 a certain space was shattered… the space… of classical perspective and geometry…."

— Page 25 of The Production of Space (Blackwell Publishing, 1991)

This suggests, for those who prefer Harvard's past glories to its current state, a different Raum from the Zeit 1910.

In January 1910 Annals of Mathematics, then edited at Harvard, published George M. Conwell's "The 3-space PG(3, 2) and Its Group." This paper, while perhaps neither epoch-making nor shattering, has a certain beauty. For some background, see this journal on February 24, 2009.†

    * Ending on Stephen King's birthday, 2008
     † Mardi Gras

Friday February 27, 2009

Filed under: General — Tags: — m759 @ 10:12 AM
Lasting Significance


Wittgenstein's Lasting Significance
, edited by Max Kölbel and Bernhard Weiss, published by Routledge, 2004–

Page 168:

"Wittgenstein told Norman Malcolm that 'a serious and good philosophical work could be written that would consist entirely of jokes (without being facetious)' (Malcolm 1999: 64)."

Malcolm, N. (1999) "Wittgenstein: A Memoir," in F.A. Flowers (ed.) Portraits of Wittgenstein, vol. 3, Bristol: Thoemmes Press, pp. 60-112

The lasting significance here is perhaps in the page numbers.
 

Or perhaps in a name…

Roger Cohen, Ash Wednesday, 2009
 

Sunday, February 22, 2009

Sunday February 22, 2009

Filed under: General — Tags: , — m759 @ 11:00 AM

Design at Harvard:
Natural or Unnatural?

Logo of Harvard Graduate School of Design compared to the 'natural' sign

From the Harvard Graduate School of Design

Call for Entries: The Space of Representation

DEADLINE FEBRUARY 27, 2009 8PM EST

"According to Henri Lefebvre, the social production of space has three components: spatial practice, the representation of space, and the space of representation. The latter two are integral to both design and the review process."

Also according to Henri Lefebvre:

An 'epoch-making event' from Lefebvre, 'The Production of Space'

This is clearly nonsense.
It is also, like much else at Harvard,
damned Marxist  nonsense.

I recommend instead  
James Joyce on space —

Dagger Definitions

From 'Ulysses,' 1922 first edition, page 178-- 'dagger definitions'

Friday, December 19, 2008

Friday December 19, 2008

Filed under: General,Geometry — Tags: , , , — m759 @ 1:06 PM
Inside the
White Cube

Part I: The White Cube

The Eightfold Cube

Part II: Inside
 
The Paradise of Childhood'-- Froebel's Third Gift

Part III: Outside

Mark Tansey, 'The Key' (1984)

Click to enlarge.

Mark Tansey, The Key (1984)

For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).

For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion
at
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.

For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation

(U. of Chicago Press, 1999):

From Chapter 3,
"Sutures* of Structures," p. 58:

"What, then, is a frame, and what is frame work?

This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'"

* P. 61:
"… the frame forms the suture of structure. A suture is 'a seamless [sic**] joint or line of articulation,' which, while joining two surfaces, leaves the trace of their separation."

 ** A dictionary says "a seamlike joint or line of articulation," with no mention of "trace," a term from Derrida's jargon.

Monday, July 21, 2008

Monday July 21, 2008

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Knight Moves:

The Relativity Theory
of Kindergarten Blocks

(Continued from
January 16, 2008)

"Hmm, next paper… maybe
'An Unusually Complicated
Theory of Something.'"

Garrett Lisi at
Physics Forums, July 16

Something:

From Friedrich Froebel,
who invented kindergarten:

Froebel's Third Gift: A cube made up of eight subcubes

Click on image for details.

An Unusually
Complicated Theory:

From Christmas 2005:

The Eightfold Cube: The Beauty of Klein's Simple Group

Click on image for details.

For the eightfold cube
as it relates to Klein's
simple group, see
"A Reflection Group
of Order 168
."

For an even more
complicated theory of
Klein's simple group, see

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

Click on image for details.

Saturday, May 10, 2008

Saturday May 10, 2008

Filed under: General,Geometry — Tags: , , , — m759 @ 8:00 AM
MoMA Goes to
Kindergarten

"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."

— "Was Modernism Born
     in Toddler Toolboxes?"
     by Trip Gabriel, New York Times,
     April 10, 1997
 

RELATED MATERIAL

Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in a book from 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

Footnotes:
 
1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

Wednesday, January 16, 2008

Wednesday January 16, 2008

Filed under: General,Geometry — Tags: , — m759 @ 12:25 PM
Knight Moves:
Geometry of the
Eightfold Cube

Actions of PSL(2, 7) on the eightfold cube

Click on the image for a larger version
and an expansion of some remarks
quoted here on Christmas 2005.

Monday, July 23, 2007

Monday July 23, 2007

Filed under: General,Geometry — Tags: , , , , — m759 @ 8:00 AM
Daniel Radcliffe
is 18 today.
Daniel Radcliffe as Harry Potter

Greetings.

“The greatest sorcerer (writes Novalis memorably)
would be the one who bewitched himself to the point of
taking his own phantasmagorias for autonomous apparitions.
Would not this be true of us?”

Jorge Luis Borges, “Avatars of the Tortoise”

El mayor hechicero (escribe memorablemente Novalis)
sería el que se hechizara hasta el punto de
tomar sus propias fantasmagorías por apariciones autónomas.
¿No sería este nuestro caso?”

Jorge Luis Borges, “Los Avatares de la Tortuga

Autonomous Apparition

At Midsummer Noon:

“In Many Dimensions (1931)
Williams sets before his reader the
mysterious Stone of King Solomon,
an image he probably drew from
a brief description in Waite’s
The Holy Kabbalah (1929) of
a supernatural cubic stone
on which was inscribed
‘the Divine Name.’”
The image “http://www.log24.com/log/pix07/070624-Waite.gif” cannot be displayed, because it contains errors.
Related material:
It is not enough to cover the rock with leaves.
We must be cured of it by a cure of the ground
Or a cure of ourselves, that is equal to a cure 

Of the ground, a cure beyond forgetfulness.
And yet the leaves, if they broke into bud,
If they broke into bloom, if they bore fruit,

And if we ate the incipient colorings
Of their fresh culls might be a cure of the ground.

– Wallace Stevens, “The Rock”

See also
as well as
Hofstadter on
his magnum opus:
“… I realized that to me,
Gödel and Escher and Bach
were only shadows
cast in different directions by
some central solid essence.
I tried to reconstruct
the central object, and
came up with this book.”
Goedel Escher Bach coverHofstadter’s cover.

Here are three patterns,
“shadows” of a sort,
derived from a different
“central object”:
Faces of Solomon's Cube, related to Escher's 'Verbum'

Click on image for details.

Thursday, March 9, 2006

Thursday March 9, 2006

Filed under: General,Geometry — m759 @ 2:56 PM

Finitegeometry.org Update

(Revised May 21, 2006)

Finitegeometry.org now has permutable JavaScript views of the 2x2x2 and 4x4x4 design cubesSolomon’s Cube presented a claim that the 4x4x4 design cube retains symmetry under a group of about 1.3 trillion transformations.  The JavaScript version at finitegeometry.org/sc/64/view/ lets the reader visually verify this claim.  The reader should first try the Diamond 16 Puzzle.  The simpler 2x2x2 design cube, with its 1,344 transformations, was described in Diamonds and Whirls; the permutable JavaScript version is at finitegeometry.org/sc/8/view/.

Saturday, November 5, 2005

Saturday November 5, 2005

Filed under: General,Geometry — Tags: — m759 @ 4:24 PM

Contrapuntal Themes
in a Shadowland

 
(See previous entry.)

Douglas Hofstadter on his magnum opus:

"… I realized that to me, Gödel and Escher and Bach were only shadows cast in different directions by some central solid essence. I tried to reconstruct the central object, and came up with this book."

The image “http://www.log24.com/theory/images/GEBcover.jpg” cannot be displayed, because it contains errors.
Hofstadter's cover

Here are three patterns,
"shadows" of a sort,
derived from a different
"central object":

The image “http://www.log24.com/theory/images/GEB.jpg” cannot be displayed, because it contains errors.

For details, see
Solomon's Cube.

Related material:
The reference to a
"permutation fugue"
(pdf) in an article on
Gödel, Escher, Bach.

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