Log24

Friday, March 4, 2016

Cube Bricks 1984

Filed under: General,Geometry — Tags: , — m759 @ 12:06 pm

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

Related aesthetics —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Monday, April 4, 2016

Cube for Berlin

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

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 —

The Eightfold 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

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

Related remark from the literature —

http://www.log24.com/log/pix11B/110918-Felsner.jpg

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.)

Wednesday, November 30, 2022

Three Representations

Filed under: General — Tags: , — m759 @ 12:05 pm

'Cube Bricks, 1984,' by Steven H. Cullinane

Cube Bricks, 1984

See also Impenetrability .

Saturday, September 3, 2022

1984 Revisited

Filed under: General — m759 @ 2:46 pm

Cube Bricks 1984 —

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

Related material

Note the three quadruplets of parallel edges  in the 1984 figure above.

Further Reading

The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —

Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.

Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —

      And then, more simply, there is the Galois tesseract

For parts of my own  world in June 2010, see this journal for that month.

The above Galois tesseract appears there as follows:

Image-- The Dream of the Expanded Field

See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post

Friday, February 11, 2022

For Space Groupies

Filed under: General — Tags: , — m759 @ 5:31 pm

A followup to Wednesday's post Deep Space

Related material from this journal on July 9, 2019

Cube Bricks 1984 —

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

From "Tomorrowland" (2015) —

From other posts tagged 1984 Cubes

Tuesday, October 8, 2019

Also* in 1984

Filed under: General — Tags: — m759 @ 11:32 am
 

American Mathematical Monthly , June-July 1984 

MISCELLANEA, 129

Triangles are square

"Every triangle consists of  n congruent copies of itself"
is true if and only if  n is a square. (The proof is trivial.)
— Steven H. Cullinane

* See Cube Bricks 1984  in previous post.

Monday, October 7, 2019

Berlekamp Garden vs. Kinder Garten

Filed under: General — Tags: — m759 @ 11:00 pm

Stevens's Omega and Alpha (see previous post) suggest a review.

Omega — The Berlekamp Garden.  See Misère Play (April 8, 2019).
Alpha  —  The Kinder Garten.  See Eighfold Cube.

Illustrations —

The sculpture above illustrates Klein's order-168 simple group.
So does the sculpture below.

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

Cube Bricks 1984 —

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

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.

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

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 —

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, June 20, 2017

Epic

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, June 7, 2017

Three Things at Once

Filed under: General,Geometry — Tags: , — m759 @ 1:28 pm

Rosalind Krauss in 1979

Nanavira Thera in 1959

Cambridge University Press in 1999 —

See also Cube Bricks.

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  

Thursday, November 3, 2016

Triple Cross

(Continued See the title in this journal, as well as Cube Bricks.)

Cube Bricks 1984 —

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

Thursday, July 14, 2016

Meditation from an April 1

Filed under: General,Geometry — Tags: — m759 @ 9:38 am

Related material from the same day —

See also

Cube Bricks 1984 —

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

The above bricks appeared in some earlier Log24 posts.

Saturday, March 12, 2016

Masonic Melody

Filed under: General — Tags: , , — m759 @ 1:31 am

"Button your lip baby
Button your coat
Let's go out dancing
Go for the throat"

Read more: Rolling Stones – Mixed Emotions Lyrics | MetroLyrics 

This melody was suggested by a post of February 25, 2016,
by tonight's previous post "Brick-Perfect," and by
the post "Cube Bricks 1984" of March 4, 2016.

"Only connect." — E. M. Forster.

Friday, August 14, 2020

Exercise

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

I prefer the boom box above to the one in Old Wives’ Tale (Aug. 10).

Monday, June 4, 2018

The Trinity Stone Defined

Filed under: General,Geometry — Tags: , , — m759 @ 8:56 pm

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

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

Some small Galois spaces (the Cullinane models)

Wednesday, April 27, 2016

Kubrick’s Rube

Filed under: General — m759 @ 9:23 pm

In memory of a culture jammer *—

* "Mr. Lyons … made a living partly by buying,
reconditioning and selling used cars." —
— Ben Ratliff in The New York Times  this evening.

See also the previous post and, from Feb. 14 in
this  journal, the phrase "more global than local."

Saturday, June 25, 2011

The Fano Entity

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

The New York Times  at 9 PM ET June 23, 2011

ROBERT FANO: I’m trying to think briefly how to put it.

GINO FANO: "On the Fundamental Postulates"—

"E la prova di questo si ha precisamente nel fatto che si è potuto costruire (o, dirò meglio immaginare) un ente per cui sono verificati tutti i postulati precedenti…."

"The proof of this is precisely the fact that you could build (or, to say it better, imagine) an entity by which are verified all previous assumptions…."

Also from the Times  article quoted above…

"… like working on a cathedral. We laid our bricks and knew that others might later replace them with better bricks. We believed in the cause even if we didn’t completely understand the implications.”

— Tom Van Vleck

Some art that is related, if only by a shared metaphor, to Van Vleck's cathedral—

http://www.log24.com/log/pix11A/110624-1984-Bricks-Sm.jpg

The art is also related to the mathematics of Gino Fano.

For an explanation of this relationship (implicit in the above note from 1984),
see "The Fano plane revisualized—or: the eIghtfold cube."

Sunday, April 22, 2007

Sunday April 22, 2007

Filed under: General,Geometry — Tags: — m759 @ 8:31 pm
Built
continued from
March 25, 2006

In honor of Scarlett Johansson's recent London films "Match Point" and "Scoop," here is a link to an entry of Women's History Month, 2006, with a discussion of an exhibition of the works of artist Liza Lou at London's White Cube Gallery.  That entry includes the following illustrations:


The image “http://www.log24.com/log/pix06/060325-WhiteCube.jpg” cannot be displayed, because it contains errors.

 

This work might aptly be
  retitled "Brick Shithouse."

Related material:

The artist's self-portrait

The image “http://www.log24.com/log/pix06/060325-LizaLouSelfPortrait.jpg” cannot be displayed, because it contains errors.

See also this morning's entry

"She's a brick… house…
The lady's stacked
   and that's a fact,
Ain't holdin' nothin' back."

— and last year's entry
on this date:

The image “http://www.log24.com/log/pix06/060422-Johansson1.jpg” cannot be displayed, because it contains errors.

"Her wall is filled with pictures,
She gets 'em one by one."

The bricks and "white cube"
above and in this morning's entry
may be contrasted with the
bricks of Diamonds and Whirls
and the cube of On Beauty.

  Poetic allusions such as these
may help provide
entertainment in the afterlife
for Beavis, Butt-Head, and
other inmates of Plato's Cave:

The image “http://www.log24.com/log/pix07/070328-PlatoCave.jpg” cannot be displayed, because it contains errors.

"The Garden of Eden is behind us
and there is no road back to innocence;
we can only go forward."

— Anne Morrow Lindbergh,
Earth Shine, p. xii

 

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