Log24

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.

Monday, January 4, 2021

The Purloined Art

Filed under: General — Tags: — m759 @ 11:18 AM

Sunday, September 27, 2020

Gleaming the Cube

Filed under: General — Tags: , , — m759 @ 1:30 AM

From a search in this journal for “Paradise of Childhood” —

Page from 'The Paradise of Childhood,' 1906 edition

Saturday, September 19, 2020

Cube School

Filed under: General — Tags: — m759 @ 3:16 AM

The new domain http://cube.school
points to posts tagged Cube School here.

Friday, September 18, 2020

Adoration of the Cube

Filed under: General — Tags: — m759 @ 3:25 AM

“WHEN I IMAGINE THE CUBE, I see a structure in motion.
I see the framework of its edges, its corners, and its flexible joints,
and the continuous transformations in front of me (before you start
to worry, I assure you that I can freeze it anytime I like). I don’t see
a static object but a system of dynamic relations. In fact, this is only
half of that system. The other half is the person who handles it.
Just like everything else in our world, a system is defined by
its place
within a network of relations—to humans, first of all.”

— Rubik, Erno.  Cubed   (p. 165). Flatiron Books. Kindle Ed., 2020.

Compare and contrast — Adoration of the Blessed Sacrament.

Sunday, July 5, 2020

The Enigma Cube

Filed under: General — Tags: , — m759 @ 5:03 AM

Promotional material —

“Did you buckle up?” —  Harlan Kane

The publication date of The Enigma Cube  reported above was February 13, 2020.

Related material — Log24 posts around that date now tagged The Reality Bond.

Friday, June 12, 2020

Art Icons

Filed under: General — Tags: — m759 @ 11:01 PM

Liz Elverenli on the writers of “I Am Not Okay with This
(Instagram, Jan. 18, 2019) —

“We went full Twin Peaks.”

Perhaps not  full  Twin Peaks . . .

Update of 11:22 PM ET —

A “Not Okay” line from Episode 1.03, “The Party’s Over”

00:00:31,573 –> 00:00:34,159
Something just didn’t feel right.

I prefer the Jacky St. James illustration of this sentiment:

Egon Schiele fans other than Elverenli may agree.

“It’s showtime, folks!” — Opening scenes of “All That Jazz.”

Thursday, May 14, 2020

Art Issue*

Filed under: General — Tags: — m759 @ 11:25 AM

“… the beautiful object
that stood in
for something else.”

— Holland Cotter quoting an art historian
in The New York Times  on May 13

From a post of April 27, 2020 —

“The yarns of seamen have a direct simplicity,
the whole meaning of which lies within the shell
of a cracked nut. But Marlow was not typical
(if his propensity to spin yarns be excepted),
and to him the meaning of an episode was not inside
like a kernel but outside….”

— Joseph Conrad in Heart of Darkness

The beautiful object —

Something else —

* The title is a reference to other posts now also tagged Art Issue.

Tuesday, April 28, 2020

The Art Issue

Filed under: General — Tags: — m759 @ 10:47 AM

A sort of pointillist mosaic” 

— Author David Mitchell on the film adaptation of 
his novel Cloud Atlas .

See also the previous post, Cloud Atlas of Unknowing.

Art isn’t easy.” — Sondheim.

Friday, April 24, 2020

Art at Cologne

Filed under: General — Tags: , — m759 @ 10:53 PM

This post was suggested by a New York Review of Books  article
on Cologne artist Gerhard Richter in the May 14, 2020, issue —

“The Master of Unknowing,” by Susan Tallman.

Some less random art

Monday, April 13, 2020

Cubes and Axes

Filed under: General — Tags: — m759 @ 12:55 PM

See also this  journal on November 29, 2011 —The Flight from Ennui.

Related illustration from earlier in 2011 —

See also this  journal on 20 Sept. 2011 — Relativity Problem Revisited —
as well as Congregated Light.

Monday, February 24, 2020

For “Time Cube” Fans

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

See also Time Cube elsewhere in this  journal.

Monday, January 6, 2020

Art for Optimus Prime

Filed under: General — Tags: — m759 @ 8:35 PM

Sunday, December 22, 2019

M24 from the Eightfold Cube

Filed under: General — Tags: , — m759 @ 12:01 PM

Exercise:  Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.

Click image below to download a Guitart PowerPoint presentation.

See as well earlier posts also tagged Triangles, Spreads, Mathieu.

Wednesday, October 23, 2019

Art-Historical Narrative*

Filed under: General — Tags: , — m759 @ 9:25 PM

"Leonardo was something like what we now call a Conceptual artist,
maybe the original one.   Ideas —  experiments, theories —  were
creative ends in themselves."

— Holland Cotter in the online New York TImes  this evening

From other Log24 posts tagged Tetrahedron vs. Square —

* Phrase from the previous post, "Overarching Narrative."

Monday, October 7, 2019

Berlekamp Garden vs. Kinder Garten

Filed under: General — 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

Thursday, September 12, 2019

Art Logos …

Filed under: General — Tags: — m759 @ 10:19 AM

… Continued from previous posts now tagged Art Logos.

"Logos," a Greek word used in philosophy and theology,
is, in modern usage, also a brief form of "logotypes,"
a name for the branding symbols used by businesses.

For some less commercial aspects of the philosophical
concept, see Logo in this journal.

Monday, September 9, 2019

ART WARS at Harvard: The Wertham Professorship

Filed under: General — m759 @ 8:38 PM

See as well an obituary for Mrs. Wertham from 1987.

Related art

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM 

See Cube Symbology.

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

Da hats ein Eck 

For further details, search the Web for "Wertham Professor" + Eck.

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 . . .

Monday, July 1, 2019

Inside the Exploded Cube

Filed under: General — m759 @ 10:15 PM
 

Metaphysical conceit | literature | Britannica.com

https://www.britannica.com/art/metaphysical-conceit

The metaphysical conceit, associated with the Metaphysical poets of the 17th century, is a more intricate and intellectual device. It usually sets up an analogy between one entity's spiritual qualities and an object in the physical world and sometimes controls the whole structure of the poem.…

This post's title refers to a metaphysical conceit 
in the previous post, Desperately Seeking Clarity.

Related material —

The source of the above mystical octahedron —

'Becoming Whole,' by Leslie Stein

      See also Jung's Imago Dei  in this journal.

Friday, June 21, 2019

Cubehenge

Filed under: General — m759 @ 1:15 AM

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 .

Wednesday, June 5, 2019

Time Cube

Filed under: General — m759 @ 11:02 AM

The opening lines of Eliot's Four Quartets

"Time present and time past
Are both perhaps present in time future,
And time future contained in time past."

Perhaps.

Those who prefer geometry to rhetoric may also prefer
to Eliot's lines the immortal opening of the Transformers  saga —

"Before time began, there was the Cube."

One version of the Cube

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

Saturday, January 5, 2019

Raiders of the Inarticulate

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

Friday, December 7, 2018

The Angel Particle

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 7:15 PM

(Continued from this morning)

Majorana spinors and fermions at ncatlab

The Gibbons paper on the geometry of Majorana spinors and the Kummer configuration

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Geometric incarnation and the Kummer configuration

See also other Log24 posts tagged Kummerhenge.

The Angel Particle

Filed under: General — Tags: — m759 @ 4:15 AM
 

https://newatlas.com/angel-particle-own-antiparticle/50579/

Scientists discover "angel particle"
that is its own antiparticle

Michael Irving
July 21st, 2017

. . . .

"Our team predicted exactly where to find the Majorana fermion and what to look for as its 'smoking gun' experimental signature," says Shoucheng Zhang, one of the senior authors of the research paper. "This discovery concludes one of the most intensive searches in fundamental physics, which spanned exactly 80 years."

. . . .

Zhang proposes that the team's discovery be named the "angel particle" after the Dan Brown novel Angels and Demons , which features a bomb powered by the meeting of matter and antimatter. In the long run, Majoranas could find practical application in making quantum computers more secure.

The research was published in the journal Science  . . . .

See as well Stanford News  yesterday  —

Shoucheng Zhang died on Dec. 1. He was 55. 

Zhang’s death was unexpected and followed
a “battle with depression,” according to his family. 

Thursday, November 29, 2018

The White Cube

Filed under: G-Notes,General,Geometry — m759 @ 9:57 AM

Clicking on Zong in the above post leads to a 2005 article
in the Bulletin of the American Mathematical Society .

See also the eightfold  cube and interality .

Friday, September 21, 2018

ABC Art

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 4:36 AM

Monday, July 23, 2018

Eightfold Cube for Furey*

Filed under: General,Geometry — m759 @ 10:31 PM

Click to enlarge:

Above are the 7 frames of an animated gif from a Wikipedia article.

* For the Furey of the title, see a July 20 Quanta Magazine  piece

See also the eightfold cube in this  journal.

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

Friday, June 29, 2018

Triangles in the Eightfold Cube

Filed under: General,Geometry — Tags: , — m759 @ 9:10 PM

From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube

Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eight-triangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”

Related material from 1975 —

More recently

Monday, June 11, 2018

Arty Fact

Filed under: General,Geometry — Tags: , , — m759 @ 10:35 PM

The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.

The Eye of ARTI

See also a post of May 19, "Uh-Oh" —

— and a post of June 6, "Geometry for Goyim" — 

Mystery box  merchandise from the 2011  J. J. Abrams film  Super 8 

An arty fact I prefer, suggested by the triangular computer-eye forms above —

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

This is from the July 29, 2012, post The Galois Tesseract.

See as well . . .

Friday, May 4, 2018

Art & Design

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

http://www.log24.com/log/pix11/110219-SquareRootQuaternion.jpg

A star figure and the Galois quaternion.

The square root of the former is the latter.

See also a passage quoted here a year ago today
(May the Fourth, "Star Wars Day") —

Cube symmetry subgroup of order 8 from 'Geometry and Symmetry,' Paul B. Yale, 1968, p.21

Saturday, March 31, 2018

Cube Theory

Filed under: General — m759 @ 9:00 PM

For Greta Gerwig and Saoirse Ronan —

See also a Log24 post from the above Cube Theory date —
April 12, 2016 — Lyrics for a Cartoon Graveyard — as well as . . .

'Loop De Loop,' Johnny Thunder, Diamond Records, 1962

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.

Saturday, November 18, 2017

Cube Space Continued

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

James Propp in the current Math Horizons  on the eightfold cube

James Propp on the eightfold cube

For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).

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).

Thursday, April 27, 2017

Partner, Anchor, Decompose

Filed under: General,Geometry — Tags: — m759 @ 12:31 PM

See also a figure from 2 AM ET April 26 

" Partner, anchor, decompose. That's not math.
That's the plot to 'Silence of the Lambs.' "

Greg Gutfeld, September 2014

Tuesday, April 4, 2017

White Cube

Filed under: General — Tags: , , — m759 @ 12:21 PM

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  

Wednesday, March 29, 2017

Art Space Illustrated

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

Another view of the previous post's art space  —

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

More generally, see Solomon's Cube in Log24.

See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above 
cube arrangement from "Affine groups on small binary spaces" (1984).

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Art Space, Continued

Filed under: General — Tags: , , — m759 @ 4:35 AM

"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."

—Staff writer Kamila Czachorowski, Harvard Crimson  today

From Log24 posts tagged Art Space

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Sunday, October 23, 2016

Quartet

Filed under: General,Geometry — Tags: — m759 @ 12:48 PM

“The man who lives in contact with what he believes to be a living Church
is a man always expecting to meet Plato and Shakespeare to-morrow
at breakfast.”

— G. K. Chesterton

Or Sunday dinner.

The Eightfold Cube

Platonic
solid

Jack in the Box, Natasha Wescoat, 2004
Natasha Wescoat, 2004

Shakespearean
Fool

Not to mention Euclid and Picasso.

 

The image “http://www.log24.com/theory/images/Pythagoras-I47.gif” cannot be displayed, because it contains errors.


The image “http://www.log24.com/log/pix06A/RobertFooteAnimation.gif” cannot be displayed, because it contains errors.

In the above pictures, Euclid is represented by 
Alexander Bogomolny, Picasso by Robert Foote.

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 —

Tuesday, August 30, 2016

The Eightfold Cube in Oslo

Filed under: General,Geometry — Tags: — m759 @ 8:04 PM

A KUNSTforum.as article online today (translation by Google) —

The eightfold cube at the Vigeland Museum in Oslo

Update of Sept. 7, 2016: The corrections have been made,
except for the misspelling "Cullinan," which was caused by 
Google translation, not by KUNSTforum.

Monday, August 22, 2016

Minimal ABC Art

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

Two portions of a post from Guy Fawkes Day 2015

 

Other art for Guy Fawkes Day

Cloak and Dagger

Tuesday, June 7, 2016

Art and Space…

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

Continues, in memory of chess grandmaster Viktor Korchnoi,
who reportedly died at 85 yesterday in Switzerland —

IMAGE- Spielfeld (1982-83), by Wolf Barth

The coloring of the 4×4 "base" in the above image
suggests St. Bridget's cross.

From this journal on St. Bridget's Day this year —

"Possible title: 

A new graphic approach 
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24
"

The narrative leap from image to date may be regarded as
an example of "knight's move" thinking.

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.

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.)

Tuesday, March 22, 2016

The Barth Spielfeld

Filed under: General — Tags: , — m759 @ 2:23 PM

For some backstory, search Log24 for "Wolf Barth."

Thursday, March 17, 2016

On the Eightfold Cube

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

The following page quotes "Raiders of the Lost Crucible,"
a Log24 post from Halloween 2015.

Discussion of Cullinane's eightfold cube as exhibited by Josefine Lyche at the Vigeland Museum in Oslo

From KUNSTforum.as, a Norwegian art quarterly, issue no. 1 of 2016.

Related posts — See Lyche Eightfold.

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

Thursday, November 5, 2015

ABC Art or: Guitart Solo

Filed under: General,Geometry — Tags: , — m759 @ 5:55 PM

“… the A B C of being….” — Wallace Stevens

Scholia —

Compare to my own later note, from March 4, 2010 —

“It seems that Guitart discovered these ‘A, B, C’ generators first,
though he did not display them in their natural setting,
the eightfold cube.” — Borromean Generators (Log24, Oct. 19)

See also Raiders of the Lost Crucible (Halloween 2015)
and “Guitar Solo” from the 2015 CMA Awards on ABC.

Friday, October 9, 2015

Eightfold Cube in Oslo

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

An eightfold cube appears in this detail 
of a photo by Josefine Lyche of her
installation "4D Ambassador" at the 
Norwegian Sculpture Biennial 2015

Sculpture by Josefine Lyche of Cullinane's eightfold cube at Vigeland Museum in Oslo

(Detail from private Instagram photo.)

Catalog description of installation —

Google Translate version —

In a small bedroom to Foredragssalen populate
Josefine Lyche exhibition with a group sculptures
that are part of the work group 4D Ambassador
(2014-2015). Together they form an installation
where she uses light to amplify the feeling of
stepping into a new dimension, for which the title
suggests, this "ambassadors" for a dimension we
normally do not have access to. "Ambassadors"
physical forms presents nonphysical phenomena.
Lyches works have in recent years been placed
in something one might call an "esoteric direction"
in contemporary art, and defines itself this
sculpture group humorous as "glam-minimalist."
She has in many of his works returned to basic
geometric shapes, with hints to the occult,
"new space-age", mathematics and where
everything in between.

See also Lyche + "4D Ambassador" in this journal and
her website page with a 2012 version of that title.

Cube Design

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

For Aaron Sorkin and Walter Isaacson

Related material — 
Bauhaus CubeDesign Cube, and
Nabokov's Transparent Things .

Monday, September 28, 2015

Hypercube Structure

Filed under: General,Geometry — m759 @ 1:01 AM

Click to enlarge:

Two views of tesseracts as 4D vector spaces over GF(2)

For the hypercube as a vector space over the two-element field GF(2),
see a search in this journal for Hypercube + Vector + Space .

For connections with the related symplectic geometry, see Symplectic
in this journal and Notes on Groups and Geometry, 1978-1986.

For the above 1976 hypercube (or tesseract ), see "Diamond Theory,"
by Steven H. Cullinane, Computer Graphics and Art , Vol. 2, No. 1,
Feb. 1977, pp. 5-7.

Friday, August 28, 2015

Art and Space

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

IMAGE- Spielfeld (1982-83), by Wolf Barth
 

            Observatory scene from "Magic in the Moonlight"

"The sixteen nodes… can be parametrized
by the sixteen points in affine four-space
over the tiny field F2 with two elements."

Wolf Barth

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                       

Monday, July 13, 2015

The Omega Cube

Filed under: General — m759 @ 12:00 PM

Why "Omega?"

Omega is a Greek letter, Ω , used in
mathematics to denote 
a set on which
a group acts. 

Wednesday, April 22, 2015

ART WARS continued

Filed under: General — Tags: — m759 @ 8:29 PM

The previous post mentioned a new mobile, "Triangle Constellation,"
commissioned for the Harvard Art Museums.

Related material (click to enlarge) —

The above review is of an exhibition by the "Constellation" artist,
Carlos Amorales, that opened on Sept. 26, 2008 — "just in time for
Halloween and the Day of the Dead."

See also this  journal on that date.

Thursday, March 26, 2015

The Möbius Hypercube

Filed under: General,Geometry — Tags: , — m759 @ 12:31 AM

The incidences of points and planes in the
Möbius 8 configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.* 
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —

Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his 
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots.  The figure,
"Gallucci's version of Möbius's 84," is shown below.
The hollow dots, representing the 8 points  (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.

Here a plane  (represented by a dotless intersection) contains
the four points  that are represented in the square array as lying
in the same row or same column as the plane. 

The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube

In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.

Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in 
Coxeter's labels above may be viewed as naming the positions 
of the 1's in (0,1) vectors (x4, x3, x2, x1) over the two-element
Galois field.  In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .

*  "Self-Dual Configurations and Regular Graphs," 
    Bulletin of the American Mathematical Society,
    Vol. 56 (1950), pp. 413-455

The subscripts' usual 1-2-3-4 order is reversed as a reminder
    that such a vector may be viewed as labeling a binary number 
    from 0  through 15, or alternately as labeling a polynomial in
    the 16-element Galois field GF(24).  See the Log24 post
     Vector Addition in a Finite Field (Jan. 5, 2013).

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)

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

A Raid on the Inarticulate

Filed under: General — Tags: , — m759 @ 6:00 PM

                                “… And so each venture
Is a new beginning, a raid on the inarticulate….”

— T. S. Eliot, “East Coker V” in Four Quartets

Monday, May 19, 2014

Cube Space

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

A sequel to this afternoon’s Rubik Quote:

“The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos.”

— Professor Ernő Rubik at Chrome Cube Lab

IMAGE- Weyl on symmetry

(Click image below to enlarge.)

Tuesday, April 1, 2014

Kindergarten Geometry

Filed under: General,Geometry — Tags: , — m759 @ 11:22 PM

(Continued)

A screenshot of the new page on the eightfold cube at Froebel Decade:

IMAGE- The eightfold cube at Froebel Decade

Click screenshot to enlarge.

Friday, March 28, 2014

Art School Confidential

Filed under: General — Tags: — m759 @ 4:01 AM

For Reba McEntire on her birthday:
Complex Reflection and Naturalized Epistemology.

Thursday, April 11, 2013

Naked Art

Filed under: General,Geometry — m759 @ 9:48 PM

The New Yorker  on Cubism:

"The style wasn’t new, exactly— or even really a style,
in its purest instances— though it would spawn no end
of novelties in art and design. Rather, it stripped naked
certain characteristics of all pictures. Looking at a Cubist
work, you are forced to see how you see. This may be
gruelling, a gymnasium workout for eye and mind.
It pays off in sophistication."

Online "Culture Desk" weblog, posted today by Peter Schjeldahl

Non-style from 1911:

IMAGE- Britannica 11th edition on the symmetry axes and planes of the cube

See also Cube Symmetry Planes  in this  journal.

A comment at The New Yorker  related to Schjeldahl's phrase "stripped naked"—

"Conceptualism is the least seductive modern-art movement."

POSTED 4/11/2013, 3:54:37 PM BY CHRISKELLEY

(The "conceptualism" link was added to the quoted comment.)

Sunday, February 3, 2013

The Gospel According to Cartier

Filed under: General — Tags: , , , — m759 @ 10:30 AM

Yesterday's 11 AM post Mad Day concluded
with a link to a 2001 American Mathematical Society
article by Pierre Cartier that sums up the religion and
politics of many mathematicians

"Here ends the infancy narrative of the gospel…."

"… while Simone Weil's Catholicism was violently
anti-Semitic (in 1942!), Grothendieck's Buddhism
bears a strong resemblance to the practices of
his Hasidic ancestors."

See also Simone Weil in this journal.

Note esp. a post of April 6, 2004 that provides
a different way of viewing Derrida's notion of
inscription .

Thursday, January 24, 2013

Cube Space

Filed under: General — Tags: , , — m759 @ 12:24 PM

For the late Cardinal Glemp of Poland,
who died yesterday, some links:

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…"

Monday, November 5, 2012

Design Cubes

Filed under: General,Geometry — m759 @ 9:48 PM

Continued from April 2, 2012.

Some predecessors of the Cullinane design cubes of 1984
that lack the Cullinane cubes' symmetry properties

Kohs cubes (see 1920 article)
Wechsler cubes (see Wechsler in this journal), and
Horowitz  cubes (see links below).

Horowitz Design Cubes Package

Horowitz Design Cubes (1971)

1973 Horowitz Design Cubes Patent

Horowitz Biography

Monday, June 4, 2012

Cube to Tesseract

Filed under: General — m759 @ 10:30 AM

Yesterday's post Child's Play displayed a cube formed
by a Hasse diagram of the 8 subsets of a 3-set.*

This suggests a review of a post from last January

IMAGE- Tesseract (i.e., hypercube) formed by a Hasse diagram of the 16 subsets of a 4-element set

* See a comment on yesterday's post relating it to earlier,
  very similar, remarks by Margaret Masterman.
  I was unaware yesterday that those remarks exist. 

Monday, April 9, 2012

Eightfold Cube Revisited

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

A search today (Élie Cartan's birthday) for material related to triality*

Dynkin diagram D4 for triality

yielded references to something that has been called a Bhargava cube .

Two pages from a 2006 paper by Bhargava—

Bhargava's reference [4] above for "the story of the cube" is to…

Higher Composition Laws I:
A New View on Gauss Composition,
and Quadratic Generalizations

Manjul Bhargava

The Annals of Mathematics
Second Series, Vol. 159, No. 1 (Jan., 2004), pp. 217-250
Published by: Annals of Mathematics
Article Stable URL: http://www.jstor.org/stable/3597249

A brief account in the context of embedding problems (click to enlarge)—

For more ways of slicing a cube,
see The Eightfold Cube —

* Note (1) some remarks by Tony Smith
   related to the above Dynkin diagram
   and (2) another colorful variation on the diagram.

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)

Sunday, September 18, 2011

Anatomy of a Cube

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

R.D. Carmichael’s seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following

“… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked.”
— D. A. Sprott, U. of Toronto, 1955

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

The figure by Cullinane included above shows a way to visualize Sprott’s remarks.

For the group actions described by Cullinane, see “The Eightfold Cube” and
A Simple Reflection Group of Order 168.”

Update of 7:42 PM Sept. 18, 2011—

From a Summer 2011 course on discrete structures at a Berlin website—

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—

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 (as above) by Cullinane. For remarks on
such group actions in the literature, see “Cube Space, 1984-2003.”

Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: , , — m759 @ 6:29 PM

Yesterday’s midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik’s mechanical contrivance as a rather absurd “Cosmic Cube.”

A simpler candidate for the “Cubepart of that phrase:

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

“Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions.”

Alexandre V. Borovik in “Coxeter Theory: The Cognitive Aspects

Borovik has a such a diagram—

http://www.log24.com/log/pix11B/110828-BorovikM.jpg

The planes in Borovik’s figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik’s remarks, the eightfold cube might serve to illustrate the “Cosmic” part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine’s Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

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.

Saturday, July 9, 2011

Gleaming the Cube (continued)

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

The New York Times  has a skateboarder obit with a URL date of July 9.

Here is an earlier version from the LA Times

July 4, 2011

By Keith Thursby, Los Angeles Times

Chris Cahill, one of the original Dogtown Z-Boys
who brought seismic changes to skateboarding
with their style and attitude, has died. He was 54.

Cahill was found June 24 at his Los Angeles home,
said Larry Dietz of the Los Angeles County
coroner's office. A cause of death has not been
determined and tests are ongoing, Dietz said.

More…

Related material from Midsummer Day, June 24, the day Cahill was found dead—

The Gleaming and The Cube.

    An illustration from the latter—

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

    The above was adapted from a 1996 cover

IMAGE- PyrE on the 1996 Vintage Books cover of 'The Stars My Destination'

 Vintage Books, July 1996. Cover: Evan Gaffney.

For the significance of the flames,
see PyrE in the book. For the significance
of the cube in the altered cover, see
The 2×2×2 Cube and The Diamond Archetype.

Tuesday, June 28, 2011

ART WARS continued

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

See the signature link in last night's post for a representation of Madison Avenue.

For a representation by  Madison Avenue, see today's New York Times—

IMAGE- Butter-Cow Lady, NY Math Museum, and World-as-Rubik-Cube ad

"As a movement Pop Art came and went in a flash, but it was the kind of flash that left everything changed. The art public was now a different public— larger, to be sure, but less serious, less introspective, less willing or able to distinguish between achievement and its trashy simulacrum. Moreover, everything connected with the life of art— everything, anyway, that might have been expected to offer some resistance to this wholesale vulgarization and demoralization— was now cheapened and corrupted. The museums began their rapid descent into show biz and the retail trade. Their exhibitions were now mounted like Broadway shows, complete with set designers and lighting consultants, and their directors pressed into service as hucksters, promoting their wares in radio and television spots and selling their facilities for cocktail parties and other entertainments, while their so-called education programs likewise degenerated into sundry forms of entertainment and promotion. The critics were co-opted, the art magazines commercialized, and the academy, which had once taken a certain pride in remaining aloof from the blandishments of the cultural marketplace, now proved eager to join the crowd— for there was no longer any standard in the name of which a sellout could be rejected. When the boundary separating art and fashion was breached, so was the dividing line between high art and popular culture, and upon all those institutions and professions which had been painstakingly created to preserve high art from the corruptions of popular culture. The effect was devastating. Some surrendered their standards with greater alacrity than others, but the drift was unmistakable and all in the same direction— and the momentum has only accelerated with the passage of time."

— Hilton Kramer, The Triumph of Modernism: The Art World, 1985-2005 , publ. by Ivan R. Dee on Oct. 26, 2006, pp. 146-147

Related material— Rubik in this journal, Exorcist in this journal, and For the Class of '11.

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.

Thursday, May 26, 2011

Prime Cubes

Filed under: General,Geometry — m759 @ 11:09 AM

The title refers not to numbers  of the form p 3, p  prime, but to geometric  cubes with p 3 subcubes.

Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation  groups of degree  (not order ) p 3.

IMAGE- From preface to Larry C. Grove, 'Classical Groups and Geometric Algebra

For the case p =2, see The Eightfold Cube.

For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")

For symmetries of group actions for larger primes, see the related 1985 remark* on two -dimensional linear groups—

"Actions of GL(2,p )  on a p ×p  coordinate-array
have the same sorts of symmetries,
where p  is any odd prime."

* Group Actions, 1984-2009

Wednesday, April 20, 2011

Romancing the Cube

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

It was a dark and stormy night…

http://www.log24.com/log/pix11/110420-DarkAndStormy-Logicomix.jpg

— Page 180, Logicomix

“… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field.”

– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups

For some context, see the small cube in “A Simple Reflection Group of Order 168.”

See also the larger cube in “Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).

That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix  page above—

http://www.log24.com/log/pix11/110420-Whitehead1906Axioms.jpg

A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night

“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”

— Gian-Carlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.

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, November 23, 2010

Art Object

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

There is more than one way
to look at a cube.

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

 From Cambridge U. Press on Feb. 20, 2006 —

IMAGE- 'Cambridge Tracts in Mathematics 168: The Cube'

and from this journal on June 30, 2010 —

In memory of Wu Guanzhong, Chinese artist
who died in Beijing on June 25, 2010

Image-- The Dream of the Expanded Field

See also this journal on Feb. 20, 2006
(the day The Cube  was published).

Sunday, October 31, 2010

ART WARS —

Filed under: General — m759 @ 2:00 AM

Keanu vs. the Devil, continued

IMAGE- Still from 'Devil's Advocate' (also starring Charlize Theron)

Al Pacino and Keanu Reeves in Devil's Advocate

For Keanu —

IMAGE- 'Cambridge Tracts in Mathematics 168: The Cube'

(Click for some background.)

For Keanu's mentor —

                                  …    There is a Cave
Within the Mount of God, fast by his Throne,
Where light and darkness in perpetual round
Lodge and dislodge by turns, which makes through Heav'n
Grateful vicissitude, like Day and Night….

Paradise Lost , by John Milton

http://www.log24.com/log/pix09A/091024-RayFigure.jpg

Click on figure for details.

http://www.log24.com/log/pix10B/101031-Pacino.jpg

Al Pacino in Devil's Advocate
as attorney John Milton

Friday, June 25, 2010

ART WARS continued

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

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

See The Klein Correspondence.

Monday, June 21, 2010

Cube Spaces

Filed under: General,Geometry — Tags: , — m759 @ 11:30 AM

Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.

 

Example 1— The 2×2×2 Cube

also known as the eightfold  cube

2x2x2 cube

Group actions on the eightfold cube, 1984—

http://www.log24.com/log/pix10A/100621-diandwh-detail.GIF

Version by Laszlo Lovasz et al., 2003—

http://www.log24.com/log/pix10A/100621-LovaszCubeSpace.gif

Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.

Example 2— The 3×3×3 Cube

A note from 1985 describing group actions on a 3×3 plane array—

http://www.log24.com/log/pix10A/100621-VisualizingDetail.gif

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

Pegg gives no reference to the 1985 work on group actions.

Example 3— The 4×4×4 Cube

A note from 27 years ago today—

http://www.log24.com/log/pix10A/100621-Cube830621.gif

As far as I know, this version of the
group-actions theorem has not yet been ripped off.

Saturday, October 24, 2009

Chinese Cubes Continued

Filed under: General,Geometry — m759 @ 8:28 AM

A search for “Chinese Cube” (based on the the previous entry’s title) reveals the existence of a most interesting character, who…

“… has attempted in his books to produce a Science and Art of Reasoning using the simplest of the Platonic solids, the Cube. [His] model also parallels, in some ways, the Cube of Space constructed from the Sepher Yetzirah’s attributions for the Hebrew letters and their direction. [He] elucidated his theories at great length….”

More…

For related remarks, see the link to Solomon’s Cube from the previous entry.

Then of course there is…

http://www.log24.com/log/pix09A/091024-RayFigure.jpg

Click on figure for details.

Thursday, October 22, 2009

Chinese Cubes

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

From the Bulletin of the American Mathematical Society, Jan. 26, 2005:

What is known about unit cubes
by Chuanming Zong, Peking University

Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in n-dimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all….

From Log24, now:

What is known about the 4×4×4 cube
by Steven H. Cullinane, unaffiliated

Abstract: The 4×4×4 cube, from one point of view, is among the simplest and the most important objects in n-dimensional binary space. In fact, as one will see from the links below, it is not simple at all.

Solomon’s Cube

The Klein Correspondence, Penrose Space-Time, and a Finite Model

Non-Euclidean Blocks

Geometry of the I Ching

Related material:

Monday’s entry Just Say NO and a poem by Stevens,

The Well Dressed Man with a Beard.”

Friday, July 24, 2020

Social Prisms

Filed under: General — Tags: — m759 @ 5:37 AM

IMAGE- 'American Hustle' and Art Cube

Saturday, May 30, 2020

GitHub Identity

Filed under: General — m759 @ 12:00 AM

Click the image below for some related material.

IMAGE- 'American Hustle' and Art Cube

Sunday, December 6, 2015

Form and Idea

Filed under: General,Geometry — Tags: , , , — m759 @ 3:24 PM

"Those early works are succinct and uncompromising
in how they give shape to the philosophical perplexities
of form and idea…."

J. J. Charlesworth, artnet news, Dec. 16, 2014

"Form" and "idea" are somewhat synonymous, 
as opposed to "form" and "substance." A reading:

IMAGE- 'American Hustle' and Art Cube

Discuss.

Tuesday, April 1, 2014

For April 1

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

IMAGE- 'American Hustle' and Art Cube

Or:  Extremely Gray Code

Related material:  Spaces as Hypercubes

Friday, March 7, 2014

Kummer Varieties

Filed under: General,Geometry — Tags: , , — m759 @ 11:20 AM

The Dream of the Expanded Field continues

Image-- The Dream of the Expanded Field

From Klein's 1893 Lectures on Mathematics —

"The varieties introduced by Wirtinger may be called Kummer varieties…."
E. Spanier, 1956

From this journal on March 10, 2013 —

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Update of 10 PM ET March 7, 2014 —

The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7(E7):

The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik  87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .

Tuesday, January 21, 2014

Class of 64

Filed under: General — Tags: — m759 @ 7:31 AM

NY Times  researcher from this morning’s previous post
tweeted last fall about art forgery and China.

Related material — Art Cube.

IMAGE- 'American Hustle' and Art Cube

Illustration from December 25, 2013.

Wednesday, December 25, 2013

Rotating the Facts

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

IMAGE- 'American Hustle' and Art Cube

"She never looked up while her mind rotated the facts,
trying to see them from all sides, trying to piece them
together into theory. All she could think was that she
was flunking an IQ test."

— Steve Martin, An Object of Beauty

"So you should not feel so all alone…"
— Adapted song lyric

Monday, January 4, 2021

The Purloined Joke

Filed under: General — Tags: , , — m759 @ 11:53 AM

See also the phrase “Beautiful Mathematics” in this  journal.

The Purloined Title

Filed under: General — Tags: , , — m759 @ 10:02 AM

When  logic and proportion have fallen sloppy dead . . . .”

The Mystery of 'Monomial Representations and Symmetric Presentations'

See as well  Symmetric Generation  in this  journal.

“Feed your head.” — Grace Slick

Sunday, January 3, 2021

Joke

Filed under: General — Tags: , , — m759 @ 12:31 PM

George Clooney to Matt Damon —

“There’s Michelangelo joke to be made.”

A search in this journal for Michelangelo suggests . . .

How about “Bach meets Bochner“?

Tuesday, December 29, 2020

I Ching  Geometry

Filed under: General — Tags: — m759 @ 11:04 AM

“Before time began, there was the Cube.”
Hassenfeld Brothers cinematic merchandising slogan

Sunday, December 27, 2020

Knight Move for Trevanian

Filed under: General — Tags: — m759 @ 11:42 PM

Knight move” remark from The Eiger Sanction

“I like to put people on myself by skipping logical steps
in the conversation until they’re dizzy.”

The following logical step — a check of the date Nov. 18, 2017
was omitted in the post Futon Dream  on this year’s St. Stephen’s Day.

For further context, see James Propp in this journal.

Friday, December 25, 2020

Change Arises: Mathematical Examples

Filed under: General — Tags: — m759 @ 12:59 AM

From old posts tagged Change Arises

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 rather 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.

The phrase “change arises” is from Arkani-Hamed in 2013, describing
calculations in physics related to properties of the positive Grassmannian

 

A related recent illustration from Quanta Magazine —

The above illustration of seven cells is not unrelated to
the eightfold-cube model of the seven projective points in
the Fano plane.

Tuesday, December 22, 2020

Small Venues

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

“… her art was rarely exhibited until the 1970s,
and then only sporadically and in small venues . . . .”

— New York Times  obituary suggested by
today’s review,

https://www.nytimes.com/2020/12/22/
arts/artists-who-died-2020.html

“No ordinary venue.” — Song lyric

Related material now linked to in the previous post

David Carradine displays a yellow book-- the Princeton I Ching.

Click on the Yellow Book.

Sunday, December 20, 2020

Looking Firmly

Filed under: General — m759 @ 2:39 PM

“… and the song of love’s recision is the music of the spheres.”
— E. L. Doctorow, City of God

Doctorow’s remark was quoted here earlier, on February 5, 2009

The central aim of Western religion–

"Each of us has something to offer the Creator...
the bridging of
 masculine and feminine,
 life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)

The central aim of Western philosophy–

 Dualities of Pythagoras
 as reconstructed by Aristotle:
  Limited Unlimited
  Odd Even
  Male Female
  Light Dark
  Straight Curved
  ... and so on ....

“Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy.”

— Jamie James in The Music of the Spheres (1993)

“In the garden of Adding
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  

Tuesday, December 8, 2020

Best Meets Bester

Filed under: General — Tags: — m759 @ 1:40 PM

The New York Times  online today —

Menu bar above a book review: “The Best of 2020.”

Alfred Bester —

Related search results

Tuesday, November 24, 2020

Scientism vs. Pure Mathematics

Filed under: General — Tags: , — m759 @ 9:18 PM

In his weblog today, Peter Woit quotes “a remarkable article
entitled Contemplating the End of Physics  posted today at
Quanta magazine [by] Robbert Dijkgraaf (the director of the IAS)”

An excerpt from the quoted remarks by the Institute for
Advanced Study director —

“All of this is part of a much larger shift in
the very scope of science, from studying what is
to what could be. In the 20th century, scientists
sought out the building blocks of reality:
the molecules, atoms and elementary particles
out of which all matter is made;
the cells, proteins and genes
that make life possible;
the bits, algorithms and networks
that form the foundation of information and intelligence,
both human and artificial. This century, instead,
we will begin to explore all there is to be made with
these building blocks.”

Then there are, of course, the building blocks of mathematical  reality:
unit cubes. See building-block.space.

Sunday, September 27, 2020

Remarks on Gordon Baker’s Death Day*

Filed under: General — Tags: , , — m759 @ 12:59 PM

* Baker was a writer on philosophy.
See a memorial by the Harvard Class of 1960.

Narratives

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

Wednesday, September 23, 2020

Geometry of Even Subsets

Filed under: General — Tags: , — m759 @ 12:06 AM

Various posts here on the geometry underlying the Mathieu group M24
are now tagged with the phrase “Geometry of Even Subsets.”

For example, a post with this diagram . . .

Monday, September 21, 2020

Zelig-Like?

Filed under: General — Tags: , — m759 @ 10:46 PM

“On their way to obscurity, the Simulmatics people
played minor parts in major events, appearing Zelig-like
at crucial moments of 1960s history.”

James Gleick reviewing a new book by Jill Lepore

Saturday, September 19, 2020

The Summerfield Prize

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

“Like Coleridge” . . .

Related material:  Bloomsday 2006.

Thursday, September 17, 2020

Structure and Mutability . . .

Filed under: General — Tags: , , , — m759 @ 9:51 PM

Continues in The New York Times :

“One day — ‘I don’t know exactly why,’ he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.

Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical:
a solid, static object that is also fluid….” — Alexandra Alter

Another such object: the eightfold cube .

Thursday, September 10, 2020

Raiders of . . .

Filed under: General — Tags: , — m759 @ 11:34 PM

Wednesday, September 9, 2020

Portrait with Holocron

Filed under: General — Tags: , , — m759 @ 11:08 PM

Novus Ordo Seclorum — Harold Bloom and the Tetrahedral Model of PG(3,2)

Sith Holocron in 'Star Wars Rebels'

For a Jedi  holocron of sorts, see this  journal on the above YouTube date

Friday, September 4, 2020

Vox Lux

Filed under: General — Tags: , — m759 @ 9:47 PM

An illustration from the Vox  article

Another approach to Nolan theory —

Or Matt Helm by way of a Jedi cube.

Thursday, August 20, 2020

Milking the Sixteenths

Filed under: General — Tags: — m759 @ 8:13 PM

“In the main belly cabin he discovered the reason
for the tropical heat; a naked woman was sweating
and swearing over the maintenance gear surrounding
a transparent incubator. She was tinkering and crawling
over and under the complications like an octopus.
It was his assistant, Dr. Cluny Decco, and Krupp had
never seen her nude before, but his controlled voice
did not betray his delighted amazement.”

— Alfred Bester, The Deceivers . Kindle Edition.

From a post, Dharma Fabric , of January 7, 2020 —

Thursday, July 9, 2020

The Enigma Glyphs

Filed under: General — Tags: , , — m759 @ 5:53 AM

IMAGE- The Diamond Theorem

For those who  prefer fiction —

“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.

This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”

— “Robert Ludlum’s”  The Bourne Enigma , published on June 21, 2016

Passing, on June 21, 2016, into a higher dimension —

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