Log24

Tuesday, July 11, 2017

Dialogue from Plato’s Cave

Filed under: Uncategorized — m759 @ 10:15 AM

At  scifi.stackexchange.com

Why was the Cosmic Cube named the Tesseract 
in the Marvel movie series? Is there any specific reason 
for the name change? According to me, Cosmic Cube
seems a nice and cooler name.

— Asked March 14, 2013, by Dhwaneet Bhatt
    
At least it wasn't called 'The AllSpark.' 
It's not out of the realm of possibility. 

— Solemnity, March 14, 2013

Monday, July 10, 2017

Under Bleu Cup

Filed under: Uncategorized — m759 @ 10:00 AM

Publishers Weekly  on a Nov. 1, 2011, book, Under Blue Cup

"Krauss’s core argument (what she deems a 'crusade')
is that the 'white cube,' which conceptual and installation
artists have deemed obsolete, actually thrives."

For other "core arguments," see Satuday's post "Common Core"
and the Art Space posts "Odd Core" and "Even Core."

Friday, July 7, 2017

A Prime Radiant for Krugman

Filed under: Uncategorized — m759 @ 6:00 AM

Paul Krugman:
Asimov's Foundation  novels grounded my economics

In the Foundation  novels of Isaac Asimov

"The Prime Radiant can be adjusted to your mind, and all
corrections and additions can be made through mental rapport.
There will be nothing to indicate that the correction or addition
is yours. In all the history of the Plan there has been no
personalization. It is rather a creation of all of us together.
Do you understand?"  

"Yes, Speaker!"

— Isaac Asimov, Second Foundation , Ch. 8: Seldon's Plan

"Before time began, there was the Cube."

See also Transformers in this journal.

Thursday, July 6, 2017

A Pleasing Situation

Filed under: Uncategorized — m759 @ 9:20 PM

The 4x4x4 cube is the natural setting
for the finite version of the Klein quadric
and the eight "heptads" discussed by
Conwell in 1910.

As R. Shaw remarked in 1995, 
"The situation is indeed quite pleasing."

Sunday, July 2, 2017

Practically Cubist

Filed under: Uncategorized — m759 @ 3:45 AM

From an Anthony Lane movie review in the April 8, 2013,
issue of The New Yorker

"When the Lord God forbade his worshippers to bow down
before any graven image, [Rosario] Dawson’s face was
exactly the kind of thing He had in mind. No other star can
boast such sculptured features—except Vincent Cassel,
who is pretty damn graven himself. When the two of them
make love, in 'Trance,' one strong bone structure pressed
against another, it’s like a clash of major religions. What if
they had a family? The kids would be practically Cubist."

As for the other film Lane reviewed in that issue, "Blancanieves" —

See Snow White + Cube in this  journal.

See as well a related cartoon graveyard, also from April 8, 2013.

Thursday, June 22, 2017

Face Henge

Filed under: Uncategorized — Tags: — m759 @ 3:07 PM

With a hat tip to Vinnie Mancuso

Wednesday, June 21, 2017

Detail for Hopkins

Filed under: Uncategorized — Tags: — m759 @ 2:56 PM

Detail from the previous post

See Space Cross in this journal.

See also Anthony Hopkins' new film
"Transformers: The Last Knight" and
 

Concept and Realization

Filed under: Uncategorized — Tags: — m759 @ 1:12 PM

Remark on conceptual art quoted in the previous post

"…he’s giving the concept but not the realization."

A concept See a note from this date in 1983:

IMAGE- 'Solomon's Cube'

A realization  

Webpage demonstrating symmetries of 'Solomon's Cube'

Not the best possible realization, but enough for proof of concept .

Tuesday, June 20, 2017

All-Spark Notes

Filed under: Uncategorized — Tags: — m759 @ 1:55 PM

(Continued)

"For years, the AllSpark rested, sitting dormant
like a giant, useless art installation."

— Vinnie Mancuso at Collider.com yesterday

Related material —

Dormant cube

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

Giant, useless art installation —

Sol LeWitt at MASS MoCA.  See also LeWitt in this journal.

Epic

Filed under: Uncategorized — 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.

Friday, June 16, 2017

Rough Night at the Museum

Filed under: Uncategorized — Tags: — m759 @ 2:56 PM

See the previous three posts and the Nobel flashback titled Cuber.

All-Spark Notes*

Filed under: Uncategorized — Tags: , — m759 @ 12:30 PM

* See All Spark in this journal.

Wednesday, June 7, 2017

Three Things at Once

Filed under: Uncategorized — m759 @ 1:28 PM

Rosalind Krauss in 1979

Nanavira Thera in 1959

Cambridge University Press in 1999 —

See also Cube Bricks.

Tuesday, June 6, 2017

Cultist Drama

Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

Rosalind Krauss on the drama of 'the cultist space of modern art'

From a post of last Friday, June 2

See also Transformers in this journal.

"Before time began, there was the Cube."

  — Transformers  (2007)

Sunday, June 4, 2017

In Memory of the Time Cube Page*

Filed under: Uncategorized — 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).

Sequel to “The Square”* —

Filed under: Uncategorized — m759 @ 3:03 PM

The Cube

CodePen logo, pictured here on May 28, 2017

From YouTube, "The Cube," published on April 6, 2016

Meanwhile, also on April 6, 2016, at 2:01 AM ET 

* See The Pinterest Directive and Expanding the Spielraum.

Thursday, May 4, 2017

In Memory of Burton Watson

Filed under: Uncategorized — m759 @ 12:00 AM

From the NYT obit of Burton Watson, a translator of classical Chinese and Japanese literature

From a post on April 1, the reported date of his death —

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

Tuesday, May 2, 2017

Image Albums

Filed under: Uncategorized — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Wednesday, April 26, 2017

Again

Filed under: Uncategorized — Tags: — m759 @ 6:18 PM

See also "Cornerstone" in this journal and

A sidebar from a Google search today —

'The Square of Opposition: A Cornerstone of Thought'

This suggests a review of posts now tagged Obelisk,
which include

Thursday, April 20, 2017

Stone Logic

Filed under: Uncategorized — m759 @ 9:48 PM

See also "Romancing the Omega" —

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Related mathematics — Guitart in this journal —

From 'Moving Logic, from Boole to Galois,' by René Guitart, 2005

See also Weyl + Palermo in this journal —

http://www.log24.com/log/pix11B/110922-TriquetrumCube.jpg

Sunday, April 16, 2017

Homily

Filed under: Uncategorized — m759 @ 12:25 PM

See also Plan 9.

Wednesday, April 12, 2017

Contracting the Spielraum

Filed under: Uncategorized — 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, April 4, 2017

Plan 9 Continues

Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

"Plan 9 deals with the resurrection of the dead."

— Bill Murray in "Ed Wood"
 

For The Church of Plan 9

(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)

White Cube

Filed under: Uncategorized — 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  

Monday, April 3, 2017

Even Core

Filed under: Uncategorized — Tags: — m759 @ 9:16 PM

4x4x4 gray cube

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

Odd Core

Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

 

3x3x3 Galois cube, gray and white

Saturday, April 1, 2017

Beyond All Recognition

Filed under: Uncategorized — m759 @ 10:45 AM

Prequel —

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

Note that Yale's die design and use of the phrase "rigid motions"
differ from those in the webpage "Solomon's Cube."

Friday, March 31, 2017

Women’s History Month

Filed under: Uncategorized — m759 @ 11:00 AM

http://www.log24.com/log/pix11/110301-Inception256w.jpg

Show me all  the blueprints.”
— Howard Hughes, according to Hollywood

Wednesday, March 29, 2017

Art Space Illustrated

Filed under: Uncategorized — 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: Uncategorized — 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 

Tuesday, March 28, 2017

Bit by Bit

Filed under: Uncategorized — 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  

Backstory

Filed under: Uncategorized — Tags: — m759 @ 12:06 AM

Click here to enlarge.  Click the image for the source page.

The "this page" reference is to …

Finite Geometry of the Square and Cube.

Also from March 14, 2017 —

Related material

'Children of the Central Structure,' adapted from 'Children of the Damned'

Saturday, March 11, 2017

Transformer Group*

Filed under: Uncategorized — m759 @ 12:45 PM

"Before time began, there was the Cube." — Transformers  (2007)

Plot summary — "An ancient struggle between two Cybertronian races,
the heroic Autobots and the evil Decepticons, comes to Earth, with
a clue to the ultimate power held by a teenager."

* A post suggested by J. D. Salinger's phrase "a fresh impetus"
  in "Raise High the Roof Beam, Carpenters" (1955)

Friday, March 10, 2017

Transformers

Filed under: Uncategorized — m759 @ 1:00 PM

Or:  Y  for Yale  continued

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

See also Transformers in this journal and Y for Yale.

Thursday, March 9, 2017

One Eighth

Filed under: Uncategorized — m759 @ 1:16 PM

From Wikipedia's Iceberg Theory

Related material: 

The Eightfold Cube and The Quantum Identity

See also the previous post.

Saturday, March 4, 2017

At 74

Filed under: Uncategorized — m759 @ 12:00 AM

New York Times  headline about a death
on Friday, March 3, 2017 —

René Préval, President of Haiti
in 2010 Quake, Dies at 74

See also

This way to the egress.

Thursday, March 2, 2017

Stories

Filed under: Uncategorized — Tags: — m759 @ 12:48 PM

"We tell ourselves stories in order to live." — Joan Didion

The New York Times Magazine  online today —

"As a former believer and now a nonbeliever, Carrère,
seeking answers, sets out, in The Kingdom , to tell
the story of the storytellers. He is trying to understand
what it takes to be able to tell a story, any story.
And what he finds, once again, is that you have to find
your role in it."

Wyatt Mason in The New York Times Magazine ,
     online March 2, 2017 

Like Tom Hanks?

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

Click image for related posts.

Sunday, February 26, 2017

Poetic Order

Filed under: Uncategorized — m759 @ 1:06 PM

Transformations acting on Solomon's Cube
furnish a model of poetic order.

Some backstory for Hollywood —

Hollywood analogue to Solomon's Cube in 'Transformers'

Saturday, February 25, 2017

Pronouncements

Filed under: Uncategorized — m759 @ 9:45 AM

"The deepest strain in a religion is
the particular and particularistic doctrine
it asserts at its heart,
in the company of such pronouncements as
‘Thou shalt have no other Gods before me.’
Take the deepest strain of religion away…
and what remains are the surface pieties — 
abstractions without substantive bite — 
to which everyone will assent
because they are empty, insipid, and safe."

— Stanley Fish, quoted here on July 3, 2007
The opening date of the film "Transformers."

The opening pronouncement of "Transformers" —

"Before time began, there was the Cube."

Those who prefer Fish's abstractions  may consult 
the previous post.

Tuesday, February 21, 2017

Arrow Economics

Filed under: Uncategorized — m759 @ 11:29 PM

Broken Symmetries  in  Diamond Space 

Saturday, February 18, 2017

Solid Symmetry (continued)

Filed under: Uncategorized — m759 @ 12:00 PM

See Hanks + Cube in this journal For instance

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 

Verbum

Filed under: Uncategorized — m759 @ 3:00 AM

The Log24 version  (Nov. 9, 2005, and later posts) —

VERBUM
SAT
SAPIENTI

 

Escher's 'Verbum'

Escher's Verbum


Solomon's Cube

Solomon's Cube
 

I Ching hexagrams as parts of 4x4x4 cube

Geometry of the I Ching

The Warner Brothers version

The Paramount version

See also related material in the previous post, Transformers.

Tuesday, February 14, 2017

Bee Season Continued

Filed under: Uncategorized — m759 @ 11:55 PM

"Spells are used for breaking enchantments as well as for inducing them."

— C. S. Lewis, quoted here during November 2005.

A review of that month was suggested by the following search

A search result for 'Sefer Yetzirah'+ Cube + Chelsea + Hotel

which itself was suggested by an obituary in tonight's online
New York Times  and by this morning's post "Rhyme."

Rhyme

Filed under: Uncategorized — m759 @ 11:00 AM

For the late poet Thomas Lux ,
who reportedly died on February 5 
(See posts tagged Bewitchment) —

"A darker riddle with no answer looms . . . .
For those who linger on among the tombs"

John Hollander, from a poem in A Draft of Light

Those who enjoy dark riddles with no answers
may search the Web for "Sefer Yetzirah" + Cube.

I prefer a purely mathematical approach to the cube.

Sunday, February 5, 2017

Primitive Principle

Filed under: Uncategorized — Tags: , , — m759 @ 12:20 PM

"Rilke's insistent quest for a purity associated with primitive origins
took a Russian turning, but could also be seen as an extension of
traditions developed in Europe in the modern era. When Balzac
takes a metaphysical look at modern life, he exclaims: 'There is a
primitive principle! . . . .' "

— The late Dore Ashton,
     "Art Critic Who Embraced and Inhabited Modernism"

Putting aside the wild inaccuracy of Ashton's remarks,
the words "modernism" and "turning" suggest a review of

Thursday, January 19, 2017

A Time Cube for Silvia

Filed under: Uncategorized — m759 @ 12:13 AM

The Silvia of the title is from the previous post.

For the Time Cube, see …

Thursday, January 12, 2017

The Cherished Gift

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

From "Solomon's Cube" —

Related material —

"Is this a dagger I see before me?

"No." (A line suggested by Polanski's 2010 "The Ghost Writer")

Changes

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

The I Ching's underlying group has 1,290,157,424,640 permutations.

Monday, January 9, 2017

Analogical Extension at Cornell

Filed under: Uncategorized — m759 @ 1:30 PM

Click to enlarge the following (from Cornell U. Press in 1962) —

For a more recent analogical extension at Cornell, see the
Epiphany 2017 post on the eightfold cube and yesterday
evening's post "A Theory of Everything."

Sunday, January 8, 2017

A Theory of Everything

Filed under: Uncategorized — m759 @ 7:11 PM

The title refers to the Chinese book the I Ching ,
the Classic of Changes .

The 64 hexagrams of the I Ching  may be arranged
naturally in a 4x4x4 cube. The natural form of transformations
("changes") of this cube is given by the diamond theorem.

A related post —

The Eightfold Cube, core structure of the I Ching

Saturday, January 7, 2017

Conceptualist Minimalism

Filed under: Uncategorized — 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, January 6, 2017

Eightfold Cube at Cornell

Filed under: Uncategorized — m759 @ 7:35 PM

The assignments page for a graduate algebra course at Cornell
last fall had a link to the eightfold cube:

Sunday, January 1, 2017

Like the Horizon

Filed under: Uncategorized — m759 @ 1:00 PM

(Continued from a remark by art critic Peter Schjeldahl quoted here
last  year on New Year's Day in the post "Art as Religion.")

"The unhurried curve got me. 
It was like the horizon of a world
that made a non-world of
all of the space outside it."

— Peter Schjeldahl, "Postscript: Ellsworth Kelly,"
The New Yorker , December 30, 2015

This suggests some further material from the paper 
that was quoted here yesterday on New Year's Eve —

"In teaching a course on combinatorics I have found
students doubting the existence of a finite projective
plane geometry with thirteen points on the grounds
that they could not draw it (with 'straight' lines)
on paper although they had tried to do so. Such a
lack of appreciation of the spirit of the subject is but
a consequence of the elements of formal geometry
no longer being taught in undergraduate courses.
Yet these students were demanding the best proof of
existence, namely, production of the object described."

— Derrick Breach (See his obituary from 1996.)

A related illustration of the 13-point projective plane 
from the University of Western Australia:

Projective plane of order 3

(The four points on the curve
at the right of the image are
the points on the line at infinity .)

The above image is from a post of August 7, 2012,
"The Space of Horizons."  A related image — 

Click on the above image for further remarks.

Tuesday, December 27, 2016

Bright Star

Filed under: Uncategorized — m759 @ 8:15 AM

See instances of the title in this journal.

Material related to yesterday evening's post
"Bright and Dark at Christmas" —

The Buddha of Rochester:

See also the Gelman (i.e., Gell-Mann) Prize
in the film "Dark Matter" and the word "Eightfold"
in this journal.

" A fanciful mark is a mark which is invented
for the sole purpose of functioning as a trademark,
e.g., 'Kodak.' "

"… don't take my Kodachrome away." — Paul Simon

Monday, December 26, 2016

Fanciful (continued)

Filed under: Uncategorized — m759 @ 1:00 PM

From "Plato Thanks the Academy," March 19, 2014 —

IMAGE- Art Jeffries (Bruce Willis) and Simon Lynch (Miko Hughes), 'Mercury Rising' (1998)

“Click on fanciful .”

A possible result —

See also "Triple Cross."

Friday, December 23, 2016

Requiem for a Mathematician

Filed under: Uncategorized — m759 @ 2:10 PM

From a Dec. 21 obituary posted by the
University of Tennessee at Knoxville —

"Wade was ordained as a pastor and served
at Oakwood Baptist Church in Knoxville."

Other information —

In a Log24 post, "Seeing the Finite Structure,"
of August 16, 2008, Wade appeared as a co-author
of the Walsh series book mentioned above —

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
,
by F. Schipp et al.,
Taylor & Francis, 1990

From the 2008 post —

The patterns on the faces of the cube on the cover
of Walsh Series above illustrate both the 
Walsh functions of order 3 and the same structure
in a different guise, subspaces of the affine 3-space 
over the binary field. For a note on the relationship
of Walsh functions to finite geometry, see 
Symmetry of Walsh Functions.

Friday, December 2, 2016

Images from a Lumber Room*

Filed under: Uncategorized — Tags: — m759 @ 7:20 AM

An image search today for
"Design Cube" + Cullinane:
Click to enlarge (5.3 MB) —
    

* For the title, see St. Andrew's Day.

Thursday, December 1, 2016

What’s in a Name

Filed under: Uncategorized — m759 @ 11:23 PM

Design Cube 2x2x2 for demonstrating Galois geometry

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

Sunday, November 27, 2016

A Machine That Will Fit

Filed under: Uncategorized — Tags: — m759 @ 8:00 AM

Or:  Notes for the Metaphysical Club

Northrop Frye on Wallace Stevens:

"He… stands in contrast to the the dualistic
approach of Eliot, who so often speaks of poetry
as though it were an emotional and sensational
soul looking for a 'correlative' skeleton of
thought to be provided by a philosopher, a
Cartesian ghost trying to find a machine that
will fit."

Ralph Waldo Emerson on "vacant and vain" knowledge:

"The new position of the advancing man has all
the powers of the old, yet has them all new. It
carries in its bosom all the energies of the past,
yet is itself an exhalation of the morning. I cast
away in this new moment all my once hoarded
knowledge, as vacant and vain." 

Harold Bloom on Emerson:

"Emerson may not have invented the American
Sublime, yet he took eternal possession of it." 

Wallace Stevens on the American Sublime:

"And the sublime comes down
To the spirit itself,

The spirit and space,
The empty spirit
In vacant space."

A founding member of the Metaphysical Club:

See also the eightfold cube.

Thursday, November 3, 2016

Triple Cross

Filed under: Uncategorized — Tags: — m759 @ 1:30 PM

(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.)

Sunday, October 23, 2016

Voids

Filed under: Uncategorized — m759 @ 8:24 PM

From mathematician Izabella Laba today —

From Harry T. Antrim’s 1967 thesis on Eliot —

“That words can be made to reach across the void
left by the disappearance of God (and hence of all
Absolutes) and thereby reestablish some basis of
relation with forms existing outside the subjective
and ego-centered self has been one of the chief
concerns of the first half of the twentieth century.”

And then there is the Snow White void  —

A logo that may be interpreted as one-eighth of a 2x2x2 array
of cubes —

The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).

Quartet

Filed under: Uncategorized — 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.

Saturday, October 15, 2016

Twelve and Twelve

Filed under: Uncategorized — m759 @ 1:12 PM

See All Saints 2014 in this journal and listen to 
the new Stevie Nicks reissue of Bella Donna.

Related religious imagery —

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points (The 6 cube faces are mapped to the 6 hexagram lines.)

Thursday, October 6, 2016

Cuber

Filed under: Uncategorized — m759 @ 12:00 PM

Nobel Flashback:

Wednesday, January 29, 2014

Nobel Note

Filed under: Uncategorized — m759 @ 7:59 PM 

"It's going to be accomplished in steps,
this establishment of the Talented
​in the scheme of things."

— To Ride Pegasus ,
     by Anne McCaffrey (Radcliffe '47)

From a post of Jan. 11, 2012 —

Tension in the Common Room

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

Mirror Play

Filed under: Uncategorized — Tags: — m759 @ 4:00 AM

See posts tagged Spiegel-Spiel.

"Mirror, Mirror …." —

A logo that may be interpreted as one-eighth of
a 2x2x2 array of cubes —

The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).

Tuesday, October 4, 2016

Westworld

Filed under: Uncategorized — m759 @ 10:27 AM

The title refers to a Log24 post of 9:45 AM ET Sunday, Oct. 2.

From the "Westworld" post of Sunday, Oct. 2 —

"It was rather like watching a play."

QED.

Square Ice

Filed under: Uncategorized — m759 @ 9:00 AM

The discovery of "square ice" is discussed in
Nature 519, 443–445 (26 March 2015).

Remarks related, if only by squareness —
this  journal on that same date, 26 March 2015

The above figure is part of a Log24 discussion of 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.  When
this fact was first  observed, I do not know. It is implicit, although
not stated explicitly, in the 1950 paper by H.S.M. Coxeter from
which the above figure is adapted (blue dots added).

Monday, October 3, 2016

Ein Eck

Filed under: Uncategorized — m759 @ 5:05 PM
 

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 

Sunday, October 2, 2016

Westworld

Filed under: Uncategorized — m759 @ 9:45 AM

On a new HBO series that opens at 9 PM ET tonight —

Watching Westworld , you can sense a grand mythology unfolding before your eyes. The show’s biggest strength is its world-building, an aspect of screenwriting that many television series have botched before. Often shows will rush viewers into plot, forgetting to instill a sense of place and of history, that you’re watching something that doesn’t just exist in a vacuum but rather is part of some larger ecosystem. Not since Lost  can I remember a TV show so committed to immersing its audience into the physical space it inhabits. (Indeed, Westworld  can also be viewed as a meta commentary on the art of screenwriting itself: brainstorming narratives, building characters, all for the amusement of other people.)

Westworld  is especially impressive because it builds two worlds at once: the Western theme park and the futuristic workplace. The Western half of Westworld  might be the more purely entertaining of the two, with its shootouts and heists and chases through sublime desert vistas. Behind the scenes, the theme park’s workers show how the robot sausage is made. And as a dystopian office drama, the show does something truly original.

Adam Epstein at QUARTZ, October 1, 2016

"… committed to immersing its audience
  into the physical space it inhabits…."

See also, in this journal, the Mimsy Cube

"Mimsy Were the Borogoves,"
classic science fiction story:

"… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."

A Crystal Block —

Cube, 4x4x4

Saturday, September 24, 2016

The Seven Seals

Filed under: Uncategorized — m759 @ 7:23 AM

From Hermann Weyl's 1952 classic Symmetry —

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

Some Galois geometry —

See the previous post for more narrative.

Core Structure

Filed under: Uncategorized — m759 @ 6:40 AM

For the director of "Interstellar" and "Inception"

At the core of the 4x4x4 cube is …

 


                                                      Cover modified.

The Eightfold Cube

Thursday, September 22, 2016

Binary Opposition Illustrated

Filed under: Uncategorized — m759 @ 12:00 PM

Click the above image for remarks on
"deep structure" and binary opposition.

See also the eightfold cube.

Saturday, September 17, 2016

Interior/Exterior

Filed under: Uncategorized — m759 @ 12:25 AM


3x3x3 Galois cube, gray and white

Thursday, September 15, 2016

Metaphysics at Notre Dame

Filed under: Uncategorized — Tags: — m759 @ 11:07 PM

Recommended reading —

"When Analogies Fail," by Alexander Stern,
a doctoral candidate in philosophy at Notre Dame, in
The Chronicle of Higher Education  online September 11, 2016.

Related material —

That same Alexander Stern in this  journal on April 17, 2016:

See also the eightfold cube in the previous post,
Metaphysics at Scientific American:

Monday, September 12, 2016

The Kummer Lattice

Filed under: Uncategorized — Tags: , — m759 @ 2:00 PM

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Friday, September 2, 2016

Raiders of the Lost Birthday

Filed under: Uncategorized — m759 @ 10:00 AM

Some images from the posts of last July 13
(Harrison Ford's birthday) may serve as funeral
ornaments for the late Prof. David Lavery.

IMAGE- Massimo Vignelli, his wife Lella, and cube

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

See as well posts on "Silent Snow" and "Starlight Like Intuition."

Wednesday, August 31, 2016

The Lost Crucible

Filed under: Uncategorized — m759 @ 4:56 PM

Yesterday's post The Eightfold Cube in Oslo suggests a review of
posts that mention The Lost Crucible.

(The crucible in question is from a book by Katherine Neville, 
The Eight . Any connection with Arthur Miller's play  "The Crucible" 
is purely coincidental.)

Tuesday, August 30, 2016

The Eightfold Cube in Oslo

Filed under: Uncategorized — 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 29, 2016

Roll Credits

Filed under: Uncategorized — m759 @ 9:29 AM

Click images for some backstories.

  

  

    

Related material: The Wet Hot Summa.

Saturday, August 27, 2016

Incarnation

Filed under: Uncategorized — Tags: — m759 @ 1:06 AM

See a search for the title in this journal.

Related material:

The incarnation of three permutations,
named A, B, and C,
on the 7-set of digits {1, 2, 3, 4, 5, 6, 7}
as  permutations on the eightfold cube.

See Minimal ABC Art, a post of August 22, 2016.

Monday, August 22, 2016

Minimal ABC Art

Filed under: Uncategorized — Tags: — m759 @ 7:00 PM

Two portions of a post from Guy Fawkes Day 2015

 

Other art for Guy Fawkes Day

Cloak and Dagger

Sunday, August 21, 2016

Imperium Emporium

Filed under: Uncategorized — m759 @ 11:30 PM

Design Cube 2x2x2 for demonstrating Galois geometry

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

Wednesday, August 10, 2016

Logos and Logic

Filed under: Uncategorized — m759 @ 8:11 PM

"Logos and logic, crystal hypothesis,
 Incipit and a form to speak the word
 And every latent double in the word…."

— Wallace Stevens,
    "Notes Toward a Supreme Fiction,"
     Section I, Canto VIII

    

Narratives

Filed under: Uncategorized — m759 @ 1:00 PM

The novel Blood on Snow , set in Oslo, was published
by Knopf on April 7, 2015.  This journal on that date —

Log24 on Tuesday, April 7, 2015

Logic

Filed under: Uncategorized — m759 @ 7:00 PM 

Seven years ago in this journal —

The above links:  the Stone,  the rules.

A related image —

Monday, August 8, 2016

A Point of Identity

Filed under: Uncategorized — m759 @ 6:00 PM

For a  Monkey Grammarian  (Viennese Version)

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

A logo that may be interpreted as one-eighth of a 2x2x2 array
of cubes —

The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).

Shown below are a few variations on the figure by VCQ,
the Vienna Center for Quantum Science and Technology —
 

(Click image to enlarge.)

Saturday, August 6, 2016

Mystic Correspondence:

Filed under: Uncategorized — m759 @ 12:00 PM

The Cube and the Hexagram

The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic 
hexagram, or Star of David, with the six cube faces 
mapped to the six hexagram lines and the twelve  
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large). 

Exercise:  Is this noteworthy mapping** of faces to lines, 
edges to points, and vertices to triangles an isolated 
phenomenon, or can it be viewed in a larger context?

* See the discussion at magic-squares.net of
   "perimeter-magic cubes"

** Apparently derived from the Cube + Hexagon figure
    discussed here in various earlier posts. See also
    "Diamonds and Whirls," a note from 1984.

Friday, August 5, 2016

Sleight of Post

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

From an earlier Log24 post —

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 

From a post of the next day, July 12, 2014 —

"So there are several different genres and tones
jostling for prominence within Lexicon :
a conspiracy thriller, an almost abstract debate
about what language can do, and an ironic
questioning of some of the things it’s currently used for."

Graham Sleight in The Washington Post 
     a year earlier, on July 15, 2013

For the Church of Synchronology, from Log24 on the next day — 

From a post titled Circles on the date of Marc Simont's death —

See as well Verhexung  in this journal.

Tuesday, August 2, 2016

Notes Toward a Dark Tower*

Filed under: Uncategorized — m759 @ 1:00 PM

Or:  Shema, SXSW

The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelve-foot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralow-frequency vibration of a massive engine. I drift toward the box, mesmerized.

— Crouch, Blake. Dark Matter: A Novel
(Kindle Locations 2004-2010).
Crown/Archetype. Kindle Edition. 

Related reading —

"Do you know there is a deliberate sinister conspiracy at work?"

"No, but hum a few bars and I'll fake it."

A few bars —

* Not the Dark Tower of Stephen King, but that of the
   University of Texas at Austin, back in time 50 years and a day.

Monday, August 1, 2016

Cube

Filed under: Uncategorized — m759 @ 10:28 PM

From this journal —

See (for instance) Sacred Order, July 18, 2006 —

The finite Galois affine space with 64 points

From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —

The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelve-foot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralow-frequency vibration of a massive engine. I drift toward the box, mesmerized.

— Crouch, Blake. Dark Matter: A Novel
(Kindle Locations 2004-2010).
Crown/Archetype. Kindle Edition. 

See also Log24 on the publication date of Dark Matter .

Sunday, July 17, 2016

Symmetries and Correspondences

Filed under: Uncategorized — m759 @ 8:19 AM

Continued from  July 14, 2016


 

Symmetries and Correspondences in 1879 —

Cyparissos Stephanos
Sur les systèmes desmiques de trois tétraèdres
Bulletin des sciences mathématiques
et astronomiques 2e série
,
Tome 3, No. 1 (1879), pp. 424-456.
<http://www.numdam.org/item?id=BSMA_1879_2_3_1_424_1>
© Gauthier-Villars, 1879, tous droits réservés.


 

Symmetries and Correspondences in 1905 —

Friday, July 15, 2016

Autistic Enchantment*

Filed under: Uncategorized — m759 @ 10:29 AM

Robert Nye, author of the novel Falstaffreportedly died
at 77 on July 2, 2016.

Harvey D. Heinz, expert on magic squares, cubes,
tesseracts, etc., reportedly died at 82 on July 6, 2013.

In memoriam —

From the date of Nye's death:

From Nye's book:

From the date of Heinz's death:

* See also a search for the title in this journal.

Thursday, July 14, 2016

Symmetries and Correspondences

Filed under: Uncategorized — m759 @ 7:30 PM

The title is that of a large-scale British research project
in mathematics. On a more modest scale

"Hanks + Cube" in this journal —

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

Block That Metaphor

Meditation from an April 1

Filed under: Uncategorized — 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.

Wednesday, July 13, 2016

Luminosity

Filed under: Uncategorized — Tags: — m759 @ 10:00 PM
 

"At CERN the LHC has reached design luminosity,
and is breaking records with a fast pace of new
collisions. This may have something to do with the
report that the LHC is also about to tear open
a portal to another dimension
."

— Peter Woit, Thursday, June 30, 2016,
    at 1:01 PM ET 

Another sort of design luminosity —

IMAGE- Massimo Vignelli, his wife Lella, and cube

Block That Metaphor

Filed under: Uncategorized — Tags: — m759 @ 8:38 PM

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points (The 6 cube faces are mapped to the 6 hexagram lines.)

Happy dies natalis  to the late Frida Kahlo.

Friday, July 1, 2016

Transparent Core

Filed under: Uncategorized — m759 @ 2:28 PM

"At the point of convergence the play of similarities and differences
cancels itself out in order that identity alone may shine forth.
The illusion of motionlessness, the play of mirrors of the one:
identity is completely empty; it is a crystallization and
in its transparent core the movement of analogy begins all over
once again." — The Monkey Grammarian  by Octavio Paz,
translated by Helen Lane 

A more specific "transparent core" —

See all references to this figure
in this journal.

For a more specific "monkey grammarian," 
see W. Tecumseh Fitch in this journal.

Thursday, June 30, 2016

Design Luminosity

Filed under: Uncategorized — Tags: — m759 @ 3:13 PM

Peter Woit today

"At CERN the LHC has reached design luminosity,* and is
breaking records with a fast pace of new collisions. This may
have something to do with the report that the LHC is also 
about to tear open a portal to another dimension." 

See also the following figure from the Log24 Bion posts

— and Greg Egan's short story "Luminous":

"The theory was, we’d located part of the boundary
between two incompatible systems of mathematics –
both of which were physically true, in their respective
domains. Any sequence of deductions which stayed
entirely on one side of the defect – whether it was the
'near side', where conventional arithmetic applied, or
the 'far side', where the alternative took over – would
be free from contradictions. But any sequence which
crossed the border would give rise to absurdities –
hence S could lead to not-S."

Greg Egan, Luminous
   (Kindle Locations 1284-1288). 

* See a definition.

Rubik vs. Galois: Preconception vs. Pre-conception

Filed under: Uncategorized — Tags: — m759 @ 1:20 PM

From Psychoanalytic Aesthetics: The British School ,
by Nicola Glover, Chapter 4  —

In his last theoretical book, Attention and Interpretation  (1970), Bion has clearly cast off the mathematical and scientific scaffolding of his earlier writings and moved into the aesthetic and mystical domain. He builds upon the central role of aesthetic intuition and the Keats's notion of the 'Language of Achievement', which

… includes language that is both
a prelude to action and itself a kind of action;
the meeting of psycho-analyst and analysand
is itself an example of this language.29.

Bion distinguishes it from the kind of language which is a substitute  for thought and action, a blocking of achievement which is lies [sic ] in the realm of 'preconception' – mindlessness as opposed to mindfulness. The articulation of this language is possible only through love and gratitude; the forces of envy and greed are inimical to it..

This language is expressed only by one who has cast off the 'bondage of memory and desire'. He advised analysts (and this has caused a certain amount of controversy) to free themselves from the tyranny of the past and the future; for Bion believed that in order to make deep contact with the patient's unconscious the analyst must rid himself of all preconceptions about his patient – this superhuman task means abandoning even the desire to cure . The analyst should suspend memories of past experiences with his patient which could act as restricting the evolution of truth. The task of the analyst is to patiently 'wait for a pattern to emerge'. For as T.S. Eliot recognised in Four Quartets , 'only by the form, the pattern / Can words or music reach/ The stillness'.30. The poet also understood that 'knowledge' (in Bion's sense of it designating a 'preconception' which blocks  thought, as opposed to his designation of a 'pre -conception' which awaits  its sensory realisation), 'imposes a pattern and falsifies'

For the pattern is new in every moment
And every moment is a new and shocking
Valuation of all we have ever been.31.

The analyst, by freeing himself from the 'enchainment to past and future', casts off the arbitrary pattern and waits for new aesthetic form to emerge, which will (it is hoped) transform the content of the analytic encounter.

29. Attention and Interpretation  (Tavistock, 1970), p. 125

30. Collected Poems  (Faber, 1985), p. 194.

31. Ibid., p. 199.

See also the previous posts now tagged Bion.

Preconception  as mindlessness is illustrated by Rubik's cube, and
"pre -conception" as mindfulness is illustrated by n×n×n Froebel  cubes
for n= 1, 2, 3, 4. 

Suitably coordinatized, the Froebel  cubes become Galois  cubes,
and illustrate a new approach to the mathematics of space .

Wednesday, June 29, 2016

Space Jews

Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

For the Feast of SS. Peter and Paul

In memory of Alvin Toffler and Simon Ramo,
a review of figures from the midnight that began
the date of their deaths, June 27, 2016 —

http://www.log24.com/log/pix11A/110427-Cube27.jpg

   The 3×3×3 Galois Cube

See also Rubik in this journal.

Monday, June 27, 2016

Interplay

Filed under: Uncategorized — Tags: — m759 @ 12:00 AM

From a search in this journal for Euclid + Galois + Interplay

http://www.log24.com/log/pix11A/110427-Cube27.jpg

   The 3×3×3 Galois Cube
 

A tune suggested by the first image above —

Sunday, June 26, 2016

Common Core versus Central Structure

Filed under: Uncategorized — Tags: , — m759 @ 2:56 PM

Rubik's Cube Core Assembly — Swarthmore Cube Project, 2008 —

"Children of the Common Core" —

There is also a central structure within Solomon's  Cube

'Children of the Central Structure,' adapted from 'Children of the Damned'

For a more elaborate entertainment along these lines, see the recent film

"Midnight Special" —

Tuesday, June 21, 2016

The Central Structure

Filed under: Uncategorized — m759 @ 8:00 AM

"The central poem is the poem of the whole,
The poem of the composition of the whole"

— Wallace Stevens, "A Primitive like an Orb"

The symmetries of the central four squares in any pattern
from the 4×4 version of the diamond theorem  extend to
symmetries of the entire pattern.  This is true also of the
central eight cubes in the 4×4×4  Solomon's cube .

Wednesday, June 15, 2016

Rubik’s Deathtrap

Filed under: Uncategorized — Tags: — m759 @ 7:20 PM

The previous post suggests a search in this journal
for "Deathtrap."

"Rubik’s Cube® used by permission
Rubik’s Brand Ltd. www.rubiks.com."

— Bernd Sturmfels, June-July 2016 Notices
of the American Mathematical Society
,
Volume 63, Number 6, page 605

"Tenser, said the Tensor …." — The Demolished Man
 

Max von Sydow in Branded  (2012)

Expanding the Spielraum …

Filed under: Uncategorized — Tags: — m759 @ 2:40 PM

Continues

Rubik cube in the heading of the homepage of the Notices of the American Mathematical Society

"My AMS invited address at the SIAM Annual Meeting July 11–15
in Boston discusses the extension of eigenvectors and singular
vectors from matrices to higher order tensors."

Bernd Sturmfels in the June-July 2016 AMS Notices

See also Sturmfels in this  journal — for instance, in
"Expanding the Spielraum," a post of Feb. 3, 2015 —

Tuesday, June 14, 2016

Model Kit

Filed under: Uncategorized — 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 

Saturday, June 11, 2016

Café Society

Filed under: Uncategorized — m759 @ 9:00 PM

A post linked to here at 9 AM on May 27, 2016 —

Monday, June 6, 2016

Structure and Sense

Filed under: Uncategorized — Tags: — m759 @ 2:01 PM

"… the war of 70-some years ago
has already become something like the Trojan War
had been for the Homeric bards:
a major event in the mythic past
that gives structure and sense to our present reality."

— Justin E. H. Smith, a professor of philosophy at
     the University of Paris 7–Denis Diderot,
     in the New York Times  column "The Stone"
     (print edition published Sunday, June 5, 2016)

In memory of a British playwright who reportedly
died at 90 this morning —

Structure

Sense

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

Sunday, June 5, 2016

Sunday School: Seven Seals

Filed under: Uncategorized — m759 @ 7:00 AM

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

Click image for some background.

See also Standard Disclaimer.

Wednesday, May 25, 2016

Framework

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

"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng  and H. van Dam
February 20, 2009

For one such framework,* see posts from that same date 
four years earlier — February 20, 2005.

* A 4×4 array. See the 19771978, and 1986 versions by 
Steven H. Cullinane,   the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Cullinane, 1978

Cullinane, 1986

Curtis, 1987

Update of 10:42 PM ET on Sunday, June 19, 2016 —

The above images are precursors to

Conway and Sloane, 1988

Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.

Friday, May 13, 2016

Geometry and Kinematics

Filed under: Uncategorized — Tags: — m759 @ 10:31 PM

"Just as both tragedy and comedy can be written
by using the same letters of the alphabet, the vast
variety of events in this world can be realized by
the same atoms through their different arrangements
and movements. Geometry and kinematics, which
were made possible by the void, proved to be still
more important in some way than pure being."

— Werner Heisenberg in Physics and Philosophy

For more about geometry and kinematics, see (for instance)

"An introduction to line geometry with applications,"
by Helmut Pottmann, Martin Peternell, and Bahram Ravani,
Computer-Aided Design  31 (1999), 3-16.

The concepts of line geometry (null point, null plane, null polarity,
linear complex, Klein quadric, etc.) are also of interest in finite  geometry.
Some small finite spaces have as their natural models arrays of cubes .

Sunday, May 8, 2016

The Three Solomons

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

Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.

A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."

From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:

THE SQUARE AND THE CUBE
by Sol LeWitt

"The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed."

"Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America  55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)"

See also the Cullinane models of some small Galois spaces

 Some small Galois spaces (the Cullinane models)

Wednesday, May 4, 2016

Solomon Golomb, 1932-2016

Filed under: Uncategorized — m759 @ 4:00 AM

Material related to the previous post, "Symmetry" —

This is the group of "8 rigid motions
generated by reflections in midplanes"
of "Solomon's Cube."

Material from this journal on May 1, the date of Golomb's death —

"Weitere Informationen zu diesem Themenkreis
finden sich unter http://​www.​encyclopediaofma​th.​org/
​index.​php/​Cullinane_​diamond_​theorem
und
http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

Tuesday, May 3, 2016

Symmetry

Filed under: Uncategorized — m759 @ 12:00 PM

A note related to the diamond theorem and to the site
Finite Geometry of the Square and Cube —

The last link in the previous post leads to a post of last October whose
final link leads, in turn, to a 2009 post titled Summa Mythologica .

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above web page* —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

Update of Sept. 5, 2016 — See also a related remark
by Lévi-Strauss in 1955: "…three different readings
become possible: left to right, top to bottom, front 
to back."

* For the underlying mathematics, see a June 21, 1983, research note

Wednesday, April 27, 2016

Kubrick’s Rube

Filed under: Uncategorized — 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."

Monday, April 25, 2016

Peirce’s Accounts of the Universe

Filed under: Uncategorized — m759 @ 8:19 PM

Compare and contrast Peirce's seven systems of metaphysics with
the seven projective points in a post of March 1, 2010 —

Wikipedia article 'Group theory' with Rubik Cube and quote from Nathan Carter-- 'What is symmetry?'

From my commentary on Carter's question —

Labelings of the eightfold cube

Sunday, April 24, 2016

Prima Materia as Cubes

Filed under: Uncategorized — m759 @ 4:04 AM

Wednesday, April 20, 2016

Symmetric Generation of a Simple Group

Filed under: Uncategorized — m759 @ 9:00 PM

The reference in the previous post to the work of Guitart and 
The Road to Universal Logic  suggests a fiction involving
the symmetric generation of the simple group of order 168.

See The Diamond Archetype and a fictional account of the road to Hell 

'PyrE' in Bester's 'The Stars My Destination'

The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.

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

For related symmetric generation of a much larger group, see Solomon's Cube.

Tuesday, April 19, 2016

The Folding

Filed under: Uncategorized — m759 @ 2:00 PM

(Continued

A recent post about the eightfold cube  suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :

As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding  a 1×8 array from Turyn's
1967 construction of the Golay code.

Folding a 2×4 Curtis array yet again  yields the 2x2x2 eightfold cube .

Those who prefer an entertainment  approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space." 

Monday, April 18, 2016

Toy

Filed under: Uncategorized — m759 @ 9:01 PM

"Finnegan's waterlogged epiphany" 

— Phrase by John Lancaster in a Washington Post  review of an
autobiography by William Finnegan, a book that today won a Pulitzer Prize.

The review is dated August 7, 2015. This  journal on that date

Sunday, April 17, 2016

The Thing and I

Filed under: Uncategorized — m759 @ 2:01 PM

The New York Times  philosophy column yesterday —

The Times's philosophy column "The Stone" is named after the legendary
"philosophers' stone." The column's name, and the title of its essay yesterday
"Is that even a thing?" suggest a review of the eightfold cube  as "The object
most closely resembling a 'philosophers' stone' that I know of" (Page 51 of
the current issue of a Norwegian art quarterly, KUNSTforum.as).

The eightfold cube —

Definition of Epiphany

From James Joyce’s Stephen Hero , first published posthumously in 1944. The excerpt below is from a version edited by John J. Slocum and Herbert Cahoon (New York: New Directions Press, 1959).

Three Times:

… By an epiphany he meant a sudden spiritual manifestation, whether in the vulgarity of speech or of gesture or in a memorable phase of the mind itself. He believed that it was for the man of letters to record these epiphanies with extreme care, seeing that they themselves are the most delicate and evanescent of moments. He told Cranly that the clock of the Ballast Office was capable of an epiphany. Cranly questioned the inscrutable dial of the Ballast Office with his no less inscrutable countenance:

— Yes, said Stephen. I will pass it time after time, allude to it, refer to it, catch a glimpse of it. It is only an item in the catalogue of Dublin’s street furniture. Then all at once I see it and I know at once what it is: epiphany.

— What?

— Imagine my glimpses at that clock as the gropings of a spiritual eye which seeks to adjust its vision to an exact focus. The moment the focus is reached the object is epiphanised. It is just in this epiphany that I find the third, the supreme quality of beauty.

— Yes? said Cranly absently.

— No esthetic theory, pursued Stephen relentlessly, is of any value which investigates with the aid of the lantern of tradition. What we symbolise in black the Chinaman may symbolise in yellow: each has his own tradition. Greek beauty laughs at Coptic beauty and the American Indian derides them both. It is almost impossible to reconcile all tradition whereas it is by no means impossible to find the justification of every form of beauty which has ever been adored on the earth by an examination into the mechanism of esthetic apprehension whether it be dressed in red, white, yellow or black. We have no reason for thinking that the Chinaman has a different system of digestion from that which we have though our diets are quite dissimilar. The apprehensive faculty must be scrutinised in action.

— Yes …

— You know what Aquinas says: The three things requisite for beauty are, integrity, a wholeness, symmetry and radiance. Some day I will expand that sentence into a treatise. Consider the performance of your own mind when confronted with any object, hypothetically beautiful. Your mind to apprehend that object divides the entire universe into two parts, the object, and the void which is not the object. To apprehend it you must lift it away from everything else: and then you perceive that it is one integral thing, that is a  thing. You recognise its integrity. Isn’t that so?

— And then?

— That is the first quality of beauty: it is declared in a simple sudden synthesis of the faculty which apprehends. What then? Analysis then. The mind considers the object in whole and in part, in relation to itself and to other objects, examines the balance of its parts, contemplates the form of the object, traverses every cranny of the structure. So the mind receives the impression of the symmetry of the object. The mind recognises that the object is in the strict sense of the word, a thing , a definitely constituted entity. You see?

— Let us turn back, said Cranly.

They had reached the corner of Grafton St and as the footpath was overcrowded they turned back northwards. Cranly had an inclination to watch the antics of a drunkard who had been ejected from a bar in Suffolk St but Stephen took his arm summarily and led him away.

— Now for the third quality. For a long time I couldn’t make out what Aquinas meant. He uses a figurative word (a very unusual thing for him) but I have solved it. Claritas is quidditas . After the analysis which discovers the second quality the mind makes the only logically possible synthesis and discovers the third quality. This is the moment which I call epiphany. First we recognise that the object is one  integral thing, then we recognise that it is an organised composite structure, a thing  in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that  thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany.

Having finished his argument Stephen walked on in silence. He felt Cranly’s hostility and he accused himself of having cheapened the eternal images of beauty. For the first time, too, he felt slightly awkward in his friend’s company and to restore a mood of flippant familiarity he glanced up at the clock of the Ballast Office and smiled:

— It has not epiphanised yet, he said.

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