Log24

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

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.

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.

Sunday, December 28, 2014

Cube of Ultron

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

The Blacklist “Pilot” Review

"There is an element of camp to this series though. Spader is
quite gleefully channeling Anthony Hopkins, complete with being
a well educated, elegant man locked away in a super-cell.
Speaking of that super-cell, it’s kind of ridiculous. They’ve got him
locked up in an abandoned post office warehouse on a little
platform with a chair inside  a giant metal cube that looks like
it could have been built by Tony Stark. And as Liz approaches
to talk to him, the entire front of the cube  opens and the whole
thing slides back to leave just the platform and chair. Really? 
FUCKING REALLY ? "

Kate Reilly at Geekenstein.com (Sept. 27, 2013)

Friday, December 28, 2012

Cube Koan

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

From Don DeLillo's novel Point Omega —

I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create."

"What reality?"

"This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional. The reality stands, it walks, it squats. Except when it doesn't."

He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku."

I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely.

"Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear."

What's there—

This view of a die's faces 3, 6, and 5, in counter-
clockwise order (see previous post) suggests a way
of labeling the eight corners  of a die (or cube):

123, 135, 142, 154, 246, 263, 365, 456.

Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counter-clockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertex-labeling may be used in describing 
the automorphisms of the order-8 quaternion group.

For a more literary approach to quaternions, see
Pynchon's novel Against the Day .

* From Peter J. Cameron's weblog:

  "The big name associated with this is Major MacMahon,
   an associate of Hardy, Littlewood and Ramanujan,
   of whom Robert Kanigel said,

His expertise lay in combinatorics, a sort of
glorified dice-throwing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.

   Glorified dice-throwing, indeed…"

Sunday, August 5, 2012

Cube Partitions

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

The second Logos  figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—

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

Click for further details.

Thursday, July 26, 2012

Solomon’s Seal

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

(Mathematics and Narrative, continued)

Narrative—

The Ring and The Stone from yesterday's post, and…

"In Medieval Jewish, Christian and Islamic legends,
the Seal of Solomon was a magical signet ring
said to have been possessed by King Solomon…."

— Wikipedia article, Seal of Solomon

Mathematics—

IMAGE- Eric Temple Bell on the mathematics of 'Solomon's Seal' (in his 'Development of Mathematics')

A fact related to the mathematical
"Solomon's seal" described above by Bell:

IMAGE- J.W.P. Hirschfeld on the mathematics of 'Solomon's Seal', with reference to Edge on the same topic

The reference to Edge is as follows—

[3] Edge, W. L., Quadrics over GF(2) and
their relevance for the cubic surface group
,
Canadian J. Maths. 11 (1959) ….

(This reference relates Hirschfeld's remarks
quoted above to the 64-point affine space
illustrated below (via the associated
63-point projective  space PG (5, 2)).

As for the narrative's Stone… 

See Solomon's Cube.

IMAGE- 'Solomon's Cube'

Sunday, February 5, 2012

Cuber

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

(Continued from January 11, 2012)

Wednesday, January 11, 2012

Cuber

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

"Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing.  And once you have made or acquired a new 'cube'… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube.  What is the essence of each operator?  One senses a deep invariant lying somehow 'down underneath' it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment.  In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….

… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube.  It is the  answer; it simply has the right spirit ."

— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern  (Kindle edition, locations 11557-11572)

See also Many Dimensions in this journal and Solomon's Cube.

Friday, December 30, 2011

Quaternions on a Cube

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

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

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

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

See also…

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

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

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

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

Saturday, August 27, 2011

Cosmic Cube*

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

IMAGE- Anthony Hopkins exorcises a Rubik cube

Prequel (Click to enlarge)

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

Background —

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

See also Rubik in this journal.

* For the title, see Groups Acting.

Friday, June 24, 2011

The Cube

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

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

Click the above image for some background.

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

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.”

Monday, November 11, 2019

Time and Chance

Filed under: General — m759 @ 2:49 PM

http://www.log24.com/log/pix10B/101202-DreidelAndStone.jpg

The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel 
Many Dimensions  published in 1993 by Wm. B. Eerdmans.

Compare and constrast —

Goedel Escher Bach cover

Cover of a book by Douglas Hofstadter

IMAGE- 'Solomon's Cube'

An Invariance of Symmetry

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.

Sunday, July 15, 2018

Jewish Oases

Filed under: General,Geometry — m759 @ 10:06 PM

"… Lincoln Plaza Cinemas, the Juilliard String Quartet,
and the Strand Book Store remained  oases
for cultural and intellectual stimulation."

John S. Friedman in The Forward , Jan. 21, 2018

Read more: 

https://forward.com/culture/392483/
how-fred-bass-dan-talbot-robert-mann
-shaped-new-york-culture/

From  the Oasis  in Steven Spielberg's "Ready Player One" (2018) —

I prefer, from a Log24 search for Flux Capacitor

Symbologist Robert Langdon views a corner of Solomon's Cube

From "Raiders of the Lost Images" —

"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."

Thursday, June 7, 2018

For Dan Brown

Filed under: General,Geometry — Tags: — m759 @ 1:09 PM

See also Eightfold Trinity in this  journal.

Symbologist Robert Langdon views a corner of Solomon's Cube

Saturday, May 19, 2018

Flux Capacitor

Filed under: General,Geometry — Tags: , — m759 @ 4:13 PM

For Tom Hanks and Dan Brown —

Symbologist Robert Langdon views a corner of Solomon's Cube

From "Raiders of the Lost Images" —

"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."

See as well a Google search for flux philosophy
https://www.google.com/search?q=flux+philosophy.

Saturday, April 7, 2018

Sides

Filed under: General,Geometry — Tags: — m759 @ 11:47 AM

The FBI holding cube in "The Blacklist" —

" 'The Front' is not the whole story . . . ."

— Vincent Canby, New York Times  film review, 1976,
     as quoted in Wikipedia.

See also Solomon's Cube in this  journal.

IMAGE- 'Solomon's Cube'

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 4x4x4 design cube —

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

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

Tuesday, January 16, 2018

The Pentagram Papers

Filed under: General,Geometry — Tags: — m759 @ 9:16 AM

Other intersection-points-counting material —

The Finkelstein Talisman:

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

See also Hanks + Cube in this journal —

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

Thursday, December 21, 2017

For Winter Solstice 2017

Filed under: G-Notes,General,Geometry — m759 @ 10:30 AM

A review —

Some context —

Webpage demonstrating symmetries of 'Solomon's Cube'

Monday, July 24, 2017

Penguin Classics Deluxe Edition

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

The above title was suggested by a film trailer quoted here Saturday

" Jeremy Irons' dry Alfred Pennyworth:
'One misses the days when one's biggest concerns
were exploding wind-up penguins.' "

"Penguin Classics Deluxe Edition" describes, among other books,
an edition of the I Ching  published on December 1, 2015.

Excerpt from this journal on that date

Tuesday, December 1, 2015

Verhexung

Filed under: Uncategorized — m759 @ 9:00 PM 

(Continued)

"The positional meaning of a symbol derives from
its relationship to other symbols in a totality, a Gestalt,
whose elements acquire their significance from the
system as a whole."

— Victor Turner, The Forest of Symbols , Ithaca, NY,
Cornell University Press, 1967, p. 51, quoted by
Beth Barrie in "Victor Turner."

(Turner pioneered the use of the term "symbology,"
a term later applied by Dan Brown to a fictional
scholarly pursuit at Harvard.)

. . . .

Related material —

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

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

Thursday, June 22, 2017

Face Henge

Filed under: General,Geometry — Tags: — m759 @ 3:07 PM

With a hat tip to Vinnie Mancuso

Wednesday, June 21, 2017

Concept and Realization

Filed under: General,Geometry — 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 .

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'

Friday, March 10, 2017

Transformers

Filed under: General,Geometry — 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 2, 2017

Stories

Filed under: General,Geometry — 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: General,Geometry — 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 18, 2017

Solid Symmetry (continued)

Filed under: General,Geometry — 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: General,Geometry — 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.

Monday, October 3, 2016

Ein Eck

Filed under: General,Geometry — 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 

Friday, August 5, 2016

Sleight of Post

Filed under: General,Geometry — 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.

Thursday, July 14, 2016

Symmetries and Correspondences

Filed under: General,Geometry — Tags: — 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

Tuesday, May 3, 2016

Symmetry

Filed under: General,Geometry — 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, December 2, 2015

Symbology

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

Symbologist Robert Langdon views a corner of Solomon's Cube

Click image to search Log24
for Solomon + Stone.

Saturday, April 25, 2015

Ghosts and Shadows

Filed under: General,Geometry — m759 @ 5:31 PM

For Poetry Month

From the home page of Alexandre Borovik:

Book in progress: Shadows of the Truth

This book (to be published soon) can be viewed
as a sequel to Mathematics under the Microscope ,
but with focus shifted on mathematics as it was
experienced by children (well, by children who
became mathematicians). The cover is designed
by Edmund Harriss.

See also Harriss's weblog post of Dec. 27, 2008, on the death
of Harold Pinter: "The Search for the Truth Can Never Stop."

This suggests a review of my own post of Dec. 3, 2012,
"The Revisiting." A figure from that post:

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

Wednesday, September 17, 2014

Raiders of the Lost Articulation

Filed under: General,Geometry — Tags: , , , , — m759 @ 6:14 PM

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

Saturday, July 12, 2014

Mars Package

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

For Ursula K. Le Guin

“For me it is a sign that we have fundamentally different
conceptions of the work of the intelligence services.”

— Germany’s Chancellor Angela Merkel in
theguardian.com, Saturday, 12 July 2014, 14.32 EDT

Another sort of service, thanks to Dan Brown and Tom Hanks:

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM 

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

Sequel

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

A sequel to the 1974 film
Thunderbolt and Lightfoot :

Contingent and Fluky

Some variations on a thunderbolt  theme:

Design Cube 2x2x2 for demonstrating Galois geometry

These variations also exemplify the larger
Verbum  theme:

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

A search today for Verbum  in this journal yielded
a Georgetown 
University Chomskyite, Professor
David W. Lightfoot.

"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."

Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.

See Tecumseh Fitch  in this journal.

For other material related to the large Verbum  cube,
see posts for the 18th birthday of Harry Potter.

That birthday was also the upload date for the following:

See esp. the comments section.

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

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

See Cube Symbology.

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

Da hats ein Eck 

Friday, May 9, 2014

Models of Everything

Filed under: General,Geometry — m759 @ 11:16 AM

“The About page contains detailed descriptions of the project….”

The Illustris project on constructing a model of the universe

For the mathematics of a simpler traditional Chinese model
of everything, see

Thursday, March 27, 2014

Diamond Space

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

(Continued)

Definition:  A diamond space  — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.

The reason for the name:

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

Click to enlarge.

Saturday, April 21, 2012

Finding a Form

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


In "Contact," Dr. Arroway  is shown the key to the Primer

In this journal, fictional symbologist Robert Langdon is shown a cube

Symbologist Robert Langdon views a corner of Solomon's Cube

"Confusion is nothing new." — Song lyric

Wednesday, April 18, 2012

Adam in Eden

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

…. and John Golding, an authority on Cubism who "courted abstraction"—

"Adam in Eden was the father of Descartes." — Wallace Stevens

Fictional symbologist Robert Langdon and a cube

Symbologist Robert Langdon views a corner of Solomon's Cube

From a Log24 post, "Eightfold Cube Revisited,"
on the date of Golding's death—

Dynkin diagram D4 for triality

A related quotation—

"… quaternions provide a useful paradigm
  for studying the phenomenon of 'triality.'"

  — David A. Richter's webpage Zometool Triality

See also quaternions in another Log24 post
from the date of Golding's death— Easter Act.

Monday, April 16, 2012

Carroll Thanks the Academy

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

Gary Gutting, "Arguing About Language," in "The Stone,"
The New York Times  philosophy column, yesterday—

There's a sense in which we speak language
and a sense in which, in Mallarmé's famous phrase,
“language itself speaks.”

Famous? A Google Book Search for

"language itself speaks" Mallarmé

yields 2 results, neither helpful.

But a Google Book Search for

"language itself speaks" Heidegger

yields "about 312 results."

A related search yields the following

Paul Valéry, encountering Un Coup de Dés  in Mallarmé’s worksheets in 1897, described the text as tracing the pattern of thought itself:

It seemed to me that I was looking at the form and pattern of a thought, placed for the first time in finite space. Here space itself truly spoke, dreamed, and gave birth to temporal forms….

… there in the same void with them, like some new form of matter arranged in systems or masses or trailing lines, coexisted the Word! (Leonardo  309*)

* The page number is apparently a reference to The Collected Works of Paul Valéry: Leonardo, Poe, Mallarmé , translated by Malcolm Cowley and James R. Lawler, Princeton University Press, 1972. (As a temporal  form, "309" might be interpreted as a reference to 3/09, March 9, the date of a webpage on the Void.)

For example—

Symbologist Robert Langdon views a corner of Solomon's Cube

Background:
Deconstructing Alice
and Symbology.

Wednesday, September 1, 2010

September Morn

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

For Alyssa Milano —

http://www.log24.com/log/pix10B/100901-MilanoFork.jpg

The Forking

(Click here for cheesy Neil Diamond background music.)

For some related philosophical remarks, see Deconstructing Alice

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

and the new Pythagorean thriller The Thousand.

Wednesday, June 16, 2010

Brightness at Noon

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

David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —

Image-- Arthur Koestler by David Levine, NY Review of Books, Dec. 17, 1964, review of 'The Act of Creation'

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

Image-- The 64 I Ching hexagrams in the 4 layers of the Cullinane cube

Geometry of the I Ching

See also this morning's post as well as
Monday's post quoting George David Birkhoff

"If I were a Leibnizian mystic… I would say that…
God thinks multi-dimensionally — that is,
uses multi-dimensional symbols beyond our grasp."

Geometry of Language

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

(Continued from April 23, 2009, and February 13, 2010.)

Paul Valéry as quoted in yesterday’s post:

“The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (Cahiers, 15:170 [2: 315])

The geometric example discussed here yesterday as a Self symbol may seem too small to be really impressive. Here is a larger example from the Chinese, rather than European, tradition. It may be regarded as a way of representing the Galois field GF(64). (“Field” is a rather ambiguous term; here it does not, of course, mean what it did in the Valéry quotation.)

From Geometry of the I Ching

Image-- The 64 hexagrams of the I Ching

The above 64 hexagrams may also be regarded as
the finite affine space AG(6,2)— a larger version
of the finite affine space AG(4,2) in yesterday’s post.
That smaller space has a group of 322,560 symmetries.
The larger hexagram  space has a group of
1,290,157,424,640 affine symmetries.

From a paper on GL(6,2), the symmetry group
of the corresponding projective  space PG(5,2),*
which has 1/64 as many symmetries—

(Click to enlarge.)

Image-- Classes of the Group GL(6,)

For some narrative in the European  tradition
related to this geometry, see Solomon’s Cube.

* Update of July 29, 2011: The “PG(5,2)” above is a correction from an earlier error.

Tuesday, May 4, 2010

Mathematics and Narrative, continued

Filed under: General,Geometry — Tags: — m759 @ 8:28 PM

Romancing the
Non-Euclidean Hyperspace

Backstory
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937

The cast of 1937's 'King Solomon's Mines' goes back to the future

For the 1937 grid, see Diamond Theory, 1937.

The grid is, as Mere Geometry points out, a non-Euclidean hyperspace.

For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.

Friday, March 19, 2010

Garden of Forking Paths

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

For Alyssa

 

 An Old Magic Symbol

http://www.log24.com/log/pix10/100319-Palermo.gif

… and for Dan Brown —

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

Saturday, March 6, 2010

Deconstructing Alice

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

Alyssa is  Wonderland

Manohla Dargis in The New York Times  yesterday

"Of course the character of Carroll’s original Alice is evident in each outrageous creation she dreams up in 'Wonderland' and in the sequel, 'Through the Looking-Glass,' which means that she’s a straight man to her own imagination. (She is  Wonderland.)"

Alyssa Milano as a child, with fork

From Inside the White Cube

"The sacramental nature of the space becomes clear, and so does one of the great projective laws of modernism: as modernism gets older, context becomes content. In a peculiar reversal, the object introduced into the gallery 'frames' the gallery and its laws."

From Yogi Berra–

"When you come to a fork in the road, take it."

Related material:  For Baron Samedi and…

Symbology
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube
Jacques Derrida on the Looking-Glass garden, 'The Time before First,' and Solomon's seal

Thursday, February 18, 2010

Theories: An Outline

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

Truth, Geometry, Algebra

The following notes are related to A Simple Reflection Group of Order 168.

1. According to H.S.M. Coxeter and Richard J. Trudeau

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

— Coxeter, 1987, introduction to Trudeau’s The Non-Euclidean Revolution

1.1 Trudeau’s Diamond Theory of Truth

1.2 Trudeau’s Story Theory of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 Coxeter Theory according to Borovik

2.1.1 The Geometry–

Mirror Systems in Coxeter Theory

2.1.2 The Algebra–

Coxeter Languages in Coxeter Theory

2.2 Diamond Theory according to Cullinane

2.2.1 The Geometry–

Examples: Eightfold Cube and Solomon’s Cube

2.2.2 The Algebra–

Examples: Cullinane and (rather indirectly related) Gerhard Grams

Summary of the story thus far:

Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side.  Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)

Tuesday, January 26, 2010

Symbology

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

From this journal:

Friday December 5, 2008

m759 @ 1:06 PM
 
Mirror-Play of
the Fourfold

For an excellent commentary
 on this concept of Heidegger,

View selected pages
from the book

Dionysus Reborn:

Play and the Aesthetic Dimension
in Modern Philosophical and
Scientific Discourse

(Mihai I. Spariosu,
Cornell U. Press, 1989)

Related material:
the logo for a
web page

Logo for 'Elements of Finite Geometry'

– and Theme and Variations.

Transition to the
Garden of Forking Paths–

(See For Baron Samedi)–

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

and Dissemination, by Jacques Derrida,
translated by Barbara Johnson,
London, Athlone Press, 1981–

Pages 354-355
On the mirror-play of the fourfold

Pages 356-357
Shaking up a whole culture

Pages 358-359
Cornerstone and crossroads

Pages 360-361
A deep impression embedded in stone

Pages 362-363
A certain Y, a certain V

Pages 364-365
The world is Zeus's play

Page 366
It was necessary to begin again

 

Saturday, January 23, 2010

For Baron Samedi

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

Yesterday's Times —

NY Times banner with Eve and apple

Today's Times —

NY Times ad for Goldstein's '36 Arguments'-- 'Deconstruct the Arguments'

   Annals of Deconstruction —

Click on image for background.

New Yorker cover on Haiti featuring Baron Samedi

Related material
   for Baron Samedi

The Found Symbol
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube
Jacques Derrida on the Looking-Glass garden, 'The Time before First,' and Solomon's seal

Saturday, November 14, 2009

Mathematics and Narrative, continued:

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

A graphic novel reviewed in the current Washington Post  features Alfred North Whitehead and Bertrand Russell–

Whitehead and Russell, 'Logicomix' page 181

Related material:

Whitehead on Fano’s finite projective three-space:

“This is proved by the consideration of a three dimensional geometry in which there are only fifteen points.”

The Axioms of Projective Geometry , Cambridge University Press, 1906

A related affine six-space:

Grey cube, 4x4x4

Further reading:

See Solomon’s Cube and the link at the end of today’s previous entry, then compare and contrast the above portraits of Whitehead and Russell with Charles Williams’s portraits of Sir Giles Tumulty and Lord Arglay in the novel Many Dimensions .

It was a dark and stormy night….

Thursday, September 17, 2009

Thursday September 17, 2009

Filed under: General,Geometry — m759 @ 11:07 AM
Symbologist Robert Langdon and a corner of Solomon's Cube

Patrick Swayze and Jennifer Grey in  'Dirty Dancing'

“Nobody puts Baby in a corner.”

Wednesday, September 16, 2009

Wednesday September 16, 2009

Filed under: General,Geometry — m759 @ 11:07 AM
The Found Symbol
Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube
Jacques Derrida on the Looking-Glass garden, 'The Time before First,' and Solomon's seal

Saturday, September 5, 2009

Saturday September 5, 2009

Filed under: General,Geometry — Tags: — m759 @ 10:31 PM
For the
Burning Man

'The Stars My Destination,' current edition (with cover slightly changed)

(Cover slightly changed.)

 
Background —

 
SAT
 
Part I:

Sophists (August 20th)

Part II:

VERBUM
SAT
SAPIENTI

Escher's 'Verbum'

Escher's Verbum


Solomon's Cube



Part III:

From August 25th

Equilateral triangle on a cube, each side's length equal to the square root of two

"Boo, boo, boo,
  square root of two.
"

Saturday, December 6, 2008

Saturday December 6, 2008

Filed under: General,Geometry — m759 @ 2:01 PM
Another Opening,
Another Show

“While feasts of Saint Nicholas are not observed nationally, cities with strong German influences like Milwaukee, Cincinnati, and St. Louis celebrate St. Nick’s Day on a scale similar to the German custom.” —Wikipedia

A footprint from Germany:

Germany
Python-urllib
/504856559/item.html 12/6/2008
1:21 PM

The link in the above footprint leads
to an entry of July 5, 2006.

The access method:

The urllib Module

“The Python urllib module implements a fairly high-level abstraction for making any web object with a URL act like a Python file: i.e., you open it, and get back an object….”

For more pictures and discussion
of the object fetched by Python,
see Anti-Christmas 2007.

For a larger and more sophisticated
relative of that object,
 see Solomon’s Cube and
the related three presents
from the German link’s target:

Spellbound: A trinity of Christmas presents

1. Many Dimensions
2. Boggle
3. My Space

Saturday, May 10, 2008

Saturday May 10, 2008

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

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

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

RELATED MATERIAL

Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

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

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in a book from 1906

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

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

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

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

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

For some mathematical background, see

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

Monday, July 23, 2007

Monday July 23, 2007

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

Greetings.

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

Jorge Luis Borges, “Avatars of the Tortoise”

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

Jorge Luis Borges, “Los Avatares de la Tortuga

Autonomous Apparition
 
 

At Midsummer Noon:

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

 

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

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

– Wallace Stevens, “The Rock”

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

Hofstadter’s cover.

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

Click on image for details.

Sunday, December 10, 2006

Sunday December 10, 2006

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

Time and Chance
on the 90th Birthday
of Kirk Douglas,
star of
The Garden of Allah

The Lottery 12/9/06 Mid-day Evening
New York 036

See

The Quest
for the 36

331

See 3/31

“square crystal” and “the symbolism could not have been more perfect.”

Pennsylvania 602

See 6/02

Walter Benjamin
on
“Adamic language.”

111

See 1/11

“Related material:
Jung’s Imago and Solomon’s Cube.”

See also

The image “http://www.log24.com/log/pix05B/051209-Douglas1.jpg” cannot be displayed, because it contains errors.

Diamonds

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

Thursday, October 5, 2006

Thursday October 5, 2006

Filed under: General,Geometry — m759 @ 9:11 AM
In Touch with God

(Title of an interview with
the late Paul Halmos, mathematician)

Since Halmos died on Yom Kippur, his thoughts on God may be of interest to some.

From a 1990 interview:

“What’s the best part of being a mathematician? I’m not a religious man, but it’s almost like being in touch with God when you’re thinking about mathematics. God is keeping secrets from us, and it’s fun to try to learn some of the secrets.”

I personally prefer Annie Dillard on God:

“… if Holy the Firm is matter at its dullest, Aristotle’s materia prima, absolute zero, and since Holy the Firm is in touch with the Absolute at base, then the circle is unbroken.  And it is…. Holy the Firm is in short the philosopher’s stone.”

Some other versions of
the philosopher’s stone:

The image �http://www.log24.com/log/pix06/060101-SixOfOne.jpg� cannot be displayed, because it contains errors.

And, more simply,
April 28, 2004:

This last has the virtue of
being connected with Halmos
via his remarks during the
“In Touch with God” interview:

“…at the root of all deep mathematics there is a combinatorial insight… the really original, really deep insights are always combinatorial….”
 
“Combinatorics, the finite case, is where the genuine, deep insight is.”

See also the remark of Halmos that serves as an epigraph to Theme and Variations.

Finally, it should be noted that
the 4×9 black rectangle

The image “http://www.log24.com/log/pix06A/061004-Halmos100x225.jpg” cannot be displayed, because it contains errors.

has also served
at least one interpreter
as a philosopher’s stone,
and is also the original
“Halmos tombstone.”

(See previous entry.)

Wednesday, September 20, 2006

Wednesday September 20, 2006

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

Public Space

“… the Danish cartoons crisis last March showed ‘two world views colliding in public space with no common point of reference.'”

George Carey, Archbishop of Canterbury from 1991 to 2002, quoted in today’s London Times.

Related material:

Geometry and Christianity
   (Google search yielding
    “about 1,540,000” results)

Geometry and Islam
   (Google search yielding
    “about 1,580,000” results)

MySpace.com/affine

The image “http://www.log24.com/log/pix06A/060828-Cube.jpg” cannot be displayed, because it contains errors.

A Public Space

The image “http://www.log24.com/log/pix06A/060920-Motto.jpg” cannot be displayed, because it contains errors.

— Motto of 
Plato’s Academy

Background from
Log24 on Feb. 15, 2006:

  

Hellmut Wilhelm on the Tao

If we replace the Chinese word “I” (change, transformation) with the word “permutation,” the relevance of Western mathematics (which some might call “the Logos“) to the I Ching (“Changes Classic”) beomes apparent.

For the relevance of Plato to
Islam, see David Wade’s
Pattern in Islamic Art
and a Google search on
Plato and Islam
(“about 1,680,000” results).

“We should let ourselves be guided by what is common to all. Yet although the Logos is common to all, most men live as if each had a private intelligence of his own.”

Heraclitus of Ephesus, about 500 B.C.

Wednesday, July 5, 2006

Wednesday July 5, 2006

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

And now, from
the author of Sphere

CUBE

He beomes aware of something else… some other presence.
“Anybody here?” he says.
I am here.
He almost jumps, it is so loud. Or it seems loud. Then he wonders if he has heard anything at all.
“Did you speak?”
No.
How are we communicating? he wonders.
The way everything communicates with everything else.
Which way is that?
Why do you ask if you already know the answer?

Sphere, by Michael Crichton, Harvard ’64

“… when I went to Princeton things were completely different. This chapel, for instance– I remember when it was just a clearing, cordoned off with sharp sticks.  Prayer was compulsory back then, and you couldn’t just fake it by moving your lips; you had to know the words, and really mean them.  I’m dating myself, but this was before Jesus Christ.”

Baccalaureate address at Princeton, Pentecost 2006, reprinted in The New Yorker, edited by David Remnick, Princeton ’81

Related figures:

The image “http://www.log24.com/log/pix06A/060617-Spellbound.jpg” cannot be displayed, because it contains errors.

For further details,
see Solomon’s Cube
and myspace.com/affine.

The image “http://www.log24.com/log/pix06A/060705-Cube.jpg” cannot be displayed, because it contains errors.

For further details,
see Jews on Buddhism
and
Adventures in Group Theory.

“In this way we are offered
a formidable lesson
for every Christian community.”

Pope Benedict XVI
on Pentecost,
June 4, 2006,
St. Peter’s Square
.

Saturday, June 17, 2006

Saturday June 17, 2006

Filed under: General,Geometry — m759 @ 7:59 AM
In memory of
Barbara Epstein:
 

Spellbound

“Breaking the spell of religion is a
 game that many people can play.”
— Freeman Dyson in the current
   New York Review of Books

Part I:
The Game

The image “http://www.log24.com/log/pix06A/060617-Boggle.jpg” cannot be displayed, because it contains errors.

Part II:
Many People

The image “http://www.log24.com/log/pix06A/060617-Spellbound.jpg” cannot be displayed, because it contains errors.

For further details,
see Solomon’s Cube
and myspace.com/affine.

“The rock cannot be broken.
It is the truth.”
— Wallace Stevens     

Wednesday, May 10, 2006

Wednesday May 10, 2006

Filed under: General,Geometry — m759 @ 4:29 PM
My Space

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

“… we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections.”
I closed my eyes and thought hard. “Zebbie, I don’t think it can be done. Maybe Escher could have done it.”

— Robert A. Heinlein,
The Number of the Beast

The image “http://www.log24.com/theory/images/Screenshot-64cube.gif” cannot be displayed, because it contains errors.

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the 2-element field).

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

“I laugh because I dare not cry.
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”

— Robert A. Heinlein,
The Number of the Beast

And so…

Compare and contrast:

Solomon’s Cube, the five
Log24 entries ending on 3/14,
and the
American Mathematical Society
on Mathematical Imagery.

Related material:

A more extensive excerpt from
The Number of the Beast, and

Story Theory and
the Number of the Beast.

Thursday, March 9, 2006

Thursday March 9, 2006

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

Finitegeometry.org Update

(Revised May 21, 2006)

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

Wednesday, January 4, 2006

Wednesday January 4, 2006

Filed under: General,Geometry — m759 @ 4:04 AM
Dragon School

In memory of Humphrey Carpenter, author of The Inklings, who attended The Dragon School.  Carpenter died a year ago today.

From Log24 on Nov. 16, 2005:

Images


Adam Gopnik on C. S. Lewis in the New Yorker:

“Lewis began with a number of haunted images….”

“The best of the books are the ones… where the allegory is at a minimum and the images just flow.”

“‘Everything began with images,’ Lewis wrote….”

The image “http://www.log24.com/log/pix05B/051116-Time.jpg” cannot be displayed, because it contains errors.

From Paul Preuss,
Broken Symmetries
(see previous entry):

“Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once.  It was a thin nectar, but he was convinced it was the nectar of the gods….”


From
Verbum Sat Sapienti?

The image “http://www.log24.com/theory/images/EscherVerbum2.jpg” cannot be displayed, because it contains errors.

Escher’s Verbum

The image “http://www.log24.com/theory/images/DTinvar246.jpg” cannot be displayed, because it contains errors.

Solomon’s Cube

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

Geometry of the I Ching

Wednesday, November 9, 2005

Wednesday November 9, 2005

Filed under: General,Geometry — m759 @ 3:09 PM
In honor of the 120th anniversary
of the birth of Hermann Weyl:

Sunday, September 29, 2019

Stage Direction: “Comments Off.”

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

The previous post dealt with "magic" cubes, so called because of the
analogous "magic" squares. Douglas Hofstadter has written about a 
different, physical , object, promoted as "the  Magic Cube," that Hofstadter
felt embodied "a deep invariant":

Wednesday, July 10, 2019

Artifice* of Eternity …

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

… and Schoolgirl Space

"This poem contrasts the prosaic and sensual world of the here and now
with the transcendent and timeless world of beauty in art, and the first line,
'That is no country for old men,' refers to an artless world of impermanence
and sensual pleasure."

— "Yeats' 'Sailing to Byzantium' and McCarthy's No Country for Old Men :
Art and Artifice in the New Novel,"
Steven Frye in The Cormac McCarthy Journal ,
Vol. 5, No. 1 (Spring 2005), pp. 14-20.

See also Schoolgirl Space in this  journal.

* See, for instance, Lewis Hyde on the word "artifice" and . . .

Tuesday, July 9, 2019

Perception of Space

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

(Continued)

The three previous posts have now been tagged . . .

Tetrahedron vs. Square  and  Triangle vs. Cube.

Related material —

Tetrahedron vs. Square:

Labeling the Tetrahedral Model  (Click to enlarge) —

Triangle vs. Cube:

and, from the date of the above John Baez remark —

Dreamtimes

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

“I am always the figure in someone else’s dream. I would really rather
sometimes make my own figures and make my own dreams.”

— John Malkovich at squarespace.com, January 10, 2017

Also on that date . . .

.

Monday, July 8, 2019

Exploring Schoolgirl Space

Filed under: General — Tags: , , , — m759 @ 9:48 AM

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

Sunday, July 7, 2019

Schoolgirl Problem

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

Anonymous remarks on the schoolgirl problem at Wikipedia —

"This solution has a geometric interpretation in connection with 
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing
."

See also Polster + Tetrahedron in this  journal.

There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square  model rather
than a tetrahedral  model. The square  model of PG(3,2) last
appeared in the schoolgirl-problem article on Feb. 11, 2017, just
before a revision that removed it.

Saturday, June 8, 2019

Art Object, continued and continued

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

Notes on a remark by Chuanming Zong

See as well posts mentioning "An Object of Beauty."

Update of 12 AM June 11 — A screenshot of this post 
is now available at  http://dx.doi.org/10.17613/hqk7-nx97 .

Monday, May 13, 2019

Doris Day at the Hudson Rock

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

" 'My public image is unshakably that of
America’s wholesome virgin, the girl next door,
carefree and brimming with happiness,' 
she said in Doris Day: Her Own Story
a 1976 book . . . ."

From "Angels & Demons Meet Hudson Hawk" (March 19, 2013) —

From the March 1 post "Solomon and the Image," a related figure —

Friday, March 1, 2019

Solomon and the Image

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

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

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

Saturday, March 24, 2018

Slight?

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

Sure, Whatever.

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

The search for Langlands in the previous post
yields the following Toronto Star  illustration —

From a review of the recent film "Justice League" —

"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."

For other cubic adventures, see yesterday's post on A Piece of Justice 
and the block patterns in posts tagged Design Cube.

Friday, March 23, 2018

Reciprocity

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

Copy editing — From Wikipedia

"Copy editing (also copy-editing or copyediting, sometimes abbreviated ce)
is the process of reviewing and correcting written material to improve accuracy,
readability, and fitness for its purpose, and to ensure that it is free of error,
omission, inconsistency, and repetition. . . ."

An example of the need for copy editing:

Related material:  Langlands and Reciprocity in this  journal.

Piece Prize

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

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

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

From the Personal to the Platonic

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

On the Oslo artist Josefine Lyche —

"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."

Ann Cathrin Andersen
    http://bryggmagasin.no/2017/behind-the-glitter/

Personal —

The Rushkoff Logo

— From a 2016 graphic novel by Douglas Rushkoff.

See also Rushkoff and Talisman in this journal.

Platonic —

The Diamond Cube.

Compare and contrast the shifting hexagon logo in the Rushkoff novel above 
with the hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).

Thursday, March 22, 2018

In Memoriam

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

Also on March 18, 2015 . . .

Wednesday, March 7, 2018

Unite the Seven.

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


Related material —

The seven points of the Fano plane within 

The Eightfold Cube.
 

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


"Before time began . . . ."

  — Optimus Prime

Monday, January 22, 2018

Hollywood Moment

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

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

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

'Love Story' director dies

An Hispanic Hollywood moment:

Ojo de Dios —

Click for related material.

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

Tuesday, September 19, 2017

Time and Chance

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

(For Qohen Leth)

Wednesday, September 13, 2017

Summer of 1984

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

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

Group actions on partitions —

Cube Bricks 1984 —

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

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

Tuesday, September 12, 2017

Think Different

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

The New York Times  online this evening

"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."

James Poniewozik 

Review —

Thursday, September 1, 2011

How It Works

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

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

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

. . . .

See also 1984 Bricks in this journal.

Chin Music

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

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

Wednesday, April 12, 2017

Contracting the Spielraum

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

The contraction of the title is from group actions on
the ninefold square  (with the center subsquare fixed)
to group actions on the eightfold cube.

From a post of June 4, 2014

At math.stackexchange.com on March 1-12, 2013:

Is there a geometric realization of the Quaternion group?” —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).

Friday, September 30, 2016

Desmic Midrash

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

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

(The term "desmic," as noted earlier, is relevant to the structure of
Heidegger's Sternwürfel .)

The Horn midrash —

(The "medieval philosopher" here is not the remembered pre-Christian
Ben Sirah (Ecclesiasticus ) but the philosopher being read — Maimonides:  
Guide for the Perplexed , 3:51.)

Here of course "that bond" may be interpreted as corresponding to the
Greek desmos  above, thus also to the desmic  structure of the
stellated octahedron, a sort of three-dimensional Star of David.

See "desmic" in this journal.

Thursday, September 29, 2016

Articulation

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

Cassirer vs. Heidegger at Harvard —

A remembrance for Michaelmas —

A version of Heidegger's "Sternwürfel " —

From Log24 on the upload date for the above figure —

Wednesday, September 28, 2016

Star Wars

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

See also in this journal "desmic," a term related 
to the structure of Heidegger's Sternwürfel .

Scholia

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

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

Scholia —

D. H. Lawrence quote from 'Kangaroo'

South Australia goes dark

Thursday, December 17, 2015

Hint of Reality

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

From an article* in Proceedings of Bridges 2014

As artists, we are particularly interested in the symmetries of real world physical objects.

Three natural questions arise:

1. Which groups can be represented as the group of symmetries of some real-world physical object?

2. Which groups have actually  been represented as the group of symmetries of some real-world physical object?

3. Are there any glaring gaps – small, beautiful groups that should have a physical representation in a symmetric object but up until now have not?

The article was cited by Evelyn Lamb in her Scientific American  
weblog on May 19, 2014.

The above three questions from the article are relevant to a more
recent (Oct. 24, 2015) remark by Lamb:

" finite projective planes [in particular, the 7-point Fano plane,
about which Lamb is writing] 
seem like a triumph of purely 
axiomatic thinking over any hint of reality…."

For related hints of reality, see Eightfold Cube  in this journal.

* "The Quaternion Group as a Symmetry Group," by Vi Hart and Henry Segerman

Friday, August 7, 2015

Parts

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

Spielerei  —

"On the most recent visit, Arthur had given him
a brightly colored cube, with sides you could twist
in all directions, a new toy that had just come onto
the market."

— Daniel Kehlmann, F: A Novel  (2014),
     translated from the German by
     Carol Brown Janeway

Nicht Spielerei  —

A figure from this journal at 2 AM ET
on Monday, August 3, 2015

Also on August 3 —

FRANKFURT — "Johanna Quandt, the matriarch of the family
that controls the automaker BMW and one of the wealthiest
people in Germany, died on Monday in Bad Homburg, Germany.
She was 89."

MANHATTAN — "Carol Brown Janeway, a Scottish-born
publishing executive, editor and award-winning translator who
introduced American readers to dozens of international authors,
died on Monday in Manhattan. She was 71."

Related material —  Heisenberg on beauty, Munich, 1970                       

Thursday, May 7, 2015

Paradigm for Pedagogues

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

Illustrations from a post of Feb. 17, 2011:

Plato’s paradigm in the Meno —

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

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

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

Ultron: By the Book

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

If The New York Times interviewed Ultron for its
Sunday Book Review "By the Book" column —

What books are currently on your night stand?

Steve Fuller's Thomas Kuhn: A Philosophical History for Our Times

Gerald Holton's Thematic Origins of Scientific Thought

John Gray's The Soul of the Marionette

Lede

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

"Who is Ultron? What is he?"

See too the previous post and Cube of Ultron.

Wednesday, May 6, 2015

Soul

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

Nonsense…

See Gary Zukav, Harvard '64, in this journal.

and damned  nonsense —

"Every institution has a soul."

— Gerald Holton in Harvard Gazette  today

Commentary —

"The Ferris wheel came into view again…."
Malcom Lowry, Under the Volcano

See also Holton in a Jan. 1977 interview:

"If people have souls, and I think a few have, it shows…."

Wednesday, April 1, 2015

Würfel-Märchen

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

Continued from yesterday, the date of death for German
billionaire philanthropist Klaus Tschira —

For Tschira in this journal, see Stiftung .

For some Würfel  illustrations, see this morning's post
Manifest O.  A related webpage —

Manifest O

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

The title was suggested by
http://benmarcus.com/smallwork/manifesto/.

The "O" of the title stands for the octahedral  group.

See the following, from http://finitegeometry.org/sc/map.html —

83-06-21 An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.
83-10-01 Portrait of O  A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem.
83-10-16 Study of O  A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem.
84-09-15 Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O.

Friday, August 29, 2014

Raum

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

A possible answer to the 1923 question of Walter Gropius, "Was ist Raum?"—

See also yesterday's Source of the Finite and the image search
on the Gropius question in last night's post.

Thursday, August 28, 2014

Brutalism Revisited

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

Yesterday's 11 AM post was a requiem for a brutalist architect.

Today's LA Times  has a related obituary:

"Architectural historian Alan Hess, who has written several books on
Mid-Century Modern design, said Meyer didn't have a signature style,
'which is one reason he is not as well-known as some other architects
of the period. But whatever style he was working in, he brought a real
sense of quality to his buildings.'

A notable example is another bank building, at South Beverly Drive
and Pico Boulevard, with massive concrete columns, a hallmark of
the New Brutalism style. 'This is a really good example of it,' Hess said."

— David Colker, 5:43 PM LA time, Aug. 28, 2014

A related search, suggested by this morning's post Source of the Finite:

(Click to enlarge.)

Source of the Finite

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

"Die Unendlichkeit  ist die uranfängliche Tatsache: es wäre nur
zu erklären, woher das Endliche  stamme…."

— Friedrich Nietzsche, Das Philosophenbuch/Le livre du philosophe
(Paris: Aubier-Flammarion, 1969), fragment 120, p. 118

Cited as above, and translated as "Infinity is the original fact;
what has to be explained is the source of the finite…." in
The Production of Space , by Henri Lefebvre. (Oxford: Blackwell,
1991 (1974)), p.  181.

This quotation was suggested by the Bauhaus-related phrase
"the laws of cubical space" (see yesterday's Schau der Gestalt )
and by the laws of cubical space discussed in the webpage
Cube Space, 1984-2003.

For a less rigorous approach to space at the Harvard Graduate
School of Design, see earlier references to Lefebvre in this journal.

Wednesday, August 27, 2014

Not Quite

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

Click image to enlarge.

Altar

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

"To every man upon this earth,
Death cometh soon or late.
And how can man die better
Than facing fearful odds,
For the ashes of his fathers,
and the temples of his gods…?"

— Macaulay, quoted in the April 2013 film "Oblivion"

"Leave a space." — Tom Stoppard, "Jumpers"

Related material: The August 16, 2014, sudden death in Scotland
of an architect of the above Cardross seminary, and a Log24 post,
Plato's Logos, from the date of the above photo: June 26, 2010.

See also…

IMAGE- T. Lux Feininger on 'Gestaltung'

Here “eidolon” should instead be “eidos .”

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

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

Schau der Gestalt

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

(Continued from Aug. 19, 2014)

"Christian contemplation is the opposite
of distanced consideration of an image:
as Paul says, it is the metamorphosis of
the beholder into the image he beholds
(2 Cor 3.18), the 'realisation' of what the
image expresses (Newman). This is
possible only by giving up one's own
standards and being assimilated to the
dimensions of the image."

— Hans Urs von Balthasar,
The Glory of the Lord:
A Theological Aesthetics,

Vol. I: Seeing the Form
[ Schau der Gestalt ],
Ignatius Press, 1982, p. 485

A Bauhaus approach to Schau der Gestalt :

I prefer the I Ching 's approach to the laws of cubical space.

Wednesday, June 4, 2014

Monkey Business

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

The title refers to a Scientific American weblog item
discussed here on May 31, 2014:

Some closely related material appeared here on
Dec. 30, 2011:

IMAGE- Quaternion group acting on an eightfold cube

A version of the above quaternion actions appeared
at math.stackexchange.com on March 12, 2013:

"Is there a geometric realization of Quaternion group?" —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note "GL(2,3) actions on a cube" of April 5, 1985).

Saturday, May 31, 2014

Quaternion Group Models:

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

The ninefold square, the eightfold cube, and monkeys.

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

For posts on the models above, see quaternion
in this journal. For the monkeys, see

"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American  weblog, May 19, 2014:

The Scientific American  item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):

See also  Finite Geometry and Physical Space.

Monday, December 9, 2013

Heaven Descending

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

An I Ching  study quoted in Waiting for Ogdoad (St. Andrew’s Day, 2013)—

(Click for clearer image.)

The author of the above I Ching  study calls his lattice “Arising Heaven.”

The following lattice might, therefore, be called “Heaven Descending.”

IMAGE- Construction of 'Heaven Descending' lattice

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

Saturday, May 11, 2013

Core

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

Promotional description of a new book:

"Like Gödel, Escher, Bach  before it, Surfaces and Essences  will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking."

"Like Gödel, Escher, Bach  before it…."

Or like Metamagical Themas

Rubik core:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Of the odd  nxnxn cube:

Of the even  nxnxn cube:

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

Related material: The Eightfold Cube and

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

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

Tuesday, March 19, 2013

Mathematics and Narrative (continued)

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

Angels & Demons meet Hudson Hawk

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

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

The Leonardo Crystal from Hudson Hawk :

Hudson:

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

"Giving himself a head rub, Hawk bears down on
the three oddly malleable objects. He TANGLES 
and BENDS and with a loud SNAP, puts them together,
forming the Crystal from the opening scene."

— A screenplay of Hudson Hawk

Happy birthday to Bruce Willis.

Thursday, December 27, 2012

Object Lesson

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

Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 1910-1925" suggests a renewed look at
abstraction and a fundamental building block: the cube.

From a recent Harvard University Press philosophical treatise on symmetry—

The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.

For greater depth see Cassirer's 1910 passage on Vorstellung :

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

This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.

For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.

"The sacramental nature of the space becomes clear…." — Brian O'Doherty

Monday, December 24, 2012

Eternal Recreation

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

Memories, Dreams, Reflections
by C. G. Jung

Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989

From pages 195-196:

"Only gradually did I discover what the mandala really is:
'Formation, Transformation, Eternal Mind's eternal recreation.'*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions."

* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:

                   … Gestaltung, Umgestaltung, 
  Des ewigen Sinnes ewige Unterhaltung….

Jung's "Formation, Transformation" quote is from the realm of
the Mothers (Faust Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.

See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:

"In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust's next encounter with
the feminine, but it's obviously of a very different kind than his
relationship with Gretchen."

The phrase "Gestaltung, Umgestaltung " suggests a more mathematical
approach to the Unterhaltung . Hence

Part I: Mothers

"The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time."

— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity 
,
Johns Hopkins Press, 1969, page 37

Part II: Matrices

        

Part III: Spaces and Hypercubes

Click image for some background.

Part IV: Forms

Forms from the I Ching :

Click image for some background.

Forms from Diamond Theory :

Click image for some background.

Wednesday, November 14, 2012

Group Actions

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

The December 2012 Notices of the American
Mathematical Society  
has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on "Low-dimensional
Topology, Geometry, and Dynamics"—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture 
of the Providence building housing ICERM, is not unlike
a picture of Froebel's Third Gift—

 

Froebel's third gift, the eightfold cube

© 2005 The Institute for Figuring

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

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops' topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and 
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—

.

Friday, July 27, 2012

Raiders of the Lost Ring

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

Wikipedia on a magical ring

IMAGE- Wikipedia article, 'Seal of Solomon'

Background—  The Ring and the Stone, a story linked to here Wednesday.

"By then he was familiar with the work of the Vienna Actionists….
He once said that he had his first taste of the movement
when he heard the screams of his mother’s dental patients
from her office next door to the family’s apartment."

Obituary of a Viennese artist who reportedly died Wednesday

"Is it safe?"

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