Log24

Monday, July 24, 2017

Penguin Classics Deluxe Edition

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

With a hat tip to Vinnie Mancuso

Wednesday, June 21, 2017

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 .

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'

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 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 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.

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 

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.

Thursday, July 14, 2016

Symmetries and Correspondences

Filed under: Uncategorized — 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: 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, December 2, 2015

Symbology

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: 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, May 9, 2014

Models of Everything

Filed under: Uncategorized — 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: Uncategorized — 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.

Sunday, August 5, 2012

Cube Partitions

Filed under: Uncategorized — 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: Uncategorized — 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'

Saturday, April 21, 2012

Finding a Form

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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, October 22, 2009

Chinese Cubes

Filed under: Uncategorized — 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.”

Thursday, September 17, 2009

Thursday September 17, 2009

Filed under: Uncategorized — m759 @ 8:00 PM
Jennifer’s Body

The following remark this evening by Ann Hornaday of The Washington Post serves as an instant review of today’s previous cinematic Log24 offering starring the late Patrick Swayze:

“Watch it, forget it, move on.”

A perhaps more enduring tribute:

Patrick Swayze in 'King Solomon's Mines'

Related material:

Solomon’s Cube,
Solomon and Sheba,
and
Raiders of the Lost Stone.

Ready when you are, C.B.

Thursday September 17, 2009

Filed under: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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:

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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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — 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: Uncategorized — m759 @ 2:56 PM

Finitegeometry.org Update

(Revised May 21, 2006)

Finitegeometry.org now has permutable JavaScript views of the 2x2x2 and 4x4x4 design cubes.  Solomon’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, February 15, 2006

Wednesday February 15, 2006

Filed under: Uncategorized — m759 @ 11:07 AM
Anthony Hopkins
Writes Screenplay
About God, Life & Death

These topics may be illuminated
by a study of the Chinese classics.

The image “http://www.log24.com/theory/images/GF64-63cycleA495.gif” cannot be displayed, because it contains errors.

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

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.

Related material:

Hitler’s Still Point
,
Jung’s Imago,
Solomon’s Cube,
Geometry of the I Ching,
and Globe Award.

Yesterday’s Valentine
may also have some relevance.

Wednesday, January 4, 2006

Wednesday January 4, 2006

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

Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: Uncategorized — m759 @ 1:00 PM

The Fano Plane
Revisualized:

The Eightfold Cube

or, The Eightfold Cube

Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):

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

Every permutation of the plane’s points that preserves collinearity is a symmetry of the  plane.  The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group  PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)

The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle.  It does not, however, indicate where the other 162 symmetries come from.  

Shown below is a new model of this same projective plane, using partitions of cubes to represent points:

Fano plane with cubes as points

The cubes’ partitioning planes are added in binary (1+1=0) fashion.  Three partitioned cubes are collinear if and only if their partitioning planes’ binary sum equals zero.

The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.

Fano plane group - generating permutations

For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.

(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations.  But the group’s action on the diagram’s seven partitions of the subcubes yields only 168 distinct results.  This illustrates the difference between affine and projective spaces over the binary field GF(2).  In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose “points” are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices.  This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices.  Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model’s seven partitions– the “points” of the plane– invariant.)

To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane’s symmetry group in another guise, see John Baez on Klein’s Quartic Curve and the online book The Eightfold Way.  For more on the mathematics of cubic models, see Solomon’s Cube.

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

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