Other intersectionpointscounting material —
See also Hanks + Cube in this journal —
Other intersectionpointscounting material —
See also Hanks + Cube in this journal —
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 windup 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

Related material —
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:
A realization —
Not the best possible realization, but enough for proof of concept .
Another view of the previous post's art space —
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).
"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?
Click image for related posts.
Transformations acting on Solomon's Cube
furnish a model of poetic order.
Some backstory for Hollywood —
See Hanks + Cube in this journal … For instance …
Friday, July 11, 2014

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

See also related material in the previous post, Transformers.
Friday, July 11, 2014

From an earlier Log24 post —
Friday, July 11, 2014

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.
The title is that of a largescale British research project
in mathematics. On a more modest scale …
"Hanks + Cube" in this journal —
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 .
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 allmeaningful, 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 LéviStrauss 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.
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:
Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :
An unarticulated (but colored) cube:
A 2x2x2 articulated cube:
A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:
“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 
A sequel to the 1974 film
Thunderbolt and Lightfoot :
Contingent and Fluky
Some variations on a thunderbolt theme:
These variations also exemplify the larger
Verbum theme:
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.
“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
Definition: A diamond space — informal phrase denoting
a subspace of AG(6, 2), the sixdimensional affine space
over the twoelement Galois field.
The reason for the name:
Click to enlarge.
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)—
Click for further details.
(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—
A fact related to the mathematical
"Solomon's seal" described above by Bell:
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 64point affine space
illustrated below (via the associated
63point projective space PG (5, 2)).
In "Contact," Dr. Arroway is shown the key to the Primer—
In this journal, fictional symbologist Robert Langdon is shown a cube—
"Confusion is nothing new." — Song lyric
…. 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—
From a Log24 post, "Eightfold Cube Revisited,"
on the date of Golding's death—
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.
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—
Background:
Deconstructing Alice
and Symbology.
For Alyssa Milano —
(Click here for cheesy Neil Diamond background music.)
For some related philosophical remarks, see Deconstructing Alice
and the new Pythagorean thriller The Thousand.
David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —
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 multidimensionally — that is,
uses multidimensional symbols beyond our grasp."
(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—
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—
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.
Romancing the
NonEuclidean Hyperspace
Backstory —
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937
For the 1937 grid, see Diamond Theory, 1937.
The grid is, as Mere Geometry points out, a nonEuclidean hyperspace.
For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.
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 LookingGlass,' which means that she’s a straight man to her own imagination. (She is Wonderland.)"
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…
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 NonEuclidean 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.)
From this journal:
Friday December 5, 2008MirrorPlay of
the Fourfold For an excellent commentary View selected pages Play and the Aesthetic Dimension (Mihai I. Spariosu, Related material: – and Theme and Variations. 
Transition to the
Garden of Forking Paths–
(See For Baron Samedi)–
The Found Symbol
and Dissemination, by Jacques Derrida,
translated by Barbara Johnson,
London, Athlone Press, 1981–
Pages 354355
On the mirrorplay of the fourfold
Pages 356357
Shaking up a whole culture
Pages 358359
Cornerstone and crossroads
Pages 360361
A deep impression embedded in stone
Pages 362363
A certain Y, a certain V
Pages 364365
The world is Zeus's play
Page 366
It was necessary to begin again
Annals of Deconstruction —
Click on image for background.
Related material
for Baron Samedi —
The Found Symbol
A graphic novel reviewed in the current Washington Post features Alfred North Whitehead and Bertrand Russell–
Related material:
Whitehead on Fano’s finite projective threespace:
“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
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 .
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 ndimensional 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 ndimensional binary space. In fact, as one will see from the links below, it is not simple at all.
The Klein Correspondence, Penrose SpaceTime, and a Finite Model
Related material:
Monday’s entry Just Say NO and a poem by Stevens,
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:
(Cover slightly changed.)
Background —
SAT
Part I:
Part II:

Part III:
From August 25th —
"Boo, boo, boo,
square root of two."
A footprint from Germany:
Germany Pythonurllib 
/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 highlevel 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 a larger and more sophisticated
relative of that object,
see Solomon’s Cube and
the related three presents
from the German link’s target:
1. Many Dimensions 2. Boggle 3. My Space 
"… 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
Figure 1 —
Concept from 1819:
(Footnotes 1 and 2)
Figure 2 —
The Third Gift, 1837:
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:
Figure 4 —
Solomon's Cube,
1981 and 1983:
Figure 5 —
Design Cube, 2006:
The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the twoelement field).
(To see how the display works,
try the Kaleidoscope Puzzle first.)
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“
At Midsummer Noon:

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 if we ate the incipient colorings – Wallace Stevens, “The Rock” 
Hofstadter’s cover.
Time and Chance
on the 90th Birthday
of Kirk Douglas,
star of
“The Garden of Allah“
The Lottery 12/9/06  Midday  Evening 
New York  036
See 
331
See 3/31— “square crystal” and “the symbolism could not have been more perfect.” 
Pennsylvania  602
See 6/02— Walter Benjamin 
111
See 1/11— “Related material: 
(Title of an interview with
the late Paul Halmos, mathematician)
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:
This last has the virtue of
being connected with Halmos
via his remarks during the
“In Touch with God” interview:
See also the remark of Halmos that serves as an epigraph to Theme and Variations.
has also served
at least one interpreter
as a philosopher’s stone,
and is also the original
“Halmos tombstone.”
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)
— Motto of
Plato’s Academy
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. 
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).
— Heraclitus of Ephesus, about 500 B.C.
And now, from
the author of Sphere…
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
For further details,
see Solomon’s Cube
and myspace.com/affine.
For further details,
see Jews on Buddhism
and
Adventures in Group Theory.
Pope Benedict XVI
on Pentecost,
June 4, 2006,
St. Peter’s Square.
Part I:
The Game
Part II:
Many People
For further details,
see Solomon’s Cube
and myspace.com/affine.
“The rock cannot be broken.
It is the truth.”
— Wallace Stevens
“… 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.”
The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the 2element field).
(To see how the display works,
try the Kaleidoscope Puzzle first.)
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
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/.
These topics may be illuminated
by a study of the Chinese classics.
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.
In memory of Humphrey Carpenter, author of The Inklings, who attended The Dragon School. Carpenter died a year ago today.
“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….”
From Paul Preuss,
Broken Symmetries
(see previous entry):
From
Verbum Sat Sapienti?
or, The Eightfold Cube
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:
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.
(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 3space over GF(2) whose “points” are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cubeslices. This is clearly a subgroup of the group generated by permuting 1x1x2 cubeslices. Such translations in the affine 3space have no effect on the projective plane, since they leave each of the plane model’s seven partitions– the “points” of the plane– invariant.)
For another application of the pointsaspartitions technique, see LatinSquare 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.
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