Tuesday, July 31, 2012
than cipher: a mask rather than a revelation
in the romantic sense. Does love meet with love?
Do we receive but what we give? The answer is
surely a paradox, the paradox that there are
Platonic universals beyond, but that the glass
is too dark to see them. Is there a light beyond
the glass, or is it a mirror only to the self?
The Platonic cave is even darker than Plato
made it, for it introduces the echo, and so
leaves us back in the world of men, which does
not carry total meaning, is just a story of events."
Monday, July 30, 2012
— and, on Jan. 1, 2005, on beverage marketing:
Happy birthday to Hilary Swank.
A Necessary Truth—
James Singer, "A Theorem in Finite Projective Geometry
and Some Applications to Number Theory," Transactions
of the American Mathematical Society 43 (1938), 377-385.
A Contingent Truth—
In his memory, some references to a "Singer 7-Cycle."
See also this journal 7 years prior to Martin's death.
"I think the best thing the AMS does by far is the Notices .
It could easily be in all doctors’ and dentists’ offices."
Notices : "Really?"
Friedlander: "It could be."
Related material from this journal:
— Annals of Art Education:
Geometry and Death
"There is no question about what arithmetic is for
or why it is supported. Society cannot proceed
without it. Addition, subtraction, multiplication,
division, percentages: though not all citizens can
deal fluently with all of them, we make the
assumption that they can when necessary.
Those who cannot are sometimes at a disadvantage.
Algebra, though, is another matter."
— Underwood Dudley in the Notices of the
American Mathematical Society, May 2010:
"What Is Mathematics For?"
A less nuanced remark from the American
Mathematical Society (AMS) today—
"The answer to the recent Op-Ed piece
in The New York Times entitled
'Is Algebra Necessary?'
is resoundingly YES!"
— Eric Friedlander, AMS president
* A review of philosophical terminology—
"The distinction between necessary truth
and contingent truth is a version of Leibniz 's
distinction between truths of reason and truths
of fact. A necessary truth must be true and
could not be false, whatever way the world is.
It is true in itself. A contingent truth, on the other
hand, depends upon the empirical world and might
have been false had the world been different."
Sunday, July 29, 2012
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vector-space structure:
|0||c||d||c + d|
|a||a + c||a + d||a + c + d|
|b||b + c||b + d||b + c + d|
|a + b||a + b + c||a + b + d|| a + b +
c + d
(This vector-space a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.
Background: Square-Triangle Theorem.
Saturday, July 28, 2012
He played them music
and everything was concentrated and timeless
and all were artists 'til the bell rang.
Another remark from Claremont—
"'Once upon a time' used to be a gateway to
a land that was inviting precisely because
it was timeless, like the stories it introduced
and their ageless lessons about the human condition."
– Dorothea Israel Wolfson,
Claremont Review of Books, Summer 2006
The end of the beginning of the London Games
suggests other games —
Shadows (July 14) —
A Game of Shadows — "You know my methods."
Related religious material —
The Feast of Saint Jude, 2011.
Friday, July 27, 2012
Wikipedia on a magical ring—
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."
Thursday, July 26, 2012
(Mathematics and Narrative, continued)
"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
A fact related to the mathematical
"Solomon's seal" described above by Bell:
The reference to Edge is as follows—
 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)).
Wednesday, July 25, 2012
Tuesday, July 24, 2012
(Continued from July 22)
Manhattan, July 22, 2012 — "Once upon a time, in a quiet corner
of the Middle East, there lived a shepherd named Gyges. Despite
the hardships in his life Gyges was relatively satisfied with his meager
existence. Then, one day, he found a ring buried in a nearby cave."
"I want you on the Swansea lathe today."
— Boss of the Christ figure in "The Machinist" (2004)
Related material in this journal—
IMDb trivia page on "The Machinist" (2004)—
"The time of 1:30 AM is significant throughout the movie.
Trevor often notices something out of the ordinary at this time.
During the 1 hour 30 minute mark in the movie,
the major plot twist is revealed."
As for the date 1/30… See Tolkien on telepathy.
(Backstory: The Gospel According to Father Hardon )
Christian Bale as a Spanish Christ-parody in "The Machinist" (2004)
"The expression 'the devil is in the details'
is turned on its head in the exhibit 'The Sacred Made Real'
at Washington's National Gallery of Art…."
— Catholic News Service, 2010
Tonight's New York Times obituaries
"My goodness, there must be a hole in this glass."
— Maria in "The Machinist" (2004)
Monday, July 23, 2012
For those who like puzzles—
What film's page at IMDb recommends
the following "also liked" choices?
A similar puzzle: Related Books
Sunday, July 22, 2012
For art collector Herbert Vogel,
who reportedly died today
Philip Kennicott in The Washington Post , July 3, 2009—
"The Vogels help allay deep cultural fears
within the art world— fears that art is elitist,
or some kind of confidence game,
or not a serious endeavor (a fear that has
dogged art since at least the time of Plato)."
Some related material from finitegeometry.org,
offered without comment—
"One ring to bring them all…"
— J. R. R. Tolkien, Catholic author
Today in History, July 22, by The Associated Press—
"In 1934, bank robber John Dillinger was shot to death
by federal agents outside Chicago's Biograph Theater,
where he had just seen the Clark Gable movie
From a Manhattan Melodrama—
"Follow the Ring"
Piatigorsky died on Sunday, July 15. Notes in this journal from that date—
Saturday, July 21, 2012
"His attachment to left-wing journalism— and controversy—
was forged very early on. His father, Claud Cockburn, while
covering the Spanish Civil War for The Daily Worker , joined
the Republican forces fighting the rebellion of Francisco
Franco. (Claud Cockburn, under a pseudonym, also wrote
novels, including Beat the Devil , which was made into a
film with Humphrey Bogart and which his son used as the
title of his column in The Nation .)"
Here's your ticket, pack your bags,
time for jumpin' overboard
Transportation is here
The update's time suggests a check
of this journal's most recent post
with the date 10/11. It turns out
to be a meditation on art and
the speed of perception.
A linked-to post, Twenty-Four.
"A Saturday morning cartoon is the colloquial term
for the animated television programming that has
typically been scheduled on Saturday mornings
on the major American television networks from
the 1960s to the present…." —Wikipedia
“I did a column in Scientific American
on minimal art, and I reproduced one of
Ed Rinehart’s [sic ] black paintings.
Of course, it was just a solid square of
Click on picture for details.
For a cartoon graveyard—
Friday, July 20, 2012
http://www.latimes.com/news/politics/ la-pn-romney-colorado-shooting-is- unspeakable-tragedy-20120720,0,1226295.story Romney: Colorado shooting is 'unspeakable tragedy' By Seema Mehta July 20, 2012, 11:15 a.m. BOW, N.H.—Hours after a shooter killed a dozen people in a Colorado cinema, Mitt Romney scrubbed a scheduled campaign rally Friday and instead offered his somber condolences and prayers to the victims and their families. “Our hearts break with the sadness of this unspeakable tragedy. Ann and I join the president and first lady and all Americans in offering our deepest condolences to those whose lives were shattered in a few moments, a few moments of evil in Colorado,” Romney told a few hundred people gathered at a lumber yard here. “I stand before you today not as a man running for office but as a father and grandfather, a husband, an American. This is a time for each of us to look into our hearts and remember how much we love one another and how much we love and how much we care for our great country. There’s so much love and goodness in the heart of America.” Romney, wearing a navy blue suit and blue tie, spoke for four minutes. The trademark campaign banners with slogans such as “Believe in America” or “Obama’s Upside- Down Economy” were gone, leaving a handful of American flags as the backdrop. Before Romney spoke, Father Christian Tutor, an Anglican Catholic priest, led a prayer. ...
Thursday, July 19, 2012
For those who prefer fiction:
"Many Dimensions (1931) — An evil antiquarian illegally purchases
the fabled Stone of Suleiman (Williams uses this Muslim form
rather than the more familiar King Solomon) from its Islamic guardian
in Baghdad and returns to England to discover not only that the Stone
can multiply itself infinitely without diminishing the original, but that it
also allows its possessor to transcend the barriers of space and time."
— Wikipedia article on the author Charles Williams
A Passage to India… With Slides and Chanting
"Why art thou here,
Come from the farthest Steppe of India?"
— Midsummer Night's Dream
"After graduating, Mr. Franken headed for Harvard,
while Mr. Davis chose the University of the Pacific
in Stockton, Calif., because, he said, he had heard
that it had a foreign study program in India, where
he hoped to smoke opium. (They did, and he did.)"
— Obituary of Saturday Night Live writer Tom Davis
by Douglas Martin in this evening's online New York Times
"Frances Alenikoff, a dancer, choreographer and visual artist
whose performances often interwove movement with slides,
film, speaking, tape recordings and chant, died on June 23
in Southampton, N.Y. She was 91."
— Margalit Fox, online NY Times of July 8, 2012
Connoisseurs of hermeneutics will find that interpretations
of three of the above numbers are not hard to come by.
Consider, say, dates, post numbers in Log24, times, and so forth.
For the fourth number, 0131, try the following search:
Wednesday, July 18, 2012
For a black widow —
(Continued from Midsummer Eve)
"At times, bullshit can only be countered with superior bullshit."
— Norman Mailer, March 3, 1992, PBS transcript
"Just because it is a transition between incommensurables, the transition between competing paradigms cannot be made a step at a time, forced by logic and neutral experience. Like the gestalt switch, it must occur all at once (though not necessarily in an instant) or not at all."
"In the spiritual traditions from which Jung borrowed the term, it is not the SYMMETRY of mandalas that is all-important, as Jung later led us to believe. It is their capacity to reveal the asymmetry that resides at the very heart of symmetry."
I have little respect for Enneagram enthusiasts, but they do at times illustrate Mailer's maxim.
My own interests are in the purely mathematical properties of the number nine, as well as those of the next square, sixteen.
Those who prefer bullshit may investigate non-mathematical properties of sixteen by doing a Google image search on MBTI.
For bullshit involving nine, see (for instance) Einsatz in this journal.
For non-bullshit involving nine, sixteen, and "asymmetry that resides at the very heart of symmetry," see Monday's Mapping Problem continued. (The nine occurs there as the symmetric figures in the lower right nine-sixteenths of the triangular analogs diagram.)
For non-bullshit involving psychological and philosophical terminology, see James Hillman's Re-Visioning Psychology .
In particular, see Hillman's "An Excursion on Differences Between Soul and Spirit."
Tuesday, July 17, 2012
— Paul Simon
Monday, July 16, 2012
Jaws for Frank
Part I: October 8, 2010
Above: Frank Langella in
Right: Johnny Depp in
Part II: Noon Today
"Merit has been traditionally equated with intelligence, industriousness, educational attainment, creativity and competency. In a meritocracy, formal qualifications provide opportunity, position is no longer ascribed by birth, and rewards flow to those who excel.
The rise of meritocratic competition as the preeminent means of social stratification in America has been hailed as a welcome advance because it replaced a society dominated by an upper class dependent on inherited wealth and status. The transition to meritocracy has, however, had unintended consequences. In the business sector, particularly, other less benign qualities emerge as essential to meritocratic success: aggressiveness, ruthlessness, dominance-seeking, victimizing behavior, acquisitiveness and the disciplined pursuit of self-interest."
— Journalism professor Thomas B. Edsall discussing remarks last December by Mitt Romney
Note the subtle shift here from "merit" to "meritocracy." Romney used the former word, not the latter.
Note also this sentence, aimed particularly at meritocratic New York Times readers—
"In a meritocracy, formal qualifications provide opportunity… and rewards flow to those who excel."
Edsall lies. In a meritocracy, rewards flow to those who rubber-stamp "formal qualifications." See particularly Walter Kirn on meritocracy.
Edsall is pandering to Times readers. Romney was pandering to a different group—
For the square model referred to in the above picture, see (for instance)
- Picturing the Smallest Projective 3-Space,
- The Relativity Problem in Finite Geometry, and
- Symmetry of Walsh Functions.
Coordinates for the 16 points in the triangular arrays
of the corresponding affine space may be deduced
from the patterns in the projective-hyperplanes array above.
This should solve the inverse problem of mapping,
in a natural way, the triangular array of 16 points
to the square array of 16 points.
Update of 9:35 AM ET July 16, 2012:
Note that the square model's 15 hyperplanes S
and the triangular model's 15 hyperplanes T —
— share the following vector-space structure —
|0||c||d||c + d|
|a||a + c||a + d||a + c + d|
|b||b + c||b + d||b + c + d|
|a + b||a + b + c||a + b + d|| a + b +
c + d
(This vector-space a b c d diagram is from
Chapter 11 of Sphere Packings, Lattices
and Groups , by John Horton Conway and
N. J. A. Sloane, first published by Springer
Sunday, July 15, 2012
A trial solution to the
square-to-triangle mapping problem—
Problem: Is there any good definition of "natural"
square-to-triangle mappings according to which
the above mapping is natural (or, for that matter,
"A figurate number… is a number
that can be represented by
a regular geometrical arrangement
of equally spaced points."
Call a convex polytope P an n-replica if P consists of n
mutually congruent polytopes similar to P packed together.
"Every triangle is an n-replica"
is true if and only if n is a square.
The positive integer n is a square
if and only if every triangle is an n-replica.
(I.e., squares are triangular.)
This supplies the converse to the saying that
Saturday, July 14, 2012
A letter to the editor of the American Mathematical Monthly
from the June-July 1985 issue has—
… a "square-triangle" lemma:
(∀ t ∈ T , t is an n -replica )
[I.e., "Every triangle is an n -replica"
For definitions, see the 1985 letter in Triangles Are Square.
(The 1984 lemma discussed there has now, in response to an article
in Wolfram MathWorld, been renamed the square-triangle theorem .)
A search today for related material yielded the following—
"Suppose that one side of a triangle
has length n . Then it can be cut
into n 2 congruent triangles which
are similar to the original one and
whose corresponding sides to the
side of length n have lengths 1."
This was supplied, without attribution, as part of the official solution
to Problem 3 in the 17th Asian Pacific Mathematics Olympiad
from March 2005. Apparently it seemed obvious to the composer
of the problem. As the 1985 letter notes, it may be not quite obvious.
A tribute to Richard D. Zanuck in the style of Tim Burton—
The above Zanuck interview on Dark Shadows
was published on Midsummer Eve, June 23, 2012.
Also from June 23, 2012—
Related material— Russell on Hardy.
Friday, July 13, 2012
"… Conceptualism — suddenly art
could be nothing more than an idea,
a thought on a piece of paper
that played in your head."
— Michael Kimmelman,
"The Dia Generation,"
The New York Times Magazine ,
Sunday, April 6, 2003
Thursday, July 12, 2012
An example of lines in a Galois space * —
The 35 lines in the 3-dimensional Galois projective space PG(3,2)—
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3-set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.
The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.
* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488-499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
Wednesday, July 11, 2012
For Pete Rustan, space recon expert, who died on June 28—
Tuesday, July 10, 2012
From this journal on June 19, 2012—
Walter Gropius on space—
"Was ist Raum, wie können wir ihn
erfassen und gestalten?"
A book published on the same date—
June 19, 2012:
"… what Chalmers called the convergence of coincidence—
a force majeure of unrelated events that shaped one's life,
that perhaps defined the concept of life itself.
He believed in the power of that force."
— The Cryptos Conundrum , by Chase Brandon
See also Chase Brandon in Sunday's Huffington Post .
"I wrote another book."
— Robert De Niro as Harlan Kane
Monday, July 9, 2012
“There is the dark, eternally silent, unknown universe;
and lastly, there is lonely, story-telling, wonder-questing,
Sunday, July 8, 2012
"That n 2 points fall naturally into a triangular array
is a not-quite-obvious fact which may have applications…
and seems worth stating more formally."
— Steven H. Cullinane, letter in the
American Mathematical Monthly 1985 June-July issue
A search for occurrences of the phrase
"n2 [i.e., n 2 ] congruent triangles"
indicates only fairly recent (i.e., later than 1984) results.*
Some related material, updated this morning—
This suggests a problem—
What mappings of a square array of n 2 points to
In the figure above, whether
* Update of July 15, 2012 (11:07 PM ET)—
Theorem on " rep-n 2 " (Golomb's terminology)
triangles from a 1982 book—
Saturday, July 7, 2012
"Euclid (Ancient Greek: Εὐκλείδης Eukleidēs), fl. 300 BC,
also known as Euclid of Alexandria, was a Greek
mathematician, often referred to as the 'Father of Geometry.'"
A Euclidean quartet (see today's previous post)—
Perfect means, among other things, completed .
See, for instance, the life of another Alexandrian who reportedly
died on the above date—
"Gabriel Georges Nahas was born in Alexandria, Egypt, on
March 4, 1920…."
— This afternoon's online New York Times
For remarks related by logic, see the square-triangle theorem.
For remarks related by synchronicity, see Log24 on
the above publication date, June 15, 2010.
Whether young readers should be captivated is open to question.
Update of 9:48 the same morning—
Thursday, July 5, 2012
A post suggested by an article on The Shard of London
in this morning's Wall Street Journal—
As for the "Personal Jesus" song that accompanies the above video tribute,
listen to Liam Neeson as the voice of Aslan in recent Narnia films
and consider the saying of C. S. Lewis that Aslan is not a tame lion.
Here Lewis may, if one likes, be regarded as the "inkling" of Heidegger
in last night's post—
Or: Night of Lunacy
From 9 PM Monday —
Note that the last line, together with the page number, forms
a sort of key—
The rest of the story—
The linked-to sermon itself has a link, based on a rereading
of 304 as 3/04, to a post of March 4, 2004, with…
WW and ZZ
as rendered by figures from the Kaleidoscope Puzzle—
Yesterday morning the same letter-combinations occurred
in a presentation at CERN of a newly discovered particle—
(Click for context.)
Since the particle under discussion may turn out to be the
God particle, it seems fitting to interpret WW and ZZ as part
of an imagined requiem High Mass.
For some background on the Holy Cross, see posts of
Sept. 14 (Holy Cross Day) and Sept. 15, 2010—
For more lunacy, see…
Continue a search for thirty-three and three
— Katherine Neville, The Eight
Wednesday, July 4, 2012
For a White House daughter…
From this evening's Capitol Fourth—
Related material— Big Bang in this journal—
Detective Cruz enters Planck's Constant Café in "The Big Bang."
Tuesday, July 3, 2012
—Rhodes, I want you to get to know people like that.
I'd like to sort of take you under my wing and educate you.
—Shucks, General, I'm just a country boy.
"And when I think about the values
that are important to me today,
I think first about meritocracy."
Other remarks on that Sunday —
Related material from Colby—
See also an MAA report on Gouvea from June 6, 2012.
Monday, July 2, 2012
Heidegger, "Hölderlin and the Essence of Poetry,"
translated by Douglas Scott, in Existence and Being ,
Regnery, 1949, pp. 291-316—
Images from a Google search suggested by
last night's post Coming to Meet, by the recent
film "Archie's Final Project," and by a Thursday,
June 28, 2012, Times Higher Education piece,
"Raiders of the Lost Archives"—
Log24, December 8, 2008 —
Sunday, July 1, 2012
This evening's online New York Times—
Doris Sams, Pro Baseball Star, Dies at 85
By RICHARD GOLDSTEIN
Published: July 1, 2012
Doris Sams, who pitched a perfect game and set a single-season home run record in the women’s professional baseball world of the 1940s and 50s that inspired the movie “A League of Their Own,” died Thursday in Knoxville, Tenn. She was 85. more>>
"High summer holds the earth."
* A phrase from the quoted Agee poem
Coming to Meet
"This hexagram indicates a situation in which
the principle of darkness, after having been eliminated,
furtively and unexpectedly obtrudes again from within
and below. Of its own accord the female principle
comes to meet the male. It is an unfavourable and
dangerous situation, and we must understand and
promptly prevent the possible consequences.
The hexagram is linked with the fifth month
[June-July], because at the summer solstice
the principle of darkness gradually becomes
— Richard Wilhelm
To counteract the principle of darkness—
The Uploading (Friday— St. Peter's Day, 2012),
Thor's Light Bulb Joke, and …