Here is a link to a copy of the home page of a Turkish
author quoted here on May 4, 2012… in honor of
archaeologist Crawford Greenewalt Jr., who reportedly
died on that date. Greenewalt was an expert on the
ancient city of Sardis, in what is now western Turkey.
The May 4 quote was about
"Heraclitus’s Aion and His Transformations."
(A post suggested by an ad in this evening's online New York Times )
"After being brought to the village's Patriarch… Mick learns
the intent of the colony and how they operate."
— Summary of a story by Orson Scott Card, a Latter-Day Saint.
For some context, see Saints Have Powers in this journal.
Related material —
The Saturday Evening Post and tonight's Saturday Night Live .
"The group of 8" is a phrase from politics, not mathematics.
Of the five groups of order 8 (see today's noon post),
the one pictured* in the center, Z2 × Z2 × Z2 , is of particular
interest. See The Eightfold Cube. For a connection of this
group of 8 to the last of the five pictured at noon, the
quaternion group, see Finite Geometry and Physical Space.
* The picture is of the group's cycle graph.
38, 23, 7B
May 19, 2012 9:06 AM
Expert: Facebook targeting all 7B people on Earth
(CBS News) NEW YORK — After all the hype, Facebook's
stock fell flat on its first day of trading. Shares in the
social networking giant opened at 38 dollars, shot up briefly,
then fell— and finished just 23 cents higher.
Midrash— "Fullness… Multitude"
Continued from Banderas (Aug. 18, 2011)—
Balakrishnan's Banners
See also The Colors of Halloween and Smiling Buddha.
"… this campaign, relatively speaking, will not be
fierce or hotly contested. Instead it'll be disappointing,
embarrassing, and over very quickly, like a hand job
in a Bangkok bathhouse." — Matt Taibbi in Rolling Stone
From a film released Friday, April 13th, 2012—
"Time for you to see the field." — Bagger Vance, as quoted here yesterday.
* Title courtesy of David Foster Wallace.
From "Crude Foyer," a Wallace Stevens poem—
In which we read the critique of paradise
And say it is the work
Of a comedian, this critique….
Related comedy — Finality and Cleavage.
* TV series starring the above actors. See Wikipedia.
(Continued from yesterday evening)
On Max Bialystock's Spider-Man Godspell Seminar—
"… for surrealism to be entertaining
onstage, it must be shaped into
some kind of satisfying form."
— Charles Isherwood
in today's New York Times
(RSS: Wed, 16 May 2012 00:37:17 GMT)
From Fritz Leiber's 1959 story "Damnation Morning" —
She drew from her handbag a pale grey gleaming
implement that looked by quick turns to me
like a knife, a gun, a slim sceptre, and a delicate
branding iron— especially when its tip sprouted
an eight-limbed star of silver wire.
“The test?” I faltered, staring at the thing.
“Yes, to determine whether you can live
in the fourth dimension or only die in it.”
(Continued from Sunday, April 22, 2012)
Xbox Background—
Design Sermon from Sunday, November 6, 2011, and
The X Box from Monday, November 7, 2011.
Six PM EDT is midnight in Paris.
Así Que Pasen Cinco Años
Theater review from The Guardian—
" 'Impossible' was how the Spanish playwright
Lorca described his own 1931 'legend of time
in three acts and five scenes,' which draws
strongly on the surrealist influences and experiments
of his close friends Salvador Dali and Luis Buñuel."
Related material—
This afternoon's previous post Murió Fuentes and,
from this date five years ago…
From the NY Times philosophy column "The Stone"
yesterday at 5 PM—
Timothy Williamson, Wykeham Professor of Logic at Oxford,
claims that all the theorems of mathematics
"… are ultimately derived from a few simple axioms
by chains of logical reasoning, some of them
hundreds of pages long…."
Williamson gives as an example recent (1986-1995)
work on Fermat's conjecture.
He does not, however, cite any axioms or "chains of
logical reasoning" in support of his claim that
a proof of Fermat's conjecture can be so derived.
Here is a chain of reasoning that forms a crucial part
of recent arguments for the truth of Fermat's conjecture—
K. A. Ribet, "On modular representations of Gal(Q̄/Q)
arising from modular forms," Invent. Math. 100 (1990), 431-476.
Whether this chain of reasoning is in fact logical is no easy question.
It is not the sort of argument easily reduced to a series of purely
logical symbol-strings that could be checked by a computer.
Few mathematicians, even now, can follow each step
in the longer chain of reasoning that led to a June 1993 claim
that Fermat's conjecture is true.
Williamson is not a mathematician, and his view of
Fermat's conjecture as a proven fact is clearly based
not on logic, but on faith.
Steve Cropper on the late Donald Dunn—
"He's in the hotel right now."
Click image for a related post.
An earlier verse in 1 John—
1 John 1:5 "This then is the message
which we have heard of him,
and declare unto you, that God is light,
and in him is no darkness at all."
Catechism from a different cult—
"Who are you, anyway?"
— Question at 00:41 of 15:01,
Rainbow Bridge (Part 5 of 9) at YouTube
See also the video accompanying artist Josefine Lyche's version
of the 2×2 case of the diamond theorem.
* Title of a Robert Stone novel
Mormon Mitt Romney at the Baptist school Liberty University today:
"The task set before you four years ago
is now completed in full."
I do not know what that task was. In this journal four years ago,
the task was lottery hermeneutics… a subject I doubt is taught
at Liberty University.
The New York lottery numbers from Sunday, May 11, 2008,
in a May 12 post four years ago could be interpreted as
pointing to the date 3/13—
Say, 3/13, 2006— a date on which this journal quoted some
remarks on the biblical phrase "the fullness of time."
Those remarks were neither Baptist nor
Mormon, but rather Presbyterian.
Today's previous post was "Midnight in Oslo (continued)."
The link "a 4-element set" in "Midnight"
was to a more elaborate structure in a post titled "Tesseract."
In memory of an Oslo "hero of midnight"
(a phrase quoted here last September 1)—
A search for material that is more entertaining—
Odin 's Tesseract.
See also a related Hollywood story in The Washington Post .
Last evening's Geometry of the Dance discussed
a book on the Norwegian mathematician
Niels Henrik Abel. The post dealt with the group
S4 of 24 permutations of a 4-element set.
"In that open field
If you do not come too close, if you do not come too close,
On a summer midnight, you can hear the music…."
— The dance in Four Quartets
For a summer midnight related to the group S4,
see Midnight in Oslo from last August.
"At the still point…." — T. S. Eliot
"…a dance results." — Marie-Louise von Franz
Peter Pesic uses a dance metaphor to explain
finite group theory, with permutations of four elements
represented by symmetries of a tetrahedron—
For a different approach to the dance metaphor, see
the dance in Four Quartets and Poetry's Bones.
In that open field
If you do not come too close, if you do not come too close,
On a summer midnight, you can hear the music
Of the weak pipe and the little drum
And see them dancing around the bonfire
The association of man and woman
In daunsinge, signifying matrimonie—
A dignified and commodiois sacrament.
Two and two, necessarye coniunction….
In memory of author Maurice Sendak,
who has died at 83—
"President Obama and his family read from
Where the Wild Things Are at this year’s
White House Easter Egg Roll." —ABC News
See also Easter Act and Shaggy Dance.
| SPOILER ALERT
This post links to a column that |
The title is from a column by Stanley Fish
on The Hunger Games books in today's
online New York Times . The column
was posted at 9 PM EDT on May 7th, but I
did not see it until this morning.
Fish says—
"In the end… [spoiler details omitted]…
children… 'don’t know they play
on a graveyard'…."
For some literary background, see last night's post
on the May 7th, 2012, NY Times obituaries as well
as the May 7th, 2006, Log24 post featuring 24 squares
arranged in a rectangular frame.
See also Frame Tales and, more generally,
The King and the Corpse.
"Et cetera, et cetera, et cetera." — Yul Brynner
Leading The New York Times obituaries on the evening of
May 7th, 2012, was "Bob Stewart, Inventor of Game Shows"—
From a publication linked to here on May 4th,
the reported date of Stewart's death—
A PASSAGE TO INDIA—
For Eastern illusion involving a (presumably different)
"Bob Stewart," see this journal on May 7th six years ago.
John Baez wrote in 1996 ("Week 91") that
"I've never quite seen anyone come right out
and admit that triality arises from the
permutations of the unit vectors i, j, and k
in 3d Euclidean space."
Baez seems to come close to doing this with a
somewhat different i , j , and k — Hurwitz
quaternions— in his 2005 book review
quoted here yesterday.
See also the Log24 post of Jan. 4 on quaternions,
and the following figures. The actions on cubes
in the lower figure may be viewed as illustrating
(rather indirectly) the relationship of the quaternion
group's 24 automorphisms to the 24 rotational
symmetries of the cube.
This post continues the April 9 post
commemorating Élie Cartan's birthday.
That post mentioned triality .
Here is John Baez reviewing
On Quaternions and Octonions:
Their Geometry, Arithmetic, and Symmetry
by John H. Conway and Derek A. Smith
(A.K. Peters, Ltd., 2003)—
"In this context, triality manifests itself
as the symmetry that cyclically permutes
the Hurwitz integers i , j , and k ."
Related material— Quaternion Acts in this journal
as well as Finite Geometry and Physical Space.
“… Which makes it a gilt-edged priority that one of us
gets into that Krell lab and takes that brain boost.”
— American adaptation of Shakespeare's Tempest , 1956
From "The Onto-theological Origin of Play:
Heraclitus and Plato," by Yücel Dursun, in
Lingua ac Communitas Vol 17 (October 2007)—
"Heraclitus’s Aion and His Transformations
The saying is as follows:
αἰὼν παῖς ἐστι παίζων, πεττεύων·
παιδὸς ἡ βασιληίη
(Aion is a child playing draughts;
the kingship is the child’s)
(Krell 1972: 64).*
* KRELL, David Farrell.
“Towards an Ontology of Play:
Eugen Fink’s Notion of Spiel,”
Research in Phenomemology ,
2, 1972: 63-93.
This is the translation of the fragment in Greek by Krell.
There are many versions of the translation of the fragment….."
See also Child's Play and Froebel's Magic Box.
Update of May 5— For some background
from the date May 4 seven years ago, see
The Fano Plane Revisualized.
For some background on the word "aion,"
see that word in this journal.
"… a long seat, or a seat with a back,
or a throne for the Queen;
or again, a cross, a doorway, etc."
"… etc., etc." — Yul Brynner
“The key is the cocktail that begins the proceedings.”
– Brian Harley, Mate in Two Moves
See also yesterday's Endgame , as well as Play and Interplay
from April 28… and, as a key, the following passage from
an earlier April 28 post—
| Euclidean geometry has long been applied to physics; Galois geometry has not. The cited webpage describes the interplay of both sorts of geometry— Euclidean and Galois, continuous and discrete— within physical space— if not within the space of physics . |
In memory of actress Patricia Medina—
RxQ
Medina, who died at 92 on April 28, starred in the 1954 Alan Ladd film The Black Knight .
April 28 is also the date the above photo appeared in this journal. See Play and Interplay.
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, 1987
Returning to the Walpurgisnacht posts
Decomposition (continued) and
Decomposition– Part III —
Some further background…
SAT
(Not a Scholastic Aptitude Test)
"In computer science, satisfiability (often written
in all capitals or abbreviated SAT) is the problem
of determining if the variables of a given Boolean
formula can be assigned in such a way as to
make the formula evaluate to TRUE."
— Wikipedia article Boolean satisfiability problem
For the relationship of logic decomposition to SAT,
see (for instance) these topics in the introduction to—
Advanced Techniques in Logic Synthesis,
Optimizations and Applications* —
Click image for a synopsis.
* Edited by Sunil P. Khatri and Kanupriya Gulati
From this journal last Christmas—
Sunday, December 25, 2011Frage"Woher dieser Sprung von Endlichen zum Unendlichen? " — Wittgenstein, Zettel , § 273 Antwort— Accomplished in Steps and For 34th Street. See also Boundary Method. |
The "Boundary Method" link above leads to a Christmas Day obituary
for Maurice Jaswon, co-author of a book on color symmetry.
Those who prefer entertainment may consult the previous Christmas.
Tony Award Nominations
"The losers? 'Spider-Man: Turn Off the Dark,'
the $75 million blockbuster that received just
two nominations. 'Seminar' and 'Godspell,'
which have some strong fans but were
shut out of the nominations."
— Patrick Healy in this morning's New York Times
A thought for Max Bialystock—
The Spider-Man Godspell Seminar!
Jeff Goldblum in "Seminar"
Update of 12:25 PM —
The reviews are in!
"A version of this article appeared in print on May 1, 2012, on page A1 of the New York edition…."
(Continued from Part I and Part II.)
The paper excerpted below supplies some badly needed technical
background for the Wikipedia article on functional decomposition.
The preprint above gives the precise definitions and technical references
that are completely absent from Wikipedia's Functional decomposition.
For some related material on 4×4 arrays like those in the above figure
see Decomposition Part I and Geometry of the 4×4 Square.
Compare and contrast
1. The following excerpt from Wikipedia—
2. A webpage subtitled "Function Decomposition Over a Finite Field."
Related material—
“Birthday, death-day — what day is not both?”
— John Updike, The New Yorker dated August 5, 2002, page 63
Today's date— Poincaré's birth, Wittgenstein's death.
A Saint for Clark University—
Today is also the birth date of William Edward Story,
a mathematician who taught at Clark University
in Worcester, Mass.
Story's date of death was April 10, 1930.
See the Log24 posts for that date in 2012.
“Oh, Sara!” she whispered joyfully. “It is like a story!”
“It is a story,” said Sara. “Everything's a story.
You are a story— I am a story.”
— Frances Hodgson Burnett, A Little Princess
The last paragraph of the previous post
(as updated at about 7:20 PM today)
suggests a search for the phrase
"play and interplay" that yields…
"He had accepted the world as the world,
but now he was comprehending the
organization of it, the play and interplay
of force and matter."
— Martin Eden by Jack London
This in turn suggests a review of the film "Queen to Play" —
(Background: Nabokov + Patterns.)
The review announces showings of the film at Clark University
in Worcester, Mass., on Sunday, October 30, 2011.
See also this journal on that date— "The Idea Idea"— and
references to a knight figure from today's date in 1985.
A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.
From that essay—
"The idea of quantization—
the discreteness of physical quantities—
turned out to be immensely fruitful."
Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic—
|
"Is Space Digital?" — Cover story, Scientific American "The idea that space may be digital — Physicist Sabine Hossenfelder "A quantization of space/time — Peter Woit in a comment |
It seems some clarification is in order.
Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.
Scientific American on the concept of digital space—
"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."
Wikipedia on the concept of quantization—
Causal sets, loop quantum gravity, string theory,
and black hole thermodynamics all predict
a quantized spacetime….
For a purely mathematical approach to the
continuous-vs.-discrete issue, see
Finite Geometry and Physical Space.
The physics there is somewhat tongue-in-cheek,
but the geometry is serious.The issue there is not
continuous-vs.-discrete physics , but rather
Euclidean-vs.-Galois geometry .
Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to
physics; Galois geometry has not. The cited
webpage describes the interplay of both sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.
This post was suggested by Paradigms Lost
(a post cited here a year ago today),
by David Weinberger's recent essay "Shift Happens,"
and by today's opening of "The Raven."
David Weinberger in The Chronicle of Higher Education , April 22—
"… Kuhn was trying to understand how Aristotle could be such a brilliant natural scientist except when it came to understanding motion. Aristotle's idea that stones fall and fire rises because they're trying to get to their natural places seems like a simpleton's animism.
Then it became clear to Kuhn all at once. Ever since Newton, we in the West have thought movement changes an object's position in neutral space but does not change the object itself. For Aristotle, a change in position was a change in a quality of the object, and qualitative change tended toward an asymmetric actualization of potential: an acorn becomes an oak, but an oak never becomes an acorn. Motion likewise expressed a tendency for things to actualize their essence by moving to their proper place. With that, 'another initially strange part of Aristotelian doctrine begins to fall into place,' Kuhn wrote in The Road Since Structure ."
Dr. John Raven (of Raven's Progressive Matrices)—
"… these tools cannot be immediately applied within our current workplaces, educational systems, and public management systems because the operation of these systems is determined, not by personal developmental or societal needs, but by a range of latent, rarely discussed, and hard to influence sociological forces.
But this is not a cry of despair: It points to another topic which has been widely neglected by psychologists: It tells us that human behaviour is not mainly determined by internal properties— such as talents, attitudes, and values— but by external social forces. Such a transformation in psychological thinking and theorising is as great as the transformation Newton introduced into physics by noting that the movement of inanimate objects is not determined by internal, 'animistic,' properties of the objects but by invisible external forces which act upon them— invisible forces that can nevertheless be mapped, measured, and harnessed to do useful work for humankind.
So this brings us to our fourth conceptualisation and measurement topic: How are these social forces to be conceptualised, mapped, measured, and harnessed in a manner analogous to the way in which Newton made it possible to harness the destructive forces of the wind and the waves to enable sailing boats to get to their destinations?"
Before Newton, boats never arrived?
"These lowbrow, popular cultural references bring the possibilities
of interpretation down to an everyday level, forcing us to acknowledge
that not every painting that looks like a splatter is necessarily a
homage/anti-homage to Jackson Pollock."
— Stina Högqvist, review of the art of Josefine Lyche
The 3×3×3 Galois Cube
Backstory— The Talented, from April 26 last year,
and Atlas Shrugged, from April 27 last year.
(Mythopoetic continued)
Voice of America today—
Thousands of Norwegians Defy Confessed Killer Breivik in Song
"The demonstrators waved roses and flags
Thursday as they and Norwegian folk singer
Lillebjoern Nilsen sang an adaptation
of the children's song, 'My Rainbow Race,'
which Breivik in court last week called
an example of Marxist brainwashing."
[See also PETE SEEGER AND LILLEBJØRN NILSEN.
Click on the image below for Seeger's original version.]
Liberia Reacts to Taylor Conviction With Mixed Emotions
"As the verdict was read out, a rainbow was seen
in the sky, encircling the sun. For many Liberians,
superstition is a part of life. The rainbow heralded
a new era, they said, beginning with the verdict of Taylor."
["You're not the only one... with mixed emotions."]
Two narratives in memory of my seventh-grade
history teacher, who died on January 27, 2012—
1. On the history of Liberia
(subject of a paper I wrote in seventh grade), and
2. Comic-book history (from the above date)
A meditation suggested by the April 20 post Complex Reflection
and by the life and April 20 death of a scientist who worked
at Los Alamos (home of the Monte Carlo method) and at
the Santa Fe Institute (home of complexity theory).
A search for 286 in this journal yields "Yet Another Cartoon Graveyard."
That June 1, 2008, post linked to poem 286 in a 1919 anthology.
Here is that poem, together with poem 823.
Together, these poems may be regarded as a meditation on
Simone Weil and her brother André Weil or,
more abstractly, on Love and Death.
Happy birthday to Al Pacino.
(Not Olympus )
(Continued from Mythopoetic, a post of April 12)
This post was suggested by a 2010 film about fictional Olympians,
by today's New York Times obituaries, and by a bar brawl at the Olympian
New York Athletic Club in "the wee hours of April 13."
Rick Riordan in the image below advertises another large
"Demigod Gathering" on October 12, 2010.
The Riordan image is from a post at ComposersCave.com
made on October 3, 2010.
Applying Jung's principle of synchronicity to this demigod material,
we find the October 3, 2010, Log24 post Search for the Basic Picture
and the October 12, 2010, Log24 post King Solomon's Mind.
Note that the latter October date is that of the traditional
Columbus Day, and that the 2010 film of The Lightning Thief
was directed by Chris Columbus of 1492 Pictures.
The film's release was earlier in 2010, on February 12.
See also —
"Pardon me. J'adoube."
— The Consul as he fastens his fly in Under the Volcano ,
the Garden of Eden scene
Suggested by a piece in the
Melbourne Herald Sun dated August 1, 2007—
A link from Log24 on that date—
Show Business
according to Fritz Leiber:
“Sid thinks you’re ready for
some of the smaller parts.”
In memory of Mike Wallace—
See also Knight Moves.
"… this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it…."
— G. H. Hardy, A Mathematician's Apology
In Geometry and the Imagination , Hilbert and Cohn-Vossen
describe the Brianchon-Pascal configuration of 9 points
and 9 lines, with 3 points on each line and 3 lines through
each point, as being "the most important configuration of all geometry."
The Brianchon-Pascal configuration is also known as the Pappus configuration—
Some background …
"The Theorem of Pappus: A Bridge Between Algebra and Geometry"
Elena Anne Marchisotto
The American Mathematical Monthly
Vol. 109, No. 6 (Jun. – Jul., 2002), pp. 497-516
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