Tuesday, May 10, 2011
The LA Times on last weekend's film "Thor"—
"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."
Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).
A high-minded— if not Shakespearean— non-Nordic approach to groups acting—
"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394
"Groups act "… For some examples, see
Related entertainment—
High-minded— Many Dimensions—
Not so high-minded— The Cosmic Cube—
One way of blending high and low—
The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.
The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—
The above illustration is from undated software by Ed Pegg Jr.
For mathematical background, see a 1985 note, "Visualizing GL(2,p)."
For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).
For the Nordic-averse, some background on the Hebrew connection—
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Thursday, May 5, 2011
From this journal on July 23, 2007—
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” |
This quotation from Stevens (Harvard class of 1901) was posted here on when Daniel Radcliffe (i.e., Harry Potter) turned 18 in July 2007.
Other material from that post suggests it is time for a review of magic at Harvard.
On September 9, 2007, President Faust of Harvard
“encouraged the incoming class to explore Harvard’s many opportunities.
‘Think of it as a treasure room of hidden objects Harry discovers at Hogwarts,’ Faust said.”
That class is now about to graduate.
It is not clear what “hidden objects” it will take from four years in the Harvard treasure room.
Perhaps the following from a book published in 1985 will help…
The March 8, 2011, Harvard Crimson illustrates a central topic of Metamagical Themas , the Rubik’s Cube—
Hofstadter in 1985 offered a similar picture—
Hofstadter asks in his Metamagical introduction, “How can both Rubik’s Cube and nuclear Armageddon be discussed at equal length in one book by one author?”
For a different approach to such a discussion, see Paradigms Lost, a post made here a few hours before the March 11, 2011, Japanese earthquake, tsunami, and nuclear disaster—
Whether Paradigms Lost is beyond forgetfulness is open to question.
Perhaps a later post, in the lighthearted spirit of Faust, will help. See April 20th’s “Ready When You Are, C.B.“
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Saturday, April 30, 2011
Part I — Unity and Multiplicity
(Continued from The Talented and Galois Cube)
Part II — "A feeling, an angel, the moon, and Italy"—
Click for details
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Sunday, April 17, 2011
The following has rather mysteriously appeared in a search at Google Scholar for "Steven H. Cullinane."
This turns out to be a link to a search within this weblog. I do not know why Google Scholar attributes the resulting web page to a journal article by "AB Story" or why it drew the title from a post within the search and applied it to the entire list of posts found. I am, however, happy with the result— a Palm Sunday surprise with an eclectic mixture of styles that might please the late Robert de Marrais.
I hope the late George Temple would also be pleased. He appears in "Romancing" as a resident of Quarr Abbey, a Benedictine monastery.
The remarks by Martin Hyland quoted in connection with Temple's work are of particular interest in light of the Pope's Christmas remark on mathematics quoted here yesterday.
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Thursday, March 10, 2011
(Continued from February 19)
The cover of the April 1, 1970 second edition of The Structure of Scientific Revolutions , by Thomas S. Kuhn—
This journal on January 19, 2011—
If Galois geometry is thought of as a paradigm shift from Euclidean geometry,
both images above— the Kuhn cover and the nine-point affine plane—
may be viewed, taken together, as illustrating the shift. The nine subcubes
of the Euclidean 3x3x3 cube on the Kuhn cover do not form an affine plane
in the coordinate system of the Galois cube in the second image, but they
at least suggest such a plane. Similarly, transformations of a
non-mathematical object, the 1974 Rubik cube, are not Galois transformations,
but they at least suggest such transformations.
See also today's online Harvard Crimson illustration of problems of translation—
not unrelated to the problems of commensurability discussed by Kuhn.
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Wednesday, January 19, 2011
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Friday, December 17, 2010
Excerpt from a post of 8 AM May 26, 2006 —
|
A Living Church
continued from March 27, 2006
"The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast."
– G. K. Chesterton
|
A related scene from the opening of Blake Edwards's "S.O.B." —
Click for Julie Andrews in the full video.
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Monday, December 13, 2010
Leading today's New York Times obituaries —
— is that of Nassos Daphnis, a painter of geometric abstractions
who in 1995 had an exhibition at a Leo Castelli gallery
titled "Energies in Outer Space." (See pictures here.)
Daphnis died, according to the Times, on November 23.
See Art Object, a post in this journal on that date—
There is more than one way
to look at a cube.
Some context— this morning's previous post (Apollo's 13,
on the geometry of the 3×3×3 cube), yesterday's noon post
featuring the 3×3 square grid (said to be a symbol of Apollo),
and, for connoisseurs of the Ed Wood school of cinematic art,
a search in this journal for the phrase "Plan 9."
You can't make this stuff up.
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Apollo's 13: A Group Theory Narrative —
I. At Wikipedia —
II. Here —
See Cube Spaces and Cubist Geometries.
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 3×3×3 cube–
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Comments Off on Mathematics and Narrative continued…
Saturday, December 4, 2010
Bulletin of the American Mathematical Society—
"Recent Advances in the Langlands Program"
Author(s): Edward Frenkel
Journal: Bull. Amer. Math. Soc. 41 (2004), 151-184.
Posted: January 8, 2004
Item in the references:
[La5] G. Laumon, La correspondance de Langlands sur les corps de fonctions (d'après Laurent
La fforgue), Séminaire Bourbaki, Exp. No. 973, Preprint math.AG/0003131.
Correction—
Related material— Peter Woit 's post on Frenkel today—
"Math Research Institute, Art, Politics, Transgressive Sex and Geometric Langlands."
See also an item from a Google search on " 'nit-picking' + Bourbaki "—
White Cube — Jake & Dinos Chapman
Fucking Hell is not, evidently, a realistic (much less nit-picking ) account of the ….
The following link enables you to pan virtually around the Bourbaki …
www.whitecube.com/artists/chapman/texts/154/ – Cached
— as well as a search for "White Cube" in this journal.
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Thursday, December 2, 2010
The Dreidel Is Cast
The Nietzschean phrase "ruling and Caesarian spirits" occurred in yesterday morning's post "Novel Ending."
That post was followed yesterday morning by a post marking, instead, a beginning— that of Hanukkah 2010. That Jewish holiday, whose name means "dedication," commemorates the (re)dedication of the Temple in Jerusalem in 165 BC.
The holiday is celebrated with, among other things, the Jewish version of a die— the dreidel . Note the similarity of the dreidel to an illustration of The Stone* on the cover of the 2001 Eerdmans edition of Charles Williams's 1931 novel Many Dimensions—
For mathematics related to the dreidel , see Ivars Peterson's column on this date fourteen years ago.
For mathematics related (if only poetically) to The Stone , see "Solomon's Cube" in this journal.
Here is the opening of Many Dimensions—
For a fanciful linkage of the dreidel 's concept of chance to The Stone 's concept of invariant law, note that the New York Lottery yesterday evening (the beginning of Hanukkah) was 840. See also the number 840 in the final post (July 20, 2002) of the "Solomon's Cube" search.
Some further holiday meditations on a beginning—
Today, on the first full day of Hanukkah, we may or may not choose to mark another beginning— that of George Frederick James Temple, who was born in London on this date in 1901. Temple, a mathematician, was President of the London Mathematical Society in 1951-1953. From his MacTutor biography—
"In 1981 (at the age of 80) he published a book on the history of mathematics. This book 100 years of mathematics (1981) took him ten years to write and deals with, in his own words:-
those branches of mathematics in which I had been personally involved.
He declared that it was his last mathematics book, and entered the Benedictine Order as a monk. He was ordained in 1983 and entered Quarr Abbey on the Isle of Wight. However he could not stop doing mathematics and when he died he left a manuscript on the foundations of mathematics. He claims:-
The purpose of this investigation is to carry out the primary part of Hilbert's programme, i.e. to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced."
For a brief review of Temple's last work, see the note by Martin Hyland in "Fundamental Mathematical Theories," by George Temple, Philosophical Transactions of the Royal Society, A, Vol. 354, No. 1714 (Aug. 15, 1996), pp. 1941-1967.
The following remarks by Hyland are of more general interest—
"… one might crudely distinguish between philosophical and mathematical motivation. In the first case one tries to convince with a telling conceptual story; in the second one relies more on the elegance of some emergent mathematical structure. If there is a tradition in logic it favours the former, but I have a sneaking affection for the latter. Of course the distinction is not so clear cut. Elegant mathematics will of itself tell a tale, and one with the merit of simplicity. This may carry philosophical weight. But that cannot be guaranteed: in the end one cannot escape the need to form a judgement of significance."
— J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.
Here Hyland appears to be discussing semantic ("philosophical," or conceptual) and syntactic ("mathematical," or structural) approaches to proof theory. Some other remarks along these lines, from the late Gian-Carlo Rota—
(Click to enlarge.)
See also "Galois Connections" at alpheccar.org and "The Galois Connection Between Syntax and Semantics" at logicmatters.net.
* Williams's novel says the letters of The Stone are those of the Tetragrammaton— i.e., Yod, He, Vau, He (cf. p. 26 of the 2001 Eerdmans edition). But the letters on the 2001 edition's cover Stone include the three-pronged letter Shin , also found on the dreidel . What esoteric religious meaning is implied by this, I do not know.
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Tuesday, November 23, 2010
A reviewer says Steve Martin finds in his new novel An Object of Beauty "a sardonic morality tale."
From this journal on the day The Cube was published (see today's Art Object ) —
|
m759 @ 12:00 AM
The Past Revisited
From Log24 a year ago on this date, a quote from Many Dimensions (1931), by Charles Williams:
“Lord Arglay had a suspicion that the Stone would be purely logical. Yes, he thought, but what, in that sense, were the rules of its pure logic?”
For the rest of the story, see the downloadable version at Project Gutenberg of Australia.
|
See also a post on Mathematics and Narrative from Nov. 14, 2009.
That post compares characters in Many Dimensions to those in Logicomix—
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Sunday, November 7, 2010
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Sunday, October 3, 2010
(Click to enlarge.)
The above is the result of a (fruitless) image search today for a current version of Giovanni Sambin's "Basic Picture: A Structure for Topology."
That search was suggested by the title of today's New York Times op-ed essay "Found in Translation" and an occurrence of that phrase in this journal on January 5, 2007.
Further information on one of the images above—
A search in this journal on the publication date of Giaquinto's Visual Thinking in Mathematics yields the following—
For the Meno 's diamond figure in Giaquinto, see a review—
— Review by Jeremy Avigad (preprint)
Finite geometry supplies a rather different context for Plato's "basic picture."
In that context, the Klein four-group often cited by art theorist Rosalind Krauss appears as a group of translations in the mathematical sense. (See Kernel of Eternity and Sacerdotal Jargon at Harvard.)
The Times op-ed essay today notes that linguistic translation "… is not merely a job assigned to a translator expert in a foreign language, but a long, complex and even profound series of transformations that involve the writer and reader as well."
The list of four-group transformations in the mathematical sense is neither long nor complex, but is apparently profound enough to enjoy the close attention of thinkers like Krauss.
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Thursday, July 8, 2010
or: Catullus vs. Ovid
(Today's previous post, "Coxeter vs. Fano,"
might also have been titled "Toronto vs. Rome.")
ut te postremo donarem munere mortis
– Catullus 101
Explicatio
Unfolding
Image by Christopher Thomas at Wikipedia —
Unfolding of a hypercube and of a cube —
The metaphor for metamorphosis no keys unlock.
— Steven H. Cullinane, "Endgame"
The current New Yorker has a translation of
the above line of Catullus by poet Anne Carson.
According to poets.org, Carson "attended St. Michael's College
at the University of Toronto and, despite leaving twice,
received her B.A. in 1974, her M.A. in 1975 and her Ph.D. in 1981."
Carson's translation is given in a review of her new book Nox.
The title, "The Unfolding," of the current review echoes an earlier
New Yorker piece on another poet, Madeleine L'Engle—
Cynthia Zarin in The New Yorker, issue dated April 12, 2004–
“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded– or it’s a story without a book.’”
(See also the "harrow up" + Hamlet link in yesterday's 6:29 AM post.)
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Wednesday, June 30, 2010
In memory of Wu Guanzhong, Chinese artist who died in Beijing on Friday—
"Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game. Elder Brother laughed. 'Go ahead and try,' he exclaimed. 'You'll see how it turns out. Anyone can create a pretty little bamboo garden in the world. But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'"
— Hermann Hesse, The Glass Bead Game, translated by Richard and Clara Winston
"The Chinese painter Wu Tao-tzu was famous because he could paint nature in a unique realistic way that was able to deceive all who viewed the picture. At the end of his life he painted his last work and invited all his friends and admirers to its presentation. They saw a wonderful landscape with a romantic path, starting in the foreground between flowers and moving through meadows to high mountains in the background, where it disappeared in an evening fog. He explained that this picture summed up all his life’s work and at the end of his short talk he jumped into the painting and onto the path, walked to the background and disappeared forever."
— Jürgen Teichmann. Teichmann notes that "the German poet Hermann Hesse tells a variation of this anecdote, according to his own personal view, as found in his 'Kurzgefasster Lebenslauf,' 1925."
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Monday, June 28, 2010
Margaret Soltan on a summer's-day poem by D.A. Powell—
first, a congregated light, the brilliance of a meadowland in bloom
and then the image must fail, as we must fail, as we
graceless creatures that we are, unmake and befoul our beds
don’t tell me deluge. don’t tell me heat, too damned much heat
"Specifically, your trope is the trope of every life:
the organizing of the disparate parts of a personality
into a self (a congregated light), blazing youth
(a meadowland in bloom), and then the failure
of that image, the failure of that self to sustain itself."
Alternate title for Soltan's commentary, suggested by yesterday's Portrait:
Smart Jewish Girl Fwows Up.
Midrash on Soltan—
Congregated Light
Meadowland
Failure
Wert thou my enemy, O thou my friend,
How wouldst thou worse, I wonder, than thou dost
Defeat, thwart me?
Coda
"…meadow-down is not distressed
For a rainbow footing…."
Comments Off on Shall I Compare Thee
Sunday, June 27, 2010
27
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 3×3×3 cube–
Comments Off on Sunday at the Apollo
Monday, June 21, 2010
From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—
Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—
Peterson associates the four-diamond figure
with the Pythagorean theorem.
A more relevant association is the
four-diamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—
This figure is relevant because of a
tesseract sculpture by Peter Forakis—
This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008-January 18, 2009.
The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
Non-Euclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).
For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).
"There is such a thing
as a tesseract."
— A Wrinkle in TIme
Comments Off on Test
This journal’s 11 AM Sunday post was “Lovasz Wins Kyoto Prize.” This is now the top item on the American Mathematical Society online home page—
Click to enlarge.
For more background on Lovasz, see today’s
previous Log24 post, Cube Spaces, and also
Cube Space, 1984-2003.
“If the Party could thrust its hand into the past and
… say of this or that event, it never happened….”
— George Orwell, 1984
Comments Off on 1984 Story (continued)
Wednesday, June 16, 2010
David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —
Escher’s Verbum
Solomon’s 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."
Comments Off on Brightness at Noon
Tuesday, March 30, 2010
Harvard Crimson headline today–
“Deconstructing Design“
Reconstructing Design
The phrase “eightfold way” in today’s
previous entry has a certain
graphic resonance…
For instance, an illustration from the
Wikipedia article “Noble Eightfold Path” —
Adapted detail–
See also, from
St. Joseph’s Day—
Harvard students who view Christian symbols
with fear and loathing may meditate
on the above as a representation of
the Gankyil rather than of the Trinity.
Comments Off on Eightfold Symmetries
Wednesday, March 17, 2010
Rigor
“317 is a prime, not because we think so,
or because our minds are shaped in one way
rather than another, but because it is so,
because mathematical reality is built that way.”
– G. H. Hardy,
A Mathematician’s Apology
The above photo is taken from
a post in this journal dated
March 10, 2010.
This was, as the Pope might say,
the dies natalis of a master gameplayer–
| New York Times, March 16, 2010–
Tim Holland, Backgammon Master,
Dies at 79
By DENNIS HEVESI
Tim Holland, who was widely considered the world’s greatest backgammon player during that ancient board game’s modern heyday, in the 1960s and ’70s, died on March 10 at his home in West Palm Beach, Fla. He was 79. <<more>> |
In Holland's honor, a post
from Columbus Day, 2004—
11:11 PM
Time and Chance
Today’s winning lottery numbers
in Pennsylvania (State of Grace):
Midday: 373
Evening: 816.
|
A quote from Holland on backgammon–
"It’s the luck factor that seduces everyone
into believing that they are good,
that they can actually win,
but that’s just wishful thinking."
For those who are, like G.H. Hardy,
suspicious of wishful thinking,
here is a quote and a picture from
Holland's ordinary birthday, March 3—
"The die is cast." — Caesar
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Saturday, March 13, 2010
From yesterday's Seattle Times—
According to police, employees of a Second Avenue mission said the suspect, clad in black and covered in duct tape, had come into the mission "and threatened to blow the place up." He then told staffers "that he was a vampire and wanted to eat people."
The man… also called himself "a space cowboy"….
This suggests two film titles…
Plan 9 from Outer Space—
and Apollo's 13—
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–
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Tuesday, March 2, 2010
From the conclusion of Weyl's Symmetry —
One example of Weyl's "structure-endowed entity" is a partition of a six-element set into three disjoint two-element sets– for instance, the partition of the six faces of a cube into three pairs of opposite faces.
The automorphism group of this faces-partition contains an order-8 subgroup that is isomorphic to the abstract group C2×C2×C2 of order eight–
The action of Klein's simple group of order 168 on the Cayley diagram of C2×C2×C2 in yesterday's post furnishes an example of Weyl's statement that
"… one may ask with respect to a given abstract group: What is the group of its automorphisms…?"
Comments Off on Symmetry and Automorphisms
Monday, March 1, 2010
The current article on group theory at Wikipedia has a Rubik's Cube as its logo–
The article quotes Nathan C. Carter on the question "What is symmetry?"
This naturally suggests the question "Who is Nathan C. Carter?"
A search for the answer yields the following set of images…
Click image for some historical background.
Carter turns out to be a mathematics professor at Bentley University. His logo– an eightfold-cube labeling (in the guise of a Cayley graph)– is in much better taste than Wikipedia's.
Comments Off on Visual Group Theory
Saturday, February 27, 2010
"The cube has…13 axes of symmetry:
6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube
These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.
The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–
A closely related structure–
the finite projective plane
with 13 points and 13 lines–
A later version of the 13-point plane
by Ed Pegg Jr.–
A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–
The above images tell a story of sorts.
The moral of the story–
Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.
The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.
If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.
The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.
The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.
(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)
See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.
Comments Off on Cubist Geometries
Sunday, February 21, 2010
From the Wikipedia article "Reflection Group" that I created on Aug. 10, 2005— as revised on Nov. 25, 2009—
|
Historically, (Coxeter 1934) proved that every reflection group [Euclidean, by the current Wikipedia definition] is a Coxeter group (i.e., has a presentation where all relations are of the form ri2 or (rirj)k), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group [again, Euclidean], and classified finite Coxeter groups.
Finite fields
When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as −1=1 so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981).
|
Related material:
"A Simple Reflection Group of Order 168," by Steven H. Cullinane, and
"Determination of the Finite Primitive Reflection Groups over an Arbitrary Field of Characteristic Not 2,"
by Ascher Wagner, U. of Birmingham, received 27 July 1977
| Journal |
Geometriae Dedicata |
| Publisher |
Springer Netherlands |
| Issue |
Volume 9, Number 2 / June, 1980 |
[A primitive permuation group preserves
no nontrivial partition of the set it acts upon.]
Clearly the eightfold cube is a counterexample.
Comments Off on Reflections
Tuesday, February 16, 2010
From today's NY Times—
Obituaries for mystery authors
Ralph McInerny and Dick Francis
From the date (Jan. 29) of McInerny's death–
"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.'"
– Harvard University Press on Persons and Things (Walpurgisnacht, 2008), by Barbara Johnson
From the date (Feb. 14) of Francis's death–
The EIghtfold Cube
The "something missing" in the above figure is an eighth cube, hidden behind the others pictured.
This eighth cube is not, as Johnson would have it, a void and "vanishing point," but is instead the "still point" of T.S. Eliot. (See the epigraph to the chapter on automorphism groups in Parallelisms of Complete Designs, by Peter J. Cameron. See also related material in this journal.) The automorphism group here is of course the order-168 simple group of Felix Christian Klein.
For a connection to horses, see
a March 31, 2004, post
commemorating the birth of Descartes
and the death of Coxeter–
Putting Descartes Before Dehors
For a more Protestant meditation,
see The Cross of Descartes—
"I've been the front end of a horse
and the rear end. The front end is better."
— Old vaudeville joke
For further details, click on
the image below–
Notre Dame Philosophical Reviews
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Sunday, February 14, 2010
"Simplify, simplify." — Henry David Thoreau
"Because of their truly fundamental role in mathematics, even the simplest diagrams concerning finite reflection groups (or finite mirror systems, or root systems– the languages are equivalent) have interpretations of cosmological proportions."
— Alexandre Borovik, 2010 (See previous entry.)
Exercise: Discuss Borovik's remark
that "the languages are equivalent"
in light of the web page
A Simple Reflection Group
of Order 168.
Background:
Theorems 15.1 and 15.2 of Borovik's book (1st ed. Nov. 10, 2009)
Mirrors and Reflections: The Geometry of Finite Reflection Groups—
15.1 (p. 114): Every finite reflection group is a Coxeter group.
15.2 (p. 114): Every finite Coxeter group is isomorphic to a finite reflection group.
Consider in this context the above simple reflection group of order 168.
(Recall that "…there is only one simple Coxeter group (up to isomorphism); it has order 2…" —A.M. Cohen.)
Comments Off on Sunday School
Friday, January 29, 2010
Part I:
"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.' She shows deftly and delicately that the void inside Keats’s urn, Heidegger’s jug, or Wallace Stevens’s jar forms the center around which we tend to organize our worlds."
— Harvard University Press on Persons and Things (April 30, 2008), by Barbara Johnson
Part II:
Part III:
From the date of Barbara Johnson's death:
"Mathematical relationships were
enough to satisfy him, mere formal
relationships which existed at
all times, everywhere, at once."
– Broken Symmetries, 1983
The X's refer to the pattern on the
cover of a paperback edition
of Nine Stories, by J. D. Salinger.
Salinger died on Wednesday.
"You remember that book he sent me
from Germany? You know–
those German poems."
In Germany, Wednesday was
Holocaust Memorial Day, 2010.
Comments Off on More Glass
Sunday, December 20, 2009
Dies Natalis of
Emil Artin
From the September 1953 Bulletin of the American Mathematical Society—
Emil Artin, in a review of Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. I-VII–
"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits the visualization of the whole, and yet this logical structure must predominate or chaos would result."
Art Versus Chaos
From an exhibit,
"Reimagining Space"
The above tesseract (4-D hypercube)
sculpted in 1967 by Peter Forakis
provides an example of what Artin
called "the visualization of the whole."
For related mathematical details see
Diamond Theory in 1937.
"'The test?' I faltered, staring at the thing.
'Yes, to determine whether you can live
in the fourth dimension or only die in it.'"
— Fritz Leiber, 1959
See also the Log24 entry for
Nov. 26, 2009, the date that
Forakis died.
"There is such a thing
as a tesseract."
— Madeleine L'Engle, 1962
Comments Off on The Test
Thursday, December 3, 2009
Mathematics and Narrative, continued…
Out of What Chaos, a novel by Lee Oser—
"This book is more or less what one would expect if Walker Percy wrote about a cynical rock musician who converts to Catholicism, and then Nabokov added some of his verbal pyrotechnics, and then Buster Keaton and the Marquis de Sade and Lionel Trilling inserted a few extra passages. It is a loving and yet appalled description of the underground music scene in the Pacific Northwest. And it is a convincing representation of someone very, very smart."
—Matt Greenfield in The Valve
"If Evelyn Waugh had lived amid the American Northwest rock music scene, he might have written a book like this."
–Anonymous Amazon.com reviewer
A possible source for Oser's title–
"…Lytton Strachey described Pope's theme as 'civilization illumined by animosity; such was the passionate and complicated material from which he wove his patterns of balanced precision and polished clarity.' But out of what chaos did that clarity and precision come!"
—Authors at Work, by Herman W. Liebert and Robert H. Taylor, New York, Grolier Club, 1957, p. 16
Related material:
Unthought Known
and the
White Cube Gallery, 2002
Comments Off on Out of What Chaos
Saturday, November 14, 2009
A graphic novel reviewed in the current Washington Post features Alfred North Whitehead and Bertrand Russell–
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:
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….“
Comments Off on Mathematics and Narrative, continued:
Saturday, September 19, 2009
Old Year, Raus!
Also in today’s New York Times obituaries index:
John T. Elson, Editor Who Asked
“Is God Dead?” at Time, Dies at 78
Wikipedia article on George Polya:
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Use a model
- Work backward
From the date of Elson’s death:
Comments Off on Saturday September 19, 2009
Tuesday, September 8, 2009
Froebel's
Magic Box
Continued from
Dec. 7, 2008,
and from
yesterday.
Non-Euclidean
Blocks
Passages from a classic story:
… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads….
Tesseract
"Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
"Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child– especially a baby– might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."
"Hardening of the thought-arteries," Jane interjected.
Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–"
"Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–"
"Poor kid," Jane said.
Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–"
"Blocks? What kind?"
Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid."
— "Mimsy Were the Borogoves," Lewis Padgett, 1943
|
Padgett (pseudonym of a
husband-and-
wife writing team) says that alphabet blocks are the intuitive "basis of Euclid."
Au contraire; they are the basis of
Gutenberg.
For the intuitive basis of one type of non-Euclidean* geometry– finite geometry over the two-element Galois field– see the work of…
Friedrich Froebel
(1782-1852), who
invented kindergarten.
His "third gift" —
© 2005 The Institute for Figuring
Comments Off on Tuesday September 8, 2009
Monday, September 7, 2009
Magic Boxes
"Somehow it seems to fill my head with ideas– only I don't exactly know what they are!…. Let's have a look at the garden first!"
— A passage from Lewis Carroll's Through the Looking-Glass. The "garden" part– but not the "ideas" part– was quoted by Jacques Derrida in Dissemination in the epigraph to Chapter 7, "The Time before First."
Commentary
on the passage:
Part I "The Magic Box," shown on Turner Classic Movies earlier tonight
Part II: "Mimsy Were the Borogoves," a classic science fiction story:
"… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."
Part III: A Crystal Block —
Image of pencils is by
Diane Robertson Design.
Related material:
"A Four-Color Theorem."
Part IV:
"Click on the Yellow Book."
Comments Off on Monday September 7, 2009
Sunday, September 6, 2009
Magic Boxes
Part I: “The Magic Box,” shown on Turner Classic Movies tonight
Part II: “Mimsy Were the Borogoves,” a classic science fiction story:
“… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example–
They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play.”
Comments Off on Sunday September 6, 2009
Saturday, September 5, 2009
Comments Off on Saturday September 5, 2009
Wednesday, August 19, 2009
Group Actions, 1984-2009
From a 1984 book review:
"After three decades of intensive research by hundreds of group theorists, the century old problem of the classification of the finite simple groups has been solved and the whole field has been drastically changed. A few years ago the one focus of attention was the program for the classification; now there are many active areas including the study of the connections between groups and geometries, sporadic groups and, especially, the representation theory. A spate of books on finite groups, of different breadths and on a variety of topics, has appeared, and it is a good time for this to happen. Moreover, the classification means that the view of the subject is quite different; even the most elementary treatment of groups should be modified, as we now know that all finite groups are made up of groups which, for the most part, are imitations of Lie groups using finite fields instead of the reals and complexes. The typical example of a finite group is GL(n, q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."
— Jonathan L. Alperin,
review of books on group theory,
Bulletin (New Series) of the American
Mathematical Society 10 (1984) 121, doi:
10.1090/S0273-0979-1984-15210-8
The same example
at Wolfram.com:
Caption from Wolfram.com:
"The two-dimensional space Z3×Z3 contains nine points: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), and (2,2). The 48 invertible 2×2 matrices over Z3 form the general linear group known as GL(2, 3). They act on Z3×Z3 by matrix multiplication modulo 3, permuting the nine points. More generally, GL(n, p) is the set of invertible n×n matrices over the field Zp, where p is prime. With (0, 0) shifted to the center, the matrix actions on the nine points make symmetrical patterns."
Citation data from Wolfram.com:
"GL(2,p) and GL(3,3) Acting on Points"
from The Wolfram Demonstrations Project,
http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/,
Contributed by: Ed Pegg Jr"
As well as displaying Cullinane's 48 pictures of group actions from 1985, the Pegg program displays many, many more actions of small finite general linear groups over finite fields. It illustrates Cullinane's 1985 statement:
"Actions of GL(2,p) on a p×p coordinate-array have the same sorts of symmetries, where p is any odd prime."
Pegg's program also illustrates actions on a cubical array– a 3×3×3 array acted on by GL(3,3). For some other actions on cubical arrays, see Cullinane's Finite Geometry of the Square and Cube.
Comments Off on Wednesday August 19, 2009
Tuesday, August 4, 2009
Due Deference
The New York Times today
on architect Charles Gwathmey,
who died Monday:
"Mr. Gwathmey's Astor Place
condominium tower drew
criticism from those who
said it was insufficiently
deferential to its
surroundings."
Astor Place tower
(click to enlarge):
Surroundings:
The above sculpture,
popularly known as
The Borg Cube,
appeared here on
Saturday:
Comments Off on Tuesday August 4, 2009
Saturday, August 1, 2009
And the Tony
goes to…
The New York Times today:
"Tony Rosenthal, who created 'Alamo,' the eternally popular revolving black cube in Astor Place in the East Village, and many other public sculptures, died on Tuesday [July 28, 2009] in Southampton, N.Y. He was 94."
The Astor Place sculpture, near Cooper Union, is also known as The Borg Cube:
The Borg Cube, with
Cooper Union at left
Wikipedia on The Borg Queen:
"The Borg Queen is the focal point within the Borg collective consciousness."
Possible Borg-Queen candidates:
Helen Mirren, who appeared in this journal on the date of Rosenthal's death (see Monumental Anniversary), and Julie Taymor, who recently directed Mirren as Prospera in a feminist version of "The Tempest."
Both Mirren and Taymor would appreciate the work of Anita Borg, who pioneered the role of women in computer science. "Her colleagues mourned Borg's passing, even as they stressed how crucial she was in creating a kind of collective consciousness for women working in the heavily male-dominated field of computer technology." —Salon.com obituary
Anita Borg
Borg died on Sunday, April 6, 2003. See The New York Times Magazine for that date in Art Wars: Geometry as Conceptual Art—
(Cover typography revised)
I would award the Borg-Queen Tony to Taymor, who seems to have a firmer grasp of technology than Mirren.
Comments Off on Saturday August 1, 2009
Friday, April 10, 2009
Pilate Goes
to Kindergarten
“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
remarks on the “Story Theory“
of truth as opposed to the
“Diamond Theory” of truth in
The Non-Euclidean Revolution
Consider the following question in a paper cited by V. S. Varadarajan:
E. G. Beltrametti, “Can a finite geometry describe physical space-time?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.
Simplifying:
“Can a finite geometry describe physical space?”
Simplifying further:
“Yes. Vide ‘The Eightfold Cube.'”
Comments Off on Friday April 10, 2009
Thursday, April 2, 2009
Transformative
Hermeneutics
In memory of
physics historian
Martin J. Klein,
(June 25, 1924-
March 28, 2009)
"… in physics itself, there was what appeared, briefly, to be an ending, which then very quickly gave way to a new beginning: The quest for the ultimate building-blocks of the universe had been taken down to the molecular level in nineteenth-century kinetic theory… and finally to the nuclear level in the second and third decades of the twentieth century. For a moment in the 1920s the quest appeared to have ended…. However… this paradise turned out to be, if not exactly a fool's paradise, then perhaps an Eden lost."
— No Truth Except in the Details: Essays in Honor of Martin J. Klein, introduction by A.J. Kox and Daniel Siegel, June 25, 1994
New York Times obituary dated April 1, 2009:
"Martin J. Klein, a historian of modern physics…. died Saturday, [March 28, 2009] in Chapel Hill, N.C. He was 84 and lived in Chapel Hill."
Klein edited, among other things, Paul Ehrenfest: Collected Scientific Papers (publ. by North-Holland, Amsterdam, 1959).
Related material:
"Almost every famous chess game
is a well-wrought urn
in Cleanth Brooks’ sense."
— John Holbo,
Now We See
Wherein Lies the Pleasure
"The entire sequence of moves in these… chapters reminds one– or should remind one– of a certain type of chess problem where the point is not merely the finding of a mate in so many moves, but what is termed 'retrograde analysis'…."
— Vladimir Nabokov, foreword to The Defense
Comments Off on Thursday April 2, 2009
Sunday, March 1, 2009
Solomon's Cube
continued
"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."
— G. H. Hardy, A Mathematician's Apology
"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia
"One has O+(6) ≅ S8, the symmetric group of order 8! …."
— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.
"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."
— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
Comments Off on Sunday March 1, 2009
Tuesday, February 24, 2009
Hollywood Nihilism
Meets
Pantheistic Solipsism
Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
to hear about our religion
… that we made up."
|
From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:
… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer…
A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination.
|
Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."
As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein.
Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.
Heinlein:
"Sharpie, 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."
Stevens:
|
A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:
For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:
The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...
The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,
Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.
The rock is the habitation of the whole,
Its strength and measure, that which is near,
point A
In a perspective that begins again
At B: the origin of the mango's rind.
(Collected Poems, 528)
|
Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:
B. J. Leggett, "Stevens's Late Poetry" in
The Cambridge Companion to Wallace Stevens— On the poem "The Rock":
"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'…. Its subject is its speaker's sense of nothingness and his need to be cured of it."
This interpretation might appeal to Joan Didion, who, as author of the classic novel
Play It As It Lays, is perhaps the world's leading expert on
Hollywood nihilism.
More positively…
Space is, of course, also a topic
in pure mathematics…
For instance, the 6-dimensional
affine space (or the corresponding
5-dimensional projective space)
over the two-element Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."
Heinlein should perhaps have had in mind
the Klein correspondence when he discussed "some way to project six dimensions into three." While such a projection is of course trivial for anyone who has taken an undergraduate course in linear algebra, the following remarks by Philippe Cara present a much more meaningful mapping, using the Klein correspondence, of structures in six (affine) dimensions to structures in three.
Cara:
Here the 6-dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3-dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.
Comments Off on Tuesday February 24, 2009
Tuesday, February 17, 2009
Diamond-Faceted:
Transformations
of the Rock
|
A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:
For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:
The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...
The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,
Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.
The rock is the habitation of the whole,
Its strength and measure, that which is near,
point A
In a perspective that begins again
At B: the origin of the mango's rind.
(Collected Poems, 528)
|
A mathematical version of
this poetic concept appears
in a rather cryptic note
from 1981 written with
Stevens's poem in mind:
For some explanation of the
groups of 8 and 24
motions referred to in the note,
see an earlier note from 1981.
For the Perlis "diamond facets,"
see the Diamond 16 Puzzle.
For a much larger group
of motions, see
Solomon's Cube.
As for "the mind itself"
and "possibilities for
human thought," see
Geometry of the I Ching.
Comments Off on Tuesday February 17, 2009
Saturday, February 7, 2009
DENNIS OVERBYE
"From the grave, Albert Einstein poured gasoline on the culture wars between science and religion this week.
A letter the physicist wrote in 1954 to the philosopher Eric Gutkind, in which he described the Bible as 'pretty childish' and scoffed at the notion that the Jews could be a 'chosen people,' sold for $404,000 at an auction in London. That was 25 times the presale estimate."
Einstein did not, at least in the place alleged, call the Bible "childish." Proof:
The image of the letter is
from the Sept./Oct. 2008
Search Magazine.
By the way, today is
the birthday of G. H. Hardy.
Here is an excerpt from his
thoughts on childish things:
"What 'purely aesthetic' qualities can we distinguish in such theorems as Euclid's or Pythagoras's?…. In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions."
"Space: what you
damn well have to see."
— James Joyce, Ulysses
Comments Off on Saturday February 7, 2009
Thursday, February 5, 2009
Through the
Looking Glass:
A Sort of Eternity
From the new president’s inaugural address:
“… in the words of Scripture, the time has come to set aside childish things.”
The words of Scripture:
| 9 |
For we know in part, and we prophesy in part. |
| 10 |
But when that which is perfect is come, then that which is in part shall be done away. |
| 11 |
When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things. |
| 12 |
For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.
— First Corinthians 13 |
“through a glass”—
[di’ esoptrou].
By means of
a mirror [esoptron].
Childish things:
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
Not-so-childish:
Three planes through
the center of a cube
that split it into
eight subcubes:
Through a glass, darkly:
A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180-degree rotation:
(Click on image
for further details.)
But then face to face:
A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.
Galois age fifteen,
drawn by a classmate.
These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.
For some generalizations,
see Galois Geometry.
Related material:
The central aim of Western religion–
"Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)
The central aim of Western philosophy–
Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
“Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy.”
— Jamie James in The Music of the Spheres (1993)
“In the garden of Adding
live Even and Odd…
And the song of love’s recision
is the music of the spheres.”
— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)
A quotation today at art critic Carol Kino’s website, slightly expanded:
“Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history.”
— Octavio Paz,”Seeing and Using: Art and Craftsmanship,” in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52
From Brian O’Doherty’s 1976 Artforum essays– not on museums, but rather on gallery space:
“Inside the White Cube“
“We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art.”
“Space: what you
damn well have to see.”
— James Joyce, Ulysses
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Tuesday, January 6, 2009
Archetypes, Synchronicity,
and Dyson on Jung
The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:
“… we do not need to accept Jung’s theory as true in order to find it illuminating.”
The same is true of Jung’s remarks on synchronicity.
For example —
Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s four-diamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:
R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.
Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.
On these dates, the entries in this journal discussed…
Oct. 24:
Cube Space, 1984-2003
Material related to that entry:
Dec. 19:
Art and Religion: Inside the White Cube
That entry discusses a book by Mark C. Taylor:
The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).
In Chapter 3, “Sutures of Structures,” Taylor asks —
“What, then, is a frame, and what is frame work?”
One possible answer —
Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.
“Examples are the stained-glass
windows of knowledge.”
— Vladimir Nabokov
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Friday, December 19, 2008
Inside the
White Cube
Part I: The White Cube
Part II: Inside
Part III: Outside
Click to enlarge.
Mark Tansey, The Key (1984)
For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).
For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion at
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.
For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation
(U. of Chicago Press, 1999):
From Chapter 3,
"Sutures* of Structures," p. 58:
"What, then, is a frame, and what is frame work?
This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'"
* P. 61:
"… the frame forms the suture of structure. A suture is 'a seamless [sic**] joint or line of articulation,' which, while joining two surfaces, leaves the trace of their separation."
** A dictionary says "a seamlike joint or line of articulation," with no mention of "trace," a term from Derrida's jargon.
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Friday, December 12, 2008
On the Symmetric Group S8
Wikipedia on Rubik's 2×2×2 "Pocket Cube"–
"Any permutation of the 8 corner cubies is possible (8! positions)."
Some pages related to this claim–
Simple Groups at Play
Analyzing Rubik's Cube with GAP
Online JavaScript Pocket Cube.
The claim is of course trivially true for the unconnected subcubes of Froebel's Third Gift:
© 2005 The Institute for Figuring
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Monday, December 8, 2008
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Sunday, December 7, 2008
Space and
the Soul
On a book by Margaret Wertheim:
“She traces the history of space beginning with the cosmology of Dante. Her journey continues through the historical foundations of celestial space, relativistic space, hyperspace, and, finally, cyberspace.” –Joe J. Accardi, Northeastern Illinois Univ. Lib., Chicago, in Library Journal, 1999 (quoted at Amazon.com)
There are also other sorts of space.
© 2005 The Institute for Figuring
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Friday, December 5, 2008
Continued from
Monday:
A Version of
Heaven’s Gate
in memory of
Alexy II, the Russian Orthodox
patriarch who died today in Moscow:
The Pearly Gates of Cyberspace:
From Geoffrey Broadbent,
“Why a Black Square?” in Malevich
(London, Art and Design/
Academy Group, 1989, p. 49):
“Malevich’s Black Square seems to be
nothing more, nor less, than his
‘Non-Objective’ representation
of Bragdon’s (human-being-as) Cube
passing through the ‘Plane of Reality.’!”
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Monday, December 1, 2008
From Braque's birthday, 2006:
"The senses deform, the mind forms. Work to perfect the mind. There is no certitude but in what the mind conceives."
— Georges Braque,
Reflections on Painting, 1917
Those who wish to follow Braque's advice may try the following exercise from a book first published in 1937:
Hint: See the following
construction of a tesseract:
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Sunday, November 30, 2008
Abstraction and Faith
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From Sol LeWitt: A Retrospective, edited by Gary Garrels, Yale University Press, 2000, p. 376:
THE SQUARE AND THE CUBE
by Sol LeWitt
The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed.
Reprinted from Lucy R. Lippard et al., "Homage to the Square," Art in America 55, No. 4 (July-August 1967): 54. (LeWitt's contribution was originally untitled.)
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A vulgarized version
of LeWitt's remarks
appears on a webpage of
the National Gallery of Art.
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Today's Sermon
"Closing the Circle on Abstract Art"
On Kirk Varnedoe's National Gallery lectures in 2003 (Philip Kennicott, Washington Post, Sunday, May 18, 2003):
"Varnedoe's lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself."
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Sunday, November 16, 2008
Art and Lies
Observations suggested by an article on author Lewis Hyde– "What is Art For?"– in today's New York Times Magazine:
Margaret Atwood (pdf) on Lewis Hyde's
Trickster Makes This World: Mischief, Myth, and Art —
"Trickster," says Hyde, "feels no anxiety when he deceives…. He… can tell his lies with creative abandon, charm, playfulness, and by that affirm the pleasures of fabulation." (71) As Hyde says, "… almost everything that can be said about psychopaths can also be said about tricksters," (158), although the reverse is not the case. "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)
What is "the next world"? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning to join, to fit, and to make. (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
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"What happened to that… cube?"
Apollinax laughed until his eyes teared. "I'll give you a hint, my dear. Perhaps it slid off into a higher dimension."
"Are you pulling my leg?"
"I wish I were," he sighed. "The fourth dimension, as you know, is an extension along a fourth coordinate perpendicular to the three coordinates of three-dimensional space. Now consider a cube. It has four main diagonals, each running from one corner through the cube's center to the opposite corner. Because of the cube's symmetry, each diagonal is clearly at right angles to the other three. So why shouldn't a cube, if it feels like it, slide along a fourth coordinate?"
— "Mr. Apollinax Visits New York," by Martin Gardner, Scientific American, May 1961, reprinted in The Night is Large
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Comments Off on Sunday November 16, 2008
Wednesday, October 22, 2008
Euclid vs. Galois
On May 4, 2005, I wrote a note about how to visualize the 7-point Fano plane within a cube.
Last month, John Baez showed slides that touched on the same topic. This note is to clear up possible confusion between our two approaches.
From Baez’s Rankin Lectures at the University of Glasgow:
Note that Baez’s statement (
pdf) “Lines in the Fano plane correspond to planes through the origin [the vertex labeled ‘1’] in this cube” is, if taken (wrongly) as a statement about a cube in Euclidean 3-space, false.
The statement is, however, true of the eightfold cube, whose eight subcubes correspond to points of the linear 3-space over the two-element field, if “planes through the origin” is interpreted as planes within that linear 3-space, as in Galois geometry, rather than within the Euclidean cube that Baez’s slides seem to picture.
This Galois-geometry interpretation is, as an article of his from 2001 shows, actually what Baez was driving at. His remarks, however, both in 2001 and 2008, on the plane-cube relationship are both somewhat trivial– since “planes through the origin” is a standard definition of lines in projective geometry– and also unrelated– apart from the possibility of confusion– to my own efforts in this area. For further details, see The Eightfold Cube.
Comments Off on Wednesday October 22, 2008
Friday, September 26, 2008
Christmas Knotfor T.S. Eliot’s birthday
(Continued from Sept. 22–
“A Rose for Ecclesiastes.”)
From Kibler’s
“Variations on a Theme of
Heisenberg, Pauli, and Weyl,”
July 17, 2008:
“It is to be emphasized
that the 15 operators…
are underlaid by the geometry
of the generalized quadrangle
of order 2…. In this geometry,
the five sets… correspond to
a spread of this quadrangle,
i.e., to a set of 5 pairwise
skew lines….”
— Maurice R. Kibler,
July 17, 2008
For ways to visualize
this quadrangle,
see Inscapes.
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Related material
A remark of Heisenberg
quoted here on Christmas 2005:
“… die Schönheit… [ist] die
richtige Übereinstimmung
der Teile miteinander
und mit dem Ganzen.”
“Beauty is the proper conformity
of the parts to one another
and to the whole.”
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Friday, August 22, 2008
Tentative movie title:
Blockheads
The Kohs Block Design
Intelligence Test
Samuel Calmin Kohs, the designer (but not the originator) of the above intelligence test, would likely disapprove of the "Aryan Youth types" mentioned in passing by a film reviewer in today's New York Times. (See below.) The Aryan Youth would also likely disapprove of Dr. Kohs.
Other related material:
1. Wechsler Cubes (intelligence testing cubes derived from the Kohs cubes shown above). See…
Harvard psychiatry and…
The Montessori Method;
The Crimson Passion;
The Lottery Covenant.
2. Wechsler Cubes of a different sort (Log24, May 25, 2008)
3. Manohla Dargis in today's New York Times:
"… 'Momma’s Man' is a touchingly true film, part weepie, part comedy, about the agonies of navigating that slippery slope called adulthood. It was written and directed by Azazel Jacobs, a native New Yorker who has set his modestly scaled movie with a heart the size of the Ritz in the same downtown warren where he was raised. Being a child of the avant-garde as well as an A student, he cast his parents, the filmmaker Ken Jacobs and the artist Flo Jacobs, as the puzzled progenitors of his centerpiece, a wayward son of bohemia….
In American movies, growing up tends to be a job for either Aryan Youth types or the oddballs and outsiders…."
4. The bohemian who named his son
Azazel:
"… I think that the deeper opportunity, the greater opportunity film can offer us is as an exercise of the mind. But an exercise, I hate to use the word, I won't say 'soul,' I won't say 'soul' and I won't say 'spirit,' but that it can really put our deepest psychological existence through stuff. It can be a powerful exercise. It can make us think, but I don't mean think about this and think about that. The very, very process of powerful thinking, in a way that it can afford, is I think very, very valuable. I basically think that the mind is not complete yet, that we are working on creating the mind. Okay. And that the higher function of art for me is its contribution to the making of mind."
— Interview with Ken Jacobs, UC Berkeley, October 1999
5. For Dargis's "Aryan Youth types"–
Comments Off on Friday August 22, 2008
Tuesday, August 19, 2008
Three Times
"Credences of Summer," VII,
by Wallace Stevens, from
Transport to Summer (1947)
"Three times the concentred
self takes hold, three times
The thrice concentred self,
having possessed
The object, grips it
in savage scrutiny,
Once to make captive,
once to subjugate
Or yield to subjugation,
once to proclaim
The meaning of the capture,
this hard prize,
Fully made, fully apparent,
fully found." |
Stevens does not say what object he is discussing.
One possibility —
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:
"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."
Another possibility —
A more modest object —
the 4×4 square.
Update of Aug. 20-21 —
Symmetries and Facets
Kostant's poetic comparison might be applied also to this object.
The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."
More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.
For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.
* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–
The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the
number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).
If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that
"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."
The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
Comments Off on Tuesday August 19, 2008
Saturday, August 16, 2008
Seeing the Finite Structure
The following supplies some context for remarks of Halmos on combinatorics.
From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–
Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–
“Part II: In Touch with God*“– on pp. 27-28:
The Root of All Deep Mathematics
“Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.
Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.
Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.
Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”
Finite Structure
on a Book Cover:
Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp
et al.,
Taylor & Francis, 1990
Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in
Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.
Walsh Series states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”
The patterns on the faces of the cube on the cover of
Walsh Series above illustrate both the
Walsh functions of order 3 and the same structure in a different guise,
subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see
Symmetry of Walsh Functions.
Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.
* See the Log24 entries following Halmos’s death.
Comments Off on Saturday August 16, 2008
Sunday, August 3, 2008
Kindergarten
Geometry
Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein’s birthday):
The play’s title, “Every Good Boy Deserves Favour,” is a mnemonic for the notes of the treble clef EGBDF.
The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.
The “u” in favour is the British way, the Stoppard way, “EGBDF” being “a Play for Actors and Orchestra” by Tom Stoppard (words) and André Previn (music).
And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.
When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard “to write something which had the need of a live full-time orchestra onstage,” the 36-year-old playwright jumped at the chance.
One hitch: Stoppard at the time knew “very little about ‘serious’ music… My qualifications for writing about an orchestra,” he says in his introduction to the 1978 Grove Press edition of “EGBDF,” “amounted to a spell as a triangle player in a kindergarten percussion band.”
— Jerry Tallmer in The Villager, March 12-18, 2008
Review of the same play as presented at Chautauqua Institution on July 24, 2008:
“Stoppard’s modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience.”
— Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008
“The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul.”
— Dan Fogelberg
“He’s watching us all the time.”
— Lucia Joyce
Finnegans Wake,
Book II, Episode 2, pp. 296-297:
I’ll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you’d wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun!
1 The chape of Doña Speranza of the Nacion. |
| ReciprocityFrom my entry of Sept. 1, 2003:
“…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….
… E. M. Forster famously advised his readers, ‘Only connect.’ ‘Reciprocity’ would be Michael Kruger’s succinct philosophy, with all that the word implies.”
— William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994
Last year’s entry on this date:
The picture above is of the complete graph K6 … Six points with an edge connecting every pair of points… Fifteen edges in all.
Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester’s synthematic totals as they relate to constructions of the Mathieu group M24.
If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. “Reciprocity” in the sense of Lao Tzu. See
Reciprocity and Reversal in Lao Tzu.
For a sense of “reciprocity” more closely related to Michael Kruger’s alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in
Shu: Reciprocity.
Kruger’s novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:
Click on the design for details.
Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in
A Graphical Representation
of the Dirac Algebra.
The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.
Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See
The Jewel of Arithmetic and
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FinnegansWiki:
Salmonson set his seel:
“Finn MacCool ate the Salmon of Knowledge.”
Wikipedia:
“George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest.”
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Friday, July 25, 2008
56 Triangles
"This wonderful picture was drawn by Greg Egan with the help of ideas from Mike Stay and Gerard Westendorp. It's probably the best way for a nonmathematician to appreciate the symmetry of Klein's quartic. It's a 3-holed torus, but drawn in a way that emphasizes the tetrahedral symmetry lurking in this surface! You can see there are 56 triangles: 2 for each of the tetrahedron's 4 corners, and 8 for each of its 6 edges."
Exercise:
Click on image for further details.
Note that if eight points are arranged
in a cube (like the centers of the
eight subcubes in the figure above),
there are 56 triangles formed by
the 8 points taken 3 at a time.
Baez's discussion says that the Klein quartic's 56 triangles can be partitioned into 7 eight-triangle Egan "cubes" that correspond to the 7 points of the Fano plane in such a way that automorphisms of the Klein quartic correspond to automorphisms of the Fano plane. Show that the 56 triangles within the eightfold cube can also be partitioned into 7 eight-triangle sets that correspond to the 7 points of the Fano plane in such a way that (affine) transformations of the eightfold cube induce (projective) automorphisms of the Fano plane.
Comments Off on Friday July 25, 2008
Monday, July 21, 2008
Knight Moves:
The Relativity Theory
of Kindergarten Blocks
(Continued from
January 16, 2008)
Something:
From Friedrich Froebel,
who invented kindergarten:
Click on image for details.
An Unusually
Complicated Theory:
From Christmas 2005:
Click on image for details.
For the eightfold cube
as it relates to Klein's
simple group, see
"A Reflection Group
of Order 168."
For an even more
complicated theory of
Klein's simple group, see
Click on image for details.
Comments Off on Monday July 21, 2008
Wednesday, July 9, 2008
God, Time, Epiphany
8:28:32 AM
Anthony Hopkins, from
All Hallows' Eve
last year:
"For me time is God,
God is time. It's an equation,
like an Einstein equation."
James Joyce, from
June 26 (the day after
AntiChristmas) this year:
"… he glanced up at the clock
of the Ballast Office and smiled:
— It has not epiphanised yet,
he said."
Ezra Pound (from a page
linked to yesterday morning):
"It seems quite natural to me
that an artist should have
just as much pleasure in an
arrangement of planes
or in a pattern of figures,
as in painting portraits…."
From Epiphany 2008:
An arrangement of planes:
From May 10, 2008:
A pattern of figures:
See also Richard Wilhelm on
Hexagram 32 of the I Ching:
"Duration is a state whose movement is not worn down by hindrances. It is not a state of rest, for mere standstill is regression. Duration is rather the self-contained and therefore self-renewing movement of an organized, firmly integrated whole, taking place in accordance with immutable laws and beginning anew at every ending. The end is reached by an inward movement, by inhalation, systole, contraction, and this movement turns into a new beginning, in which the movement is directed outward, in exhalation, diastole, expansion."
— The Middle-English
Harrowing of Hell…
by Hulme, 1907, page 64
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Friday, July 4, 2008
Comments Off on Friday July 4, 2008
Friday, June 27, 2008
Comments Off on Friday June 27, 2008
Wednesday, June 18, 2008
CHANGE
FEW CAN BELIEVE IN |
What I Loved, a novel by Siri Hustvedt (New York, Macmillan, 2003), contains a paragraph on the marriage of a fictional artist named Wechsler–
Page 67 —
“… Bill and Violet were married. The wedding was held in the Bowery loft on June 16th, the same day Joyce’s Jewish Ulysses had wandered around Dublin. A few minutes before the exchange of vows, I noted that Violet’s last name, Blom, was only an o away from Bloom, and that meaningless link led me to reflect on Bill’s name, Wechsler, which carries the German root for change, changing, and making change. Blooming and changing, I thought.”
For Hustvedt’s discussion of Wechsler’s art– sculptured cubes, which she calls “tightly orchestrated semantic bombs” (p. 169)– see Log24, May 25, 2008.
Related material:
Wechsler cubes
(after David Wechsler,
1896-1981, chief
psychologist at Bellevue)
These cubes are used to
make 3×3 patterns for
psychological testing.
Related 3×3 patterns appear
in “nine-patch” quilt blocks
and in the following–
Don Park at docuverse.com, Jan. 19, 2007:
“How to draw an Identicon
A 9-block is a small quilt using only 3 types of patches, out of 16 available, in 9 positions. Using the identicon code, 3 patches are selected: one for center position, one for 4 sides, and one for 4 corners.
Positions and Rotations
For center position, only a symmetric patch is selected (patch 1, 5, 9, and 16). For corner and side positions, patch is rotated by 90 degree moving clock-wise starting from top-left position and top position respectively.” |
| Jared Tarbell at levitated.net, May 15, 2002:
“The nine block is a common design pattern among quilters. Its construction methods and primitive building shapes are simple, yet produce millions of interesting variations.
Figure A. Four 9 block patterns, arbitrarily assembled, show the
grid composition of the block.
Each block is composed of 9 squares, arranged in a 3 x 3 grid. Each square is composed of one of 16 primitive shapes. Shapes are arranged such that the block is radially symmetric. Color is modified and assigned arbitrarily to each new block.
The basic building blocks of the nine block are limited to 16 unique geometric shapes. Each shape is allowed to rotate in 90 degree increments. Only 4 shapes are allowed in the center position to maintain radial symmetry.
Figure B. The 16 possible shapes allowed
for each grid space. The 4 shapes allowed
in the center have bold numbers.”
|
Such designs become of mathematical interest when their size is increased slightly, from square arrays of nine blocks to square arrays of sixteen. See Block Designs in Art and Mathematics.
(This entry was suggested by examples of 4×4 Identicons in use at Secret Blogging Seminar.)
Comments Off on Wednesday June 18, 2008
Sunday, May 25, 2008
Wechsler Cubes
"Confusion is nothing new."
— Song lyric, Cyndi Lauper
Part I:
Magister Ludi
Hermann Hesse's 1943 The Glass Bead Game (Picador paperback, Dec. 6, 2002, pp. 139-140)–
"For the present, the Master showed him a bulky memorandum, a proposal he had received from an organist– one of the innumerable proposals which the directorate of the Game regularly had to examine. Usually these were suggestions for the admission of new material to the Archives. One man, for example, had made a meticulous study of the history of the madrigal and discovered in the development of the style a curved that he had expressed both musically and mathematically, so that it could be included in the vocabulary of the Game. Another had examined the rhythmic structure of Julius Caesar's Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns. Or again some fanatic had once more unearthed some new cabala hidden in the musical notation of the fifteenth century. Then there were the tempestuous letters from abstruse experimenters who could arrive at the most astounding conclusions from, say, a comparison of the horoscopes of Goethe and Spinoza; such letters often included pretty and seemingly enlightening geometric drawings in several colors."
Part II:
A Bulky Memorandum
From Siri Hustvedt, author of Mysteries of the Rectangle: Essays on Painting (Princeton Architectural Press, 2005)– What I Loved: A Novel (Picador paperback, March 1, 2004, page 168)–
A description of the work of Bill Wechsler, a fictional artist:
"Bill worked long hours on a series of autonomous pieces about numbers. Like O's Journey, the works took place inside glass cubes, but these were twice as large– about two feet square. He drew his inspiration from sources as varied as the Cabbala, physics, baseball box scores, and stock market reports. He painted, cut, sculpted, distorted, and broke the numerical signs in each work until they became unrecognizable. He included figures, objects, books, windows, and always the written word for the number. It was rambunctious art, thick with allusion– to voids, blanks, holes, to monotheism and the individual, the the dialectic and yin-yang, to the Trinity, the three fates, and three wishes, to the golden rectangle, to seven heavens, the seven lower orders of the sephiroth, the nine Muses, the nine circles of Hell, the nine worlds of Norse mythology, but also to popular references like A Better Marriage in Five Easy Lessons and Thinner Thighs in Seven Days. Twelve-step programs were referred to in both cube one and cube two. A miniature copy of a book called The Six Mistakes Parents Make Most Often lay at the bottom of cube six. Puns appeared, usually well disguised– one, won; two, too, and Tuesday; four, for, forth; ate, eight. Bill was partial to rhymes as well, both in images and words. In cube nine, the geometric figure for a line had been painted on one glass wall. In cube three, a tiny man wearing the black-and-white prison garb of cartoons and dragging a leg iron has
— End of page 168 —
opened the door to his cell. The hidden rhyme is "free." Looking closely through the walls of the cube, one can see the parallel rhyme in another language: the German word drei is scratched into one glass wall. Lying at the bottom of the same box is a tiny black-and-white photograph cut from a book that shows the entrance to Auschwitz: ARBEIT MACHT FREI. With every number, the arbitrary dance of associations worked togethere to create a tiny mental landscape that ranged in tone from wish-fulfillment dream to nightmare. Although dense, the effect of the cubes wasn't visually disorienting. Each object, painting, drawing, bit of text, or sculpted figure found its rightful place under the glass according to the necessary, if mad, logic of numerical, pictorial, and verbal connection– and the colors of each were startling. Every number had been given a thematic hue. Bill had been interested in Goethe's color wheel and in Alfred Jensen's use of it in his thick, hallucinatory paintings of numbers. He had assigned each number a color. Like Goethe, he included black and white, although he didn't bother with the poet's meanings. Zero and one were white. Two was blue. Three was red, four was yellow, and he mixed colors: pale blue for five, purples in six, oranges in seven, greens in eight, and blacks and grays in nine. Although other colors and omnipresent newsprint always intruded on the basic scheme, the myriad shades of a single color dominated each cube.
The number pieces were the work of a man at the top of his form. An organic extension of everything Bill had done before, these knots of symbols had an explosive effect. The longer I looked at them, the more the miniature constructions seemed on the brink of bursting from internal pressure. They were tightly orchestrated semantic bombs through which Bill laid bare the arbitrary roots of meaning itself– that peculiar social contract generated by little squiggles, dashes, lines, and loops on a page."
Part III:
Wechsler Cubes
(named not for
Bill Wechsler, the
fictional artist above,
but for the non-fictional
David Wechsler) –
Part IV:
A Magic Gallery
|
ZZ
WW
Figures from the
Kaleidoscope Puzzle
of Steven H. Cullinane:
Poem by Eugen Jost:
Zahlen und Zeichen,
Wörter und Worte
Mit Zeichen und Zahlen
vermessen wir Himmel und Erde
schwarz
auf weiss
schaffen wir neue Welten
oder gar Universen
Numbers and Names,
Wording and Words
With numbers and names
we measure heaven and earth
black
on white
we create new worlds
and universes
English translation
by Catherine Schelbert
A related poem:
Alphabets
by Hermann Hesse
From time to time
we take our pen in hand
and scribble symbols
on a blank white sheet
Their meaning is
at everyone's command;
it is a game whose rules
are nice and neat.
But if a savage
or a moon-man came
and found a page,
a furrowed runic field,
and curiously studied
lines and frame:
How strange would be
the world that they revealed.
a magic gallery of oddities.
He would see A and B
as man and beast,
as moving tongues or
arms or legs or eyes,
now slow, now rushing,
all constraint released,
like prints of ravens'
feet upon the snow.
He'd hop about with them,
fly to and fro,
and see a thousand worlds
of might-have-been
hidden within the black
and frozen symbols,
beneath the ornate strokes,
the thick and thin.
He'd see the way love burns
and anguish trembles,
He'd wonder, laugh,
shake with fear and weep
because beyond this cipher's
cross-barred keep
he'd see the world
in all its aimless passion,
diminished, dwarfed, and
spellbound in the symbols,
and rigorously marching
prisoner-fashion.
He'd think: each sign
all others so resembles
that love of life and death,
or lust and anguish,
are simply twins whom
no one can distinguish…
until at last the savage
with a sound
of mortal terror
lights and stirs a fire,
chants and beats his brow
against the ground
and consecrates the writing
to his pyre.
Perhaps before his
consciousness is drowned
in slumber there will come
to him some sense
of how this world
of magic fraudulence,
this horror utterly
behind endurance,
has vanished as if
it had never been.
He'll sigh, and smile,
and feel all right again.
— Hermann Hesse (1943),
"Buchstaben," from
Das Glasperlenspiel,
translated by
Richard and Clara Winston
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Comments Off on Sunday May 25, 2008
Thursday, May 22, 2008
The Undertaking:
An Exercise in
Conceptual Art
Hexagram 54:
THE JUDGMENT
Undertakings bring misfortune.
Nothing that would further.
“Brian O’Doherty, an Irish-born artist,
before the [Tuesday, May 20] wake
of his alter ego* ‘Patrick Ireland’
on the grounds of the
Irish Museum of Modern Art.”
— New York Times, May 22, 2008
THE IMAGE
Thus the superior man
understands the transitory
in the light of
the eternity of the end.
Another version of
the image:
See 2/22/08
and 4/19/08.
Related material:
Michael Kimmelman in today’s New York Times—
“An essay from the ’70s by Mr. O’Doherty, ‘Inside the White Cube,’ became famous in art circles for describing how modern art interacted with the gallery spaces in which it was shown.”
Brian O’Doherty, “Inside the White Cube,” 1976 Artforum essays on the gallery space and 20th-century art:
“The history of modernism is intimately framed by that space. Or rather the history of modern art can be correlated with changes in that space and in the way we see it. We have now reached a point where we see not the art but the space first…. An image comes to mind of a white, ideal space that, more than any single picture, may be the archetypal image of 20th-century art.”
An archetypal image
THE SPACE:
A non-archetypal image
THE ART:
Natasha Wescoat, 2004
See also
Epiphany 2008:
“Nothing that would further.”
— Hexagram 54
|
Lear’s fool:
…. Now thou art an 0
without a figure. I am better
than thou art, now. I am a fool;
thou art nothing….
|
“…. in the last mystery of all the single figure of what is called the World goes joyously dancing in a state beyond moon and sun, and the number of the Trumps is done. Save only for that which has no number and is called the Fool, because mankind finds it folly till it is known. It is sovereign or it is nothing, and if it is nothing then man was born dead.”
— The Greater Trumps,
by Charles Williams, Ch. 14
Comments Off on Thursday May 22, 2008
Sunday, May 18, 2008
From the Grave
DENNIS OVERBYE
in yesterday's New York Times:
"From the grave, Albert Einstein
poured gasoline on the culture wars
between science and religion this week…."
An announcement of a
colloquium at Princeton:
Above: a cartoon,
"Coxeter exhuming Geometry,"
with the latter's tombstone inscribed
"GEOMETRY
600 B.C. —
1900 A.D.
R.I.P."
The above is from
The Paradise of Childhood,
a work first published in 1869.
"I need a photo-opportunity,
I want a shot at redemption.
Don't want to end up a cartoon
In a cartoon graveyard."
— Paul Simon
Albert Einstein,
1879-1955:
"It is quite clear to me that the religious paradise of youth, which was thus lost, was a first attempt to free myself from the chains of the 'merely-personal,' from an existence which is dominated by wishes, hopes and primitive feelings. Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking. The contemplation of this world beckoned like a liberation…."
— Autobiographical Notes, 1949
Related material:
A commentary on Tom Wolfe's
"Sorry, but Your Soul Just Died"–
"The Neural Buddhists," by David Brooks,
in the May 13 New York Times:
"The mind seems to have
the ability to transcend itself
and merge with a larger
presence that feels more real."
A New Yorker commentary on
a new translation of the Psalms:
"Suddenly, in a world without
Heaven, Hell, the soul, and
eternal salvation or redemption,
the theological stakes seem
more local and temporal:
'So teach us to number our days.'"
and a May 13 Log24 commentary
on Thomas Wolfe's
"Only the Dead Know Brooklyn"–
"… all good things — trout as well as
eternal salvation — come by grace
and grace comes by art
and art does not come easy."
— A River Runs Through It
"Art isn't easy."
— Stephen Sondheim,
quoted in
Solomon's Cube.
For further religious remarks,
consult Indiana Jones and the
Kingdom of the Crystal Skull
and The Librarian:
Return to King Solomon's Mines.
Comments Off on Sunday May 18, 2008
Saturday, May 10, 2008
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:
(Footnotes 1 and 2)
Figure 2 —
The Third Gift, 1837:
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:
Figure 4 —
Solomon's Cube,
1981 and 1983:
Figure 5 —
Design Cube, 2006:
For some mathematical background, see
Footnotes:
Comments Off on Saturday May 10, 2008
Monday, April 28, 2008
Religious Art
The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.
One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.
The following
figure does
allow such
an epiphany.
One approach to
the epiphany:
"Transformations play
a major role in
modern mathematics."
– A biography of
Felix Christian Klein
The above 2×4 array
(2 columns, 4 rows)
furnishes an example of
a transformation acting
on the parts of
an organized whole:
For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose Space-Time, and
a Finite Model."
For a related structure–
not rectangle but cube–
see Epiphany 2008.
Comments Off on Monday April 28, 2008
Tuesday, April 8, 2008
Comments Off on Tuesday April 8, 2008
Thursday, March 6, 2008
This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.
"The historical road
from the Platonic solids
to the finite simple groups
is well known."
— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture
Euclid is said to have remarked that "there is no royal road to geometry." The road to the end of the four-color conjecture may, however, be viewed as a royal road
from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, "Map-Colouring Problems," in
Mathematical Recreations and Essays, by
W. W. Rouse Ball and
H. S. M. Coxeter.) That road
ended in 1976 at the AMS-MAA summer meeting in Toronto– home of
H. S. M. Coxeter, a.k.a. "the king of geometry."
See also Log24, May 21, 2007.
A different road– from Plato to the finite simple groups– is, as I noted in November 2000, well known. But new roadside attractions continue to appear. One such attraction is the role played by a Platonic solid– the icosahedron– in design theory, coding theory, and the construction of the sporadic simple group M
24.
"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of
Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))
This Steiner system is closely connected to M24 and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of M24):
"Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix (I J-N) generate the extended binary Golay code." [Here I is the identity matrix and J is the matrix of all 1's.]
— Op. cit., p. 719
Related material:
Finite Geometry of
the Square and Cube
and
Jewel in the Crown
"There is a pleasantly discursive
treatment of Pontius Pilate's
unanswered question
'What is truth?'"
— H. S. M. Coxeter, 1987,
introduction to Trudeau's
"story theory" of truth
Those who prefer stories to truth
may consult the Log24 entries
of March 1, 2, 3, 4, and 5.
They may also consult
the poet Rubén Darío:
… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
Comments Off on Thursday March 6, 2008
Monday, February 25, 2008
Comments Off on Monday February 25, 2008
Saturday, February 16, 2008
Bridges
Between Two Worlds
From the world of mathematics…
“… my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.'”
— Scott Carnahan at Secret Blogging Seminar, December 14, 2007
Carnahan’s remark in context:
“About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GLr over function fields…. In the beginning, he drew a diagram….
If we remove all of the explanatory text, the diagram looks like this:
I was a bit hesitant to draw this, because my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.’ Anyway, I’ll explain roughly how it works.
Langlands correspondence is a ‘bridge between two worlds,’ or more specifically, an assertion of a bijection….”
Compare and contrast the above…
… to the world of Rudolf Kaehr:
The above reference to “diamond theory” is from Rudolf Kaehr‘s paper titled Double Cross Playing Diamonds.
Another bridge…
Carnahan’s advisor, referring to “meaningless diagrams with arrows connecting different areas of mathematics,” probably did not have in mind diagrams like the two above, but rather diagrams like the two below–
From the world of mathematics…
“A rough sketch of
how diamond theory is
related to some other
fields of mathematics”
— Steven H. Cullinane
Related material:
For further details on
the “diamond theory” of
Cullinane, see
Finite Geometry of the
Square and Cube.
For further details on
the “diamond theory” of
Kaehr, see
Rudy’s Diamond Strategies.
Those who prefer entertainment
may enjoy an excerpt
from Log24, October 2007:
Comments Off on Saturday February 16, 2008
Sunday, September 2, 2007
Re: This Week’s Finds in Mathematical Physics (Week 251)
On Spekkens’ toy system and finite geometry
Background–
- In “Week 251” (May 5, 2007), John wrote:
“Since Spekkens’ toy system resembles a qubit, he calls it a “toy bit”. He goes on to study systems of several toy bits – and the charming combinatorial geometry I just described gets even more interesting. Alas, I don’t really understand it well: I feel there must be some mathematically elegant way to describe it all, but I don’t know what it is…. All this is fascinating. It would be nice to find the mathematical structure that underlies this toy theory, much as the category of Hilbert spaces underlies honest quantum mechanics.”
- In the n-Category Cafe ( May 12, 2007, 12:26 AM, ) Matt Leifer wrote:
“It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the state-space is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.”
- In the n-category Cafe ( May 12, 2007, 6:25 AM) John Baez wrote:
“Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.”
On finite geometry:
The actions of permutations on a 4 × 4 square in Spekkens’ paper (quant-ph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4-space over the two-element field.
Related material:
Update of
Sept. 5, 2007
See also arXiv:0707.0074v1 [quant-ph], June 30, 2007:
A fully epistemic model for a local hidden variable emulation of quantum dynamics,
by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
Hypercube from the Skotiniotis paper:
Reference:
Comments Off on Sunday September 2, 2007
Sunday, August 12, 2007
In the context of quantum information theory, the following structure seems to be of interest–
"… the full two-by-two matrix ring with entries in GF(2), M2(GF(2))– the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."
— "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007
This ring is another way of looking at the 16 elements of the affine space A4(GF(2)) over the 2-element field. (Arrange the four coordinates of each element– 1's and 0's– into a square instead of a straight line, and regard the resulting squares as matrices.) (For more on A4(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.) Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring. This system of 35 objects has a subconfiguration isomorphic to the (2,2) generalized quadrangle W2 (which occurs naturally as a subconfiguration of PG(3,2)– see Inscapes.)
Saniga concludes:
"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus
-
not only offers a principally new geometrically-underlined insight into their intrinsic nature,
-
but also gives their applications a wholly new perspective
-
and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."
is not without relevance to
the physics of quantum theory.
Comments Off on Sunday August 12, 2007
Monday, June 25, 2007
Object Lesson
"… the best definition
I have for Satan
is that it is a real
spirit of unreality."
M. Scott Peck,
People of the Lie
"Far in the woods they sang
their unreal songs,
Secure. It was difficult
to sing in face
Of the object. The singers
had to avert themselves
Or else avert the object."
— Wallace Stevens,
"Credences of Summer"
|
Today is June 25,
anniversary of the
birth in 1908 of
Willard Van Orman Quine.
Quine died on
Christmas Day, 2000.
Today, Quine's birthday, is,
as has been noted by
Quine's son, the point of the
calendar opposite Christmas–
i.e., "AntiChristmas."
If the Anti-Christ is,
as M. Scott Peck claims,
a spirit of unreality, it seems
fitting today to invoke
Quine, a student of reality,
and to borrow the title of
Quine's Word and Object…
Word:
An excerpt from
"Credences of Summer"
by Wallace Stevens:
"Three times the concentred
self takes hold, three times
The thrice concentred self,
having possessed
The object, grips it
in savage scrutiny,
Once to make captive,
once to subjugate
Or yield to subjugation,
once to proclaim
The meaning of the capture,
this hard prize,
Fully made, fully apparent,
fully found."
— "Credences of Summer," VII,
by Wallace Stevens, from
Transport to Summer (1947)
|
Object:
From Friedrich Froebel,
who invented kindergarten:
From Christmas 2005:
Click on the images
for further details.
For a larger and
more sophisticaled
relative of this object,
see yesterday's entry
At Midsummer Noon.
The object is real,
not as a particular
physical object, but
in the way that a
mathematical object
is real — as a
pure Platonic form.
"It's all in Plato…."
— C. S. Lewis
Comments Off on Monday June 25, 2007
Sunday, June 24, 2007
Raiders of
the Lost Stone
(Continued from June 23)
Charles Williams:
"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
Comments Off on Sunday June 24, 2007
Wednesday, June 20, 2007
Kernel
Mathematical Reviews citation:
MR2163497 (2006g:81002) 81-03 (81P05)
Gieser, Suzanne The innermost kernel. Depth psychology and quantum physics. Wolfgang Pauli's dialogue with C. G. Jung. Springer-Verlag, Berlin, 2005. xiv+378 pp. ISBN: 3-540-20856-9
A quote from MR at Amazon.com:
"This revised translation of a Swedish Ph. D. thesis in philosophy offers far more than a discussion of Wolfgang Pauli's encounters with the psychoanalyst Carl Gustav Jung…. Here the book explains very well how Pauli attempted to extend his understanding beyond superficial esotericism and spiritism…. To understand Pauli one needs books like this one, which… seems to open a path to a fuller understanding of Pauli, who was seeking to solve a quest even deeper than quantum physics." (Arne Schirrmacher, Mathematical Reviews, Issue 2006g)
An excerpt:
I do not yet know what Gieser means by "the innermost kernel." The following is my version of a "kernel" of sorts– a diagram well-known to students of anthropologist
Claude Levi-Strauss and art theorist
Rosalind Krauss:
The four-group is also known as the Vierergruppe or Klein group. It appears, notably, as the translation subgroup of A, the group of 24 automorphisms of the affine plane over the 2-element field, and therefore as the kernel of the homomorphism taking A to the group of 6 automorphisms of the projective line over the 2-element field. (See Finite Geometry of the Square and Cube.)
Comments Off on Wednesday June 20, 2007
Friday, June 15, 2007
A Study in
Art Education
Rudolf Arnheim, a student of Gestalt psychology (which, an obituary notes, emphasizes "the perception of forms as organized wholes") was the first Professor of the Psychology of Art at Harvard. He died at 102 on Saturday, June 9, 2007.
The conclusion of yesterday's New York Times obituary of Arnheim:
"… in The New York Times Book Review in 1986, Celia McGee called Professor Arnheim 'the best kind of romantic,' adding, 'His wisdom, his patient explanations and lyrical enthusiasm are those of a teacher.'"
A related quotation:
"And you are teaching them a thing or two about yourself. They are learning that you are the living embodiment of two timeless characterizations of a teacher: 'I say what I mean, and I mean what I say' and 'We are going to keep doing this until we get it right.'"
— Tools for Teaching
Here, yet again, is an illustration that has often appeared in Log24– notably, on the date of Arnheim's death:
Related quotations:
"We have had a gutful of fast art and fast food. What we need more of is slow art: art that holds time as a vase holds water: art that grows out of modes of perception and whose skill and doggedness make you think and feel; art that isn't merely sensational, that doesn't get its message across in 10 seconds, that isn't falsely iconic, that hooks onto something deep-running in our natures. In a word, art that is the very opposite of mass media. For no spiritually authentic art can beat mass media at their own game."
— Robert Hughes, speech of June 2, 2004
"Whether the 3×3 square grid is fast art or slow art, truly or falsely iconic, perhaps depends upon the eye of the beholder."
— Log24, June 5, 2004
If the beholder is Rudolf Arnheim, whom we may now suppose to be viewing the above figure in the afterlife, the 3×3 square is apparently slow art. Consider the following review of his 1982 book The Power of the Center:
"Arnheim deals with the significance of two kinds of visual organization, the concentric arrangement (as exemplified in a bull's-eye target) and the grid (as exemplified in a Cartesian coordinate system)….
It is proposed that the two structures of grid and target are the symbolic vehicles par excellence for two metaphysical/psychological stances. The concentric configuration is the visual/structural equivalent of an egocentric view of the world. The self is the center, and all distances exist in relation to the focal spectator. The concentric arrangement is a hermetic, impregnable pattern suited to conveying the idea of unity and other-worldly completeness. By contrast, the grid structure has no clear center, and suggests an infinite, featureless extension…. Taking these two ideal types of structural scaffold and their symbolic potential (cosmic, egocentric vs. terrestrial, uncentered) as given, Arnheim reveals how their underlying presence organizes works of art."
— Review of Rudolf Arnheim's The Power of the Center: A Study of Composition in the Visual Arts (Univ. of Calif. Press, 1982). Review by David A. Pariser, Studies in Art Education, Vol. 24, No. 3 (1983), pp. 210-213
Arnheim himself says in this book (pp. viii-ix) that "With all its virtues, the framework of verticals and horizontals has one grave defect. It has no center, and therefore it has no way of defining any particular location. Taken by itself, it is an endless expanse in which no one place can be distinguished from the next. This renders it incomplete for any mathematical, scientific, and artistic purpose. For his geometrical analysis, Descartes had to impose a center, the point where a pair of coordinates [sic] crossed. In doing so he borrowed from the other spatial system, the centric and cosmic one."
Students of art theory should, having read the above passages, discuss in what way the 3×3 square embodies both "ideal types of structural scaffold and their symbolic potential."
We may imagine such a discussion in an afterlife art class– in, perhaps, Purgatory rather than Heaven– that now includes Arnheim as well as Ernst Gombrich and Kirk Varnedoe.
Such a class would be one prerequisite for a more advanced course– Finite geometry of the square and cube.
Comments Off on Friday June 15, 2007
Thursday, June 14, 2007
Scholarly Notes
In memory of
Rudolf Arnheim,
who died on
Saturday, June 9
“Originally trained in Gestalt psychology, with its emphasis on the perception of forms as organized wholes, he was one of the first investigators to apply its principles to the study of art of all kinds.” —
Today’s New York Times
From the Wikipedia article on Gestalt psychology prior to its modification on May 31, 2007:
“Emergence, reification, multistability, and invariance are not separable modules to be modeled individually, but they are different aspects of a single unified dynamic mechanism.
For a mathematical example of such a mechanism using the cubes of psychologists’ block design tests, see Block Designs in Art and Mathematics and The Kaleidoscope Puzzle.”
The second paragraph of the above passage refers to my own work.
Some Gestalt-related work of Arnheim:
Comments Off on Thursday June 14, 2007
Sunday, June 3, 2007
Haunting Time
"Macquarrie remains one of the most
important commentators [on] …
Heidegger's work. His co-translation
of Being and Time into English is
considered the canonical version."
— Wikipedia
The Rev. Macquarrie died on
May 28. The Log24 entry
for that date contains the
following illustration:
The part of the illustration
relevant to the death of
Macquarrie is the color.
From my reply to
a comment on the
May 28 entry:
"I checked out [Terence] McKenna and found this site on the aging druggie. I didn't like the hippie scene in the sixties and I don't like it now. Booze was always my drug of choice. Still, checking further, I found that McKenna's afterword to Dick's In Pursuit of Valis was well written."
From McKenna's afterword:
"Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirl-wind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey 'haunts time like a ghost.'"
I can find no source for
any remarks of Whitehead
on the color "dove grey"
(or "gray") but Whitehead
did say that
"A colour is eternal. It haunts time like a spirit. It comes and it goes. But where it comes it is the same colour. It neither survives nor does it live. It appears when it is wanted." —Science and the Modern World, 1925
The poetic remark of
McKenna on the color
"dove grey" may be
taken, in a schizophrenic
(or, similarly, a Christian) way,
as a reference to the Holy Spirit.
My own remarks on the hippie
scene seem appropriate as a
response to media celebration
of today's 40th anniversary of
the beginning of the 1967
"summer of love."
Comments Off on Sunday June 3, 2007
Monday, May 28, 2007
and a Finite Model
Notes by Steven H. Cullinane
May 28, 2007
Part I: A Model of Space-Time
The following paper includes a figure illustrating
Penrose’s model of “complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space.”
Click on picture to enlarge.
For some background on the Klein quadric and space-time, see Roger Penrose, “
On the Origins of Twistor Theory,” from
Gravitation and Geometry:
A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.
Part II: A Corresponding Finite Model
The Klein quadric also occurs in a finite model of projective 5-space. See a 1910 paper:
G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.
Conwell discusses the quadric, and the related Klein correspondence, in detail. This is noted in a more recent paper by Philippe Cara:
As Cara goes on to explain, the Klein correspondence underlies Conwell’s discussion of eight
heptads. These play an important role in another correspondence, illustrated in the
Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M
24.
Related material:
The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.
The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.
There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube. This correspondence leads to a natural way to generate the affine group AGL(6,2). This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.
“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game. Elder Brother laughed. ‘Go ahead and try,’ he exclaimed. ‘You’ll see how it turns out. Anyone can create a pretty little bamboo garden in the world. But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”
translated by Richard and Clara Winston
Monday, April 30, 2007
Structure and Logic
The phrase “structural logic” in yesterday’s entry was applied to Bach’s cello suites. It may equally well be applied to geometry. In particular:
“The aim of this thesis is to classify certain structures which are, from a certain point of view, as homogeneous as possible, that is which have as many symmetries as possible.”
— Alice Devillers, “Classification of Some Homogeneous and Ultrahomogeneous Structures,” Ph.D. thesis, Université Libre de Bruxelles, academic year 2001-2002
Related material:
The above models for the corresponding projective spaces may be regarded as illustrating the phrase “structural logic.”
For a possible application of the 16-point space’s “many symmetries” to logic proper, see The Geometry of Logic.
Comments Off on Monday April 30, 2007
Sunday, April 22, 2007
Built
continued from
March 25, 2006
In honor of Scarlett Johansson's recent London films "Match Point" and "Scoop," here is a link to an entry of Women's History Month, 2006, with a discussion of an exhibition of the works of artist Liza Lou at London's White Cube Gallery. That entry includes the following illustrations:
Comments Off on Sunday April 22, 2007
Saturday, April 7, 2007
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Tuesday, April 3, 2007
Our Judeo-Christian
Heritage –
Lottery
Hermeneutics
Part II: Christian
Part III:
Imago Dei
Click on picture
for details.
Related material:
It is perhaps relevant to
this Holy Week that the
date 6/04 (2006) above
refers to both the Christian
holy day of Pentecost and
to the day of the
facetious baccalaureate
of the Class of 2006 in
the University Chapel
at Princeton.
For further context for the
Log24 remarks of that same
date, see June 1-15, 2006.
Comments Off on Tuesday April 3, 2007
Wednesday, February 28, 2007
Elements
of Geometry
The title of Euclid’s Elements is, in Greek, Stoicheia.
From Lectures on the Science of Language,
by Max Muller, fellow of All Souls College, Oxford.
New York: Charles Scribner’s Sons, 1890, pp. 88-90 –
Stoicheia
“The question is, why were the elements, or the component primary parts of things, called stoicheia by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist.
Stoichos, from which stoicheion, means a row or file, like stix and stiches in Homer. The suffix eios is the same as the Latin eius, and expresses what belongs to or has the quality of something. Therefore, as stoichos means a row, stoicheion would be what belongs to or constitutes a row….
Hence stoichos presupposes a root stich, and this root would account in Greek for the following derivations:–
- stix, gen. stichos, a row, a line of soldiers
- stichos, a row, a line; distich, a couplet
- steicho, estichon, to march in order, step by step; to mount
- stoichos, a row, a file; stoichein, to march in a line
In German, the same root yields steigen, to step, to mount, and in Sanskrit we find stigh, to mount….
Stoicheia are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order.”
Comments Off on Wednesday February 28, 2007
Tuesday, February 20, 2007
Symmetry
Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”
Some relevant quotations:
“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
Describing the branch of mathematics known as Galois theory, Weyl says that it
“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”
— Weyl, Symmetry, Princeton University Press, 1952, p. 138
Weyl’s set Sigma is a finite set of complex numbers. Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes. For illustrations, see Finite Geometry of the Square and Cube. What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations. For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:
“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of space-time].‘[22]
22. The significance of the notion of invariance and its group-theoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).
References:
Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139-149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155-167.
Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]
Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410-421.
Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.
See also
Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–
“Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86–
“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”
References:
Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.
Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen Mathematiker-Vereinigung 19: 281-300. [Reprinted: Physikalische Zeitschrift 12 (1911): 17-27].
Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press. See Invariances: The Structure of the Objective World, by Robert Nozick.
Comments Off on Tuesday February 20, 2007
Tuesday, February 6, 2007
The Poetics of Space
The title is from Bachelard.
I prefer Stevens:
The rock is the habitation of the whole,
Its strength and measure, that which is near, point A
In a perspective that begins again
At B: the origin of the mango's rind.
It is the rock where tranquil must adduce
Its tranquil self, the main of things, the mind,
The starting point of the human and the end,
That in which space itself is contained, the gate
To the enclosure, day, the things illumined
By day, night and that which night illumines,
Night and its midnight-minting fragrances,
Night's hymn of the rock, as in a vivid sleep.
— Wallace Stevens,
"The Rock," 1954
Joan Ockman in Harvard Design Magazine (Fall 1998):
"'We are far removed from any reference to simple geometrical forms,' Bachelard wrote…."
No, we are not. See Log24, Christmas 2005:
Compare and contrast:
(Click on pictures for details.)
More on Bachelard from Harvard Design Magazine:
"The project of discerning a loi des quatre éléments would preoccupy him until his death…."
For such a loi, see Theme and Variations and…
(Click on design for details.)
Thought for Today:
"If you can talk brilliantly
about a problem, it can create
the consoling illusion that
it has been mastered."
— Stanley Kubrick, American
movie director (1928-1999).
(AP, "Today in History,"
February 6, 2007)
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Thursday, February 1, 2007
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Saturday, December 23, 2006
Black Mark
Bernard Holland in The New York Times on Monday, May 20, 1996:
“Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday….”
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Lottery on Friday,
Dec. 22, 2006:
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Analysis of the structure
of a 2x2x2 cube
via trinities of
projective points
in a Fano plane.
Comments Off on Saturday December 23, 2006
Sunday, October 8, 2006
Today’s Birthday:
Matt Damon
“The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast.”
— G. K. Chesterton
See also works by the late Arthur Loeb of Harvard’s Department of Visual and Environmental Studies.
“I don’t want to be a product of my environment. I want my environment to be a product of me.” — Frank Costello in The Departed
For more on the Harvard environment,
see today’s online Crimson:
The Harvard Crimson,
Online Edition |
Sunday,
Oct. 8, 2006 |
|
POMP AND
CIRCUS-STANCE
CRIMSON/ MEGHAN T. PURDY
Friday, Oct. 6:
The Ringling Bros. Barnum & Bailey Circus has come to town, and yesterday the animals were disembarked near MIT and paraded to their temporary home at the Banknorth Garden.
|
OPINION
At Last, a
Guiding Philosophy
The General Education report is a strong cornerstone, though further scrutiny is required.
By THE CRIMSON STAFF
After four long years, the Curricular Review has finally found its heart.
The Trouble
With the Germans
The College is a little under-educated these days.
By SAHIL K. MAHTANI
Harvard College– in the best formulation I’ve heard– promulgates a Japanese-style education, where the professoriate pretend to teach, the students pretend to learn, and everyone is happy.
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Comments Off on Sunday October 8, 2006
Thursday, October 5, 2006
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:
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.
Comments Off on Thursday October 5, 2006
Tuesday, October 3, 2006
Serious
"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."
— Charles Matthews at Wikipedia, Oct. 2, 2006
"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."
— G. H. Hardy, A Mathematician's Apology
Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:
Affine group
Reflection group
Symmetry in mathematics
Incidence structure
Invariant (mathematics)
Symmetry
Finite geometry
Group action
History of geometry
This would appear to be a fairly large complex of mathematical ideas.
See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:
Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.
Comments Off on Tuesday October 3, 2006
Sunday, October 1, 2006
Tales of Philosophy:
Recipe for Disaster
according to Jerome Kagan,
Harvard psychologist emeritus
|
From Log24 —
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The Line
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The Cube
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From Harvard's
Jerome Kagan —
|
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"'Humans demand that there be a clear right and wrong,' he said. 'You've got to believe that the track you've taken is the right track. You get depressed if you're not certain as to what it is you're supposed to be doing or what's right and wrong in the world.'"
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"People need to divide the world into good and evil, us and them, Kagan continued. To do otherwise– to entertain the possibility that life is not black and white, but variously shaded in gray– is perhaps more honest, rational and decent. But it's also, psychically, a recipe for disaster."
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Black and White:
Log24 in
May 2005
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Shades of Gray:
An affine space
and
Harvard's
Jerome Kagan
|
The above Kagan quotes are taken
from a New York Times essay by
Judith Warner as transcribed by
Mark Finkelstein on Sept. 29.
See also Log24 on
Sept. 29 and 30.
Related material:
Kagan's book
Surprise, Uncertainty,
and Mental Structures
(Harvard U. Press, April 2002)
and Werner Heisenberg–
discoverer of the
uncertainty principle—
as Anakin Skywalker
being tempted by
the Dark Side:
George Lucas, who has profited
enormously from public depictions
of the clash between
good and evil, light and dark,
may in private life be inclined
"It is the brain, the little gray cells
on which one must rely.
One must seek the truth
within– not without."
(This is another version of the
"Descartes before dehors" principle–
Wednesday, September 20, 2006
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
A Public Space
— Motto of
Plato's Academy
Background from
Log24 on Feb. 15, 2006:
|
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.
Comments Off on Wednesday September 20, 2006
Sunday, July 9, 2006
Today’s birthday:
Tom Hanks, star of
“The Da Vinci Code”
Ben Nicholson
and the Holy Grail
Part I:
A Current Exhibit
A. Diamond Theory, a 1976 preprint containing, in the original version, the designs on the faces of Nicholson’s “Kufi blocks,” as well as some simpler traditional designs, and
B. “
Block Designs,” a web page illustrating design blocks based on the 1976 preprint.
Part III:
The Leonardo Connection
Part IV:
Nicholson’s Grail Quest
“I’m interested in locating the holy grail of the minimum means to express the most complex ideas.”
— Ben Nicholson in a 2005 interview
Nicholson’s quest has apparently lasted for some time. Promotional material for a 1996 Nicholson exhibit in Montreal says it “invites visitors of all ages to experience a contemporary architect’s search for order, meaning and logic in a world of art, science and mystery.” The title of that exhibit was “Uncovering Geometry.”
For web pages to which this same title might apply, see Quilt Geometry, Galois Geometry, and Finite Geometry of the Square and Cube.
* “Square Kufi” calligraphy is used in Islamic architectural ornament. I do not know what, if anything, is signified by Nicholson’s 6×12 example of “Kufi blocks” shown above.
Comments Off on Sunday July 9, 2006
Friday, June 23, 2006
Binary Geometry
There is currently no area of mathematics named “binary geometry.” This is, therefore, a possible name for the geometry of sets with 2n elements (i.e., a sub-topic of Galois geometry and of algebraic geometry over finite fields– part of Weil’s “Rosetta stone” (pdf)).
Examples:
- Charles Sanders Peirce, “The Simplest Mathematics.”
- Donald E. Knuth’s discussion of binary hypercubes in “Boolean Basics,” a draft of section 7.1.1 in The Art of Computer Programming, Volume 4: Combinatorial Algorithms
- My own discussion of a binary hypercube in Geometry of the 4x4x4 Cube
- A more sophisticated example: the geometry of elliptic curves over a binary Galois field. For an excellent introduction, see the Certicom online elliptic curve tutorial. This has an applet illustrating elliptic curves in a space of 256 points (256=16×16, with the x and y variables of a curve each having 16 possible values).
- In summary, apart from the fact that the native language of computers has characteristic 2, “binary” mathematics, i.e. mathematics in characteristic 2, is of special interest both in the study of finite geometry (Finite Geometry of the Square and Cube) and in algebraic geometry (see, for instance, the work of Brian Conrad).
Comments Off on Friday June 23, 2006
Saturday, June 17, 2006
“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
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
Comments Off on Saturday June 17, 2006
Friday, May 26, 2006
A Living Church
continued from March 27
"The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast."
— G. K. Chesterton
Comments Off on Friday May 26, 2006
Wednesday, May 10, 2006
“… 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.”
Comments Off on Wednesday May 10, 2006
Friday, April 7, 2006
Comments Off on Friday April 7, 2006
Sunday, March 26, 2006
Clint Eastwood on the
“Midnight in the Garden
of Good and Evil”
soundtrack CD—
“Accentuate the positive”–
and an entry from last Christmas:
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Compare and contrast:
(Click on pictures
for details.)
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“Recollect what I have said to you,
that this world is a comedy
to those who think,
a tragedy to those who feel.
This is the quint-essence of all
I have learnt in fifty years!”
— Horace Walpole,
letter to Horace Mann,
5 March, 1772
Comments Off on Sunday March 26, 2006
Saturday, March 25, 2006
Built
In memory of Rolf Myller,
who died on Thursday,
March 23, 2006, at
Mount Sinai Hospital
in Manhattan:
Myller was,
according to the
New York Times,
an architect
whose eclectic pursuits
included writing
children’s books,
The Bible Puzzle Book, and
Fantasex: A Book of Erotic Games.
He also wrote, the Times says,
“Symbols and Their Meaning
(1978), a graphic overview of
children’s nonverbal communication.”
This is of interest in view of the
Log24 reference to “symbol-mongers”
on the date of Myller’s death.
In honor of Women’s History Month
and of Myller’s interests in the erotic
and in architecture, we present
the following work from a British gallery.
This work might aptly be
retitled “Brick Shithouse.”
Related material:
(1) the artist’s self-portrait
and, in view of the cover
illustration for Myller’s
The Bible Puzzle Book,
(2) the monumental treatise
by Leonard Shlain
The Alphabet Versus
the Goddess: The Conflict
Between Word and Image.
For devotees of women’s history
and of the Goddess,
here are further details from
the
White Cube gallery:
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Liza Lou
03.03.06 – 08.04.06
White Cube is pleased to present the first UK solo exhibition by Los Angeles-based artist Liza Lou.
Combining visionary, conceptual and craft approaches, Lou makes mixed-media sculptures and room-size installations that are suggestive of a transcendental reality. Lou’s work often employs familiar, domestic forms, crafted from a variety of materials such as steel, wood, papier-mâché and fibreglass, which is then covered with tiny glass beads that are painstakingly applied, one at a time, with tweezers. Dazzling and opulent and constantly glistening with refracted light, her sculptures bristle with what Peter Schjeldahl has aptly described as ‘surreal excrescence’.
This exhibition, a meditation on the vulnerability of the human body and the architecture of confinement, will include several new figurative sculptures as well as two major sculptural installations. Security Fence (2005) is a large scale cage made up of four steel, chain link walls, topped by rings of barbed wire and Cell (2004-2006), as its name suggests, is a room based on the approximate dimensions of a death row prison cell, a kind of externalized map of the prisoner’s mind. Both Security Fence and Cell, like Lou’s immense earlier installations Kitchen (1991-1995) and Back Yard (1995-1999) are characterized by the absence of their real human subject. But whereas the absent subject in Kitchen and Back Yard could be imagined through the details and accessories carefully laid out to view, in Lou’s two new installations the human body is implied simply through the empty volume created by the surrounding architecture. Both Cell and Security Fence are monochromatic and employ iconic forms that make direct reference to Minimalist art in its use of repetition, formal perfection and materiality. In contrast to this, the organic form of a gnarled tree trunk, Scaffold (2005-2006), its surface covered with shimmering golden beads, juts directly out from the wall.
Lou’s work has an immediate ‘shock’ content that works on different levels: first, an acknowledgement of the work’s sheer aesthetic impact and secondly the slower comprehension of the labour that underlies its construction. But whereas in Lou’s earlier works the startling clarity of the image is often a counterpoint to the lengthy process of its realization, for the execution of Cell, Lou further slowed down the process by using beads of the smallest variety with their holes all facing up in an exacting hour-by-hour approach in order to ‘use time as an art material’.
Concluding this body of work are three male figures in states of anguish. In The Seer (2005-2006), a man becomes the means of turning his body back in on himself. Bent over double, his body becomes an instrument of impending self-mutilation, the surface of his body covered with silver-lined beads, placed with the exactitude and precision of a surgeon. In Homeostasis (2005-2006) a naked man stands prostrate with his hands up against the wall in an act of surrender. In this work, the dissolution between inside and outside is explored as the ornate surface of Lou’s cell-like material ‘covers’ the form while exposing the systems of the body, both corporeal and esoteric. In The Vessel (2005-2006), Christ, the universal symbol of torture and agony holds up a broken log over his shoulders. This figure is beheaded, and bejewelled, with its neck carved out, becoming a vessel into which the world deposits its pain and suffering.
Lou has had numerous solo exhibitions internationally, including Museum Kunst Palast, Düsseldorf, Henie Onstad Kunstsenter, Oslo and Fondació Joan Miró, Barcelona. She was a 2002 recipient of the MacArthur Foundation Fellowship.
Liza Lou’s film Born Again (2004), in which the artist tells the compelling and traumatic story* of her Pentecostal upbringing in Minnesota, will be screened at 52 Hoxton Square from 3 – 25 March courtesy of Penny Govett and Mick Kerr.
Liza Lou will be discussing her work following a screening of her film at the ICA, The Mall, London on Friday 3 March at 7pm. Tickets are available from the ICA box office (+ 44 (0) 20 7930 3647).
A fully illustrated catalogue, with a text by Jeanette Winterson and an interview with Tim Marlow, will accompany the exhibition.
White Cube is open Tuesday to Saturday, 10.00 am to 6.00 pm.
For further information please contact Honey Luard or Susannah Hyman on + 44 (0) 20 7930 5373
* Warning note from Adrian Searle
in The Guardian of March 21:
“How much of her story is
gospel truth we’ll never know.”
|
Comments Off on Saturday March 25, 2006
Thursday, February 23, 2006
Cubist Epiphany
“In The Painted Word, a rumination on the state of American painting in the 1970s, Tom Wolfe described an epiphany….”
— Peter Berkowitz, “Literature in Theory”
“I had an epiphany.”
— Apostolos Doxiadis, organizer of last summer’s conference on mathematics and narrative. See the Log24 entry of 1:06 PM last August 23 and the four entries that preceded it.
“… das Durchleuchten des ewigen Glanzes des ‘Einen’ durch die materielle Erscheinung“
— A definition of beauty from Plotinus, via Werner Heisenberg
“By groping toward the light we are made to realize how deep the darkness is around us.”
— Arthur Koestler, The Call Girls: A Tragi-Comedy, Random House, 1973, page 118, quoted in The Shining of May 29
“Perhaps we are meant to see the story as a cubist retelling of the crucifixion….”
— Adam White Scoville, quoted in Cubist Crucifixion, on Iain Pears’s novel, An Instance of the Fingerpost
Related material:
Log24 entries of
Feb. 20, 21, and 22.
Comments Off on Thursday February 23, 2006
Friday, January 13, 2006
Beyond the Fire
“Who Needs a White Cube These Days?”
— Headline in today’s New York Times
“That Nature is a Heraclitean Fire…“
— Poem title, Gerard Manley Hopkins
“… Sleep realized
Was the whiteness that is the ultimate intellect,
A diamond jubilance beyond the fire,
That gives its power to the wild-ringed eye.”
— Wallace Stevens,
“The Owl in the Sarcophagus” III 13-16,
from The Auroras of Autumn, 1950
Related material:
The five entries ending on Christmas, 2005.
Comments Off on Friday January 13, 2006
Wednesday, January 11, 2006
Time in the Rock
"a world of selves trying to remember the self
before the idea of self is lost–
Walk with me world, upon my right hand walk,
speak to me Babel, that I may strive to assemble
of all these syllables a single word
before the purpose of speech is gone."
— Conrad Aiken, "Prelude" (1932),
later part of "Time in the Rock,
or Preludes to Definition, XIX" (1936),
in Selected Poems, Oxford U. Press
paperback, 2003, page 156
"The rock is the habitation of the whole,
Its strength and measure, that which is near, point A
In a perspective that begins again
At B: the origin of the mango's rind.
It is the rock where tranquil must adduce
Its tranquil self, the main of things, the mind,
The starting point of the human and the end,
That in which space itself is contained, the gate
To the enclosure, day, the things illumined
By day, night and that which night illumines,
Night and its midnight-minting fragrances,
Night's hymn of the rock, as in a vivid sleep."
— Wallace Stevens in The Rock (1954)
"Poetry is an illumination of a surface,
the movement of a self in the rock."
— Wallace Stevens, introduction to
The Necessary Angel, 1951
Related material:
Jung's Imago and
Solomon's Cube.
The following may help illuminate the previous entry:
"I want, as a man of the imagination, to write poetry with all the power of a monster equal in strength to that of the monster about whom I write. I want man's imagination to be completely adequate in the face of reality."
— Wallace Stevens, 1953 (Letters 790)
The "monster" of the previous entry is of course not Reese Witherspoon, but rather Vox Populi itself.
Comments Off on Wednesday January 11, 2006
Sunday, December 25, 2005
Eight is a Gate
(continued)
Compare and contrast:
Click on pictures for details.
"… die Schönheit… [ist] die
richtige Übereinstimmung
der Teile miteinander
und mit dem Ganzen."
"Beauty is the proper conformity
of the parts to one another
and to the whole."
— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974
Comments Off on Sunday December 25, 2005
Wednesday, December 14, 2005
From Here
to Eternity
For Loomis Dean
See also
For Rita Moreno
on Her Birthday
(Dec. 11, 2005)
|
Los Angeles Times
Tuesday, Dec. 13, 2005
OBITUARIES
LOOMIS DEAN
After many years at Life magazine,
he continued to find steady work
as a freelancer and as a still
photographer on film sets.
(Dean Family) Loomis Dean, 88;
Life Magazine Photographer
Known for Pictures of
Celebrities and Royalty
By Jon Thurber, Times Staff Writer
Loomis Dean, a Life magazine photographer who made memorable pictures of the royalty of both Europe and Hollywood, has died. He was 88.
Dean died Wednesday [December 7, 2005] at Sonoma Valley Hospital in Sonoma, Calif., of complications from a stroke, according to his son, Christopher.
In a photographic career spanning six decades, Dean's leading images included shirtless Hollywood mogul Darryl F. Zanuck trying a one-handed chin-up on a trapeze bar, the ocean liner Andrea Doria listing in the Atlantic and writer Ernest Hemingway in Spain the year before he committed suicide. One of his most memorable photographs for Life was of cosmopolitan British playwright and composer Noel Coward in the unlikely setting of the Nevada desert.
Dean shot 52 covers for Life, either as a freelance photographer or during his two stretches as a staffer with the magazine, 1947-61 and 1966-69. After leaving the magazine, Dean found steady freelance work in magazines and as a still photographer on film sets, including several of the early James Bond movies starring Sean Connery.
Born in Monticello, Fla., Dean was the son of a grocer and a schoolteacher.
When the Dean family's business failed during the Depression, they moved to Sarasota, Fla., where Dean's father worked as a curator and guide at the John and Mable Ringling Museum of Art.
Dean studied engineering at the University of Florida but became fascinated with photography after watching a friend develop film in a darkroom. He went off to what is now the Rochester Institute of Technology, which was known for its photography school.
After earning his degree, Dean went to work for the Ringling circus as a junior press agent and, according to his son, cultivated a side job photographing Ringling's vast array of performers and workers.
He worked briefly as one of Parade magazine's first photographers but left after receiving an Army Air Forces commission during World War II. During the war, he worked in aerial reconnaissance in the Pacific and was along on a number of air raids over Japan.
His first assignment for Life in 1946 took him back to the circus: His photograph of clown Lou Jacobs with a giraffe looking over his shoulder made the magazine's cover and earned Dean a staff job.
In the era before television, Life magazine photographers had some of the most glamorous work in journalism. Life assigned him to cover Hollywood. In 1954, the magazine published one of his most memorable photos, the shot of Coward dressed for a night on the town in New York but standing alone in the stark Nevada desert.
Dean had the idea of asking Coward, who was then doing a summer engagement at the Desert Inn in Las Vegas, to pose in the desert to illustrate his song "Mad Dogs and Englishmen Go Out in the Midday Sun."
As Dean recalled in an interview with John Loengard for the book "Life Photographers: What They Saw," Coward wasn't about to partake of the midday sun. "Oh, dear boy, I don't get up until 4 o'clock in the afternoon," Dean recalled him saying.
But Dean pressed on anyway. As he related to Loengard, he rented a Cadillac limousine and filled the back seat with a tub loaded with liquor, tonic and ice cubes — and Coward.
The temperature that day reached 119 as Coward relaxed in his underwear during the drive to a spot about 15 miles from Las Vegas. According to Dean, Coward's dresser helped him into his tuxedo, resulting in the image of the elegant Coward with a cigarette holder in his mouth against his shadow on the dry lake bed.
"Splendid! Splendid! What an idea! If we only had a piano," Coward said of the shoot before hopping back in the car and stripping down to his underwear for the ride back to Las Vegas.
In 1956, Life assigned Dean to Paris. While sailing to Europe on the Ile de France, he was awakened with the news that the Andrea Doria had collided with another liner, the Stockholm.
The accident occurred close enough to Dean's liner that survivors were being brought aboard.
His photographs of the shaken voyagers and the sinking Andrea Doria were some of the first on the accident published in a U.S. magazine.
During his years in Europe, Dean photographed communist riots and fashion shows in Paris, royal weddings throughout Europe and noted authors including James Jones and William S. Burroughs.
He spent three weeks with Hemingway in Spain in 1960 for an assignment on bullfighting. In 1989, Dean published "Hemingway's Spain," about his experiences with the great writer.
In 1965, Dean won first prize in a Vatican photography contest for a picture of Pope Paul VI. The prize included an audience with the pope and $750. According to his son, it was Dean's favorite honor.
In addition to his son, he is survived by a daughter, Deborah, and two grandsons.
Instead of flowers, donations may be made to the American Child Photographer's Charity Guild (www.acpcg.com) or the Make-A-Wish Foundation. |
Related material:
The Big Time
(Log 24, July 29, 2003):
A Story That Works |
Comments Off on Wednesday December 14, 2005
Saturday, November 5, 2005
Contrapuntal Themes
in a Shadowland
(See previous entry.)
Douglas 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."
Hofstadter's cover
Here are three patterns,
"shadows" of a sort,
derived from a different
"central object":
For details, see
Solomon's Cube.
Related material:
The reference to a
"permutation fugue"
(pdf) in an article on
Gödel, Escher, Bach.
Comments Off on Saturday November 5, 2005
Tuesday, August 2, 2005
Today's birthday:
Peter O'Toole
"What is it, Major Lawrence,
that attracts you personally
to the desert?"
"It's clean."
Visible Mathematics,
continued —
From May 18:
Lindbergh's Eden
"The Garden of Eden is behind us
and there is no road
back to innocence;
we can only go forward."
— Anne Morrow Lindbergh,
Earth Shine, p. xii
"Beauty is the proper conformity
of the parts to one another
and to the whole."
— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974
Related material:
The Eightfold Cube
(in Arabic, ka'b)
and
Comments Off on Tuesday August 2, 2005
Tuesday, June 7, 2005
“A SINGLE VERSE by Rimbaud,”
writes Dominique de Villepin,
the new French Prime Minister,
“shines like a powder trail
on a day’s horizon.
It sets it ablaze all at once,
explodes all limits,
draws the eyes
to other heavens.”
— Ben Macintyre,
The London Times, June 4:
When Rimbaud Meets Rambo
“Room 101 was the place where
your worst fears were realised
in George Orwell’s classic
Nineteen Eighty-Four.
[101 was also]
Professor Nash’s office number
in the movie ‘A Beautiful Mind.'”
— Prime Curios
Classics Illustrated —
Comments Off on Tuesday June 7, 2005
Wednesday, May 18, 2005
“Beauty is the proper conformity
of the parts to one another
and to the whole.”
— Werner Heisenberg,
“
Die Bedeutung des Schönen in der exakten Naturwissenschaft,”
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg’s
Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974
Related material:
The Eightfold Cube
Comments Off on Wednesday May 18, 2005
Friday, May 6, 2005
Fugues
"To improvise an eight-part fugue
is really beyond human capability."
— Douglas R. Hofstadter,
Gödel, Escher, Bach
Order of a projective
automorphism group:
168
"There are possibilities of
contrapuntal arrangement
of subject-matter."
— T. S. Eliot, quoted in
Origins of Form in Four Quartets.
Order of a projective
automorphism group:
20,160
Comments Off on Friday May 6, 2005
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