Log24

Wednesday, May 23, 2007

Wednesday May 23, 2007

Filed under: General,Geometry — m759 @ 7:00 am
 
Strong Emergence Illustrated:
 
The Beauty Test
 
"There is no royal road
to geometry"

— Attributed to Euclid

There are, however, various non-royal roads.  One of these is indicated by yesterday's Pennsylvania lottery numbers:

PA Lottery May 22, 2007: Mid-day 515, Evening 062

The mid-day number 515 may be taken as a reference to 5/15. (See the previous entry, "Angel in the Details," and 5/15.)

The evening number 062, in the context of Monday's entry "No Royal Roads" and yesterday's "Jewel in the Crown," may be regarded as naming a non-royal road to geometry: either U. S. 62, a major route from Mexico to Canada (home of the late geometer H.S.M. Coxeter), or a road less traveled– namely, page 62 in Coxeter's classic Introduction to Geometry (2nd ed.):

The image “http://www.log24.com/log/pix07/070523-Coxeter62.jpg” cannot be displayed, because it contains errors.

The illustration (and definition) is
of regular tessellations of the plane.

This topic Coxeter offers as an
illustration of remarks by G. H. Hardy
that he quotes on the preceding page:

The image “http://www.log24.com/log/pix07/070523-Hardy.jpg” cannot be displayed, because it contains errors.

One might argue that such beauty is strongly emergent because of the "harmonious way" the parts fit together: the regularity (or fitting together) of the whole is not reducible to the regularity of the parts.  (Regular triangles, squares, and hexagons fit together, but regular pentagons do not.)

The symmetries of these regular tessellations of the plane are less well suited as illustrations of emergence, since they are tied rather closely to symmetries of the component parts.

But the symmetries of regular tessellations of the sphere— i.e., of the five Platonic solids– do emerge strongly, being apparently independent of symmetries of the component parts.

Another example of strong emergence: a group of 322,560 transformations acting naturally on the 4×4 square grid— a much larger group than the group of 8 symmetries of each component (square) part.

The lottery numbers above also supply an example of strong emergence– one that nicely illustrates how it can be, in the words of Mark Bedau, "uncomfortably like magic."

(Those more comfortable with magic may note the resemblance of the central part of Coxeter's illustration to a magical counterpart– the Ojo de Dios of Mexico's Sierra Madre.)

Tuesday, May 22, 2007

Tuesday May 22, 2007

Filed under: General,Geometry — m759 @ 7:11 am
 
Jewel in the Crown

A fanciful Crown of Geometry

The Crown of Geometry
(according to Logothetti
in a 1980 article)

The crown jewels are the
Platonic solids, with the
icosahedron at the top.

Related material:

"[The applet] Syntheme illustrates ways of partitioning the 12 vertices of an icosahedron into 3 sets of 4, so that each set forms the corners of a rectangle in the Golden Ratio. Each such rectangle is known as a duad. The short sides of a duad are opposite edges of the icosahedron, and there are 30 edges, so there are 15 duads.

Each partition of the vertices into duads is known as a syntheme. There are 15 synthemes; 5 consist of duads that are mutually perpendicular, while the other 10 consist of duads that share a common line of intersection."

— Greg Egan, Syntheme

Duads and synthemes
(discovered by Sylvester)
also appear in this note
from May 26, 1986
(click to enlarge):

 

Duads and Synthemes in finite geometry

The above note shows
duads and synthemes related
to the diamond theorem.

See also John Baez's essay
"Some Thoughts on the Number 6."
That essay was written 15 years
ago today– which happens
to be the birthday of
Sir Laurence Olivier, who,
were he alive today, would
be 100 years old.

Olivier as Dr. Christian Szell

The icosahedron (a source of duads and synthemes)

"Is it safe?"

Monday, May 21, 2007

Monday May 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 4:00 pm
No Royal Roads
Illustration from a
1980 article at JSTOR:

Coxeter as King of Geometry

A more recent royal reference:

"'Yau wants to be the king of geometry,' Michael Anderson, a geometer at Stony Brook, said. 'He believes that everything should issue from him, that he should have oversight. He doesn't like people encroaching on his territory.'" –Sylvia Nasar and David Gruber in The New Yorker, issue dated Aug. 28, 2006

Wikipedia, Cultural references to the Royal Road:

"Euclid is said to have replied to King Ptolemy's request for an easier way of learning mathematics that 'there is no royal road to geometry.' Charles S. Peirce, in his 'How to Make Our Ideas Clear' (1878), says 'There is no royal road to logic, and really valuable ideas can only be had at the price of close attention.'"

Related material:

Day Without Logic
(March 8, 2007)

and
The Geometry of Logic
(March 10, 2007)
:

The image “http://www.log24.com/log/pix07/070521-Tesseract.gif” cannot be displayed, because it contains errors.

There may be
no royal roads to
geometry or logic,
but…

"There is such a thing
as a tesseract."
— Madeleine L'Engle, 
A Wrinkle in Time

Sunday, May 20, 2007

Sunday May 20, 2007

Filed under: General,Geometry — m759 @ 8:00 am
Plato and Shakespeare:
Solid and Central

"I have another far more solid and central ground for submitting to it as a faith, instead of merely picking up hints from it as a scheme. And that is this: that the Christian Church in its practical relation to my soul is a living teacher, not a dead one. It not only certainly taught me yesterday, but will almost certainly teach me to-morrow. Once I saw suddenly the meaning of the shape of the cross; some day I may see suddenly the meaning of the shape of the mitre. One free morning I saw why windows were pointed; some fine morning I may see why priests were shaven. Plato has told you a truth; but Plato is dead. Shakespeare has startled you with an image; but Shakespeare will not startle you with any more. But imagine what it would be to live with such men still living, to know that Plato might break out with an original lecture to-morrow, or that at any moment Shakespeare might shatter everything with a single song. 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. He is always expecting to see some truth that he has never seen before."

— G. K. Chesterton, Orthodoxy, Ch. IX

From Plato, Pegasus, and the Evening Star (11/11/99):
 

"Nonbeing must in some sense be, otherwise what is it that there is not? This tangled doctrine might be nicknamed Plato's beard; historically it has proved tough, frequently dulling the edge of Occam's razor…. I have dwelt at length on the inconvenience of putting up with it. It is time to think about taking steps."
— Willard Van Orman Quine, 1948, "On What There Is," reprinted in From a Logical Point of View, Harvard University Press, 1980

"The Consul could feel his glance at Hugh becoming a cold look of hatred. Keeping his eyes fixed gimlet-like upon him he saw him as he had appeared that morning, smiling, the razor edge keen in sunlight. But now he was advancing as if to decapitate him."
— Malcolm Lowry, Under the Volcano, 1947, Ch. 10

 

"O God, I could be
bounded in a nutshell
and count myself
a king of infinite space,
were it not that
I have bad dreams."
Hamlet

Coxeter: King of Infinite Space

Coxeter exhuming geometry

From today's newspaper:

Dilbert on space, existence, and the dead

Notes:

For an illustration of
the phrase "solid and central,"
see the previous entry.

For further context, see the
five Log24 entries ending
on September 6, 2006
.

For background on the word
"hollow," see the etymology of
 "hole in the wall" as well as
"The God-Shaped Hole" and
"Is Nothing Sacred?"

For further ado, see
Macbeth, V.v
("signifying nothing")
and The New Yorker,
issue dated tomorrow.

Wednesday, November 8, 2006

Wednesday November 8, 2006

Filed under: General,Geometry — m759 @ 8:00 pm
Grave Matters

See Log24 four years ago
on this date:
Religious Symbolism
at Princeton
.

Compare and contrast
with last month’s entries
related to a Princeton
Coxeter colloquium:

Geometry’s Tombstones
and
Birth, Death, and Symmetry.

Thursday, October 19, 2006

Thursday October 19, 2006

Filed under: General,Geometry — Tags: — m759 @ 7:59 pm
King of Infinite Space
 
  (continued from Sept. 5):

The image “http://www.log24.com/log/pix06A/061019-Coxeter.jpg” cannot be displayed, because it contains errors.

Thanks to Peter Woit’s weblog
for a link to the above illustration.

This picture of
“Coxeter Exhuming Geometry”
suggests the following comparison:

The image “http://www.log24.com/log/pix06A/061019-Tombstones.jpg” cannot be displayed, because it contains errors.

For the second tombstone,
see this morning’s entry,
Birth, Death, and Symmetry.

Further details on the geometry
underlying the second tombstone:

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

The above is from
Variable Resolution 4–k Meshes:
Concepts and Applications
(pdf),
by Luiz Velho and Jonas Gomes.

See also Symmetry Framed
and The Garden of Cyrus.

 “That corpse you planted
          last year in your garden,
  Has it begun to sprout?
          Will it bloom this year? 
  Or has the sudden frost
          disturbed its bed?”

— T. S. Eliot, “The Waste Land

Tuesday, September 5, 2006

Tuesday September 5, 2006

Filed under: General,Geometry — m759 @ 1:00 am
The King of
Infinite Space
 
was published today.

The image “http://www.log24.com/log/pix06A/KingOfInfiniteSpace.jpg” cannot be displayed, because it contains errors.

Click on picture for details.

Tuesday, April 25, 2006

Tuesday April 25, 2006

Filed under: General,Geometry — m759 @ 3:09 pm

“There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?'”

— H. S. M. Coxeter, 1987, introduction to
Richard J. Trudeau’s remarks on
the “Story Theory” of truth
as opposed to
the “Diamond Theory” of truth
in The Non-Euclidean Revolution

A Serious Position

“‘Teitelbaum,’ in German,
is ‘date palm.'”
Generations, Jan. 2003   

“In Hasidism, a mystical brand
of Orthodox Judaism, the grand rabbi
is revered as a kinglike link to God….”

Today’s New York Times obituary
of Rabbi Moses Teitelbaum,
who died on April 24, 2006
(Easter Monday in the
Orthodox Church
)

From Nextbook.org, “a gateway to Jewish literature, culture, and ideas”:

NEW BOOKS: 02.16.05
Proofs and Paradoxes
Alfred Teitelbaum changed his name to Tarski in the early 20s, the same time he changed religions, but when the Germans invaded his native Poland, the mathematician was in California, where he remained. His “great achievement was his audacious assault on the notion of truth,” says Martin Davis, focusing on the semantics and syntax of scientific language. Alfred Tarski: Life and Logic, co-written by a former student, Solomon Feferman, offers “remarkably intimate information,” such as abusive teaching and “extensive amorous involvements.”

From Wikipedia, an unsigned story:

“In 1923 Alfred Teitelbaum and his brother Wacław changed their surnames to Tarski, a name they invented because it sounded very Polish, was simple to spell and pronounce, and was unused. (Years later, he met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, the national religion of the Poles. Alfred did so, even though he was an avowed atheist, because he was about to finish his Ph.D. and correctly anticipated that it would be difficult for a Jew to obtain a serious position in the new Polish university system.”

The image “http://www.log24.com/log/pix06/060425-Tarski.jpg” cannot be displayed, because it contains errors.

Alfred Tarski

The image “http://www.log24.com/log/pix06/060424-Crimson2.jpg” cannot be displayed, because it contains errors.

See also
 
The Crimson Passion.

Friday, January 20, 2006

Friday January 20, 2006

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm
Fourstone Parable

"Wherefore let it hardly… be… thought that the prisoner… was at his best a onestone parable
for… pathetically few… cared… to doubt… the canonicity of his existence as a tesseract."

Finnegans Wake, page 100, abridged

"… we have forgotten that we were angels and painted ourselves into a corner
of resource extraction and commodification of ourselves."

— A discussion, in a draft of a paper (rtf) attributed to Josh Schultz, 
of the poem "Diamond" by Attila Jozsef

Commodification of
the name Cullinane:

See the logos at
cullinane.com,
a design firm with
the motto

The image “http://www.log24.com/log/pix06/060120-Motto.jpg” cannot be displayed, because it contains errors.

(Note the 4Cs theme.)

To adapt a phrase from
Finnegans Wake, the
"fourstone parable" below
is an attempt to
decommodify my name.

Fourstone Parable:

(See also yesterday's "Logos."
The "communicate" logo is taken from
an online library at Calvin College;
the "connect" logo is a commonly
available picture of a tesseract
(Coxeter, Regular Polytopes, p. 123),
and the other two logos
are more or less original.)

For a more elegant
four-diamond figure, see
Jung and the Imago Dei.

Sunday, November 20, 2005

Sunday November 20, 2005

Filed under: General,Geometry — Tags: , — m759 @ 4:04 pm

An Exercise
of Power

Johnny Cash:
“And behold,
a white horse.”

The image “http://www.log24.com/log/pix05B/051120-SpringerLogo9.gif” cannot be displayed, because it contains errors.
Adapted from
illustration below:

The image “http://www.log24.com/log/pix05B/051120-NonEuclideanRev.jpg” cannot be displayed, because it contains errors.

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to  the “Diamond Theory” of truth in The Non-Euclidean Revolution

“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*….”

Richard J. Trudeau in
The Non-Euclidean Revolution

“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”

— Jim Holt in The New Yorker.

(Click on the box below.)

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

Exercise of Power:

Show that a white horse–

A Singer 7-Cycle

a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.

This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).

Related material

On this date:

 In 1490, The White Knight
 (Tirant lo Blanc The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. )–
a major influence on Cervantes–
was published, and in 1910

The image “http://www.log24.com/log/pix05B/051120-Caballo1.jpg” cannot be displayed, because it contains errors.

the Mexican Revolution began.

Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York

The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. Description from Amazon.com

“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of God-supplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an all-encompassing reality.'”

Wednesday, September 28, 2005

Wednesday September 28, 2005

Filed under: General,Geometry — m759 @ 4:26 am
Mathematical Narrative,
continued:

There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
“What is truth?”

— H. S. M. Coxeter, introduction to
Richard J. Trudeau’s
The Non-Euclidean Revolution

“People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only ‘truths’ strictly worthy of the name. Such truths I will call ‘diamonds’; they are highly desirable but hard to find….The happy metaphor is Morris Kline’s in Mathematics in Western Culture (Oxford, 1953), p. 430.”

— Richard J. Trudeau,
   The Non-Euclidean Revolution,
   Birkhauser Boston,
   1987, pages 114 and 117

“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory…. I concluded long ago that each enterprise contains only stories (which the scientists call ‘models of reality’). I had started by hunting diamonds; I did find dazzlingly beautiful jewels, but always of human manufacture.”

  — Richard J. Trudeau,
     The Non-Euclidean Revolution,
     Birkhauser Boston,
     1987, pages 256 and 259

An example of
the story theory of truth:

The image “http://www.log24.com/log/pix05B/050925-Proof1.jpg”  cannot be displayed, because it contains errors.

Actress Gwyneth Paltrow (“Proof”) was apparently born on either Sept. 27, 1972, or Sept. 28, 1972.   Google searches yield  “about 193” results for the 27th and “about 610” for the 28th.

Those who believe in the “story theory” of truth may therefore want to wish her a happy birthday today.  Those who do not may prefer the contents of yesterday’s entry, from Paltrow’s other birthday.

Friday, August 19, 2005

Friday August 19, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:00 pm

Mathematics and Narrative
continued

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"

H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to  the "Diamond Theory" of truth " in The Non-Euclidean Revolution

"I had an epiphany: I thought 'Oh my God, this is it! People are talking about elliptic curves and of course they think they are talking mathematics. But are they really? Or are they talking about stories?'"

An organizer of last month's "Mathematics and Narrative" conference

"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*…."

Richard J. Trudeau in The Non-Euclidean Revolution

"'Deniers' of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others."

— Jim Holt in this week's New Yorker magazine.  Click on the box below.

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

* Many stripes

   "What disciplines were represented at the meeting?"
   "Apart from historians, you mean? Oh, many: writers, artists, philosophers, semioticians, cognitive psychologists – you name it."

 

An organizer of last month's "Mathematics and Narrative" conference

Thursday, August 11, 2005

Thursday August 11, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 8:16 am

Kaleidoscope, continued

From Clifford Geertz, The Cerebral Savage:

"Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately).  And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought.  Both anecdotal and geometric, it builds coherent structures out of 'the odds and ends left over from psychological or historical process.'

These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore….  as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular.   But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort….  Levi-Strauss generalizes this permutational view of thinking to savage thought in general.  It is all a matter of shuffling discrete (and concrete) images–totem animals, sacred colors, wind directions, sun deities, or whatever–so as to produce symbolic structures capable of formulating and communicating objective (which is not to say accurate) analyses of the social and physical worlds.

…. And the point is general.  The relationship between a symbolic structure and its referent, the basis of its meaning,  is fundamentally 'logical,' a coincidence of form– not affective, not historical, not functional.  Savage thought is frozen reason and anthropology is, like music and mathematics, 'one of the few true vocations.'

Or like linguistics."

Edward Sapir on Linguistics, Mathematics, and Music:

"… linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

— Edward Sapir, "The Grammarian and his Language,"
  American Mercury 1:149-155,1924

From Robert de Marrais, Canonical Collage-oscopes:

"…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups.  You know, those things the non-professionals call . . . kaleidoscopes! *  (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' **— then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry  (A polytope is an n-dimensional analog of a polygon or polyhedron.  Chapter V of this book is entitled 'The Kaleidoscope'….)

** … contemporary with the Johns Hopkins hatchet job that won him American marketshare, Derrida was also being subjected to a series of probing interviews in Paris by the hometown crowd.  He first gained academic notoriety in France for his book-length reading of Husserl's two-dozen-page essay on 'The Origin of Geometry.'  The interviews were collected under the rubric of Positions (Chicago: U. of Chicago Press, 1981…).  On pp. 34-5 he says the following: 'the resistance to logico-mathematical notation has always been the signature of logocentrism and phonologism in the event to which they have dominated metaphysics and the classical semiological and linguistic projects…. A grammatology that would break with this system of presuppositions, then, must in effect liberate the mathematization of language…. The effective progress of mathematical notation thus goes along with the deconstruction of metaphysics, with the profound renewal of mathematics itself, and the concept of science for which mathematics has always been the model.'  Nice campaign speech, Jacques; but as we'll see, you reneged on your promise not just with the kaleidoscope (and we'll investigate, in depth, the many layers of contradiction and cluelessness you put on display in that disingenuous 'playing to the house'); no, we'll see how, at numerous other critical junctures, you instinctively took the wrong fork in the road whenever mathematical issues arose… henceforth, monsieur, as Joe Louis once said, 'You can run, but you just can't hide.'…."

Thursday, March 31, 2005

Thursday March 31, 2005

Filed under: General — m759 @ 3:16 am

“In collage, juxtaposition is everything.”

    April 2, 2004

The above material may be regarded
as commemorating the March 31
birth of René Descartes
 and death of H. S. M. Coxeter.

For material related to Descartes,
see The Line.
For material related to Coxeter,
see Art Wars.

Wednesday, March 31, 2004

Wednesday March 31, 2004

Filed under: General — Tags: , , — m759 @ 12:25 am

To Be

A Jesuit cites Quine:

"To be is to be the value of a variable."

— Willard Van Orman Quine, cited by Joseph T. Clark, S. J., in Conventional Logic and Modern Logic: A Prelude to Transition,  Woodstock, MD: Woodstock College Press, 1952, to which Quine contributed a preface.

Quine died in 2000 on Xmas Day.

From a July 26, 2003, entry,
The Transcendent Signified,
on an essay by mathematician
Michael Harris:

Kubrick's
monolith

Harris's
slab

From a December 10, 2003, entry:

Putting Descartes Before Dehors

      

"Descartes déclare que c'est en moi, non hors de moi, en moi, non dans le monde, que je pourrais voir si quelque chose existe hors de moi."

ATRIUM, Philosophie

For further details, see ART WARS.

The above material may be regarded as commemorating the March 31 birth of René Descartes and death of H. S. M. Coxeter.

For further details, see

Plato, Pegasus, and the Evening Star.

Monday, April 28, 2003

Monday April 28, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 12:07 am

ART WARS:

Toward Eternity

April is Poetry Month, according to the Academy of American Poets.  It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."

Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:

"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.

………………………
Since then–'tis Centuries–and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity– "

 

Consider the following journal entries from April 7, 2003:
 

Math Awareness Month

April is Math Awareness Month.
This year's theme is "mathematics and art."


 

An Offer He Couldn't Refuse

Today's birthday:  Francis Ford Coppola is 64.

"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."


H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

 

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:

 
The "horse's head" figure above is from a note I wrote on this date 18 years ago.  The following journal entry from April 4, 2003, gives some details:
 

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight.  Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight.  Consider eight rectangular cells arranged in an array of four rows and two columns.  Let us label these cells with coordinates, then apply a permutation.

 


 Decimal 
labeling

 
Binary
labeling


Algebraic
labeling


Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions.   See also "Pieces of Eight," by Robert L. Griess.

Monday, April 7, 2003

Monday April 7, 2003

Filed under: General,Geometry — Tags: , — m759 @ 1:17 pm

An Offer He Couldn't Refuse

Today's birthday:  Francis Ford Coppola is 64.

"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."


— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

 

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:


 

Saturday, April 5, 2003

Saturday April 5, 2003

Filed under: General,Geometry — Tags: — m759 @ 9:49 am

Art Wars:
Mathematics and the
Emperor's New Art

From Maureen Dowd's New York Times column of June 9, 2002: 

"The shape of the government is not as important as the policy of the government. If he makes the policy aggressive and pre-emptive, the president can conduct the war on terror from the National Gallery of Art."

 

NY Times, April 5, 2003:
U.S. Tanks Move Into Center of Baghdad
See also today's op-ed piece
by Patton's grandson.

Meanwhile, at the Washington Post, another example of great determination and strength of character:

 

Donald Coxeter Dies: Leader in Geometry

By Martin Weil
Washington Post Staff Writer
Saturday, April 5, 2003

"Donald Coxeter, 96, a mathematician who was one of the 20th century's foremost specialists in geometry and a man of great determination and strength of character as well, died March 31 at his home in Toronto."

From another Coxeter obituary:

In the Second World War, Coxeter was asked by the American government to work in Washington as a code-breaker. He accepted, but then backed out, partly because of his pacifist views and partly for aesthetic reasons: "The work didn't really appeal to me," he explained; "it was a different sort of mathematics."

For a differing account of how geometry is related to code-breaking, see the "Singer 7-cycle" link in yesterday's entry, "The Eight," of 3:33 PM.  This leads to a site titled

An Introduction to the
Applications of Geometry in Cryptography
.

"Now I have precisely the right instrument, at precisely the right moment of history, in exactly the right place."

 — "Patton,"
the film

Quod erat
demonstrandum
.


 

Added Sunday, April 6, 2003, 3:17 PM:

The New York Times Magazine of April 6
continues this Art Wars theme.


                 (Cover typography revised)

The military nature of our Art Wars theme appears in the Times's choice of words for its cover headline: "The Greatest Generation." (This headline appears in the paper, but not the Internet, version.)

Some remarks in today's Times Magazine article seem especially relevant to my journal entry for Michelangelo's birthday, March 6.

"…Conceptualism — suddenly art could be nothing more than an idea….

LeWitt moved between his syntax of geometric sculptures and mental propositions for images: concepts he wrote on paper that could be realized by him or someone else or not at all.  Physical things are perishable.  Ideas need not be."

— Michael Kimmelman, chief art critic of the New York Times, April 6, 2003

Compare this with a mathematician's aesthetics:

"A mathematician, like a painter or a poet, is a maker of patterns.  If his patterns are more permanent than theirs, it is because they are made with ideas."

— G. H. Hardy, A Mathematician's Apology (1940), reprinted 1969, Cambridge U. Press, p. 84 

It seems clear from these two quotations that the real conceptual art is mathematics and that Kimmelman is peddling the emperor's new clothes.

Thursday, October 24, 2002

Thursday October 24, 2002

Filed under: General — Tags: — m759 @ 6:00 am

A (Very Brief) Course of
Modern Analysis 

In honor of today's anniversary of the 1873 birth of Edmund Taylor Whittaker, here are some references to a topic that still interests some mathematicians of today.

From A Course of Modern Analysis, by E. T. Whittaker and G. N. Watson, Fourth Edition, Cambridge University Press, 1927, reprinted 1969:

Section 20.7  "…the fact, that x and y can be expressed as one-valued functions of the variable z, makes this variable z of considerable importance… z is called the uniformizing variable of the equation…. When the genus of the algebraic curve f(x,y) = 0 is greater than unity, the uniformisation can be effected by means of what are known as automorphic functions. Two classes of such functions of genus greater than unity have been constructed, the first by Weber…(1886), the second by Whittaker…(1898)…."

The topic of uniformisation of algebraic curves has appeared frequently lately in connection with Wiles's attack on Fermat's Last Theorem. See, for instance, Lang's 1995 AMS Notices article

"Shimura's… insight was that the ordinary modular functions for a congruence subgroup of SL2(Z) suffice to uniformize elliptic curves defined over the rationals."

and Charles Daney's notes

"The property of an elliptic curve [over Q] of being parameterized by modular functions is one way of defining a modular elliptic curve, and the Taniyama-Shimura conjecture asserts that every elliptic curve is modular."

For a deeper discussion of uniformisation in the context of Wiles's efforts, see "Elliptic curves and p-adic uniformisation," by H. Darmon, 1999.

For a more traditional approach to uniformisation, see "On the uniformisation of algebraic curves," by Yu. V. Brezhnev (24 May, 2002), which cites two of Whittaker's papers on automorphic functions (from 1898 and 1929) and a 1930 paper, "The uniformisation of algebraic curves," by J. M. Whittaker, apparently E. T. Whittaker's son.  

« Newer Posts

Powered by WordPress