Log24

Monday, October 21, 2013

A Gathering for Gardner

Filed under: General — Tags: — m759 @ 8:25 pm

The title was suggested by Gardner + Darkness in this journal
and by recent remarks on the Devil by Justice Scalia and the Pope.

Thursday, November 5, 2015

The Monster

Filed under: General,Geometry — Tags: — m759 @ 9:00 am

In memory of Princeton mathematician John Nash

"For the past six years all over the world 
experts in the branch of abstract algebra
called group theory have been struggling
to capture a group known as the monster."

—Martin Gardner, Scientific American ,  June 1980

"When the Hawkline Monster moved to get a better view
of what was happening, the shadow, after having checked
all the possibilities of light, had discovered a way that it
could shift itself in front of the monster, so that the monster
at this crucial time would be blinded by darkness for a few
seconds, did so, causing confusion to befall the monster.

This was all that the shadow could do and it hoped that this
would give Greer and Cameron the edge they would need
to destroy the Hawkline Monster using whatever plan they
had come up with, for it seemed that they must have a plan
if they were to have any chance at all with the monster and
they did not seem like fools.

When Cameron yelled at Greer, the shadow interpreted this
as the time to move and did so. It obscured the vision of the
Hawkline Monster for a few seconds, knowing full well that if
the monster were destroyed it would be destroyed, too, but
death was better than going on living like this, being a part of
this evil."

— Richard Brautigan, The Hawkline Monster , 1974

From the post For Scientific Witch Hunters of October 30,
an illustration from The Boston Globe —

From the post Colorful Story (All Souls' Day),  
an Illustration from Google Book Search —

Earlier in Brautigan's tale

" Everybody started to leave the parlor to go downstairs
and pour out the Hawkline Monster but just as
they reached the door and one of the Hawkline women
had her hand on the knob, Cameron said, 'Hold it for a
second. I want to get myself a little whiskey.' "

Thursday, May 15, 2014

Moonshine

Filed under: General — m759 @ 2:56 pm

“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside,
enveloping the tale which brought it out only as a glow brings out a
haze, in the likeness of one of these misty halos that sometimes
are made visible by the spectral illumination of moonshine.”

— Joseph Conrad in Heart of Darkness

Photo of full moon over Oslo last night by Josefine Lyche:

A scene from my film viewing last night:

Some background (click to enlarge):

Note:

The “I, Frankenstein” scene above should not be interpreted as
a carrying of Martin Gardner through a lyche gate.  Gardner
is, rather, symbolized by the asterisk in the first image from
the above Google search.

Thursday, May 8, 2014

Like a Bee

Filed under: General — m759 @ 11:22 am

From a Log24 search for “Boxing Day“—

(Click image for some commentary.)

The image “http://www.log24.com/log/pix05B/051227-Diebold.jpg” cannot be displayed, because it contains errors.

From The New York Times —

Correction: Jan. 16, 2006

“An obituary on Dec. 27 about John Diebold,
a businessman and engineer who helped shape
modern industrial development in America,
misstated a business venture of John Diebold Inc.,
an investment firm he founded in 1967. It did
not finance Diebold Election Systems, a maker of
polling machines that, despite its name, has no
connection to John Diebold.”

Related material:

Synchronicity and this  journal on the date of the correction.

Tuesday, November 9, 2010

Design

Filed under: General — m759 @ 5:01 pm

A Theory of Pure Design

by Denman Waldo Ross

Lecturer on the Theory of Design
in Harvard University

Boston and New York
Houghton, Mifflin and Company, 1907

PREFACE

"My purpose in this book is to elucidate, so far as I can, the
principles which underlie the practice of drawing and painting
as a Fine Art.  Art is generally regarded as the expression of
feelings and emotions which have no explanation except per-
haps in such a word as inspiration , which is expletive rather
than explanatory
.  Art is regarded as the one activity of man
which has no scientific basis, and the appreciation of Art is
said to be a matter of taste in which no two persons can be
expected to agree.  It is my purpose in this book to show how,
in the practice of Art, as in all other practices, we use certain
terms and follow certain principles.  Being defined and ex-
plained, these terms and principles may be known and under-
stood by everybody.  They are, so to speak, the form of the
language
.

While an understanding of the terms and principles of Art
will not, in itself, enable any one to produce important works,
such works are not produced without it.  It must be understood,
however, that the understanding of terms and principles
is not, necessarily, an understanding in words.  It may lie in
technical processes and in visual images and may never rise,
or shall I say fall, to any formulation in words, either spoken
or written."

_________________________________________________

One of Ross's protégés, Jack Levine, died yesterday at 95. He
is said to have remarked, "I want to paint with the dead ones."

Related material: This journal on the day of Levine's death
and on the day of Martin Gardner's death.

The latter post has an image illustrating Ross's remarks on
formulations in words—
 

Image-- The Case of the Lyche Gate Asterisk

For further details, see Finale, Darkness Visible, and Packed.

Tuesday, June 22, 2010

Mathematics and Narrative, continued

Filed under: General,Geometry — Tags: , — m759 @ 2:14 pm

"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

A 1973 review of Koestler's book—

"Koestler's 'call girls,' summoned here and there
 by this university and that foundation
 to perform their expert tricks, are the butts
 of some chilling satire."

Examples of Light—

Felix Christian Klein (1849- June 22, 1925) and Évariste Galois (1811-1832)

Klein on Galois—

"… in France just about 1830 a new star of undreamt-of brilliance— or rather a meteor, soon to be extinguished— lighted the sky of pure mathematics: Évariste Galois."

— Felix Klein, Development of Mathematics in the 19th Century, translated by Michael Ackerman. Brookline, Mass., Math Sci Press, 1979. Page 80.

"… um 1830 herum in Frankreich als ein neuer Stern von ungeahntem Glanze am Himmel der reinen Mathematik aufleuchtet, um freilich, einem Meteor gleich, sehr bald zu verlöschen: Évariste Galois."

— Felix Klein, Vorlesungen Über Die Entwicklung Der Mathematick Im 19. Jahrhundert. New York, Chelsea Publishing Co., 1967. (Vol. I, originally published in Berlin in 1926.) Page 88.

Examples of Darkness—

Martin Gardner on Galois—

"Galois was a thoroughly obnoxious nerd,
 suffering from what today would be called
 a 'personality disorder.'  His anger was
 paranoid and unremitting."

Gardner was reviewing a recent book about Galois by one Amir Alexander.

Alexander himself has written some reviews relevant to the Koestler book above.

See Alexander on—

The 2005 Mykonos conference on Mathematics and Narrative

A series of workshops at Banff International Research Station for Mathematical Innovation between 2003 and 2006. "The meetings brought together professional mathematicians (and other mathematical scientists) with authors, poets, artists, playwrights, and film-makers to work together on mathematically-inspired literary works."

Saturday, June 12, 2010

Holy Geometry

Filed under: General,Geometry — m759 @ 10:31 am

The late mathematician V.I. Arnold was born on this date in 1937.

"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

Light

Image-- AMS site screenshot of V.I. Arnold obituary, June 12, 2010

Darkness

Image-- AMS site screenshot of Martin Gardner tribute, May 25, 2010

Choosing light rather than darkness, we observe Arnold's birthday with a quotation from his 1997 Paris talk 'On Teaching Mathematics.'

"The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact…."

The "experimental fact" part, perhaps offered with tongue in cheek, is of less interest than the assertion that the Jacobi identity forces the altitude-intersection theorem.

Albert Einstein on that theorem in the "holy geometry book" he read at the age of 12—

"Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question.  This lucidity and certainty made an indescribable impression upon me.”

Arnold's much less  evident assertion about altitudes and the Jacobi identity is discussed in "Arnol'd, Jacobi identity, and orthocenters" (pdf) by Nikolai V. Ivanov.

Ivanov says, without giving a source,  that the altitudes theorem "was known to Euclid." Alexander Bogomolny, on the other hand, says it is "a matter of real wonderment that the fact of the concurrency of altitudes is not mentioned in either Euclid's Elements  or subsequent writings of the Greek scholars. The timing of the first proof is still an open question."

For other remarks on geometry, search this journal for the year of Arnold's birth.

Friday, June 4, 2010

A Better Story

Filed under: General,Geometry — Tags: , — m759 @ 7:59 am

Continued from May 8
(Feast of Saint Robert Heinlein)

“Wells and trees were dedicated to saints.  But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden.  Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy.”

— Charles Williams, Witchcraft, Faber and Faber, London, 1941

Why "Saint" Robert? See his accurate depiction of evil– the Eater of Souls in Glory Road.

For more on Williams's "other capacities," see Heinlein's story "Lost Legacy."

A related story– Fritz Leiber's "The Mind Spider." An excerpt:

The conference—it was much more a hyper-intimate
gabfest—proceeded.

"My static box bugged out for a few ticks this morning,"
Evelyn remarked in the course of talking over the
trivia of the past twenty-four hours.

The static boxes were an invention of Grandfather
Horn. They generated a tiny cloud of meaningless brain
waves. Without such individual thought-screens, there was
too much danger of complete loss of individual personality

—once Grandfather Horn had "become" his infant daughter
as well as himself for several hours and the unfledged
mind had come close to being permanently lost in its own
subconscious. The static boxes provided a mental wall be-
– hind which a mind could safely grow and function, similar
to the wall by which ordinary minds are apparently
always enclosed.

In spite of the boxes, the Horns shared thoughts and
emotions to an amazing degree. Their mental togetherness
was as real and as mysterious—and as incredible—as
thought itself . . . and thought is the original angel-cloud
dancing on the head of a pin. Their present conference
was as warm and intimate and tart as any actual family
gathering in one actual room around one actual table.
Five minds, joined together in the vast mental darkness
that shrouds all minds. Five minds hugged together for
comfort and safety in the infinite mental loneliness that
pervades the cosmos.

Evelyn continued, "Your boxes were all working, of
course, so I couldn't get your thoughts—just the blurs of
your boxes like little old dark grey stars. But this time
if gave me a funny uncomfortable feeling, like a spider
Crawling down my—Grayl! Don't feel so wildly! What
Is it?”

Then… just as Grayl started to think her answer…
something crept from the vast mental darkness and infinite
cosmic loneliness surrounding the five minds of the
Horns
.

Grayl was the first to notice. Her panicky thought had
ttie curling too-keen edge of hysteria. "There are six of
us now! There should only be five, but there are six.
Count! Count, I tell you! Six!"

To Mort it seemed that a gigantic spider was racing
across the web of their thoughts….

See also this journal on May 30– "720 in the Book"– and on May 31– "Memorial for Galois."

("Obnoxious nerds"— a phrase Martin Gardner recently applied to Galois— will note that 720 (= 6!) is one possible result of obeying Leiber's command "Count! Count, I tell you! Six!")

Saturday, May 29, 2010

Packed

Filed under: General — m759 @ 7:11 am

Significant Passage:
On the Writing Style of Visual Thinkers

"The words are filled with unstated meaning.
They are (the term is Ricoeur's) 'packed'
and need unpacking." —Gerald Grow

From the date of Ricoeur's death,
May 20, 2005

“Plato’s most significant passage
    may be found in Phaedrus  265b…."

From Sept. 30, 2004

With a little effort,
anything can be shown
to connect with anything else:
existence is infinitely

cross-referenced."Image-- 8-rayed asterisk

— Opening sentence
of Martha Cooley's
The Archivist

Image-- 8-rayed asterisk Example:
Mozart's K 265,
the page number 265,
and a story by George MacDonald.

Mozart's K 265 is variations on the theme
now known as "Twinkle, Twinkle, Little Star."

For darker variations on the Twinkle theme,
see the film "Joshua" and Martin Gardner's
Annotated Alice  (Norton, 2000, pp. 73-75).

Image-- From the film 'Joshua,' Joshua with the Alice statue in Central Park

Joshua

For darker variations on the asterisk theme,
see Darkness Visible (May 25)
and Vonnegut's Asterisk.

Tuesday, May 25, 2010

ART WARS continued

Filed under: General — Tags: — m759 @ 2:01 am

Darkness Visible

The inevitable tribute to Martin Gardner
has now appeared at the AMS website—

Image-- American Mathematical Society (AMS) tribute to Martin Gardner, May 25, 2010

Related Imagery—

The following is an image from Saturday morning—

Image-- 'Darkness Visible,' a picture from Log24 on Saturday, May 22, 2010

See also Art Wars and
Mathematics and Narrative.

Sunday, December 31, 2006

Sunday December 31, 2006

Filed under: General — m759 @ 7:15 pm
7/15, 2005:

From Darkness Visible:

“Ed Rinehart [sic] made a fortune painting canvases that were just one solid color.  He had his black period in which the canvas was totally black.  And then he had a blue period in which he was painting the canvas blue.”

— Martin Gardner interview in AMS Notices, June/July 2005

Monday, January 16, 2006

Monday January 16, 2006

Filed under: General — m759 @ 4:00 am

Mathematics and Narrative

Rebecca Goldstein, Mathematics and the Character of Tragedy:

“It was Plato who best expressed– who veritably embodied– the tension between the narrative arts and mathematics.”

Veritably.

Friday, July 15, 2005

Friday July 15, 2005

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

Feast of St. Bonaventure

From Darkness Visible:

“Ed Rinehart [sic] made a fortune painting canvases that were just one solid color.  He had his black period in which the canvas was totally black.  And then he had a blue period in which he was painting the canvas blue.”

— Martin Gardner interview in AMS Notices, June/July 2005 

From Art History:

“Art history was very personal through the eyes of Ad Reinhardt.”

— Robert Morris,
    Smithsonian Archives of American Art

From The Edge of Eternity:

Christopher Fry’s obituary
in The New York Times

“His plays radiated an optimistic faith in God and humanity, evoking, in his words, ‘a world in which we are poised on the edge of eternity, a world which has deeps and shadows of mystery, and God is anything but a sleeping partner.’ He said he wrote his plays in poetry because that was ‘the language in which man expresses his own amazement’ at the complexity both of himself and of a reality which, beneath the surface, was ‘wildly, perilously, inexplicably fantastic.'”

From
Arrangement in
Black and Blue:

 

The image “http://www.log24.com/log/pix05A/050703-Cold.jpg” cannot be displayed, because it contains errors.

Adapted from cover of
German edition of Cold Mountain

Thursday, June 23, 2005

Thursday June 23, 2005

Filed under: General,Geometry — Tags: — m759 @ 3:00 pm

Mathematics and Metaphor

The current (June/July) issue of the Notices of the American Mathematical Society has two feature articles.  The first, on the vulgarizer Martin Gardner, was dealt with here in a June 19 entry, Darkness Visible.  The second is related to a letter of André Weil (pdf) that is in turn related to mathematician Barry Mazur’s attempt to rewrite mathematical history  and to vulgarize other people’s research by using metaphors drawn, it would seem, from the Weil letter.
 
A Mathematical Lie conjectures that Mazur’s revising of history was motivated by a desire to dramatize some arcane mathematics, the Taniyama conjecture, that deals with elliptic curves and modular forms, two areas of mathematics that have been known since the nineteenth century to be closely related.

Mazur led author Simon Singh to believe that these two areas of mathematics were, before Taniyama’s conjecture of 1955, completely unrelated — 

“Modular forms and elliptic equations live in completely different regions of the mathematical cosmos, and nobody would ever have believed that there was the remotest link between the two subjects.” — Simon Singh, Fermat’s Enigma, 1998 paperback, p. 182

This is false.  See Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp, Bulletin of the American Mathematical Society, January 1994.

It now appears that Mazur’s claim was in part motivated by a desire to emulate the great mathematician André Weil’s manner of speaking; Mazur parrots Weil’s “bridge” and “Rosetta stone” metaphors —

From Peter Woit’s weblog, Feb. 10, 2005:

“The focus of Weil’s letter is the analogy between number fields and the field of algebraic functions of a complex variable. He describes his ideas about studying this analogy using a third, intermediate subject, that of function fields over a finite field, which he thinks of as a ‘bridge‘ or ‘Rosetta stone.'” 

In “A 1940 Letter of André Weil on Analogy in Mathematics,” (pdf), translated by Martin H. Krieger, Notices of the A.M.S., March 2005, Weil writes that

“The purely algebraic theory of algebraic functions in any arbitrary field of constants is not rich enough so that one might draw useful lessons from it. The ‘classical’ theory (that is, Riemannian) of algebraic functions over the field of constants of the complex numbers is infinitely richer; but on the one hand it is too much so, and in the mass of facts some real analogies become lost; and above all, it is too far from the theory of numbers. One would be totally obstructed if there were not a bridge between the two.  And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants….

On the other hand, between the function fields and the ‘Riemannian’ fields, the distance is not so large that a patient study would not teach us the art of passing from one to the other, and to profit in the study of the first from knowledge acquired about the second, and of the extremely powerful means offered to us, in the study of the latter, from the integral calculus and the theory of analytic functions. That is not to say that at best all will be easy; but one ends up by learning to see something there, although it is still somewhat confused. Intuition makes much of it; I mean by this the faculty of seeing a connection between things that in appearance are completely different; it does not fail to lead us astray quite often. Be that as it may, my work consists in deciphering a trilingual text {[cf. the Rosetta Stone]}; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary. Sometimes I worked on one column, sometimes under another.”

Here is another statement of the Rosetta-stone metaphor, from Weil’s translator, Martin H.  Krieger, in the A.M.S. Notices of November 2004,  “Some of What Mathematicians Do” (pdf):

“Weil refers to three columns, in analogy with the Rosetta Stone’s three languages and their arrangement, and the task is to ‘learn to read Riemannian.’  Given an ability to read one column, can you find its translation in the other columns?  In the first column are Riemann’s transcendental results and, more generally, work in analysis and geometry.  In the second column is algebra, say polynomials with coefficients in the complex numbers or in a finite field. And in the third column is arithmetic or number theory and combinatorial properties.”

For greater clarity, see  Armand Borel (pdf) on Weil’s Rosetta stone, where the three columns are referred to as Riemannian (transcendental), Italian (“algebraico-geometric,” over finite fields), and arithmetic (i.e., number-theoretic).
 
From Fermat’s Enigma, by Simon Singh, Anchor paperback, Sept. 1998, pp. 190-191:

Barry Mazur: “On the one hand you have the elliptic world, and on the other you have the modular world.  Both these branches of mathematics had been studied intensively but separately…. Than along comes the Taniyama-Shimura conjecture, which is the grand surmise that there’s a bridge between these two completely different worlds.  Mathematicians love to build bridges.”

Simon Singh: “The value of mathematical bridges is enormous.  They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other’s  creations…. The great potential of the Taniyama-Shimura conjecture was that it would connect two islands and allow them to speak to each other for the first time.  Barry Mazur thinks of the Taniyama-Shimura conjecture as a translating device similar to the Rosetta stone…. ‘It’s as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language,’ says Mazur.  ‘But the Taniyama-Shimura conjecture is a Rosetta stone with a certain magical power.'”

If Mazur, who is scheduled to speak at a conference on Mathematics and Narrative this July, wants more material on stones with magical powers, he might consult The Blue Matrix and The Diamond Archetype.

Sunday, June 19, 2005

Sunday June 19, 2005

Filed under: General,Geometry — Tags: , — m759 @ 4:00 am
ART WARS:
Darkness Visible
“No light, but rather darkness visible
 Serv’d only to discover sights of woe”
John Milton, Paradise Lost,
Book I,  lines 63-64

From the cover article (pdf) in the
June/July 2005 Notices of the
American Mathematical Society–

Martin Gardner


A famed vulgarizer, Martin Gardner,
summarizes the art of Ad Reinhardt
(Adolph Dietrich Friedrich Reinhardt,
  Dec. 24, 1913 – Aug. 30, 1967):

“Ed Rinehart [sic] made a fortune painting canvases that were just one solid color.  He had his black period in which the canvas was totally black.  And then he had a blue period in which he was painting the canvas blue.  He was exhibited in top shows in New York, and his pictures wound up in museums.  I did a column in Scientific American on minimal art, and I reproduced one of Ed Rinehart’s black paintings.  Of course, it was just a solid square of pure black.  The publisher insisted on getting permission from the gallery to reproduce it.”

Related material
from Log24.net,
Nov. 9-12, 2004:

Fade to Black

“…that ineffable constellation of talents that makes the player of rank: a gift for conceiving abstract schematic possibilities; a sense of mathematical poetry in the light of which the infinite chaos of probability and permutation is crystallized under the pressure of intense concentration into geometric blossoms; the ruthless focus of force on the subtlest weakness of an opponent.”

— Trevanian, Shibumi

“‘Haven’t there been splendidly elegant colors in Japan since ancient times?’

‘Even black has various subtle shades,’ Sosuke nodded.”

— Yasunari Kawabata, The Old Capital

An Ad Reinhardt painting
described in the entry of
noon, November 9, 2004
is illustrated below.

Ad Reinhardt,  Greek Cross

Ad Reinhardt,
Abstract Painting,
1960-66.
Oil on canvas, 60 x 60 inches.
Solomon R. Guggenheim Museum

The viewer may need to tilt
the screen to see that this
painting is not uniformly black,
but is instead a picture of a
Greek cross, as described below.

“The grid is a staircase to the Universal…. We could think about Ad Reinhardt, who, despite his repeated insistence that ‘Art is art,’ ended up by painting a series of… nine-square grids in which the motif that inescapably emerges is a Greek cross.

Greek Cross

There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora’s box of spiritual reference that is opened once one uses it.”

— Rosalind Krauss,
Meyer Schapiro Professor
of Modern Art and Theory
at Columbia University

(Ph.D., Harvard U., 1969),
in “Grids”

The image “http://www.log24.com/log/pix04B/041109-Krauss.jpg” cannot be displayed, because it contains errors.

Krauss

 
In memory of
St. William Golding
(Sept. 19, 1911 – June 19, 1993)

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