Log24

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

Darkness

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.

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

J. G. Ballard on “the architecture of death“:

“… a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death.”

— The Guardian, March 20, 2006

Edward Hirsch on Lorca:

“For him, writing is a struggle both with geometry and death.”

— “The Duende,” American Poetry Review, July/August 1999

SINGER, ISAAC:
“Are Children the Ultimate Literary Critics?”
 — Top of the News 29 (Nov. 1972): 32-36.
“Sets forth his own aims in writing for children
 and laments ‘slice of life’ and chaos in
children’s literature. Maintains that children
like good plots, logic, and clarity,
and that they have a concern for
‘so-called eternal questions.'”

— An Annotated Listing of Criticism
by Linnea Hendrickson

“She returned the smile, then looked
across the room to her youngest brother,
Charles Wallace, and to their father,
who were deep in concentration, bent
over the model they were building
of a tesseract: the square squared,
and squared again: a construction
of the dimension of time.”

— A Swiftly Tilting Planet,
by Madeleine L’Engle

For “the dimension of time,”
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937. 

A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

For a more adult audience —

In memory of General Augusto Pinochet, who died yesterday in Santiago, Chile, a quotation from Federico Garcia Lorca‘s lecture on “the Duende” (Buenos Aires, Argentina, 1933):

“… Philip of Austria… longing to discover the Muse and the Angel in theology, found himself imprisoned by the Duende of cold ardors in that masterwork of the Escorial, where geometry abuts with a dream and the Duende wears the mask of the Muse for the eternal chastisement of the great king.”

The Escorial

Perhaps. Or perhaps Philip, “the lonely
hermit of the Escorial,” is less lonely now.