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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.

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