“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
Or Sunday dinner.
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Platonic |
Shakespearean |
| Not to mention Euclid and Picasso. | |
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In the above pictures, Euclid is represented by |
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A check of recent tweets by Alexander Bogomolny, who was
mentioned in the previous post, yields a remark of Oct. 26, 2015…
This is not unrelated to a word from Freud:
See as well "Digging Out the Truth?" (Jerusalem Post 2/25/2010)
and Michener's The Source in this journal.
The cover of the K. O. Friedrichs book From Pythagoras to Einstein
shown in the previous post suggests a review (click the Log24
images for webpages where they can be manipulated) ….
The "more sophisticated" link in the first image above
leads to a webpage by Alexander Bogomolny,
"Pythagoras' Theorem by Tessellation," that says
"This is a subtle and beautiful proof."
Bogomolny refers us to the Friedrichs book, from which one of
the illustrations of the proof by tessellation is as follows —
For a quite different use of superposition, see
The Lindbergh Manifesto (May 19, 2015).
The Mathematical Association of America has a
submit-a-review form that apparently allows readers
to comment on previously reviewed books.
This morning I submitted the following comment on
Alexander Bogomolny's March 16, 2012, review of
Martin J. Erickson's Beautiful Mathematics :
In section 5.17, pages 106-108, "A Group of Operations,"
Erickson does not acknowledge any source. This section
is based on the Cullinane diamond theorem. See that
theorem (published in an AMS abstract in 1979) at
PlanetMath.org and EncyclopediaOfMath.org, and
elsewhere on the Web. Details of the proof given by
Erickson may be found in "Binary Coordinate Systems,"
a 1984 article on the Web at
http://finitegeometry.org/sc/gen/coord.html.
If and when the comment may be published, I do not know.
Update of about 6:45 PM ET June 7:
The above comment is now online at the MAA review site.
Update of about 7 PM ET July 29:
The MAA review site's web address was changed, and the
above comment was omitted from the page at the new address.
This has now been corrected.
"Philosophy is a battle against the bewitchment
of our intelligence by means of our language." — Wittgenstein
"You've got to pick up every stitch…
Must be the season of the witch."
— Donovan song at the end of Nicole Kidman's "To Die For"
Today's morning post, Rubric, suggests a check
of Alexander Bogomolny's tweets:
Clicking the hint leads to Bogomolny's Ambiguities in Plain Language:
See also, in this journal, alea (which appears within the derived word "aleatory").
From a mathematical review:
"The book ends with eye-opening explorations….
If pressed for an extra rubric, I would consider
a separate section on "Engaging Games," as
this is something that mathematicians are
preoccupied with— literally and metaphorically."
See in this journal Language Game, Nexus, and
posts of May 12, 2013.
A review of Beautiful Mathematics for 8 PM.
The corrected index the reviewer links to is here.
As a June 1 post shows, it is not corrected enough .
The review is dated March 16, 2012.
See this journal on that date.
See also Good Bye, Marty.
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
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.
“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.”
 Natasha Wescoat, 2004 Shakespearean Fool |
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| Not to mention Euclid and Picasso | |
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(Click on pictures for details. Euclid is represented by Alexander Bogomolny, Picasso by Robert Foote.)
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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 |
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POMP AND
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 By THE CRIMSON STAFF The Trouble By SAHIL K. MAHTANI |
The great mathematician
Robert P. Langlands
is 70 today.
In honor of his expository work–
notably, lectures at
The Institute for Advanced Study
on “The Practice of Mathematics“
and a very acerbic review (pdf) of
a book called Euclid’s Window—
here is a “Behold!” proof of
the Pythagorean theorem:
The picture above is adapted from
a sketch by Eves of a “dynamical”
proof suitable for animation.
The proof has been
described by Alexander Bogomolny
as “a variation on” Euclid I.47.
Bogomolny says it is a proof
by “shearing and translation.”
It has, in fact, been animated.
The following version is
by Robert Foote:
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Plato's |
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From The Unknowable (1999), by Gregory J. Chaitin, who has written extensively about his constant, which he calls Omega:
"What is Omega? It's just the diamond-hard distilled and crystallized essence of mathematical truth! It's what you get when you compress tremendously the coal of redundant mathematical truth…"
Charles H. Bennett has written about Omega as a cabalistic number.
Here is another result with religious associations which, historically, has perhaps more claim to be called the "diamond-hard essence" of mathematical truth: The demonstration in Plato's Meno that a diamond inscribed in a square has half the area of the square (or that, vice-versa, the square has twice the area of the diamond).
From Ivars Peterson's discussion of Plato's diamond and the Pythagorean theorem:
"In his textbook The History of Mathematics, Roger Cooke of the University of Vermont describes how the Babylonians might have discovered the Pythagorean theorem more than 1,000 years before Pythagoras.
Basing his account on a passage in Plato's dialogue Meno, Cooke suggests that the discovery arose when someone, either for a practical purpose or perhaps just for fun, found it necessary to construct a square twice as large as a given square…."
From "Halving a Square," a presentation of Plato's diamond by Alexander Bogomolny, the moral of the story:
SOCRATES: And if the truth about reality is always in our soul, the soul must be immortal….
From "Renaissance Metaphysics and the History of Science," at The John Dee Society website:
Galileo on Plato's diamond:
"Cassirer, drawing attention to Galileo's frequent use of the Meno, particularly the incident of the slave's solving without instruction a problem in geometry by 'natural' reason stimulated by questioning, remarks, 'Galileo seems to accept all the consequences drawn by Plato from this fact…..'"
Roger Bacon on Plato's diamond:
"Fastening on the incident of the slave in the Meno, which he had found reproduced in Cicero, Bacon argued from it 'wherefore since this knowledge (of mathematics) is almost innate and as it were precedes discovery and learning or at least is less in need of them than other sciences, it will be first among sciences and will precede others disposing us towards them.'"
It is perhaps appropriate to close this entry, made on All Hallows' Eve, with a link to a page on Dr. John Dee himself.
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