Gareth Owen quotes Wikipedia:
> "A poll of readers conducted by The Mathematical Intelligencer in 1990
> named Euler's identity as the "most beautiful theorem in mathematics"."
I was curious enough about what was actually asked to pursue this.
For example, were readers allowed to choose freely between all
theorems or were they given a list of candidates? Was the theorem
referred to by the alleged name of "Euler's identity" or was it
described, as by giving the equation?
The answers are that there was a list, and this and most of the other
theorems were given as equations or similar, not by name. The equation
in question was actually given in the form e^(i pi) = -1, and no name
for it was given.
The poll was actually in the fall 1988 issue (at pages 30-31) of the
Mathematical Intelligencer; the *results* were published in 1990
(summer issue, pages 37-41). It was conducted by David Wells, a
writer who, according to his author-blurb in the results article,
"won a scholarship to Cambridge University, England, but then failed
his degree, a rare achievement".
The 1988 article, the poll itself, is available online only behind a
paywall. However, courtesy of a friend at a university library, I have
now seen it. Wells began by quoting six writers on the subject of
beauty or esthetics in math: Aristotle, Hardy, von Neumann, Poincare,
Weyl, and Morris Kline. He then said:
| Beauty does seem to be an essential, if little discussed, aspect
| of mathematics and the work of mathematicians. Yet no one can say
| precisely of what beauty in mathematics consists, and professional
| mathematicians will not necessarily agree on their definitions
| of mathematical beauty, on their practical judgements of which
| theorems, proofs, concepts, or strategies are the most beautiful,
| or on the role their personal feelings for mathematical beauty
| play in their own work.
|
| This questionnaire is a simple attempt to gather some data...
He provided a list of 24 theorems and asked readers to photocopy the
page and send it in, rating the beauty of each one on a scale from
0 to 10. He also invited comments.
The paper with the full results *is* available online; see:
http://www.gwern.net/docs/math/1990-wells.pdf
To summarize, he received a number of responses with many identical
responses of either blank, 0, or 10; ignoring these for purposes of
tabulation, there were 68 usable ones. The average scores of the
24 theorems varied from 3.9 to 7.7. And the top 10 most beautiful
theorems, the ones scoring 6 or higher, were:
[1] e^(i pi) = -1
[2] Euler's formula for a polyhedron: V+F = E+2
[3] The number of primes is infinite.
[4] There are 5 regular polyhedra.
[5] 1 + 1/2^2 + 1/3^2 + 1/4^2 + ... = pi^2 / 6
[6] A continuous mapping of the closed unit disk into itself
has a fixed point.
[7] There is no rational number whose square is 2.
[8] pi is transcendental.
[9] Every plane map can be colored with 4 colors.
[10] Every prime number of the form 4*n + 1 is the sum of two
integral squares in exactly one way.
If this is interesting to you, you should certainly read the results
paper, which contains several interesting comments about what factors
contributed, or might have contributed, to people's judgments.
Two interesting points about the equation in question. First,
Wells wondered if people rated it highly because it was well known to
have been described as beautiful. And second, two readers suggested
that if the equation in question was given in the form that Stephen
used -- e^(i pi) + 1 = 0 -- then it would be even more beautiful.
To which Wells responded by asking whether a "small and 'inessential'
change" can affect a theorem's esthetic: "How would i^i = e^(-pi/2)
have scored?", he wondered.
--
Mark Brader | "What a strange field. Studying beings instead of mathematics.
Toronto | Could lead to recursive problems in logic."
m...@vex.net | -- Robert L. Forward (The Flight of the Dragonfly)