## December 5, 2011

### Beautiful Mathematics and Imaginative Words

Some years ago, David Wells surveyed readers of the Mathematical Intelligencer about what they consider to be beautiful in mathematics. The respondents ranked Euler's identity linking e, π, and i as the most beautiful expression in mathematics. See "Euler's Beauties."

In the intriguing new book Beautiful Mathematics (MAA, 2011), Martin Erickson expands on this theme, contemplating the esthetic appeal and elegance of selected mathematical words, geometric structures, formulas, theorems, proofs, solutions, and unsolved problems.

"My approach to mathematics is as an art form, like painting, sculpture, or music," Erickson writes. "While the artist works in a tangible medium, the mathematician works in a medium with numbers, shapes, and abstract patterns."

Erickson's many examples include not only familiar beauties but also underappreciated wonders. His chapter on the fascinating names of mathematical objects, for example, ranges from the figure-eight curve known as the lemniscate to the waterfall of primes, which depicts the way that prime numbers fall into two classes (primes of the form 4n + 1 and primes of the form 4n + 3).

Among these "imaginative" terms, Erickson considers centillion, golden ratio, Borromean rings, sieve of Eratosthenes, transversal of primes, triangular numbers, determinant, and complex plane.

In the language of mathematics, what terms do you find particularly evocative, peculiar, or apt?

My nominee of the moment is the term "block monoid," to which I was recently introduced by Scott Chapman, Editor-Elect of the American Mathematical Monthly. Chapman and Paul Baginski have an article titled "Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory" in the December 2011 Monthly.

References:

Wells, D.1990. Are these the most beautiful? Mathematical Intelligencer 12(September): 37-41.

______.  1988. Which is the most beautiful? Mathematical Intelligencer 10(December): 30-31.

Steven H. Cullinane said...

In "Beautiful Mathematics," pages 106-108, section 5.17, "A Group of Operations," Martin Erickson does not acknowledge any source.

That section, on a group of 322,560 permutations generated by permuting the rows, columns, and quadrants of a 4x4 array, 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.

Steven H. Cullinane said...

In "Beautiful Mathematics," pages 106-108, section 5.17, "A Group of Operations," Martin Erickson does not acknowledge any source.

That section, on a group of 322,560 permutations generated by permuting the rows, columns, and quadrants of a 4x4 array, 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.