The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter

revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University

professor Martin Erickson as a book title. See below.

In the same IAS Fall 2015 Letter appear the following remarks

by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald

discovered between two kinds of symmetry which we call modular and affine.

The two kinds of symmetry were originally found in separate parts of science,

modular in pure mathematics and affine in physics. Modular symmetry is

displayed for everyone to see in the drawings of flying angels and devils

by the artist Maurits Escher. Escher understood the mathematics and got the

details right. Affine symmetry is displayed in the peculiar groupings of particles

created by physicists with high-energy accelerators. The mathematician

Robert Langlands was the first to conjecture a connection between these and

other kinds of symmetry. . . ." (Wikipedia link added.)

**The adjective "modular"** might aptly be applied to . . .

**The adjective "affine"** might aptly be applied to . . .

The geometry of the 4×4 square combines **modular** symmetry

(i.e., related to theta functions) with the **affine** symmetry above.

Hudson's 1905 discussion of **modular** symmetry (that of Rosenhain

tetrads and Göpel tetrads) in the 4×4 square used a parametrization

of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but

did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer **modular** 2-subsets with the 16-

element **affine** space over the two-element Galois field GF(2), see my note

of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract

"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by

Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For *Macdonald's* *own* use of the words "**modular**" and "**affine**," see

Macdonald, I. G., "**Affine Lie algebras and modular forms**,"

*Séminaire N. Bourbaki* , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.