Log24

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.

The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by 
a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space  coordinates. He describes it
as simply a mapping to a set of reproducible symbols . 

(But Weyl does imply that these symbols should, like vector-space 
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)

The "points" and "lines" of finite  geometry are abstract
entities satisfying only whatever incidence requirements
yield non-contradictory and interesting results. In finite
geometry, neither the points nor the lines are required to
lie within any Euclidean (or, for that matter, non-Euclidean)
space.

Models  of finite geometries may, however, embed the
points and lines within non -finite geometries in order
to aid visualization.

For instance, the 15 points and 35 lines of PG(3,2) may
be represented by subsets of a 4×4 array of dots, or squares,
located in the Euclidean plane. These "lines" are usually finite
subsets of dots or squares and not*  lines of the Euclidean plane.

Example — See "4×4" in this journal.

Some impose on configurations from finite geometry
the rather artificial requirement that both  points and lines
must be representable as those of a Euclidean plane.

Example:  A Cremona-Richmond pentagon —

A square version of these 15 "points" —

A 1905 square version of these 15 "points" 
with digits instead of letters —

See Parametrizing the 4×4 Array
(Log24 post of Sept. 13, 2016).

Update of 8 AM ET Sunday, Sept. 25, 2016 —
For more illustrations, do a Google image search
on "the 2-subsets of a 6-set." (See one such search.)

* But in some models are subsets of the grid lines 
   that separate squares within an array.