Moonshine

A Note by Steven H. Cullinane on Sept. 4, 2013

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example is the apparent connections
between parts of complex analysis and groups related to the 
large Mathieu group M₂₄. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces.
In a classic book, Kummer's Quartic Surface (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.)

The Google search below documents the moonshine
relating Rosenhain's and Göpel's work to M₂₄, via 
the geometry of the 4×4 square.