Diamond theory
Diamond theory is the theory of affine groups over GF(2) acting on small square and cubic arrays. In the simplest case, the
symmetric group of degree 4 acts on a two-colored diamond figure like that in Plato's Meno dialogue, yielding 24 distinct patterns, each of which has some ordinary or color-interchange
symmetry.
This symmetry-invariance can be generalized to (at least) a group of order approximately 1.3 trillion acting on a 4x4x4 array of cubes.
The theory has applications to finite geometry and to the construction of the large Witt design underlying the Mathieu group of degree 24.
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