Log24

For the purpose of defining figurate geometry , a figurate space  might be
loosely described as any space consisting of finitely many congruent figures  —
subsets of Euclidean space such as points, line segments, squares, 
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them. 

Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate  space, considered as a collection of figures —  vertices, edges,
faces — seen in the nineteenth century as acted upon by a group  of symmetries .

More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator  of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M₂₄ . That group may be viewed as acting
on various subsets of a 24-set… for instance, the 759 octads  that are analogous
to the faces  of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.