For St. Andrew’s Day

“The miraculous enters…. When we investigate these problems, some fantastic things happen….”

— John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)

The remarkable Mathieu group M24, a group of permutations on 24 elements, may be studied by picturing its action on three interchangeable 8-element “octads,” as in the “Miracle Octad Generator” of R. T. Curtis.

A picture of the Miracle Octad Generator, with my comments, is available online.