From yesterday's post "Imago Review" . . .
For a natural group of 322,560 transformations
acting on this figure, see the diamond theorem.
"What remains fixed (globally, not pointwise)
under these transformations is the system of
points and hyperplanes from the diamond theorem."
For an example of a subset of points remaining fixed
pointwise under this group, embed the diamond
theorem's underlying four-by-four array of points in a
four-by-six (4 rows, six columns) array . . . as in the
R. T. Curtis Miracle Octad Generator. The 322,560
transformations of the diamond theorem are then
called by some the "octad group." It is more properly
called the "octad stabilizer group" because, within the
full group of automorphisms of the 4×6 array —
the Mathieu group M24 — it leaves an 8-point octad
fixed locally within the 4×6 array, while permuting (or not)
the eight points within the octad. Sixteen of the octad
stabilizer transformations, the translations, leave the octad
fixed pointwise. See (for instance) octad.us.
