Log24

For the bricks of the title, see other posts tagged Brick Space. 
For some cubes* and flowers, see below.

Combining features of the above two images, one might picture the 24
cells of the 4×6 array underlying the Curtis Miracle Octad Generator
(MOG) as each containing an eightfold cube, pictured as above with seven
of its subcubes showing and an eighth subcube hidden behind them.

The seven visible subcubes may be colored, as in the Curtis image of
the Klein map, with seven distinct colors… corresponding to the seven
edge-colors used in the Curtis-Klein map. Each of the seven visible
subcubes in a cell may also be labeled, on its visible faces, with a symbol
denoting one of the 24 points of the projective line over GF(23), just as the
faces in the Curtis-Klein map are labeled.  The hidden subcube in each cell
may be regarded as also so labeled, by the MOG label of the cell's position.

There is then enough information in the array's eightfold cubes' colors and
labels to construct the seven generating permutations of M₂₄ described by
Curtis, and the 24 array cells may be regarded as now containing 24 distinct
entities — which perhaps might be called "octoids."

Those desiring a more decorative approach may replace the 24 labeled cubes
with 24 labeled "flowers." Each flower — like the map's symmetric seven
"petals" and the central "infinity heptagon" they surround — forms an octad.

Related Illustrations . . .

* See as well posts tagged Mathieu Cube . . .

Related material — 

The 56 triangles of  the eightfold cube . . .

• in Aitchison's March 9, 2018, talk (slides 32-34), and
• in this journal on July 25, 2008, and later.

​

   Image from Christmas Day 2005.

Post last revised:  December 30, 2025 @ 21:30 E.S.T.