Double Triamond, w/ Hexastix!

Chaim Goodman-Strauss and Eugene Sargent G4G8 2008

The quarter symmetries of space are in a sense the sparsest and so the most pleasing aesthetically—To our limited geometric senses, these quarter symmetries appear to lie astride the boundary of order and disorder, our direct comprehension never fully enveloping the mathematical structure before our eyes.

Two interpenetrating, dual cubic lattices are as symmetrical as can be, with the full plenary symmetry type 8o:2. In this symmetry, and in all the plenary symmetry types, there are four directions of three-fold axes of rotation, and such axes intersect at the centers of any cube.

The quarter groups also have such axes, but only one-quarter as many. In particular, only one such axis passes through the center of any cube.

The most symmetrical of quarter groups (with the pencils identical on both ends) is denoted 8o⁄4 ; by breaking symmetries we have subgroups of this, ultimately reaching 1o⁄4 .