by wendy » Mon May 09, 2005 11:17 am
The prismatic product is one of the four geometric products. It produces the sequence of polytopes line, square, cube, tesseract, ..., It generalises to any combinations of bases, eg
pentagon * hexagon prism
The other three are pyramid, tegum and comb products.
The pyritohedral symmetry is one based on the symmetry of pyrities.
In mathematical terms, it means for a point (x,y,z), even permutations of positions and all change of sign. It has the orbifold 3 * 2.
It generalises to higher dimensions. The symmetry of the snub 24-choron is the point (w,x,y,z), with even permutations and all change of sign. The group in 4D is order 192, with a subgroup of order 576.
The pyritotopic group generalises this.
The cubic has these subgroups in all dimensions.
C = cubic all permutations, all change of sign
C/2 = semicubic all permutations, even change of sign
C/2 = pyritohedral even permutations, all change of sign
C/2 = cubic-rotational even (permutations + change of sign)
C/4 = semicubic - rot even permutations, even change of sign
Wendy