User:Tomruen/A5

Tetrahedral symmetry


The simplest irreducible 3-dimensional finite reflective group is tetrahedral symmetry, [3,3], order 24,. The reflection generators, from a D3=A3 construction, are matrices R0, R1, R2. R02=R12=R22=(R0×R1)3=(R1×R2)3=(R0×R2)2=Identity. [3,3]+ is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2. A trionic subgroup, isomorphic to [2+,4], is generated by S0,2 and R1. A 4-fold rotoreflection is generated by V0,1,2.

Hypertetrahedral symmetry
The simplest irreducible 4-dimensional finite reflective group is hypertetrahedral or pentachoric symmetry, [3,3,3], order 120,. The reflection generators, defined by extending D3, are matrices R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)3=(R1×R2)3=(R2×R3)3=(R0×R2)2=(R1×R3)2=(R0×R3)2=Identity. [3,3,3]+ is generated by 3 of 6 rotations: S0,1, S1,2, S2,3, S0,2, S1,3, and S0,3. There are 4 rotoreflections generated by a product of 3 reflections: S0,1,2, S0,1,3, S0,2,3 and S1,2,3. A 5-fold double rotation is generated by V0,1,2,3=R0×R1×R2×R3.