User:Tomruen/Octahedral symmetry

Square symmetry
An irreducible 2-dimensional finite reflective group is B2=[4], order 8,. The reflection generators matrices are R0, R1. R02=R12=(R0×R1)4=Identity.

Chiral square symmetry, [4]+, is generated by rotation: S0,1.

Octahedral symmetry
Another irreducible 3-dimensional finite reflective group is octahedral symmetry, [4,3], order 48,. The reflection generators matrices are R0, R1, R2. R02=R12=R22=(R0×R1)4=(R1×R2)3=(R0×R2)2=Identity. Chiral octahedral symmetry, [4,3]+, is generated by 2 of 3 rotations: S0,1, S1,2, and S0,2. Pyritohedral symmetry [4,3+], is generated by reflection R0 and rotation S1,2. A 6-fold rotoreflection is generated by V0,1,2, the product of all 3 reflections.

Hyperoctahedral symmetry
A irreducible 4-dimensional finite reflective group is hyperoctahedral group, B4=[4,3,3], order 384,. The reflection generators matrices are R0, R1, R2, R3. R02=R12=R22=R32=(R0×R1)4=(R1×R2)3=(R2×R3)3=(R0×R2)2=(R1×R3)2=(R0×R3)2=Identity.

Chiral octahedral symmetry, [4,3,3]+, is generated by 3 of 6 rotations: S0,1, S1,2, S2,3, S0,2, S1,3, and S0,3. Hyperpyritohedral symmetry [4,(3,3)+], is generated by reflection R0 and rotations S1,2 and S2,3. An 8-fold double rotation is generated by W0,1,2,3, the product of all 4 reflections.