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: S_0,1.

[4],

	Reflections		Rotations
Name	R0	R1	S0,1
Order	2	2	4
Matrix	$\left[{\begin{smallmatrix}1&0\\0&-1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1\\-1&0\\\end{smallmatrix}}\right]$

Octahedral symmetry

Reflection lines for B₃=[4,3] =

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: S_0,1, S_1,2, and S_0,2. Pyritohedral symmetry [4,3⁺], () is generated by reflection R₀ and rotation S_1,2. A 6-fold rotoreflection is generated by V_0,1,2, the product of all 3 reflections.

[4,3],

	Reflections			Rotations			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Order	2	2	2	4	3	2	6
Matrix	$\left[{\begin{smallmatrix}1&0&0\\0&1&0\\0&0&-1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0\\0&0&1\\0&-1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0\\0&0&1\\1&0&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0\\1&0&0\\0&0&-1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0\\0&0&1\\-1&0&0\\\end{smallmatrix}}\right]$
	(0,0,1)n	(0,1,-1)n	(1,-1,0)n	(1,0,0)axis	(1,1,1)axis	(1,-1,0)axis

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: S_0,1, S_1,2, S_2,3, S_0,2, S_1,3, and S_0,3. Hyperpyritohedral symmetry [4,(3,3)⁺], () is generated by reflection R₀ and rotations S_1,2 and S_2,3. An 8-fold double rotation is generated by W_0,1,2,3, the product of all 4 reflections.

[4,3,3],

	Reflections				Rotations						Rotoreflection				Double rotation
Name	R0	R1	R2	R3	S0,1	S1,2	S2,3	S0,2	S1,3	S0,3	V0,1,2	V1,2,3	V0,1,3	V0,2,3	W0,1,2,3
Order	2	2	2	2	4 (B2)	3 (A2)	3 (A2)	2	2	2	6 (B3)	4 (A3)	4	6	8 (B4)
Matrix	$\left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&1&0&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&1\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&-1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&-1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&-1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&-1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\-1&0&0&0\\\end{smallmatrix}}\right]$