User:Tomruen/A5

Tetrahedral symmetry

reflection lines for [3,3] =

Further information: Coxeter notation § 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: S_0,1, S_1,2, and S_0,2. A trionic subgroup, isomorphic to [2⁺,4], is generated by S_0,2 and R₁. A 4-fold rotoreflection is generated by V_0,1,2.

[3,3],

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

Hypertetrahedral symmetry

Further information: Point groups in four dimensions § Pentachoric_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: S_0,1, S_1,2, S_2,3, S_0,2, S_1,3, and S_0,3. There are 4 rotoreflections generated by a product of 3 reflections: S_0,1,2, S_0,1,3, S_0,2,3 and S_1,2,3. A 5-fold double rotation is generated by V_0,1,2,3=R₀×R₁×R₂×R₃.

[3,3,3],

Reflections
Name	R0 = = [ ]	R1 = = [ ]	R2 = = [ ]	R3 = = [ ]
Order	2	2	2	2
Matrix	$\left[{\begin{smallmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&0&-1&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-1}{4}}&{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {3}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-1}{4}}\\\end{smallmatrix}}\right]$
	(0,1,-1,0)n	(1,-1,0,0)n	(0,1,1,0)n	(-1,-1,-1,√5)n
Rotations
Name	S0,1 = R0×R1 [3]+	S1,2 = R1×R2 [3]+	S2,3 = R2×R3 [3]+	S0,2 = R0×R2 [2]+	S1,3 = R1×R3 [2]+	S0,3 = R0×R3 [2]+
Order	3	3	3	2	2	2
Matrix	$\left[{\begin{smallmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&0&-1&0\\1&0&0&0\\0&-1&0&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {1}{4}}&{\frac {1}{4}}&{\frac {-3}{4}}&{\frac {\sqrt {5}}{4}}\\{\frac {1}{4}}&{\frac {-3}{4}}&{\frac {1}{4}}&{\frac {\sqrt {5}}{4}}\\{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-1}{4}}\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {-1}{4}}&{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {3}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-1}{4}}\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {3}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-1}{4}}&{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-1}{4}}\\\end{smallmatrix}}\right]$
Rotoreflections					Double rotation
Name	T0,1,2 = R0×R1×R2 [4+,2+]	T0,1,3 = R0×R1×R3 [6+,2+]	T0,2,3 = R0×R2×R3 [6+,2+]	T1,2,3 = R1×R2×R3 [4+,2+]	V0,1,2,3 = R0×R1×R2×R3
Order	4	6	6	4	5
Matrix	$\left[{\begin{smallmatrix}0&0&-1&0\\0&-1&0&0\\1&0&0&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {-1}{4}}&{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {3}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-1}{4}}\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {\sqrt {5}}{4}}\\{\frac {1}{4}}&{\frac {-3}{4}}&{\frac {1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {1}{4}}&{\frac {1}{4}}&{\frac {-3}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-1}{4}}\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {1}{4}}&{\frac {1}{4}}&{\frac {-3}{4}}&{\frac {\sqrt {5}}{4}}\\{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {1}{4}}&{\frac {-3}{4}}&{\frac {1}{4}}&{\frac {\sqrt {5}}{4}}\\{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-1}{4}}\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}{\frac {1}{4}}&{\frac {1}{4}}&{\frac {-3}{4}}&{\frac {\sqrt {5}}{4}}\\{\frac {1}{4}}&{\frac {-3}{4}}&{\frac {1}{4}}&{\frac {\sqrt {5}}{4}}\\{\frac {3}{4}}&{\frac {-1}{4}}&{\frac {-1}{4}}&{\frac {-{\sqrt {5}}}{4}}\\{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-{\sqrt {5}}}{4}}&{\frac {-1}{4}}\\\end{smallmatrix}}\right]$