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Tetrahedral symmetry

[edit]
reflection lines for [3,3] =

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.

[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

(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

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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.

[3,3,3],
Reflections
Name R0 = = [ ] R1 = = [ ] R2 = = [ ] R3 = = [ ]
Order 2 2 2 2
Matrix

(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

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