Some rotation-reflection groups, like the tetrahedral and octahedral ones, can be expressed in a rather simple form.
A permutation matrix is a matrix that is a permutation of the rows or columns or the identity matrix. Each row and column contains one 1 and all the rest 0.
For those rotoreflection groups, some or all of the 1's in some of the elements may be replaced by -1. Its elements are (diagonal matrix of 1's and -1's) . (permutation matrix)
This can be expressed more abstractly as (A,P) where A is a vector of elements of some group and P is a permutation. Their multiplication law is (A,P) * (B,Q) = (A*(P.B),P.Q). This is called a "wreath product".
A wreath product has a subgroup: all elements with the form (A,I), where I is the identity permutation. This subgroup is normal, and its quotient group is the group of the permutations in the group.
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Now for the symmetry groups of the regular polytopes, generalizations of the regular polygons and polyhedra.
2 space dimensions: a regular n-gon has symmetry group Dih
, with its pure rotations being Cyc
.
For more space dimensions, there is only a finite number of regular polytopes for each one, and only three of them except for 3D and 4D. Here are those three.
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The simplex. It generalizes the 2D triangle, the 3D tetrahedron and the 4D 5-cell. For n-D, it has (n+1) vertices with constant separation between them. It is self-dual, and its symmetry group is the n-D representation of Sym(n+1) (rotoreflections) and Alt(n+1) (rotations).
The hypercube. It generalizes the 2D square, the 3D cube, and the 4D 8-cell. For n-D, it has 2
n vertices, at locations (+-1,+-1, ...,+-1).
The cross-polytope. It generalizes the 2D square, the 3D octahedron, and the 4D 16-cell. For n-D, it has 2n vertices, at locations (0,0,...,+-1,...,0,0)
The hypercube and the cross-polytope are duals, and they share a symmetry group: the wreath product of (n of +-1's) and all n-D permutation matrices (group Sym
). It thus has 2
n*n! elements. The pure rotations have all of those with determinant = +1, (+-1's multiplying to 1) . (even permutations) + (+-1's multiplying to -1) . (odd permutations)
This rotoreflection group has index-2 subgroups (pure rotations in it), (+-1's multiplying to 1), (even permutations), and an index-4 subgroup with quotient group Z2*Z2: (+-1's multiplying to 1) . (even permutations).
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In 3D, there are two additional polyhedra, duals of each other, the dodecahedron and the icosahedron. Their symmetry group is the icosahedral group, Ih (120 elements) for rotoreflections and I (60 elements) for rotations only.
In 4D, there are three additional polyhedra. One of them, the 12-cell, is self-dual and halfway between the 8-cell and the 16-cell. Its symmetry group is the 8-cell/16-cell one with elements (1/2)*(4*4 array of +-1's) added, if those elements have determinant +-1. That group has 1152 elements in total and 768 for pure rotations. This builds on the 384 elements (rotoreflection) / 192 elements (rotation) for 8-cell and 16-cell symmetry.
The other two, the 120-cell and the 600-cell, are related to the 3D dodecahedron and icosahedron. They share a symmetry group that is roughly the square of the icosahedral group, one with 14400 elements for rotoreflections and 7200 elements for pure rotations.