Symmetries of the n-cube

Part 20: Symmetries of the n-cube
The n-cube	The n-cube C_n is the n-dimensional version of the 2-cube (square) and 3-cube (cube). The most convenient way to think of it is as the set of all real n-vectors (a₁, a₂,...,a_n) with -1 < = a_i <= 1 for all i. C_n has 2ⁿ vertices (two choices +1 or -1 for each coordinate), n2^n-1 edges (n at each vertex, counting each edge twice), and 2n boundary hyperplanes (n-1-dimensional boundary elements), also called facets. Note that each facet is a C_n-1 obtained by keeping one coordinate constant. [Check how this works for n = 2 and 3] C_n has n²reflection hyperplanes, which are the solution sets of the homogeneous linear equations: x_i = 0 (the n coordinate hyperplanes) x_i = x_j (n choose 2 of these) x_i = -x_j (n choose 2 of these too) In the cube these are the 3 coordinate planes, and the 6 planes that contain the diagonals of two opposite faces and two edges of the cube.	.
Symmetries of the n-cube	Now we show by induction that C_n has at most 2ⁿ n! symmetries. We have seen this is true for n = 2 and 3. Assume true for C_n-1. Choose one of the 2n facets to go to the bottom (i.e. the facet which meets the x_n-axis at the point (0,0,...,-1)). Use a symmetry of C_n-1 as a symmetry of the bottom facet. This will permute the other facets of C_n. Now every symmetry of C_n is obtained in this way because any symmetry maps facets to facets. So there are at most 2n x 2^n-1 x (n-1)! = 2ⁿn! symmetries of C_n. Now we show that there at least 2ⁿn! symmetries of C_n. Choose a vertex v. There are n facets containing v. Each of the n! permutations of these facets gives a symmetry of C_n fixing v, the even permutations being direct (rotations about an n-2-dimensional hyperplane) and the odd ones indirect (reflections in a hyperplane of symmetry). Then choose a vertex v' to map v to. This can be done in 2ⁿ ways. Every symmetry that maps v to v' can be obtained by conjugating a permutation of facets by one of these 2ⁿ maps. Conclusion: C_n has exactly 2ⁿ n! symmetries, and we know how they act on vertices and adjacent facets.	.

Part 20: Symmetries of the n-cube

The n-cube

The n-cube C_n is the n-dimensional version of the 2-cube (square) and 3-cube (cube). The most convenient way to think of it is as the set of all real n-vectors (a₁, a₂,...,a_n) with -1 < = a_i <= 1 for all i.

C_n has 2ⁿ vertices (two choices +1 or -1 for each coordinate), n2^n-1 edges (n at each vertex, counting each edge twice), and 2n boundary hyperplanes (n-1-dimensional boundary elements), also called facets. Note that each facet is a C_n-1 obtained by keeping one coordinate constant. [Check how this works for n = 2 and 3] C_n has n²reflection hyperplanes, which are the solution sets of the homogeneous linear equations:

x_i = 0 (the n coordinate hyperplanes)
x_i = x_j (n choose 2 of these)
x_i = -x_j (n choose 2 of these too)

In the cube these are the 3 coordinate planes, and the 6 planes that contain the diagonals of two opposite faces and two edges of the cube.

Symmetries of the n-cube

Now we show by induction that C_n has at most 2ⁿ n! symmetries. We have seen this is true for n = 2 and 3. Assume true for C_n-1. Choose one of the 2n facets to go to the bottom (i.e. the facet which meets the x_n-axis at the point (0,0,...,-1)). Use a symmetry of C_n-1 as a symmetry of the bottom facet. This will permute the other facets of C_n. Now every symmetry of C_n is obtained in this way because any symmetry maps facets to facets. So there are at most 2n x 2^n-1 x (n-1)! = 2ⁿn! symmetries of C_n.

Now we show that there at least 2ⁿn! symmetries of C_n. Choose a vertex v. There are n facets containing v. Each of the n! permutations of these facets gives a symmetry of C_n fixing v, the even permutations being direct (rotations about an n-2-dimensional hyperplane) and the odd ones indirect (reflections in a hyperplane of symmetry). Then choose a vertex v' to map v to. This can be done in 2ⁿ ways. Every symmetry that maps v to v' can be obtained by conjugating a permutation of facets by one of these 2ⁿ maps.

Conclusion: C_n has exactly 2ⁿ n! symmetries, and we know how they act on vertices and adjacent facets.

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Last update: 12 January, 2000

Author: Phill Schultz, schultz@maths.uwa.edu.au