Cube_permutation_4_4.svg
Summary
Description Cube permutation 4 4.svg |
One of the 48 elements of the full octahedral group represented as
The cube has the coordinate origin in its center. Vertex 0 has coordinate , vertex 1 has , and vertex 7 has . Transformation matrices (3×3) in the same row have the same pattern of negative rows, and those in the same column have the same pattern of non-zero entries. This is left action : Permutation means that first is applied and then . The 24 permutations of the cube that leave a contained tetrahedron unchanged (those where the bottom left vertex of the resulting cube is light gray) are identified with numbers 0..23 corresponding to elements of S 4 . These elements' combinations with the inversion are identified with the same number with an added apostrophe. Transformations with dark and with green are odd (have a negative determinant ) . Permutations in these rows and columns are marked with a horizontal and vertical gray bar respectively. (Conveniently, where it looks like a minus the determinant is negative, and when it looks like a plus it is positive.) |
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