Template:Full octahedral group; single elements; signed perm mat, perm mat and cube
Description |
[[File:Cube permutation {{{1}}} {{{2}}} JF.png|thumb|Corresponding permutation of the JF compound]] [[File:Cube permutation {{{1}}} {{{2}}} JF ortho.png|thumb|Orthographic projection shown in the file]] 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 S4. 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.) |
||||||
Date | |||||||
Source | Own work | ||||||
Author |
|
||||||
FULL OCTAHEDRAL GROUP; SINGLE ELEMENTS; SIGNED PERM MAT, PERM MAT AND CUBE development | |||||||
Source code |