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April 15

Special mappings for the 24 cell?

Is there a bijective mapping between the vertices and the cells of the 24 Cell so that each vertex maps to a Cell that it is part of? Similarly, is there a bijective mapping between the 96 edges and the 96 triangles of the 24 cell so that each edge maps to a triangle that it is part of?

You could I think construct one by just listing them and pairing them off. In each case you have multiple choices (each cell has eight vertices adjoining it, etc.) so it should be possible to pair them off so each has one to pair with.

Quite possibly there's a mapping that's also a rotation, between the 24-cell and its dual as the dual is also a 24-cell. This could work in 4D as you could use a double rotation through a small angle so every vertex is offset. If the angle's the smallest possible to produce a rotation from the 24-cell to its dual then surely it rotates every vertex to an adjacent cell. I wouldn't know where to start constructing this or proving it though.--JohnBlackburne^words_deeds 16:03, 15 April 2014 (UTC)[reply]

Take the vertices of the 24-cell to be (±2, 0, 0, 0), (0, ±2, 0, 0), (0, 0, ±2, 0), (0, 0, 0, ±2), (±1, ±1, ±1, ±1). Normal vectors to the faces, which you can take to be coordinates of the dual, are (±1, ±1, 0, 0), (±1, 0, ±1, 0), (±1, 0, 0, ±1), (0, ±1, ±1, 0), (0, ±1, 0, ±1), (0, 0, ±1, ±1). Scale the second set by √2 to make them the same length, so the original and dual are the same size. The matrix

     1/√2  1/√2  0     0 
     1/√2 -1/√2  0     0 
      0     0   1/√2  1/√2  
      0     0   1/√2 -1/√2

representing a double rotation by angle π/4, now maps the original to the dual and vice versa. Combining the two 24-cells would make a compound with 48 vertices. --RDBury (talk) 04:28, 16 April 2014 (UTC)[reply]

Cool, I'd forgotten that the two standard methods of stating the vertices were each other's duals. (and in the centers (scaled) of the dual's octahedral cells. I'll take a look at the effects that that rotation has on the edges/faces.Naraht (talk) 19:37, 17 April 2014 (UTC)[reply]