by Bathsheba
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There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This one is the odd polytope out, the one without a 3D counterpart.

It has 24 octahedral cells, all shown in this Schlegel diagram. Like the pentachoron it's self-dual -- the only self-dual solid in any dimension > 2 that is not a simplex. And if that wasn't enough, it's also the only regular convex polytope in any dimension > 2 that tiles its space and is not a hypercube.


IN: 2.158 w x 2.16 d x 2.159 h
CM: 5.482 w x 5.486 d x 5.484 h


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I believe it is also the 4 dimensional 'vector equilibrium', the next step up from the hexagon and the cubeoctahedron. It is of course not obvious in its 3D projection but each of its 24 points are the same distance from their connected neighbors as they are from the center of the 4 dimensional shape and therefore all 96 triangles form tetrahedrons to the center point. In a higher dimensional geometry forum I saw it was voted the most popular higher dimensional form by a large majority!
September 19, 2013, 3:00 am


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