24-cell, stereographic projection, steel

The 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a polygon nor a simplex. Due to this singular property, it does not have a good analogue in 3 dimensions.

This is a stereographic projection of the 24-cell, or octaplex, a 4-dimensional regular solid, into 3-space.  The octaplex is  self-dual, having no real equivalent in 3-space. All of the polytope edges are circles in this projection, which preserves angles, but not straight lines or edge length. This makes the 3D projection a unique sculpture consisting of 12 large circles (4 circles each in 3 perpendicular planes which intersect to form a large outer and a small inner octahedron) and 4 small ones,which intersect at the vertices of a regular cuboctahedron. The entire structure has 24 vertices and 96 edges (6 arcs in each of the 16 circles).