This an be done -- although while all zonohedra are polyhedra, not all polyhedra are zonohedra. A zonohedron is a convex polyhedron bounded entirely by centrally-symmetrical polygons. Such polygons always have an even number of sides, just as zonohedra have an even number of faces.
Zonohedral tilings with all regular polygons certainly exist: the space-filling of cubes, or of of Truncated Octahedra, or of mixtures including the Truncated Cuboctahedron, and right regular 2n-gonal prisms.
Thanks! The way these tilings arise is convoluted: I take a set of symmetry vectors (here, the edge-centers of the Platonic Icosahedron), and then create an "arrangement of planes" perpendicular to the symmetry vectors. Wherever the planes intersect in a single point, a zonohedron arises in the tiling. So I paint tilings, as it were, but my brush is arrangements of planes. I cannot usually predict just what is going to happen! And that is part of the fun ...
It looks like the palace floor of a four dimensional king.
Yakshinian 2 years ago
This an be done -- although while all zonohedra are polyhedra, not all polyhedra are zonohedra. A zonohedron is a convex polyhedron bounded entirely by centrally-symmetrical polygons. Such polygons always have an even number of sides, just as zonohedra have an even number of faces.
Zonohedral tilings with all regular polygons certainly exist: the space-filling of cubes, or of of Truncated Octahedra, or of mixtures including the Truncated Cuboctahedron, and right regular 2n-gonal prisms.
rufus16180339887 3 years ago
Thanks! The way these tilings arise is convoluted: I take a set of symmetry vectors (here, the edge-centers of the Platonic Icosahedron), and then create an "arrangement of planes" perpendicular to the symmetry vectors. Wherever the planes intersect in a single point, a zonohedron arises in the tiling. So I paint tilings, as it were, but my brush is arrangements of planes. I cannot usually predict just what is going to happen! And that is part of the fun ...
rufus16180339887 3 years ago