The Eccentricity of Twins

Using the Generalized Dual Method, an arrangement of planes goes to a zonohedral tiling. If certain subsets of the planes are forced to intersect at two given points, twin zonohedra arise, within the tiling ...

Category:

Film & Animation

Tags:

• geometry
• tessellation
• animation
• zonohedron
• art

[rufus16180339887]

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 ...

[Yakshinian]

It looks like the palace floor of a four dimensional king.

[rufus16180339887]

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.