Polytwister

Polytwisters are a class of four-dimensional curved shapes based on polyhedra and Hopf fibration. They were invented by Jonathan Bowers circa 2007. Many properties of polytwisters are unclear and/or conjectural. As of November 2022, a paper is underway that intends to formalize Bowers' expository work.

The Hopf fibration is a function h that maps a certain set of great circles on the 3-sphere onto points of the ordinary sphere. Polytwisters arise from the preimage h-1, by converting points on the sphere into great circles on the 3-sphere.

Define a cycloplane as the Cartesian product of a closed disk and a plane. In 4D, a unique cycloplane can be constructed from any circle centered at the origin by taking the polar set of that circle. Given a convex polyhedron whose faces are tangent to a unit sphere centered at the origin, each point of tangency can be mapped to a unit circle about the origin via the preimage h-1. If a cycloplane is constructed from each unit circle and the intersection of all resulting cycloplanes is taken, the result is a convex polytwister. Cycloplanes in polytwisters are analogous to half-spaces in polyhedra. Nonconvex polytwisters are formed by other Boolean operations on cycloplanes.

It was once believed that convex polytwisters can be formed by taking a polyhedron's vertices, converting them to a set of circles via the preimage h-1, then taking the convex hull of the resulting circles. However, this is now known to be incorrect, as this process produces different shapes from cycloplane intersections. These shapes have been christened soft polytwisters, as they resemble convex polytwisters but with rounded edges.

Polytwisters are believed to have an abstract polyhedral structure. In place of vertices they have circular "rings," in place of edges they have "strips," and in place of faces they have "twisters." However, these elements are only known due to visual inspection and should be considered conjectural.

Using (currently intuitive) definitions of "regular" and "uniform," Bowers discovered 36 regular polytwisters, 186 nonregular uniform polytwisters, and an infinite family of regular "dyadic twisters" based on hosohedra.