Polytwister

Polytwisters are a class of four-dimensional curved shapes based on polyhedra and Hopf fibration and are related to the swirlchora, which are finite, non-curved polychora. They were invented by Jonathan Bowers. Many properties of polytwisters are unclear and/or conjectural.

Polytwisters are believed to have an abstract polyhedral structure. In place of vertices they have circular "rings"; in place of edges they have "strips" that look like a strip of paper twisted 360 degrees with ends attached, and in place of faces they have "twisters" that consist of a rod with a polygonal cross section, also twisted 360 degrees and bent circularly so the faces join. The twisters swirl in one direction, and as a result of this, all polytwisters are chiral. However, these properties are only known due to visual inspection and while likely to be true, 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.

Construction
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, based on complex numbers. 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 (visually, "erecting" the cycloplane from the circle with a plane orthogonal to the circle's plane). 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, although their exact definition is not entirely clear.

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 are now designated as soft polytwisters, as they resemble convex polytwisters but with rounded edges, while the ones constructed from Boolean operations on cycloplanes are likewise designated as hard polytwisters. When the hard/soft designation is not given, hard polytwisters are assumed. An interesting fact is that, at least for regular convex base polyhedra, soft polytwisters are the polar sets of convex hard polytwisters and vice versa.

A polytwister mimics some of the base polyhedron's properties. For example, if the base polyhedron is regular, then the resulting polytwister is also regular. Likewise, if the base polyhedron is isogonal or isohedral, then the resulting polytwister is also isogonal or isochoric. However, uniformity is not mimicked, such as a polytwister based on a truncated tetrahedron, because the hexagon is reflected topologically as a ditrigonal twister. Not all base polyhedra have a polytwister equivalent; an example of this is the tetrahemihexahedron, due to its non-orientability.

Generalizations
Polytwisters are not limited to four dimensions. In all even dimensions greater than 2, polytwister-like analogues exist. In 6 dimensions, one has the so-called "polyduotwisters", which have circular rings in place of vertices, but twist in two orthogonal directions. In 8 dimensions, aside from the "polytriotwisters", which are simply extensions of the polyduotwisters and polytwisters, one has the "polyswirlers", which are based on polytera and have glomeric rings instead of circular rings, and are closely related to quaternion Hopf fibration. In 16 dimensions, one has the "polymaelstroms", which are based on polyyotta and have eight-dimensional hyperspherical rings, and are closely related to octonion Hopf fibration.

While the series of "poly-n-twisters" in (n+2) dimensions and "poly-n-swirlers" in (4n+4) dimensions exist and are infinite, there are no "poly-n-maelstroms" for n > 1 because octonions are not associative, unlike complex numbers and quaternions.

History
Polytwisters were discovered and enumerated by Jonathan Bowers, with help by Mason Green. According to the Wayback Machine, the first versions of Bowers' polytwister page went up in 2007. Much of polytwister discovery was done with an unpublished POV-Ray script.

In 2022, Nathan Ho began an effort to review Bowers' expository work and port the POV-Ray code to a modern toolchain using Blender scripting. As of November 2022, a paper is underway that intends to formalize the definition of polytwisters.