Swirlchoron

A swirlchoron is a polychoron that expresses the Hopf fibration of a given polyhedron. In other words, the vertices or cells of a polychoron map to a ring of swirling circles that represents a face of a polytwister, and are therefore their polychoric approximations. For every regular spherical polyhedron (including degenerate dihedra/hosohedra), a swirlchoron can be constructed.

There are two types of swirlchora. Swirlprisms are isogonal and can be thought as duals to the swirltegums, which are isochoric. Some swirlchora, such as the bi-icositetradiminished hexacosichoron and its dual, the tri-icositetradiminished hexacosichoron, are both isogonal and isochoric.

Tetrahedron-based
1. Hexadecachoron (8 vertices, octahedron vertex figure) - tesseract (16 cubes)

2. 2-tetrahedral swirlprism (16 vertices, triakis triangular bipyramid vertex figure) - 2-tetrahedral swirltegum or tetswirl 16 (16 truncated triangular prisms)

3. Icositetrachoron (24 vertices, cube vertex figure) - dual icositetrahedron (24 octahedra)

Cube-based
1. Icositetrachoron (24 vertices, cube vertex figure) - dual icositetrahedron (24 octahedra)

2. Tetradisphenoidal diacosioctacontoctachoron (48 vertices, triakis tetrahedron vertex figure) - tetracontoctachoron (48 truncated cubes)

3. Triangular-antiprismatic enneacontahexachoron or octswirl 96 (72 vertices, tetragonal trapezohedron vertex figure) - square-antiprismatic heptacontadichoron or cubeswirl 72 (72 square antiprisms)

4. Bi-icositetradiminished hexacosichoron (96 vertices, tetragonal antiwedge vertex figure) - tri-icositetradiminished hexacosichoron (96 tetragonal antiwedges)

Octahedron-based
1. Tri-icositetradiminished hexacosichoron (48 vertices, tridiminished icosahedron dual vertex figure) - bi-icositetradiminished hexacosichoron (48 tridiminished icosahedra)

2. Square-antiprismatic heptacontadichoron (96 vertices, trigonal trapezohedron vertex figure) - triangular-antiprismatic enneacontahexachoron (96 triangular antiprisms)

Dodecahedron-based
1. Hexacosichoron (120 vertices, icosahedron vertex figure) - hecatonicosachoron (120 dodecahedra)

2. 2-dodecahedral swirlprism (240 vertices, triakis pentagonal bipyramid vertex figure) - 2-dodecahedral swirltegum or doeswirl 240 (240 truncated pentagonal prisms)

5. Swirlprismatodiminished rectified hexacosichoron (600 vertices, bidiminished pentagonal prism vertex figure) - swirlprismatodiminished rectified hexacosichoron dual (600 bidiminished pentagonal prism dual cells)

Icosahedron-based
1. Hecatonicosachoron (600 vertices, tetrahedron vertex figure) - hexacosichoron (600 tetrahedra)

Triangular dihedron-based
1. 3-3 duopyramid (6 vertices, tetragonal disphenoid vertex figure) - 3-3 duoprism (6 triangular prisms) 2. 9-3 step prism (9 vertices, tetragonal antiwedge vertex figure) - 9-3 gyrochoron (9 tetragonal antiwedges) 3. 6-6 duopyramid (12 vertices,tetragonal disphenoid vertex figure) - 6-6 duoprism (12 triangular prisms) 4. 15-4 step prism (15 vertices) - 15-4 gyrochoron (15 cells) 5. 18-5 step prism (18 vertices) - 18-5 gyrochoron (18 cells)