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 great circles within a glome 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. Their vertices can be compounded to form new swirlchora, an example being the 1-bitetrahedral swirlprism, being the compound of two hexadecachora in tetrahedral swirlprism symmetry, or the tri-icositetradiminished hexacosichoron, being the compound of two icositetrachora in cubic swirlprism symmetry.

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, and are therefore noble.

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. Square-antiprismatic heptacontadichoron or cubeswirl 72 (96 vertices, trigonal trapezohedron vertex figure) - triangular-antiprismatic enneacontahexachoron (96 triangular antiprisms)

5. Hexacosichoron (120 vertices, icosahedron vertex figure) - hecatonicosachoron (120 dodecahedra)

12. Tetracontoctachoron (288 vertices, tetragonal disphenoid vertex figure) - tetradisphenoidal diacosioctacontoctachoron (288 tetragonal disphenoids)

Octahedron-based
1. Tri-icositetradiminished hexacosichoron (48 vertices, tristellated dodecahedron 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, parabidiminished pentagonal prism vertex figure) - 5-dodecahedral swirltegum (600 parabistellated pentagonal bipyramids)

6. Rectified hexacosichoron (720 vertices, pentagonal prism vertex figure) - joined hexacosichoron (720 pentagonal bipyramids)

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

Triangular dihedron-based
1. Triangular duotegum (6 vertices, tetragonal disphenoid vertex figure) - Triangular duoprism (6 triangular prisms)

2. 9-2 step prism (9 vertices, ridge-triakis bi-apiculated tetrahedron vertex figure) - 9-2 gyrochoron (9 ridge-truncated edge-truncated tetrahedra)

3. Hexagonal duotegum (12 vertices, tetragonal disphenoid vertex figure) - Hexagonal duoprism (12 triangular prisms)

4. 15-4 step prism (15 vertices, paratetraaugmented digonal scalenohedron vertex figure) - 15-4 gyrochoron (15 paratetratruncated rhombic prisms)

5. 18-5 step prism (18 vertices, metabitriakis snub disphenoid vertex figure) - 18-5 gyrochoron (18 metabitruncated elongated gyrobifastigia)

6. 21-8 step prism (21 vertices) - 21-8 gyrochoron (21 cells)

8. Triangular triswirlprism (27 vertices) - triangular triswirltegum (27 cells)