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 isotopic. 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)

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

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

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

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

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 (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)

3. Triangular-antiprismatic hexacosichoron or doeswirl 60 (360 vertices, pentagonal trapezohedron vertex figure) - pentagonal-antiprismatic triacosihexecontachoron or doeswirl 30 (360 pentagonal antiprisms)

5. Pentagonal-antiprismatic triacosihexecontachoron or (600 vertices, trigonal trapezohedron vertex figure) - triangular-antiprismatic hexacosichoron (600 triangular antiprisms)

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)

Icosidodecahedron-based
1. Swirlprismatodiminished rectified hexacosichoron (600 vertices, parabidiminished pentagonal prism vertex figure) - 1-icosidodecahedral swirltegum (600 parabistellated pentagonal bipyramids)

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)

7. 24-7 step prism (24 vertices) - 24-7 gyrochoron (24 cells)

8. 27-8 step prism (27 vertices) - 27-8 gyrochoron (27 cells)

9. 30-11 step prism (30 vertices) - 30-11 gyrochoron (30 cells)

10. 33-10 step prism (33 vertices) - 33-10 gyrochoron (33 cells)

Square dihedron-based
1. Hexadecachoron (8 vertices, octahedron vertex figure) - tesseract (8 cubes)

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

3. 16-3 step prism (16 vertices) - 16-3 gyrochoron (16 cells)

4. Decagonal duotegum (20 vertices, decagonal bipyramid vertex figure) - decagonal duoprism (20 decagonal prisms)

5. 24-5 step prism (24 vertices) - 24-5 gyrochoron (24 cells)

6. Tetradecagonal duotegum (28 vertices, tetradecagonal bipyramid vertex figure) - tetradecagonal duoprism (28 tetradecagonal prisms)

7. 32-7 step prism (32 vertices) - 32-7 gyrochoron (32 cells)

8. Octadecagonal duotegum (36 vertices, octadecagonal bipyramid vertex figure) - octadecagonal duoprism (36 octadecagonal prisms)

9. 40-9 step prism (40 vertices) - 40-9 gyrochoron (40 cells)

10. Icosidigonal duotegum (44 vertices, icosidigonal bipyramid vertex figure) - icosidigonal duoprism (44 icosidigonal prisms)

Pentagonal dihedron-based
1. Pentagonal duotegum (10 vertices, pentagonal bipyramid vertex figure) - pentagonal duoprism (10 pentagonal prisms)

2. 15-2 step prism (15 vertices, ridge-quadritriakis bi-apiculated tetrahedron vertex figure) - 15-2 gyrochoron (12 ridge-quadritruncated edge-truncated tetrahedra)

3. 20-3 step prism (20 vertices) - 20-3 gyrochoron (20 cells)

4. 25-4 step prism (25 vertices) - 25-4 gyrochoron (20 cells)

5. 30-7 step prism (30 vertices) - 30-7 gyrochoron (30 cells)

6. 35-6 step prism (35 vertices) - 35-6 gyrochoron (35 cells)

7. 40-7 step prism (40 vertices) - 40-7 gyrochoron (40 cells)

8. 45-8 step prism (45 vertices) - 45-8 gyrochoron (45 cells)

9. 50-9 step prism (50 vertices) - 50-9 gyrochoron (50 cells)

10. 55-12 step prism (55 vertices) - 55-12 gyrochoron (55 cells)

Hexagonal dihedron-based
1. Hexagonal duotegum (12 vertices, hexagonal bipyramid vertex figure) - hexagonal duoprism (12 hexagonal prisms)

2. 18-5 step prism (18 vertices, metabitriakis snub disphenoid vertex figure) - 18-5 gyrochoron (18 metabitruncated elongated gyrobifastigia)

3. 24-5 step prism (24 vertices) - 24-5 gyrochoron (24 cells)

4. 30-11 step prism (30 vertices) - 30-11 gyrochoron (30 cells)

5. 36-5 step prism (36 vertices) - 36-5 gyrochoron (36 cells)

6. 42-13 step prism (42 vertices) - 42-13 gyrochoron (42 cells)

7. 48-7 step prism (48 vertices) - 48-7 gyrochoron (48 cells)

8. 54-17 step prism (54 vertices) - 54-17 gyrochoron (54 cells)

9. 60-11 step prism (60 vertices) - 60-11 gyrochoron (60 cells)

10. 66-23 step prism (66 vertices) - 66-23 gyrochoron (66 cells)

Digonal antiprism-based
1. Digonal double antiprismoid (16 vertices, hexakis digonal-hexagonal gyrowedge vertex figure) - digonal double trapezohedroid (16 order-6 truncated digonal-hexagonal gyronotches)

Triangular antiprism-based
1. Triangular double antiprismoid (36 vertices, sphenocorona vertex figure) - triangular double trapezohedroid (36 order-5 truncated bi-apiculated tetrahedra)

Square antiprism-based
1. Square double antiprismoid (64 vertices, octakis digonal-octagonal gyrowedge vertex figure) - square double trapezohedroid (64 order-8 truncated digonal-octagonal gyronotches)

Pentagonal antiprism-based
1. Grand antiprism (100 vertices, sphenocorona vertex figure) - pentagonal double trapezohedroid (100 order-5 truncated bi-apiculated tetrahedra)

Hexagonal antiprism-based
1. Hexagonal double antiprismoid (144 vertices, decakis digonal-decagonal gyrowedge vertex figure) - square double trapezohedroid (144 order-10 truncated digonal-decagonal gyronotches)