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.

Cubic-based swirlprisms
Coordinates for the vertices of an n-cubic swirlprism of circumradius 1, centered at the origin, are given by all permutations and sign changes of: defining an icositetrachoron, along with reflections through the x=y and z=w hyperplanes and with all sign changes of (k represents any number from 1 up to floor(n/2)): along with reflections through the x=y and z=w hyperplanes and with all even sign changes of (if n is odd, then k represents any odd number from 1 up to n; if n is even, then k represents any number from 0 up to n/2): along with reflections through the x=y and z=w hyperplanes and with all odd sign changes of:
 * (0, 0, 0, 1),
 * (1/2, 1/2, 1/2, 1/2),
 * (0, 0, cos(kπ/2n), sin(kπ/2n)),
 * (cos(kπ/2n), sin(kπ/2n), 0, 0),
 * (sin(kπ/4n)/$\sqrt{2}$, cos(kπ/4n)/$\sqrt{2}$, sin(kπ/4n)/$\sqrt{2}$, cos(kπ/4n)/$\sqrt{2}$) (if n is odd),
 * (sin(kπ/2n)/$\sqrt{2}$, cos(kπ/2n)/$\sqrt{2}$, sin(kπ/2n)/$\sqrt{2}$, cos(kπ/2n)/$\sqrt{2}$) (if n is even),
 * (sin(kπ/4n)/$\sqrt{2}$, cos(kπ/4n)/$\sqrt{2}$, cos(kπ/4n)/$\sqrt{2}$, sin(kπ/4n)/$\sqrt{2}$) (if n is odd),
 * (sin(kπ/2n)/$\sqrt{2}$, cos(kπ/2n)/$\sqrt{2}$, cos(kπ/2n)/$\sqrt{2}$, sin(kπ/2n)/$\sqrt{2}$) (if n is even).

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)

4. 4-tetrahedral swirlprism (32 vertices, vertical-bisected joined triangular prism vertex figure) - 4-tetrahedral swirltegum or tetswirl 32 (96 edge-alternate laterostellated hexagonal prisms)

5. 5-tetrahedral swirlprism (40 vertices, alternate-metatriakis hexagonal bipyramid vertex figure) - 5-tetrahedral swirltegum or tetswirl 40 (40 alternate-metatruncated hexagonal prisms)

6. 6-tetrahedral swirlprism (48 vertices, edge-vertical bisected trigonal trapezohedron vertex figure) - 6-tetrahedral swirltegum or tetswirl 48 (48 rhombistellated hexagonal antiprisms)

7. 7-tetrahedral swirlprism (56 vertices) - 7-tetrahedral swirltegum or tetswirl 56 (56 cells)

8. 8-tetrahedral swirlprism (64 vertices) - 8-tetrahedral swirltegum or tetswirl 64 (64 cells)

9. 9-tetrahedral swirlprism (72 vertices) - 9-tetrahedral swirltegum or tetswirl 72 (72 cells)

10. 10-tetrahedral swirlprism (80 vertices) - 10-tetrahedral swirltegum or tetswirl 80 (80 cells)

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

2. Tetradisphenoidal diacosioctacontoctachoron (48 vertices, triakis octahedron 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. 4-cubic swirlprism (96 vertices, bisected rhombic dodecahedron vertex figure) - 4-cubic swirltegum or cubeswirl 96 (96 edge-alternate laterostellated octagonal prisms)

5. 5-cubic swirlprism (120 vertices, alternate-metatriakis octagonal bipyramid vertex figure) - 5-cubic swirltegum or cubeswirl 120 (120 alternate-metatruncated octagonal prisms)

6. 6-cubic swirlprism (144 vertices, edge-vertical bisected tetragonal trapezohedron vertex figure) - 6-cubic swirltegum or cubeswirl 144 (144 rhombistellated octagonal antiprisms)

7. 7-cubic swirlprism (168 vertices) - 7-cubic swirltegum or cubeswirl 168 (168 cells)

8. 8-cubic swirlprism (192 vertices) - 8-cubic swirltegum or cubeswirl 192 (192 cells)

9. 9-cubic swirlprism (216 vertices) - 9-cubic swirltegum or cubeswirl 216 (216 cells)

10. 10-cubic swirlprism (240 vertices) - 10-cubic swirltegum or cubeswirl 240 (240 cells)

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

2. 2-octahedral swirlprism (192 vertices) - 2-octahedral swirltegum or octswirl 192 (192 cells)

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 ikeswirl 600 (360 vertices, pentagonal trapezohedron vertex figure) - pentagonal-antiprismatic triacosihexecontachoron or doeswirl 360 (360 pentagonal antiprisms)

4. 4-dodecahedral swirlprism (480 vertices, vertical-bisected joined pentagonal prism vertex figure) - 4-dodecahedral swirltegum or doeswirl 480 (480 edge-alternate laterostellated decagonal prisms)

5. 5-dodecahedral swirlprism (600 vertices, alternate-metatriakis decagonal bipyramid vertex figure) - 5-dodecahedral swirltegum or doeswirl 600 (600 alternate-metatruncated decagonal prisms)

6. 6-dodecahedral swirlprism (720 vertices, edge-vertical bisected pentagonal trapezohedron vertex figure) - 6-dodecahedral swirltegum or doeswirl 720 (720 rhombistellated decagonal antiprisms)

7. 7-dodecahedral swirlprism (840 vertices) - 7-dodecahedral swirltegum or doeswirl 840 (840 cells)

8. 8-dodecahedral swirlprism (960 vertices) - 8-dodecahedral swirltegum or doeswirl 960 (960 cells)

9. 9-dodecahedral swirlprism (1080 vertices) - 9-dodecahedral swirltegum or doeswirl 1080 (1080 cells)

10. 10-dodecahedral swirlprism (1200 vertices) - 10-dodecahedral swirltegum or doeswirl 1200 (1200 cells)

Icosahedron-based
1. Pentagonal-antiprismatic triacosihexecontachoron (600 vertices, trigonal trapezohedron vertex figure) - triangular-antiprismatic hexacosichoron (600 triangular antiprisms)

2. 2-icosahedral swirlprism (1200 vertices) - 2-icosahedral swirltegum or ikeswirl 1200 (1200 cells)

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. 24-5 step prism (24 vertices) - 24-5 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. Hexadecachoron (8 vertices, octahedron vertex figure) - tesseract (16 cubes)

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

3. Digonal double triswirlprism (24 vertices, paratetraaugmented hexagonal bipyramid vertex figure) - digonal double triswirltegum (24 paratetratruncated hexagonal prisms)

4. Digonal double tetraswirlprism (32 vertices) - digonal double tetraswirltegum (32 cells)

Triangular antiprism-based
1. Triangular duoantiprism (18 vertices, gyrobifastigium vertex figure) - triangular duoantitegum (18 elongated tetragonal disphenoids)

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

3. Triangular double triswirlprism (54 vertices) - triangular double triswirltegum (54 cells)

4. Triangular double tetraswirlprism (72 vertices) - triangular double tetraswirltegum (72 cells)

Square antiprism-based
1. Square duoantiprism (32 vertices, gyrobifastigium vertex figure) - square duoantitegum (32 elongated tetragonal disphenoids)

2. Square double antiprismoid (64 vertices, sphenocorona vertex figure) - square double trapezohedroid (64 order-5 truncated bi-apiculated tetrahedra)

3. Square double triswirlprism (96 vertices) - square double triswirltegum (96 cells)

4. Square double tetraswirlprism (128 vertices) - square double tetraswirltegum (128 cells)

Pentagonal antiprism-based
1. Pentagonal duoantiprism (50 vertices, gyrobifastigium vertex figure) - pentagonal duoantitegum (50 elongated tetragonal disphenoids)

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

3. Pentagonal double triswirlprism (150 vertices) - pentagonal double triswirltegum (150 cells)

4. Pentagonal double tetraswirlprism (200 vertices) - pentagonal double tetraswirltegum (200 cells)

Hexagonal antiprism-based
1. Hexagonal duoantiprism (72 vertices, gyrobifastigium vertex figure) - hexagonal duoantitegum (72 elongated tetragonal disphenoids)

2. Hexagonal double antiprismoid (144 vertices, sphenocorona vertex figure) - hexagonal double trapezohedroid (144 order-5 truncated bi-apiculated tetrahedra)

3. Hexagonal double triswirlprism (216 vertices) - hexagonal double triswirltegum (216 cells)

4. Hexagonal double tetraswirlprism (288 vertices) - hexagonal double tetraswirltegum (288 cells)