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 tetragonal-antiwedge 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)

Small rhombitetratetrahedron-based
1. Swirlprismatodiminished rectified icositetrachoron (72 vertices, triangular-ridge expanded tetragonal antiwedge vertex figure) - 1-small rhombitetratetrahedral swirltegum (72 tetragonal-ridge stellated tetragonal antiwedges)

Great rhombitetratetrahedron-based
1. Bi-icositetradiminished truncated icositetrachoron (144 vertices, mirror-symmetric triangular prism vertex figure) - 1-great rhombitetratetrahedral swirltegum (144 mirror-symmetric notches)

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, edge-vertical bisected trigonal trapezohedron vertex figure) - 2-octahedral swirltegum or octswirl 192 (192 rhombistellated hexagonal antiprisms)

3. 3-octahedral swirlprism (288 vertices) - 3-octahedral swirltegum or octswirl 288 (288 cells)

4. 4-octahedral swirlprism (384 vertices) - 4-octahedral swirltegum or octswirl 384 (384 cells)

5. 5-octahedral swirlprism (480 vertices) - 5-octahedral swirltegum or octswirl 480 (480 cells)

6. 6-octahedral swirlprism (576 vertices) - 6-octahedral swirltegum or octswirl 576 (576 cells)

7. 7-octahedral swirlprism (672 vertices) - 7-octahedral swirltegum or octswirl 672 (672 cells)

8. 8-octahedral swirlprism (768 vertices) - 8-octahedral swirltegum or octswirl 768 (768 cells)

9. 9-octahedral swirlprism (864 vertices) - 9-octahedral swirltegum or octswirl 864 (864 cells)

10. 10-octahedral swirlprism (960 vertices) - 10-octahedral swirltegum or octswirl 960 (960 cells)

Cuboctahedron-based
1. Small prismatotetracontoctachoron (144 vertices, square antiprism vertex figure) - tetragonal-trapezohedral hecatonictetracontatetrachoron (144 tetragonal trapezohedra)

Small rhombicuboctahedron-based
1. 1-small rhombicuboctahedral swirlprism (576 vertices, irregular square antiprism vertex figure) - 1-small rhombicuboctahedral swirltegum (576 irregular tetragonal trapezohedra)

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, edge-vertical bisected trigonal trapezohedron vertex figure) - 2-icosahedral swirltegum or ikeswirl 1200 (1200 rhombistellated hexagonal antiprisms)

3. 3-icosahedral swirlprism (1800 vertices) - 3-icosahedral swirltegum or ikeswirl 1800 (1800 cells)

4. 4-icosahedral swirlprism (2400 vertices) - 4-icosahedral swirltegum or ikeswirl 2400 (2400 cells)

5. 5-icosahedral swirlprism (3000 vertices) - 5-icosahedral swirltegum or ikeswirl 3000 (3000 cells)

6. 6-icosahedral swirlprism (3600 vertices) - 6-icosahedral swirltegum or ikeswirl 3600 (3600 cells)

7. 7-icosahedral swirlprism (4200 vertices) - 7-icosahedral swirltegum or ikeswirl 4200 (4200 cells)

8. 8-icosahedral swirlprism (4800 vertices) - 8-icosahedral swirltegum or ikeswirl 4800 (4800 cells)

9. 9-icosahedral swirlprism (5400 vertices) - 9-icosahedral swirltegum or ikeswirl 5400 (5400 cells)

10. 10-icosahedral swirlprism (6000 vertices) - 10-icosahedral swirltegum or ikeswirl 6000 (6000 cells)

Icosidodecahedron-based
1a. Triangular 1-icosidodecahedral swirlprism (360 vertices) - triangular 1-icosidodecahedral swirltegum (360 cells)

1b. Pentagonal 1-icosidodecahedral swirltegum (600 vertices, mirror-symmetric triangular prism vertex figure) - pentagonal 1-icosidodecahedral swirltegum (600 mirror-symmetric notches)

Small rhombicosidodecahedron-based
1. 1-small rhombicosidodecahedral swirlprism (3600 vertices, irregular square antiprism vertex figure) - 1-small rhombicosidodecahedral swirltegum (3600 irregular tetragonal trapezohedra)

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)