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.

Octahedron-based isogonal swirlprisms
Coordinates for the vertices of an 4n-fold octaswirlchoron 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 ({x,y,z,w} = {z,w,x,y} = {y,x,w,z} = {w,z,y,x}) 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)),
 * (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. Tetrafold tetraswirlchoron (16 vertices, triakis triangular bipyramid vertex figure) - tetraswirlic hexadecachoron or tetswirl 16 (16 truncated triangular prisms)

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

4. Octafold tetraswirlchoron (32 vertices, vertical-bisected joined triangular prism vertex figure) - tetraswirlic triacontadichoron or tetswirl 32 (96 edge-alternate laterostellated hexagonal prisms)

5. Decafold tetraswirlchoron (40 vertices, alternate-metatriakis hexagonal bipyramid vertex figure) - tetraswirlic tetracontachoron or tetswirl 40 (40 alternate-metatruncated hexagonal prisms)

6. Dodecafold tetraswirlchoron (48 vertices, edge-vertical bisected trigonal trapezohedron vertex figure) - tetraswirlic tetracontoctachoron or tetswirl 48 (48 rhombistellated hexagonal antiprisms)

7. Tetradecafold tetraswirlchoron (56 vertices) - tetraswirlic pentacontahexachoron or tetswirl 56 (56 cells)

8. Hexadecafold tetraswirlchoron (64 vertices) - tetraswirlic hexacontatetrachoron or tetswirl 64 (64 cells)

9. Octadecafold tetraswirlchoron (72 vertices) - tetraswirlic heptacontadichoron or tetswirl 72 (72 cells)

10. Icosafold tetraswirlchoron (80 vertices) - tetraswirlic octacontachoron 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. Square-antiprismatic heptacontadichoron (96 vertices, trigonal trapezohedron vertex figure) - triangular-antiprismatic enneacontahexachoron (96 triangular antiprisms)

2. Icositetrafold cubiswirlchoron (192 vertices, edge-vertical bisected trigonal trapezohedron vertex figure) - octaswirlic hecatonicenneacontadichoron or octswirl 192 (192 rhombistellated hexagonal antiprisms)

3. Triacontahexafold cubiswirlchoron (288 vertices) - octaswirlic diacosioctacontoctachoron or octswirl 288 (288 cells)

4. Tetracontoctafold cubiswirlchoron (384 vertices) - octaswirlic triacosioctacontatetrachoron or octswirl 384 (384 cells)

5. Hexecontafold cubiswirlchoron (480 vertices) - octaswirlic tetracosioctacontachoron or octswirl 480 (480 cells)

6. Heptacontadifold cubiswirlchoron (576 vertices) - octaswirlic pentacosiheptacontahexachoron or octswirl 576 (576 cells)

7. Octacontatetrafold cubiswirlchoron (672 vertices) - octaswirlic hexacosiheptacontadichoron or octswirl 672 (672 cells)

8. Enneacontahexafold cubiswirlchoron (768 vertices) - octaswirlic heptacosihexacontoctachoron or octswirl 768 (768 cells)

9. Hecatonicoctafold cubiswirlchoron (864 vertices) - octaswirlic octacosihexacontatetrachoron or octswirl 864 (864 cells)

10. Hecatonicosafold cubiswirlchoron (960 vertices) - octaswirlic enneacosihexecontachoron or octswirl 960 (960 cells)

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

2. Bitetracontoctachoron (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. Hexadecafold octaswirlchoron (96 vertices, bisected rhombic dodecahedron vertex figure) - cubiswirlic enneacontahexachoron or cubeswirl 96 (96 edge-alternate laterostellated octagonal prisms)

5. Icosafold octaswirlchoron (120 vertices, alternate-metatriakis octagonal bipyramid vertex figure) - cubiswirlic hecatonicosachoron or cubeswirl 120 (120 alternate-metatruncated octagonal prisms)

6. Icositetrafold octaswirlchoron (144 vertices, edge-vertical bisected tetragonal trapezohedron vertex figure) - cubiswirlic hecatonictetracontatetrachoron or cubeswirl 144 (144 rhombistellated octagonal antiprisms)

7. Icosioctafold octaswirlchoron (168 vertices) - cubiswirlic hecatonichexacontaoctachoron or cubeswirl 168 (168 cells)

8. Triacontadifold octaswirlchoron (192 vertices) - cubiswirlic hecatonicenneacontadichoron or cubeswirl 192 (192 cells)

9. Triacontahexafold octaswirlchoron (216 vertices) - cubiswirlic diacosihexadecachoron or cubeswirl 216 (216 cells)

10. Tetracontafold octaswirlchoron (240 vertices) - cubiswirlic diacositetracontachoron or cubeswirl 240 (240 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. Pentagonal-antiprismatic triacosihexecontachoron (600 vertices, trigonal trapezohedron vertex figure) - triangular-antiprismatic hexacosichoron (600 triangular antiprisms)

2. Hexecontafold dodecaswirlchoron (1200 vertices, edge-vertical bisected trigonal trapezohedron vertex figure) - icosaswirlic chilliadiacosichoron or ikeswirl 1200 (1200 rhombistellated hexagonal antiprisms)

3. Enneacontafold dodecaswirlchoron (1800 vertices) - icosaswirlic chilliaoctacosichoron or ikeswirl 1800 (1800 cells)

4. Hecatonicosafold dodecaswirlchoron (2400 vertices) - icosaswirlic dischilliatetracosichoron or ikeswirl 2400 (2400 cells)

5. Hecatonicpentacontafold dodecaswirlchoron (3000 vertices) - icosaswirlic trischilliachoron or ikeswirl 3000 (3000 cells)

6. Hecatonicoctacontafold dodecaswirlchoron (3600 vertices) - icosaswirlic trischilliahexacosichoron or ikeswirl 3600 (3600 cells)

7. Diacosidecafold dodecaswirlchoron (4200 vertices) - icosaswirlic tetrachilliadiacosichoron or ikeswirl 4200 (4200 cells)

8. Diacositetracontafold dodecaswirlchoron (4800 vertices) - icosaswirlic tetrachilliaoctacosichoron or ikeswirl 4800 (4800 cells)

9. Diacosiheptacontafold dodecaswirlchoron (5400 vertices) - icosaswirlic pentachilliatetracosichoron or ikeswirl 5400 (5400 cells)

10. Triacosifold dodecaswirlchoron (6000 vertices) - icosaswirlic hexachilliachoron or ikeswirl 6000 (6000 cells)

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

2. Icosafold icosaswirlchoron (240 vertices, triakis pentagonal bipyramid vertex figure) - dodecaswirlic diacositetracontachoron 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. Tetracontafold icosaswirlchoron (480 vertices, vertical-bisected joined pentagonal prism vertex figure) - dodecaswirlic tetracosioctacontachoron or doeswirl 480 (480 edge-alternate laterostellated decagonal prisms)

5. Pentacontafold icosaswirlchoron (600 vertices, alternate-metatriakis decagonal bipyramid vertex figure) - dodecaswirlic hexacosichoron or doeswirl 600 (600 alternate-metatruncated decagonal prisms)

6. Hexecontafold icosaswirlchoron (720 vertices, edge-vertical bisected pentagonal trapezohedron vertex figure) - dodecaswirlic heptacosicosachoron or doeswirl 720 (720 rhombistellated decagonal antiprisms)

7. Heptacontafold icosaswirlchoron (840 vertices) - dodecaswirlic octacositetracontachoron or doeswirl 840 (840 cells)

8. Octacontafold icosaswirlchoron (960 vertices) - dodecaswirlic enneacosihexecontachoron or doeswirl 960 (960 cells)

9. Enneacontafold icosaswirlchoron (1080 vertices) - dodecaswirlic chilliaoctacontachoron or doeswirl 1080 (1080 cells)

10. Hectofold icosaswirlchoron (1200 vertices) - dodecaswirlic chilliadiacosichoron or doeswirl 1200 (1200 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)