Swirlchoron

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A pentagonal-antiprismatic triacosihexecontachoron has rings of pentagonal antiprisms arranged as a larger pentagonal antiprism.

Swirlchora are polychora whose symmetry group is built up in a specific manner from lower-dimensional symmetry groups. Most commonly, swirlchora refer to polychora with symmetry groups built up from a cyclic symmetry group and a polyhedral symmetry group. They're so called because their vertices and faces appear to "swirl" around in various rings, corresponding to great circles of a glome, whose overall positioning mimics that of a polyhedron. In other words, these kinds of swirlchora can be seen as discrete versions of polytwisters.

General swirlchora can be seen as being derived from special kinds of compounds. For instance, the bitetrahedral tetracontoctachoron can be made out of either a compound of eight icositetrachora or four bitetracontoctachora in a specific arrangement. Swirlchora based on cyclic symmetry groups can thus be seen as derived from specific compounds of dihedra.

Being very closely related to quaternions, swirlchora are a phenomenon specific to 4D. Analogues of swirlchora exist in any even number of dimensions, including the eight-dimensional swirlzetta with symmetry groups built up from a polychoric symmetry group and a polyteric symmetry group, which are closely related to octonions.

Swirlchora are always isogonal, and they're often isochoric as well. Examples include the bi-icositetradiminished hexacosichoron and its dual, the tri-icositetradiminished hexacosichoron, which are both isogonal and isochoric, and are therefore noble.

Coordinates[edit | edit source]

Tetrahedron-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a 2n-fold tetraswirlchoron of circumradius 1, centered at the origin, are given by:

  • ±(0, 0, sin(kπ/n), cos(kπ/n)),

along with 120° and 240° rotations in the xy axis of:

  • ±(6sin(kπ/n)/3, 6cos(kπ/n)/3, 3cos(kπ/n)/3, 3sin(kπ/n)/3),

where k is an integer from 0 to n-1.

Ambotetrahedron-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a 4n-fold ambotetraswirlchoron of circumradius 1, centered at the origin, are given by, along with their 120° and 240° rotations in the xy axis of:

  • ±(sin(kπ/2n)/3+3, cos(kπ/2n)/3+3, cos(kπ/2n)/3-3, sin(kπ/2n)/3-3),
  • ±(sin((k+n/2)π/2n)/3-3, cos((k+n/2)π/2n)/3-3, -cos((k+n/2)π/2n)/3+3, -sin((k+n/2)π/2n)/3+3),

where k is an integer from 0 to 2n-1.

Cube-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a 12n-fold cubiswirlchoron of circumradius 1, centered at the origin, are given by, along with their 90°, 180° and 270° rotations in the xy axis of:

  • ±(sin(kπ/6n)/3+3, cos(kπ/6n)/3+3, cos(kπ/6n)/3-3, sin(kπ/6n)/3-3),
  • ±(sin((k+n/2)π/6n)/3-3, cos((k+n/2)π/6n)/3-3, cos((k+n/2)π/6n)/3+3, sin((k+n/2)π/6n)/3+3),

where k is an integer from 0 to 6n-1.

Pyritohedral cube-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a 6n-fold pyritocubiswirlchoron of circumradius 1, centered at the origin, are given by:

  • ±(0, 0, sin(kπ/3n), cos(kπ/3n)),
  • ±(sin(kπ/3n), cos(kπ/3n), 0, 0),

along with 120° and 240° rotations in the xy axis of:

  • ±(6sin(kπ/3n)/3, 6cos(kπ/3n)/3, 3cos(kπ/3n)/3, 3sin(kπ/3n)/3),
  • ±(3cos(kπ/3n)/3, 3sin(kπ/3n)/3, -6sin(kπ/3n)/3, -6cos(kπ/3n)/3),

where k is an integer from 0 to 3n-1.

Octahedron-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a 4n-fold octaswirlchoron of circumradius 1, centered at the origin, are given by:

  • ±(0, 0, sin(kπ/2n), cos(kπ/2n)),
  • ±(sin(kπ/2n), cos(kπ/2n), 0, 0),

along with 90°, 180° and 270° rotations in the xy axis of:

  • ±(sin((k+n/2)π/2n)/2, cos((k+n/2)π/2n)/2, cos((k+n/2)π/2n)/2, sin((k+n/2)π/2n)/2),

where k is an integer from 0 to 2n-1.

Cuboctahedron-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of an 8n-fold cuboctaswirlchoron of circumradius 1, centered at the origin, are given by, along with their 90°, 180° and 270° rotations in the xy axis of:

  • ±(sin(kπ/4n)/4+22, cos(kπ/4n)/4+22, cos(kπ/4n)/4-22, sin(kπ/4n)/4-22),
  • ±(sin(kπ/4n)/4-22, cos(kπ/4n)/4-22, cos(kπ/4n)/4+22, sin(kπ/4n)/4+22),
  • ±(sin((2k+n)π/8n)/2, cos((2k+n)π/8n)/2, cos((2k-n)π/8n)/2, sin((2k-n)π/8n)/2),

where k is an integer from 0 to 4n-1.

Chiral cuboctahedron-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a chiro-(8n-4)-fold cuboctaswirlchoron of circumradius 1, centered at the origin, are given by, along with their 180° rotations in the xy axis of:

  • ±(sin(kπ/(4n-2))/4+22, cos(kπ/(4n-2))/4+22, cos(kπ/(4n-2))/4-22, sin(kπ/(4n-2))/4-22),
  • ±(sin(kπ/(4n-2))/4-22, cos(kπ/(4n-2))/4-22, cos(kπ/(4n-2))/4+22, sin(kπ/(4n-2))/4+22),
  • ±(cos((2k-1)π/(8n-4))/4+22, -sin((2k-1)π/(8n-4))/4+22, cos((2k-1)π/(8n-4))/4-22, sin((2k-1)π/(8n-4))/4-22),
  • ±(cos((2k-1)π/(8n-4))/4-22, -sin((2k-1)π/(8n-4))/4-22, cos((2k-1)π/(8n-4))/4+22, sin((2k-1)π/(8n-4))/4+22),
  • ±(sin((4k+2n+1)π/(16n-8))/2, cos((4k+2n+1)π/(16n-8))/2, cos((4k+2n+3)π/(16n-8))/2, sin((4k+2n+3)π/(16n-8))/2),
  • ±(cos((4k+2n-1)π/(16n-8))/2, -sin((4k+2n-1)π/(16n-8))/2, cos((4k+2n+1)π/(16n-8))/2, sin((4k+2n+1)π/(16n-8))/2),

where k is an integer from 0 to 4n-3.

Dodecahedron-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a 30n-fold dodecaswirlchoron of circumradius 1, centered at the origin, are given by, along with 72°, 144°, 216° and 288° rotations in the xy axis of:

  • ±(c1*sin(kπ/15n), c1*cos(kπ/15n), c2*cos(kπ/15n), c2*sin(kπ/15n)),
  • ±(c2*sin(kπ/15n), c2*cos(kπ/15n), -c1*cos(kπ/15n), -c1*sin(kπ/15n)),
  • ±(c3*sin((k+n/2)π/15n), c3*cos((k+n/2)π/15n), c4*cos((k+n/2)π/15n), c4*sin((k+n/2)π/15n)),
  • ±(c4*sin((k+n/2)π/15n), c4*cos((k+n/2)π/15n), -c3*cos((k+n/2)π/15n), -c3*sin((k+n/2)π/15n)),

where c1 = 450-3075+305/30, c2 = 450+3075+305/30, c3 = 450-3075-305/30, c4 = 450+3075-305/30 and k is an integer from 0 to 15n-1.

Icosahedron-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a 10n-fold icosaswirlchoron of circumradius 1, centered at the origin, are given by:

  • ±(0, 0, sin(kπ/5n), cos(kπ/5n)),
  • ±(cos(kπ/5n), sin(kπ/5n), 0, 0),

along with 72°, 144°, 216° and 288° rotations in the xy axis of:

  • ±(2sin(kπ/5n)/10+25, 2cos(kπ/5n)/10+25, 2cos(kπ/5n)/10-25, 2sin(kπ/5n)/10-25),
  • ±(2sin(kπ/5n)/10-25, 2cos(kπ/5n)/10-25, -2cos(kπ/5n)/10+25, -2sin(kπ/5n)/10+25),

where k is an integer from 0 to 5n-1.

Dihedron-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of a n-fold dihedroswirlchoron of circumradius 1, centered at the origin, are given by, along with their m-fold rotations in the xy axis of:

  • (sin(k2π/mn)/2, cos(k2π/mn)/2, cos(k2π/mn)/2, sin(k2π/mn)/2),

where k is an integer from 0 to 'mn-1.

Prism-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of an m*n-fold m-gonal prismatoswirlchoron of circumradius 1, centered at the origin, are given by, along with their m-fold rotations in the xy axis of:

  • (a*sin(k2π/mn), a*cos(k2π/mn), b*cos(k2π/mn), b*sin(k2π/mn)),
  • (b*sin(k2π/mn), b*cos(k2π/mn), a*cos(k2π/mn), a*sin(k2π/mn)),

where a is less than 1, b is equivalent to 1-a2 and k is an integer from 0 to mn-1.

Antiprism-based isogonal swirlchora[edit | edit source]

Coordinates for the vertices of an m*n-fold m-gonal antiprismatoswirlchoron of circumradius 1, centered at the origin, are given by, along with their m-fold rotations in the xy axis of:

  • (a*sin(k2π/mn), a*cos(k2π/mn), b*cos(k2π/mn), b*sin(k2π/mn)),
  • (b*sin((k+n/2)2π/mn), b*cos((k+n/2)2π/mn), a*cos(k2π/mn), a*sin(k2π/mn)),

where a is less than 1, b is equivalent to 1-a2 and k is an integer from 0 to mn-1.

Isogonal and isochoric swirlchoron examples[edit | edit source]

Tetrahedron-based[edit | edit source]

1. Hexadecachoron (8 vertices, octahedron vertex figure) - tesseract (16 cubes)

2. Tetrafold tetraswirlchoron (16 vertices, triakis triangular tegum vertex figure) - tetraswirlic hexadecachoron or tetswirl 16 (16 truncated triangular prisms)

3. Icositetrachoron (24 vertices, cube vertex figure) - dual icositetrachoron (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 tegum vertex figure) - tetraswirlic tetracontachoron or tetswirl 40 (40 alternate-metatruncated hexagonal prisms)

6. Dodecafold tetraswirlchoron (48 vertices, edge-vertical bisected triangular antitegum 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)

Ambotetrahedron-based[edit | edit source]

1. Tetrafold ambotetraswirlchoron (24 vertices) - rhombihexaswirlic icositetrachoron (24 cells)

2. Octafold ambotetraswirlchoron (48 vertices) - rhombihexaswirlic tetracontoctachoron (48 cells)

3. Triangular-antiprismatic enneacontahexachoron or octswirl 96 (72 vertices, square antitegum vertex figure) - square-antiprismatic heptacontadichoron or cubeswirl 72 (72 square antiprisms)

4. Hexadecafold ambotetraswirlchoron (96 vertices) - rhombihexaswirlic enneacontahexachoron (96 cells)

5. Icosafold ambotetraswirlchoron (120 vertices) - rhombihexaswirlic hecatonicosachoron (120 cells)

6. Icositetrafold octaswirlchoron (144 vertices, edge-vertical bisected square antitegum vertex figure) - cubiswirlic hecatontetracontatetrachoron or cubeswirl 144 (144 rhombistellated octagonal antiprisms)

7. Icosioctafold ambotetraswirlchoron (168 vertices) - rhombihexaswirlic hecatonhexacontoctachoron (168 cells)

8. Triacontadifold ambotetraswirlchoron (192 vertices) - rhombihexaswirlic hecatonenneacontadichoron (192 cells)

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

10. Tetracontafold ambotetraswirlchoron (240 vertices) - rhombihexaswirlic diacositetracontachoron (240 cells)

Cube-based[edit | edit source]

1. Square-antiprismatic heptacontadichoron (96 vertices, triangular antitegum vertex figure) - triangular-antiprismatic enneacontahexachoron (96 triangular antiprisms)

2. Icositetrafold cubiswirlchoron (192 vertices, edge-vertical bisected triangular antitegum vertex figure) - octaswirlic hecatonenneacontadichoron 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. Hecatonoctafold cubiswirlchoron (864 vertices) - octaswirlic octacosihexacontatetrachoron or octswirl 864 (864 cells)

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

Pyritohedral cube-based[edit | edit source]

1. Hexafold pyritocubiswirlchoron (48 vertices) - pyritooctaswirlic tetracontoctachoron (48 cells)

2. Dodecafold pyritocubiswirlchoron (96 vertices) - pyritooctaswirlic enneacontahexachoron (96 cells)

3. Octadecafold pyritocubiswirlchoron (144 vertices) - pyritooctaswirlic hecatontetracontatetrachoron (144 cells)

4. Icositetrafold cubiswirlchoron (192 vertices, edge-vertical bisected triangular antitegum vertex figure) - octaswirlic hecatonenneacontadichoron (192 rhombistellated hexagonal antiprisms)

5. Triacontafold pyritocubiswirlchoron (240 vertices) - pyritooctaswirlic diacositetracontachoron (240 cells)

6. Triacontahexafold pyritocubiswirlchoron (288 vertices) - pyritooctaswirlic diacosioctacontoctachoron (288 cells)

7. Tetracontadifold pyritocubiswirlchoron (336 vertices) - pyritooctaswirlic triacositriacontahexachoron (336 cells)

8. Tetracontoctafold cubiswirlchoron (384 vertices) - octaswirlic triacosioctacontatetrachoron (384 cells)

9. Pentacontahexafold pyritocubiswirlchoron (432 vertices) - pyritooctaswirlic tetracositriacontadichoron (432 cells)

10. Hexecontafold pyritocubiswirlchoron (480 vertices) - pyritooctaswirlic tetracosioctacontachoron (480 cells)

Octahedron-based[edit | edit source]

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, square antitegum 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 tegum vertex figure) - cubiswirlic hecatonicosachoron or cubeswirl 120 (120 alternate-metatruncated octagonal prisms)

6. Icositetrafold octaswirlchoron (144 vertices, edge-vertical bisected square antitegum vertex figure) - cubiswirlic hecatontetracontatetrachoron or cubeswirl 144 (144 rhombistellated octagonal antiprisms)

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

8. Triacontadifold octaswirlchoron (192 vertices) - cubiswirlic hecatonenneacontadichoron 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[edit | edit source]

1. Octafold cuboctaswirlchoron (96 vertices) - rhombidodecaswirlic enneacontahexachoron (96 cells)

2. Hexadecafold cuboctaswirlchoron (192 vertices) - rhombidodecaswirlic hecatonenneacontadichoron (192 cells)

3. Icositetrafold cuboctaswirlchoron (288 vertices, rectangular antitegum vertex figure) - rhombidodecaswirlic diacosioctacontoctachoron (288 rhombic antiprism cells)

4. Triacontadifold cuboctaswirlchoron (384 vertices) - - rhombidodecaswirlic triacosioctacontatetrachoron (384 cells)

5. Tetracontafold cuboctaswirlchoron (480 vertices) - rhombidodecaswirlic tetracosioctacontachoron (480 cells)

6. Tetracontoctafold cuboctaswirlchoron (576 vertices) - rhombidodecaswirlic pentacosiheptacontahexachoron (576 cells)

7. Pentacontahexafold cuboctaswirlchoron (672 vertices) - rhombidodecaswirlic hexacosiheptacontadichoron (672 cells)

8. Hexacontatetrafold cuboctaswirlchoron (768 vertices) - rhombidodecaswirlic heptacosihexacontoctachoron (768 cells)

9. Heptacontadifold cuboctaswirlchoron (864 vertices) - rhombidodecaswirlic octacosihexacontatetrachoron (864 cells)

10. Octacontafold cuboctaswirlchoron (960 vertices) - rhombidodecaswirlic enneacosihexecontachoron (960 cells)

Chiral cuboctahedron-based[edit | edit source]

1. Bitetracontoctachoron (48 vertices, triakis octahedron vertex figure) - tetracontoctachoron (48 truncated cubes)

2. Chirododecafold cuboctaswirlchoron (144 vertices) - chirorhombidodecaswirlic hecatontetracontatetrachoron (144 cells)

3. Chiroicosafold cuboctaswirlchoron (240 vertices) - chirorhombidodecaswirlic diacositetracontachoron (240 cells)

4. Chiroicosioctafold cuboctaswirlchoron (336 vertices) - chirorhombidodecaswirlic triacositriacontahexachoron (336 cells)

5. Chirotriacontahexafold cuboctaswirlchoron (432 vertices) - chirorhombidodecaswirlic tetracositriacontadichoron (432 cells)

6. Chirotetracontatetrafold cuboctaswirlchoron (528 vertices) - chirorhombidodecaswirlic pentacosicosoctachoron (528 cells)

7. Chiropentacontadifold cuboctaswirlchoron (624 vertices) - chirorhombidodecaswirlic hexacosicositetrachoron (624 cells)

8. Chirohexecontafold cuboctaswirlchoron (720 vertices) - chirorhombidodecaswirlic heptacosiicosachoron (720 cells)

9. Chirohexacontoctafold cuboctaswirlchoron (816 vertices) - chirorhombidodecaswirlic octacosihexadecachoron (816 cells)

10. Chiroheptacontahexafold cuboctaswirlchoron (912 vertices) - chirorhombidodecaswirlic enneacosidodecachoron (912 cells)

Pyritohedral cuboctahedron-based[edit | edit source]

1. Hexafold pyritocuboctaswirlchoron (72 vertices) - pyritorhombidodecaswirlic heptacontadichoron (72 cells)

2. Dodecafold pyritocuboctaswirlchoron (144 vertices) - pyritorhombidodecaswirlic hecatontetracontoctachoron (144 cells)

3. Octadecafold pyritocuboctaswirlchoron (216 vertices) - pyritorhombidodecaswirlic diacosihexadecachoron (216 cells)

4. Icositetrafold cuboctaswirlchoron (288 vertices) - rhombidodecaswirlic diacosioctacontoctachoron (288 cells)

5. Triacontafold pyritocuboctaswirlchoron (360 vertices) - pyritorhombidodecaswirlic triacosihexecontachoron (360 cells)

6. Triacontahexafold pyritocuboctaswirlchoron (432 vertices) - pyritorhombidodecaswirlic tetracositriacontadichoron (432 cells)

7. Tetracontadifold pyritocuboctaswirlchoron (504 vertices) - pyritorhombidodecaswirlic pentacositetrachoron (504 cells)

8. Tetracontoctafold cuboctaswirlchoron (576 vertices) - rhombidodecaswirlic pentacosiheptacontahexachoron (576 cells)

9. Pentacontahexafold pyritocuboctaswirlchoron (648 vertices) - pyritorhombidodecaswirlic hexacositetracontoctachoron (648 cells)

10. Hexecontafold pyritocuboctaswirlchoron (720 vertices) - pyritorhombidodecaswirlic heptacosiicosachoron (720 cells)

Dipyritocuboctahedron-based[edit | edit source]

1. Difold dipyritocuboctaswirlchoron (48 vertices) - dipyritorhombidodecaswirlic tetracontoctachoron (48 cells)

2. Hexafold dipyritocuboctaswirlchoron (144 vertices) - dipyritorhombidodecaswirlic hecatontetracontatetrachoron (144 cells)

3. Decafold dipyritocuboctaswirlchoron (240 vertices) - dipyritorhombidodecaswirlic diacositetracontachoron (240 cells)

4. Tetradecafold dipyritocuboctaswirlchoron (336 vertices) - dipyritorhombidodecaswirlic triacositriacontahexachoron (336 cells)

5. Octadecafold dipyritocuboctaswirlchoron (432 vertices) - dipyritorhombidodecaswirlic tetracositriacontadichoron (432 cells)

6. Icosidifold dipyritocuboctaswirlchoron (528 vertices) - dipyritorhombidodecaswirlic pentacosicosoctachoron (528 cells)

7. Icosihexafold dipyritocuboctaswirlchoron (624 vertices) - dipyritorhombidodecaswirlic hexacosicositetrachoron (624 cells)

8. Triacontafold dipyritocuboctaswirlchoron (720 vertices) - dipyritorhombidodecaswirlic heptacosiicosachoron (720 cells)

9. Triacontatetrafold dipyritocuboctaswirlchoron (816 vertices) - dipyritorhombidodecaswirlic octacosihexadecachoron (816 cells)

10. Triacontoctafold dipyritocuboctaswirlchoron (912 vertices) - dipyritorhombidodecaswirlic enneacosidodecachoron (912 cells)

Small rhombicuboctahedron-based[edit | edit source]

1. Octafold small rhombicuboctaswirlchoron (192 vertices) - deltoicositetraswirlic hecatonenneacontadichoron (192 cells)

Dodecahedron-based[edit | edit source]

1. Pentagonal-antiprismatic triacosihexecontachoron (600 vertices, triangular antitegum vertex figure) - triangular-antiprismatic hexacosichoron (600 triangular antiprisms)

2. Hexecontafold dodecaswirlchoron (1200 vertices, edge-vertical bisected triangular antitegum vertex figure) - icosaswirlic chiliadiacosichoron or ikeswirl 1200 (1200 rhombistellated hexagonal antiprisms)

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

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

5. Hecatonpentacontafold dodecaswirlchoron (3000 vertices) - icosaswirlic trischiliachoron or ikeswirl 3000 (3000 cells)

6. Hecatonoctacontafold dodecaswirlchoron (3600 vertices) - icosaswirlic trischiliahexacosichoron or ikeswirl 3600 (3600 cells)

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

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

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

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

Icosahedron-based[edit | edit source]

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

2. Icosafold icosaswirlchoron (240 vertices, triakis pentagonal tegum vertex figure) - dodecaswirlic diacositetracontachoron or doeswirl 240 (240 truncated pentagonal prisms)

3. Triangular-antiprismatic hexacosichoron or ikeswirl 600 (360 vertices, pentagonal antitegum 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 tegum vertex figure) - dodecaswirlic hexacosichoron or doeswirl 600 (600 alternate-metatruncated decagonal prisms)

6. Hexecontafold icosaswirlchoron (720 vertices, edge-vertical bisected pentagonal antitegum vertex figure) - dodecaswirlic heptacosiicosachoron 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 chiliaoctacontachoron or doeswirl 1080 (1080 cells)

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

Icosidodecahedron-based[edit | edit source]

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

2. Bitetrahedral diacositetracontachoron (240 vertices, chiral ditriakis tetrahedron vertex figure) - ditruncated-tetrahedral diacositetracontachoron (240 chiral ditruncated tetrahedra)

3. Dodecafold icosidodecaswirlchoron (360 vertices) - rhombitriacontaswirlic triacosihexecontachoron (360 cells)

4. Hexadecafold icosidodecaswirlchoron (480 vertices) - - rhombitriacontaswirlic tetracosioctacontachoron (480 cells)

5. Icosafold icosidodecaswirlchoron (600 vertices) - rhombitriacontaswirlic hexacosichoron (600 cells)

6. Icositetrafold icosidodecaswirlchoron (720 vertices) - rhombitriacontaswirlic heptacosiicosachoron (720 cells)

7. Icosioctafold icosidodecaswirlchoron (840 vertices) - rhombitriacontaswirlic octacositetracontachoron (840 cells)

8. Triacontadifold icosidodecaswirlchoron (960 vertices) - rhombitriacontaswirlic enneacosihexecontachoron (960 cells)

9. Triacontahexafold icosidodecaswirlchoron (1080 vertices) - rhombitriacontaswirlic chiliaoctacontachoron (1080 cells)

10. Tetracontafold icosidodecaswirlchoron (1200 vertices) - rhombitriacontaswirlic chiliadiacosichoron (1200 cells)

12. Tetracontoctafold icosidodecaswirlchoron (1440 vertices) - rhombitriacontaswirlic chiliatetracositetracontachoron (1440 cells)

15. Hexecontafold icosidodecaswirlchoron (1800 vertices, rectangular antitegum vertex figure) - rhombitriacontaswirlic chiliaoctacosichoron (1800 rhombic antiprism cells)

20. Octacontafold icosidodecaswirlchoron (2400 vertices) - rhombitriacontaswirlic dischiliatetracosichoron (2400 cells)

25. Hectofold icosidodecaswirlchoron (3000 vertices) - rhombitriacontaswirlic trischiliachoron (3000 cells)

30. Hecatonicosafold icosidodecaswirlchoron (3600 vertices) - rhombitriacontaswirlic trischiliahexacosichoron (3600 cells)

Digonal dihedron-based[edit | edit source]

1. Triangular duotegum (6 vertices, triangular tegum vertex figure) - triangular duoprism (6 triangular prisms)

2. Hexadecachoron (8 vertices, octahedron vertex figure) - tesseract (8 cubes)

3. Pentagonal duotegum (10 vertices, pentagonal tegum vertex figure) - pentagonal duoprism (10 pentagonal prisms)

4. Hexagonal duotegum (12 vertices, hexagonal tegum vertex figure) - hexagonal duoprism (12 hexagonal prisms)

5. Heptagonal duotegum (14 vertices, heptagonal tegum vertex figure) - heptagonal duoprism (14 heptagonal prisms)

6. Octagonal duotegum (16 vertices, octagonal tegum vertex figure) - octagonal duoprism (16 octagonal prisms)

7. Enneagonal duotegum (18 vertices, enneagonal tegum vertex figure) - enneagonal duoprism (18 enneagonal prisms)

8. Decagonal duotegum (20 vertices, decagonal tegum vertex figure) - decagonal duoprism (20 decagonal prisms)

9. Hendecagonal duotegum (22 vertices, hendecagonal tegum vertex figure) - hendecagonal duoprism (22 triangular prisms)

10. Dodecagonal duotegum (24 vertices, dodecagonal tegum vertex figure) - dodecagonal duoprism (24 dodecagonal prisms)

Triangular dihedron-based[edit | edit source]

1. Triangular duoprism (9 vertices, tetragonal disphenoid vertex figure) - triangular duotegum (9 tetragonal disphenoids)

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

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

4. Triangular tetraswirlprism (36 vertices) - triangular tetraswirltegum (36 cells)

5. Triangular pentaswirlprism (45 vertices) - triangular pentaswirltegum (45 cells)

6. Triangular hexaswirlprism (54 vertices) - triangular hexaswirltegum (54 cells)

7. Triangular heptaswirlprism (63 vertices) - triangular heptaswirltegum (63 cells)

8. Triangular octaswirlprism (72 vertices) - triangular octaswirltegum (72 cells)

9. Triangular enneaswirlprism (81 vertices) - triangular enneaswirltegum (81 cells)

10. Triangular decaswirlprism (90 vertices) - triangular decaswirltegum (90 cells)

Square dihedron-based[edit | edit source]

1. Tesseract (16 vertices, tetrahedron vertex figure) - hexadecachoron (16 tetrahedra)

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

3. Square triswirlprism (48 vertices) - square triswirltegum (48 cells)

4. Square tetraswirlprism (64 vertices) - square tetraswirltegum (64 cells)

5. Square pentaswirlprism (80 vertices) - square pentaswirltegum (80 cells)

6. Square hexaswirlprism (96 vertices) - square hexaswirltegum (96 cells)

7. Square heptaswirlprism (112 vertices) - square heptaswirltegum (112 cells)

8. Square octaswirlprism (128 vertices) - square octaswirltegum (128 cells)

9. Square enneaswirlprism (144 vertices) - square enneaswirltegum (144 cells)

10. Square decaswirlprism (160 vertices) - square decaswirltegum (160 cells)

Pentagonal dihedron-based[edit | edit source]

1. Pentagonal duoprism (25 vertices, tetragonal disphenoid vertex figure) - pentagonal duotegum (25 tetragonal disphenoids)

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

3. Pentagonal triswirlprism (75 vertices) - pentagonal triswirltegum (75 cells)

4. Pentagonal tetraswirlprism (100 vertices) - pentagonal tetraswirltegum (100 cells)

5. Pentagonal pentaswirlprism (125 vertices) - pentagonal pentaswirltegum (125 cells)

6. Pentagonal hexaswirlprism (150 vertices) - pentagonal hexaswirltegum (150 cells)

7. Pentagonal heptaswirlprism (175 vertices) - pentagonal heptaswirltegum (175 cells)

8. Pentagonal octaswirlprism (200 vertices) - pentagonal octaswirltegum (200 cells)

9. Pentagonal enneaswirlprism (225 vertices) - pentagonal enneaswirltegum (225 cells)

10. Pentagonal decaswirlprism (250 vertices) - pentagonal decaswirltegum (250 cells)

Hexagonal dihedron-based[edit | edit source]

1. Hexagonal duoprism (36 vertices, tetragonal disphenoid vertex figure) - hexagonal duotegum (36 tetragonal disphenoids)

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

3. Hexagonal triswirlprism (108 vertices) - hexagonal triswirltegum (108 cells)

4. Hexagonal tetraswirlprism (144 vertices) - hexagonal tetraswirltegum (144 cells)

5. Hexagonal pentaswirlprism (180 vertices) - hexagonal pentaswirltegum (180 cells)

6. Hexagonal hexaswirlprism (216 vertices) - hexagonal hexaswirltegum (216 cells)

7. Hexagonal heptaswirlprism (252 vertices) - hexagonal heptaswirltegum (252 cells)

8. Hexagonal octaswirlprism (288 vertices) - hexagonal octaswirltegum (288 cells)

9. Hexagonal enneaswirlprism (324 vertices) - hexagonal enneaswirltegum (324 cells)

10. Hexagonal decaswirlprism (360 vertices) - hexagonal decaswirltegum (360 cells)

Digonal prism-based[edit | edit source]

1. Octagonal duotegum (16 vertices, octagonal tegum vertex figure) - octagonal duoprism (16 octagonal prisms)

2. 24-7 step prism (24 vertices, paratetraaugmented hexagonal tegum vertex figure) - 24-7 gyrochoron (24 paratetratruncated hexagonal prisms)

3. Digonal double tetraswirlprism (32 vertices) or square duoantiprism (32 vertices, gyrobifastigium vertex figure) - digonal double tetraswirltegum (32 cells) or square duoantitegum (32 elongated tetragonal disphenoids

4. 40-9 step prism (40 vertices) - 40-9 gyrochoron (40 cells)

5. Small 24-7 gyrostep prism (48 vertices) - small 24-7 antibigyrochoron (48 cells)

6. 56-15 step prism (56 vertices) - 56-15 gyrochoron (56 cells)

7. Digonal double octaswirlprism (64 vertices) or square tetraswirlprism (64 vertices) - digonal double octaswirltegum (64 cells) or square tetraswirltegum (64 cells)

8. 72-17 step prism (72 vertices) - 72-17 gyrochoron (72 cells)

9. Small 40-9 gyrostep prism (80 vertices) - small 40-9 antibigyrochoron (80 cells)

10. 88-23 step prism (88 vertices) - 88-23 gyrochoron (88 cells)

Triangular prism-based[edit | edit source]

1. Triangular ditetragoltriate (18 vertices, notch vertex figure) - triangular tetrambitriate (18 wedges)

2. Triangular double gyroantiprismoid (36 vertices, octakis digonal-octagonal gyrowedge vertex figure) - triangular double gyroantitegmoid (36 order-8 truncated digonal-octagonal gyronotches)

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

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

5. Triangular double pentaswirlprism (90 vertices) - triangular double pentaswirltegum (90 cells)

6. Triangular double hexaswirlprism (108 vertices) - triangular double hexaswirltegum (108 cells)

7. Triangular double heptaswirlprism (126 vertices) - triangular double heptaswirltegum (126 cells)

8. Triangular double octaswirlprism (144 vertices) - triangular double octaswirltegum (144 cells)

9. Triangular double enneaswirlprism (162 vertices) - triangular double enneaswirltegum (162 cells)

10. Triangular double decaswirlprism (180 vertices) - triangular double decaswirltegum (180 cells)

Square prism-based[edit | edit source]

1. Square ditetragoltriate (32 vertices, notch vertex figure) - square tetrambitriate (32 wedges)

2. Square double gyroantiprismoid (64 vertices, octakis digonal-octagonal gyrowedge vertex figure) - square double gyroantitegmoid (64 order-8 truncated digonal-octagonal gyronotches)

3. Square double triswirlprism (96 vertices) or square-antiprismatic heptacontadichoron (96 vertices, triangular antitegum vertex figure) - square double triswirltegum (96 cells) or triangular-antiprismatic enneacontahexachoron (96 triangular antiprisms)

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

5. Square double pentaswirlprism (160 vertices) - square double pentaswirltegum (160 cells)

6. Square double hexaswirlprism (192 vertices) or icositetrafold cubiswirlchoron (192 vertices, edge-vertical bisected triangular antitegum vertex figure) - square double hexaswirltegum (192 cells) or octaswirlic hecatonenneacontadichoron (192 rhombistellated hexagonal antiprisms)

7. Square double heptaswirlprism (224 vertices) - square double heptaswirltegum (224 cells)

8. Square double octaswirlprism (256 vertices) - square double octaswirltegum (256 cells)

9. Square double enneaswirlprism (288 vertices) or triacontahexafold cubiswirlchoron (288 vertices) - square double enneaswirltegum (288 cells) or octaswirlic diacosioctacontoctachoron (288 cells)

10. Square double decaswirlprism (320 vertices) - square double decaswirltegum (320 cells)

Pentagonal prism-based[edit | edit source]

1. Pentagonal ditetragoltriate (50 vertices, notch vertex figure) - pentagonal tetrambitriate (50 wedges)

2. Pentagonal double gyroantiprismoid (100 vertices, octakis digonal-octagonal gyrowedge vertex figure) - pentagonal double gyroantitegmoid (100 order-8 truncated digonal-octagonal gyronotches)

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

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

5. Pentagonal double pentaswirlprism (250 vertices) - pentagonal double pentaswirltegum (250 cells)

6. Pentagonal double hexaswirlprism (300 vertices) - pentagonal double hexaswirltegum (300 cells)

7. Pentagonal double heptaswirlprism (350 vertices) - pentagonal double heptaswirltegum (350 cells)

8. Pentagonal double octaswirlprism (400 vertices) - pentagonal double octaswirltegum (400 cells)

9. Pentagonal double enneaswirlprism (450 vertices) - pentagonal double enneaswirltegum (450 cells)

10. Pentagonal double decaswirlprism (500 vertices) - pentagonal double decaswirltegum (500 cells)

Hexagonal prism-based[edit | edit source]

1. Hexagonal ditetragoltriate (72 vertices, notch vertex figure) - hexagonal tetrambitriate (72 wedges)

2. Hexagonal double gyroantiprismoid (144 vertices, octakis digonal-octagonal gyrowedge vertex figure) - hexagonal double gyroantitegmoid (144 order-8 truncated digonal-octagonal gyronotches)

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

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

5. Hexagonal double pentaswirlprism (360 vertices) - hexagonal double pentaswirltegum (360 cells)

6. Hexagonal double hexaswirlprism (432 vertices) - hexagonal double hexaswirltegum (432 cells)

7. Hexagonal double heptaswirlprism (504 vertices) - hexagonal double heptaswirltegum (504 cells)

8. Hexagonal double octaswirlprism (576 vertices) - hexagonal double octaswirltegum (576 cells)

9. Hexagonal double enneaswirlprism (648 vertices) - hexagonal double enneaswirltegum (648 cells)

10. Hexagonal double decaswirlprism (720 vertices) - hexagonal double decaswirltegum (720 cells)

Digonal antiprism-based[edit | edit source]

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

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

3. Dodecafold diantiprismatoswirlchoron (48 vertices) - diantiprismatoswirlic tetracontoctachoron (48 cells)

4. Hexadecafold tetraswirlchoron (64 vertices) - tetraswirlic hexacontatetrachoron (64 cells)

5. Icosafold diantiprismatoswirlchoron (80 vertices) - diantiprismatoswirlic octacontachoron (80 cells)

6. Icositetrafold diantiprismatoswirlchoron (96 vertices) or icositetrafold tetraswirlchoron (96 vertices) - diantiprismatoswirlic enneacontahexachoron (96 cells) or tetraswirlic enneacontahexachoron (96 cells)

7. Icosioctafold diantiprismatoswirlchoron (112 vertices) - diantiprismatoswirlic hecatondodecachoron (112 cells)

8. Triacontadifold tetraswirlchoron (128 vertices) - tetraswirlic hecatonicosoctachoron (128 cells)

9. Triacontahexafold diantiprismatoswirlchoron (144 vertices) - diantiprismatoswirlic hecatontetracontatetrachoron (144 cells)

10. Tetracontafold tetraswirlchoron (160 vertices) - tetraswirlic hecatonhexecontachoron (160 cells)

Triangular antiprism-based[edit | edit source]

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

2. Dodecafold triantiprismatoswirlchoron (72 vertices) - triantitegmatoswirlic heptacontadichoron (72 cells)

3. Small octadecafold triantiprismatoswirlchoron (108 vertices) or great octadecafold triantiprismatoswirlchoron (108 vertices) - small transitional octadecafold triantiprismatoswirlchoron (108 vertices) or great transitional octadecafold triantiprismatoswirlchoron (108 vertices) - small triantitegmatoswirlic hecatonoctachoron (108 cells) or great triantitegmatoswirlic hecatonoctachoron (108 cells) - small transitional triantitegmatoswirlic hecatonoctachoron (108 cells) or great transitional triantitegmatoswirlic hecatonoctachoron (108 cells)

4. Icositetrafold triantiprismatoswirlchoron (144 vertices) or icositetrafold octaswirlchoron (144 vertices, edge-vertical bisected triangular antitegum vertex figure) - triantitegmatoswirlic hecatontetracontatetrachoron (144 vertices) or cubiswirlic hecatonictetracontatetrachoron (144 rhombistellated octagonal antiprisms)

Square antiprism-based[edit | edit source]

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

2. Small hexadecafold tetraantiprismatoswirlchoron (128 vertices) or great hexadecafold tetraantiprismatoswirlchoron (128 vertices) or small transitional hexadecafold tetraantiprismatoswirlchoron (128 vertices) or great transitional hexadecafold tetraantiprismatoswirlchoron (128 vertices) - small tetraantitegmatoswirlic hecatonicosoctachoron (128 cells) or great tetraantitegmatoswirlic hecatonicosoctachoron (128 cells) or small transitional tetraantitegmatoswirlic hecatonicosoctachoron (128 cells) or great transitional tetraantitegmatoswirlic hecatonicosoctachoron (128 cells)

Pentagonal antiprism-based[edit | edit source]

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

2. Icosafold pentaantiprismatoswirlchoron (200 vertices) - pentaantitegmatoswirlic diacosichoron (200 cells)

Hexagonal antiprism-based[edit | edit source]

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