Swirlchoron

A pentagonal-gyroprismatic triacosihexecontachoron has rings of pentagonal gyroprisms arranged as a larger pentagonal gyroprism.

Swirlchora (sometimes swirlprisms) are polychora whose symmetry group is built up in a specific manner from 3D symmetry groups, called base groups. They are so called because when one of the base groups is isomorphic to a finite cyclic group, vertices and faces appear to "swirl" around in various rings, corresponding to great circles of a glome. They are related to Hopf fibration and 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 can either be isogonal, isochoric, or both. Examples include the bi-icositetradiminished hexacosichoron and its dual, the tri-icositetradiminished hexacosichoron, which are both isogonal and isochoric, and are therefore noble.

Construction

Convex isogonal and isotopic swirlchora can be constructed as follows. Given a finite 3D rotation group G, let Q(G) be the set of all unit quaternions corresponding to members of G. As every 3D rotation corresponds to two quaternions (q and -q), the cardinality of Q(G) is twice the order of G.

Given two nontrivial finite 3D rotation groups G₁ and G₂ and an initial point $p \neq 0$, let P be the set of all points of the form $q_1pq_2$ where $q_1 \in Q(G_1)$ and $q_2 \in Q(G_2)$. The convex hull of P is a convex isogonal swirlchoron, assuming it is not degenerate. The dual of this convex isogonal swirlchoron is a convex isotopic swirlchoron.

It can be shown that the rotations of the form $p \mapsto q_1pq_2$ form a symmetry group S, which is known as swirl symmetry or a swirl group. Both the above types of swirlchora have swirl symmetry, although depending on the initial point p they may possess more symmetries (i.e. S is a subgroup of their actual symmetry group). By selecting G₁, G₂, or both as rotation groups isomorphic to cyclic groups of arbitrary order, it can be seen that there is a countably infinite number of swirl groups. However, the number of distinct convex isogonal swirlchora up to scaling and rotation is uncountable, as the choice of the initial point p is important, producing polychora with no straightforward relation to each other.

The term "swirlchoron" is subject to some minor controversy. Usually swirlchora are defined as any polychora with swirl symmetry, but some propose to restrict the term to only convex isogonal swirlchora (i.e. those made as convex hulls of point sets with swirl symmetry).

Tetrahedron-based isogonal swirlchora

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

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

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

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

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

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+2√2, cos(kπ/4n)/√4+2√2, cos(kπ/4n)/√4-2√2, sin(kπ/4n)/√4-2√2),
• ±(sin(kπ/4n)/√4-2√2, cos(kπ/4n)/√4-2√2, cos(kπ/4n)/√4+2√2, sin(kπ/4n)/√4+2√2),
• ±(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

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 and even sign changes of the first and third coordinates of:

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

Coordinates for the vertices of a 30n-fold dodecaswirlchoron of circumradius 1, centered at the origin, are given by, along with their 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 = √450-30√75+30√5/30, c2 = √450+30√75+30√5/30, c3 = √450-30√75-30√5/30, c4 = √450+30√75-30√5/30 and k is an integer from 0 to 15n-1.

Icosahedron-based isogonal swirlchora

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

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

Dihedron-based isogonal swirlchora

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

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

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

Tetrahedron-based

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 gyrotegum vertex figure) - tetraswirlic tetracontoctachoron or tetswirl 48 (48 rhombistellated triambic gyroprisms)

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)

11. Icosidifold tetraswirlchoron (88 vertices) - tetraswirlic octacontoctachoron or tetswirl 88 (88 cells)

12. Icositetrafold tetraswirlchoron (96 vertices) - tetraswirlic enneacontahexachoron or tetswirl 96 (96 cells)

Ambotetrahedron-based

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

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

3. Triangular-gyroprismatic enneacontahexachoron or octswirl 96 (72 vertices, square gyrotegum vertex figure) - square-gyroprismatic heptacontadichoron or cubeswirl 72 (72 square gyroprisms)

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 gyrotegum vertex figure) - cubiswirlic hecatontetracontatetrachoron or cubeswirl 144 (144 rhombistellated tetrambic gyroprisms)

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)

11. Tetracontatetrafold ambotetraswirlchoron (264 vertices) - rhombihexaswirlic diacosihexacontatetrachoron (264 cells)

12. Tetracontoctafold octaswirlchoron (288 vertices) - cubiswirlic diacosioctacontoctachoron or cubeswirl 288 (288 cells)

Cube-based

1. Square-gyroprismatic heptacontadichoron (96 vertices, triangular gyrotegum vertex figure) - triangular-gyroprismatic enneacontahexachoron (96 triangular gyroprisms)

2. Icositetrafold cubiswirlchoron (192 vertices, edge-vertical bisected triangular gyrotegum vertex figure) - octaswirlic hecatonenneacontadichoron or octswirl 192 (192 rhombistellated triambic gyroprisms)

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)

11. Hecatontriacontadifold cubiswirlchoron (1056 vertices) - octaswirlic chiliapentacontahexachoron or octswirl 1056 (1056 cells)

12. Hecatontetracontatetrafold cubiswirlchoron (1152 vertices) - octaswirlic chiliahecatonpentacontadichoron or octswirl 1152 (1152 cells)

Pyritohedral cube-based

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 gyrotegum vertex figure) - octaswirlic hecatonenneacontadichoron or octswirl 192 (192 rhombistellated triambic gyroprisms)

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 or octswirl 384 (384 cells)

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

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

11. Hexacontahexafold pyritocubiswirlchoron (528 vertices) - pyritooctaswirlic pentacosicosoctachoron (528 cells)

12. Heptacontadifold cubiswirlchoron (576 vertices) - octaswirlic pentacosiheptacontahexachoron or octswirl 576 (576 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-gyroprismatic enneacontahexachoron or octswirl 96 (72 vertices, square gyrotegum vertex figure) - square-gyroprismatic heptacontadichoron or cubeswirl 72 (72 square gyroprisms)

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 gyrotegum vertex figure) - cubiswirlic hecatontetracontatetrachoron or cubeswirl 144 (144 rhombistellated tetrambic gyroprisms)

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)

11. Tetracontatetrafold octaswirlchoron (264 vertices) - cubiswirlic diacosihexacontatetrachoron or cubeswirl 264 (264 cells)

12. Tetracontoctafold octaswirlchoron (288 vertices) - cubiswirlic diacosioctacontoctachoron or cubeswirl 288 (288 cells)

Cuboctahedron-based

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

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

3. Icositetrafold cuboctaswirlchoron (288 vertices, rectangular gyrotegum vertex figure) - rhombidodecaswirlic diacosioctacontoctachoron (288 rhombic gyroprism 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)

11. Octacontoctafold cuboctaswirlchoron (1056 vertices) - rhombidodecaswirlic chiliapentacontahexachoron (1056 cells)

12. Enneacontahexafold cuboctaswirlchoron (1152 vertices) - rhombidodecaswirlic chiliahecatonpentacontadichoron (1152 cells)

Chiral cuboctahedron-based

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 heptacosicosachoron (720 cells)

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

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

11. Chirooctacontatetrafold cuboctaswirlchoron (1008 vertices) - chirorhombidodecaswirlic chiliaoctachoron (1008 cells)

12. Chiroenneacontadifold cuboctaswirlchoron (1104 vertices) - chirorhombidodecaswirlic chiliahecatontetrachoron (1104 cells)

Pyritohedral cuboctahedron-based

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 heptacosicosachoron (720 cells)

11. Hexacontahexafold pyritocuboctaswirlchoron (792 vertices) - pyritorhombidodecaswirlic heptacosienneacontadichoron (792 cells)

12. Heptacontadifold pyritocuboctaswirlchoron (864 vertices) - pyritorhombidodecaswirlic octacosihexacontatetrachoron (864 cells)

Dipyritocuboctahedron-based

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 heptacosicosachoron (720 cells)

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

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

11. Tetracontadifold dipyritocuboctaswirlchoron (1008 vertices) - dipyritorhombidodecaswirlic chiliaoctachoron (1008 cells)

12. Tetracontatetrafold dipyritocuboctaswirlchoron (1104 vertices) - dipyritorhombidodecaswirlic chiliahecatontetrachoron (1104 cells)

Small rhombicuboctahedron-based

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

Dodecahedron-based

1. Pentagonal-gyroprismatic triacosihexecontachoron (600 vertices, triangular gyrotegum vertex figure) - triangular-gyroprismatic hexacosichoron (600 triangular gyroprisms)

2. Hexecontafold dodecaswirlchoron (1200 vertices, edge-vertical bisected triangular gyrotegum vertex figure) - icosaswirlic chiliadiacosichoron or ikeswirl 1200 (1200 rhombistellated triambic gyroprisms)

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)

11. Triacositriacontafold dodecaswirlchoron (6600 vertices) - icosaswirlic hexachiliahexacosichoron or ikeswirl 6600 (6600 cells)

12. Triacosihexecontafold dodecaswirlchoron (7200 vertices) - icosaswirlic heptachiliadiacosichoron or ikeswirl 7200 (7200 cells)

Icosahedron-based

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-gyroprismatic hexacosichoron or ikeswirl 600 (360 vertices, pentagonal gyrotegum vertex figure) - pentagonal-gyroprismatic triacosihexecontachoron or doeswirl 360 (360 pentagonal gyroprisms)

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 gyrotegum vertex figure) - dodecaswirlic heptacosicosachoron or doeswirl 720 (720 rhombistellated pentambic gyroprisms)

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)

11. Hecatondecafold icosaswirlchoron (1320 vertices) - dodecaswirlic chiliatriacosicosachoron or doeswirl 1320 (1320 cells)

12. Hecatonicosafold icosaswirlchoron (1440 vertices) - dodecaswirlic chiliatetracositetracontachoron or doeswirl 1440 (1440 cells)

Icosidodecahedron-based

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 heptacosicosachoron (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)

11. Tetracontatetrafold icosidodecaswirlchoron (1320 vertices) - rhombitriacontaswirlic chiliatriacosicosachoron (1320 cells)

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

15. Hexecontafold icosidodecaswirlchoron (1800 vertices, rectangular gyrotegum vertex figure) - rhombitriacontaswirlic chiliaoctacosichoron (1800 rhombic gyroprism 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

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 hendecagonal prisms)

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

11. Tridecagonal duotegum (26 vertices, tridecagonal tegum vertex figure) - tridecagonal duoprism (26 tridecagonal prisms)

12. Tetradecagonal duotegum (28 vertices, tetradecagonal tegum vertex figure) - tetradecagonal duoprism (28 tetradecagonal prisms)

Triangular dihedron-based

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)

11. Triangular hendecaswirlprism (99 vertices) - triangular hendecaswirltegum (99 cells)

12. Triangular dodecaswirlprism (108 vertices) - triangular dodecaswirltegum (108 cells)

Square dihedron-based

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)

11. Square hendecaswirlprism (176 vertices) - square hendecaswirltegum (176 cells)

12. Square dodecaswirlprism (192 vertices) - square dodecaswirltegum (192 cells)

Pentagonal dihedron-based

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)

11. Pentagonal hendecaswirlprism (275 vertices) - pentagonal hendecaswirltegum (275 cells)

12. Pentagonal dodecaswirlprism (300 vertices) - pentagonal dodecaswirltegum (300 cells)

Hexagonal dihedron-based

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)

11. Hexagonal hendecaswirlprism (396 vertices) - hexagonal hendecaswirltegum (396 cells)

12. Hexagonal dodecaswirlprism (432 vertices) - hexagonal dodecaswirltegum (432 cells)

Digonal prism-based

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)

11. Digonal double dodecaswirlprism (96 vertices) or square hexaswirlprism (96 vertices) - digonal double dodecaswirltegum (96 cells) or square hexaswirltegum (96 cells)

12. 104-25 step prism (104 vertices) - 104-25 gyrochoron (104 cells)

Triangular prism-based

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. Small triangular double triswirlprism (54 vertices) or great triangular double triswirlprism (54 vertices) or transitional triangular double triswirlprism (54 vertices) - small triangular double triswirltegum (54 cells) or great triangular double triswirltegum (54 cells) or transitional triangular double triswirltegum (54 cells)

4. Small triangular double tetraswirlprism (72 vertices) or great triangular double tetraswirlprism (72 vertices) or transitional triangular double tetraswirlprism (72 vertices) - small triangular double tetraswirltegum (72 cells) or great triangular double tetraswirltegum (72 cells) or transitional triangular double tetraswirltegum (72 cells)

Square prism-based

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. Small square double triswirlprism (96 vertices) or great square double triswirlprism (96 vertices) or transitional square double triswirlprism (96 vertices) - small square double triswirltegum (96 cells) or great square double triswirltegum (96 cells) or transitional square double triswirltegum (96 cells)

4. Small square double tetraswirlprism (128 vertices) or great square double tetraswirlprism (128 vertices) or transitional square double tetraswirlprism (128 vertices) - small square double tetraswirltegum (128 cells) or great square double tetraswirltegum (128 cells) or transitional square double tetraswirltegum (128 cells)

Pentagonal prism-based

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. Small pentagonal double triswirlprism (150 vertices) or great pentagonal double triswirlprism (150 vertices) or transitional pentagonal double triswirlprism (150 vertices) - small pentagonal double triswirltegum (150 cells) or great pentagonal double triswirltegum (150 cells) or transitional pentagonal double triswirltegum (150 cells)

4. Small pentagonal double tetraswirlprism (200 vertices) or great pentagonal double tetraswirlprism (200 vertices) or transitional pentagonal double tetraswirlprism (200 vertices) - small pentagonal double tetraswirltegum (200 cells) or great pentagonal double tetraswirltegum (200 cells) or transitional pentagonal double tetraswirltegum (200 cells)

Hexagonal prism-based

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. Small hexagonal double triswirlprism (216 vertices) or great hexagonal double triswirlprism (216 vertices) or transitional hexagonal double triswirlprism (216 vertices) - small hexagonal double triswirltegum (216 cells) or great hexagonal double triswirltegum (216 cells) or transitional hexagonal double triswirltegum (216 cells)

4. Small hexagonal double tetraswirlprism (288 vertices) or great hexagonal double tetraswirlprism (288 vertices) or transitional hexagonal double tetraswirlprism (288 vertices) - small hexagonal double tetraswirltegum (288 cells) or great hexagonal double tetraswirltegum (288 cells) or transitional hexagonal double tetraswirltegum (288 cells)

Digonal antiprism-based

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

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) or small transitional octadecafold triantiprismatoswirlchoron (108 vertices) or great transitional octadecafold triantiprismatoswirlchoron (108 vertices) - small triantitegmatoswirlic hecatonoctachoron (108 cells) or great triantitegmatoswirlic hecatonoctachoron (108 cells) or 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 square gyrotegum vertex figure) - triantitegmatoswirlic hecatontetracontatetrachoron (144 vertices) or cubiswirlic hecatonictetracontatetrachoron (144 rhombistellated tetrambic gyroprisms)

Square antiprism-based

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

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

2. Small icosafold pentaantiprismatoswirlchoron (200 vertices) or great icosafold pentaantiprismatoswirlchoron (200 vertices) or small transitional icosafold pentaantiprismatoswirlchoron (200 vertices) or great transitional icosafold pentaantiprismatoswirlchoron (200 vertices) - small pentaantitegmatoswirlic diacosichoron (200 cells) or great pentaantitegmatoswirlic diacosichoron (200 cells) or small transitional pentaantitegmatoswirlic diacosichoron (200 cells) or great transitional pentaantitegmatoswirlic diacosichoron (200 cells)

3. Small triacontafold pentaantiprismatoswirlchoron (300 vertices) or great triacontafold pentaantiprismatoswirlchoron (300 vertices) or small transitional triacontafold pentaantiprismatoswirlchoron (300 vertices) or great transitional triacontafold pentaantiprismatoswirlchoron (300 vertices) - small pentaantitegmatoswirlic triacosichoron (300 cells) or great pentaantitegmatoswirlic triacosichoron (300 cells) or small transitional pentaantitegmatoswirlic triacosichoron (300 cells) or great transitional pentaantitegmatoswirlic triacosichoron (300 cells)

Hexagonal antiprism-based

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