Isogonal polytope

An isogonal polytope or vertex-transitive polytope is a polytope whose vertices are identical under its highest symmetry group. In an isogonal polytope, there is a singular vertex figure, and all of the vertices lie on a hypersphere. The dual of an isogonal polytope is an isotopic polytope, which are made out of one facet type. All regular and uniform polytopes are isogonal.

If an isogonal polytope is also isotopic, it is called a noble polytope. Self-dual isogonal polytopes are also noble.

Polygons

 * Regular polygons (infinite, half symmetry variants exist with two alternating edge lengths)

Polyhedra

 * Platonic solids (5 total)
 * Tetrahedron (lower symmetry variants are the tetragonal disphenoid and the rhombic disphenoid)
 * Cube (lower symmetry variants are the square prism and the rectangular trapezoprism)
 * Octahedron (a lower symmetry variant is the triangular antiprism)
 * Dodecahedron (no lower symmetry variants)
 * Icosahedron (lower symmetry variants are the pyritohedral icosahedron and the snub tetrahedron)
 * Archimedean solids (13 total)
 * Truncated tetrahedron
 * Cuboctahedron (a lower symmetry variant is the small rhombitetratetrahedron)
 * Truncated cube
 * Truncated octahedron (a lower symmetry variant is the great rhombitetratetrahedron)
 * Small rhombicuboctahedron (a lower symmetry variant is the pyritohedral small rhombicuboctahedron)
 * Great rhombicuboctahedron
 * Snub cube
 * Icosidodecahedron
 * Truncated dodecahedron
 * Truncated icosahedron
 * Small rhombicosidodecahedron
 * Great rhombicosidodecahedron
 * Snub dodecahedron
 * Polygonal prisms (infinite, half symmetry variants exist for even-sided polygons with bases alternating two edge lengths, and can either be parallel or gyrated with respect to each other)
 * Polygonal antiprisms (infinite, half symmetry variants exist with the two bases rotated so that the base-first projection envelope is not a regular polygon)

Polychora

 * Regular polychora (6 total)
 * Pentachoron (a lower symmetry variant is the 5-2 step prism/5-2 gyrochoron)
 * Tesseract (lower symmetry variants are the cubic prism, square-square duoprism, rectangular-square duoprism, rectangular duoprism and the rectangular-rectangular duoprism)
 * Hexadecachoron (lower symmetry variants are the tetrahedral antiprism, rectangular duotegum, digonal-digonal duoantiprism, rhombic disphenoidal antiprism and the 8-3 step prism)
 * Icositetrachoron (a lower symmetry variant is the 1-tritetrahedral swirlprism)
 * Hecatonicosachoron
 * Hexacosichoron (a lower symmetry variant is the 1-pentatetrahedral swirlprism)
 * Non-regular uniform polychora (47 total)
 * Rectified pentachoron (a lower symmetry variant is the 5-2 double step prism)
 * Truncated pentachoron
 * Decachoron (a lower symmetry variant is the pentapentachoron)
 * Small rhombated pentachoron
 * Great rhombated pentachoron
 * Small prismatodecachoron (a lower symmetry variant is the small disprismatopentapentachoron and the triangular-prismatic 10-3 double gyrostep prism)
 * Prismatorhombated pentachoron
 * Great prismatodecachoron (a lower symmetry variant is the great disprismatopentapentachoron)
 * Rectified tesseract (a lower symmetry variant is the runcic tesseract)
 * Truncated tesseract
 * Truncated hexadecachoron (a lower symmetry variant is the cantic tesseract)
 * Tesseractihexadecachoron (a lower symmetry variant is the runcicantic tesseract)
 * Small rhombated tesseract
 * Great rhombated tesseract
 * Small disprismatotesseractihexadecachoron (a lower symmetry variant is the square duoexpandoprism)
 * Prismatorhombated tesseract
 * Prismatorhombated hexadecachoron
 * Great disprismatotesseractihexadecachoron
 * Rectified icositetrachoron (lower symmetry variants are the cantellated hexadecachoron, prismatorhombated demitesseract and the 3-tetratetrahedral swirlprism)
 * Truncated icositetrachoron (lower symmetry variants are the cantitruncated hexadecachoron and the omnitruncated demitesseract)
 * Tetracontoctachoron (a lower symmetry variant is the icositetraicositetrachoron)
 * Small rhombated icositetrachoron (a lower symmetry variant is the cantic snub disicositetrachoron)
 * Great rhombated icositetrachoron
 * Small prismatotetracontoctachoron (a lower symmetry variant is the small disprismatoicositetraicositetrachoron)
 * Prismatorhombated icositetrachoron (a lower symmetry variant is the runcicantic snub disicositetrachoron)
 * Great prismatotetracontoctachoron (a lower symmetry variant is the great disprismatoicositetraicositetrachoron)
 * Snub disicositetrachoron (lower symmetry variants are the snub rectified hexadecachoron and the omnisnub demitesseract)
 * Rectified hecatonicosachoron
 * Rectified hexacosichoron (a lower symmetry variant is the snub tetrahedral hecatonicosachoron)
 * Truncated hecatonicosachoron
 * Truncated hexacosichoron
 * Hexacosihecatonicosachoron
 * Small rhombated hecatonicosachoron
 * Small rhombated hexacosichoron
 * Great rhombated hecatonicosachoron
 * Great rhombated hexacosichoron
 * Small disprismatohexacosihecatonicosachoron
 * Prismatorhombated hecatonicosachoron
 * Prismatorhombated hexacosichoron
 * Great disprismatohexacosihecatonicosachoron
 * Grand antiprism
 * Scaliform polychora (4 total)
 * Truncated tetrahedral cupoliprism
 * Bi-icositetradiminished hexacosichoron
 * Prismatorhombisnub icositetrachoron
 * Swirlprismatodiminished rectified hexacosichoron
 * Rectified isotoxal decachoric and tetracontoctachoric polychora (4 total)
 * Rectified decachoron/Rectified tetracontoctachoron
 * Rectified small prismatodecachoron/Rectified small prismatotetracontoctachoron
 * Truncated isotoxal decachoric and tetracontoctachoric polychora (4 total)
 * Truncated decachoron/Truncated tetracontoctachoron
 * Truncated small prismatodecachoron/Truncated small prismatotetracontoctachoron
 * Decachoric and tetracontoctachoric doublings (at least 46 total)
 * Bidecachoron/Bitetracontoctachoron (the former has a lower symmetry variant as a 10-3 step prism, while the latter has a lower symmetry variant as a 3-bitetrahedral swirlprism
 * Biambodecachoron/Biambotetracontoctachoron (the former has a lower symmetry variant as a 10-3 double step prism)
 * Bitruncatodecachoron/Bitruncatotetracontoctachoron
 * Bimesotruncatodecachoron/Bimesotruncatotetracontoctachoron
 * Octahedral birhombatodecachoron/Cuboctahedral birhombatotetracontoctachoron
 * Octahedral-triangular prismatic birhombatodecachoron/Cuboctahedral-triangular prismatic birhombatotetracontoctachoron
 * Small rhombitetratetrahedral birhombatodecachoron/Small rhombicuboctahedral birhombatotetracontoctachoron
 * Truncated tetrahedral-triangular prismatic bicantitruncatodecachoron/Truncated cubic-triangular prismatic bicantitruncatotetracontoctachoron
 * Truncated tetrahedral-ditrigonal bicantitruncatodecachoron/Truncated cubic-ditrigonal bicantitruncatotetracontoctachoron
 * Truncated tetrahedral-transitional bicantitruncatodecachoron/Truncated cubic-transitional bicantitruncatotetracontoctachoron
 * Great rhombitetratetrahedral-triangular prismatic bicantitruncatodecachoron/Great rhombicuboctahedral-triangular prismatic bicantitruncatotetracontoctachoron
 * Great rhombitetratetrahedral-ditrigonal bicantitruncatodecachoron/Great rhombicuboctahedral-ditrigonal bicantitruncatotetracontoctachoron
 * Great rhombitetratetrahedral-transitional bicantitruncatodecachoron/Great rhombicuboctahedral-transitional bicantitruncatotetracontoctachoron
 * Biprismatodecachoron/Biprismatotetracontoctachoron
 * Small rhombitetratetrahedral-triangular prismatic biprismatorhombatodecachoron/Small rhombicuboctahedral-triangular prismatic biprismatorhombatotetracontoctachoron
 * Small rhombitetratetrahedral-ditrigonal prismatic biprismatorhombatodecachoron/Small rhombicuboctahedral-ditrigonal prismatic biprismatorhombatotetracontoctachoron
 * Small rhombitetratetrahedral-transitional biprismatorhombatodecachoron/Small rhombicuboctahedral-transitional biprismatorhombatotetracontoctachoron
 * Truncated tetrahedral-triangular prismatic biprismatorhombatodecachoron/Truncated octahedral-triangular prismatic biprismatorhombatotetracontoctachoron
 * Truncated tetrahedral-ditrigonal prismatic biprismatorhombatodecachoron/Truncated octahedral-ditrigonal prismatic biprismatorhombatotetracontoctachoron
 * Truncated tetrahedral-transitional biprismatorhombatodecachoron/Truncated octahedral-transitional biprismatorhombatotetracontoctachoron
 * Transitional-triangular prismatic biprismatorhombatodecachoron/Transitional-triangular prismatic biprismatorhombatotetracontoctachoron
 * Transitional-ditrigonal prismatic biprismatorhombatodecachoron/Transitional-ditrigonal prismatic biprismatorhombatotetracontoctachoron
 * Biomnitruncatodecachoron/Biomnitruncatotetracontoctachoron
 * Chiral decachoric and tetracontoctachoric doublings (at least 16 total)
 * Small biomnisnub decachoron/Small biomnisnub tetracontoctachoron
 * Medial biomnisnub decachoron/Medial biomnisnub tetracontoctachoron
 * Great biomnisnub decachoron/Great biomnisnub tetracontoctachoron
 * Gyrowedged biomnisnub decachoron/Gyrowedged biomnisnub tetracontoctachoron
 * Antipodial biomnisnub decachoron/Antipodial biomnisnub tetracontoctachoron
 * Prismatic-gyrowedged biomnisnub decachoron/Prismatic-gyrowedged biomnisnub tetracontoctachoron
 * Gyroamboprismatic biomnisnub decachoron/Gyroamboprismatic biomnisnub tetracontoctachoron
 * Gyrorhombitetratetrahedral biomnisnub decachoron/Gyrorhombicuboctahedral biomnisnub tetracontoctachoron
 * Omnisnubs (9 total)
 * Omnisnub pentachoron
 * Omnisnub bidecachoron
 * Omnisnub tesseract
 * Omnisnub icositetrachoron
 * Omnisnub bitetracontoctachoron
 * Omnisnub hecatonicosachoron
 * Pyritohedral icosahedral antiprism (a lower symmetry variant is the omnisnub tetrahedral antiprism)
 * Omnisnub cubic antiprism
 * Omnisnub dodecahedral antiprism
 * F4+/2 polychora (unknown)
 * H4+/5 polychora (at least 2 total)
 * Tetrahedral-snub tetrahedral hecatonicosachoron
 * Snub ambotetrahedral hecatonicosachoron
 * H4+×2/5 polychora (at least 6 total)
 * Bitetrahedral diacositetracontachoron
 * Snub bitetrahedral diacositetracontachoron
 * Bitetratetrahedral diacositetracontachoron
 * Omnisnub tetrahedral hecatonicosachoron
 * Antipodial omnisnub bitetrahedral diacositetracontachoron
 * Disphenoidal omnisnub bitetrahedral diacositetracontachoron
 * Edge-snubs (2 total)
 * Edge-snub hexadecachoron
 * Edge-snub octahedral hosochoron
 * Polyhedral prisms (17 total, variations are the same as the polyhedral bases)
 * Tetrahedral prism
 * Octahedral prism
 * Dodecahedral prism
 * Icosahedral prism
 * Truncated tetrahedral prism
 * Cuboctahedral prism
 * Truncated cubic prism
 * Truncated octahedral prism
 * Small rhombicuboctahedral prism
 * Great rhombicuboctahedral prism
 * Snub cubic prism
 * Icosidodecahedral prism
 * Truncated dodecahedral prism
 * Truncated icosahedral prism
 * Small rhombicosidodecahedral prism
 * Great rhombicosidodecahedral prism
 * Snub dodecahedral prism
 * Antiprismatic prisms (infinite, variations are the same as the antiprismatic bases)
 * Duoprisms (infinite, variations are the same as the polygonal bases)
 * Duotegums (infinite, the bases must have identical isogonal polygons with additional step prism variations)
 * Rectified duoprisms (infinite)
 * Truncated duoprisms (infinite)
 * Duoantiprisms (infinite, lower symmetry variants have one or both antiprism bases twisted)
 * Prismantiprismoids (infinite, contains a ring of alternating prisms and antiprisms, lower symmetry variants have twisted antiprisms)
 * Ditetragoltriates (infinite, contains two orthogonal rings of identical prisms)
 * Antiditetragoltriates (infinite, contains two antialigned orthogonal rings of two types of prisms each)
 * Duoexpandoprisms (infinite, contains two orthogonal rings of two types of prisms each)
 * Duotruncatoprisms (infinite, contains two orthogonal rings of identical prisms whose bases are truncated polygons)
 * Double antiprismoids (infinite, contains two orthogonal rings of identical antiprisms)
 * Double gyroantiprismoids (infinite, contains two antialigned orthogonal rings of identical antiprisms)
 * Double prismantiprismoids (infinite, contains two orthogonal rings of alternating prisms and antiprisms)
 * Duoprismatic swirlprisms (infinite)
 * Double duoprismatoswirlprisms (infinite)
 * Duoprismatic prismantiprismatoswirlprisms (infinite)
 * Double prismantiprismatoswirlprisms (infinite)
 * Step prisms (infinite, variations are the same as the pentachoron)
 * Step prism compound hulls (infinite)
 * Swirlprisms (infinite)
 * Swirlprism compound hulls (infinite)

Polytera

 * Regular polytera (3 total)
 * Hexateron
 * Penteract
 * Triacontaditeron
 * Non-regular uniform polytera (55 total)
 * Rectified hexateron
 * Dodecateron
 * Truncated hexateron
 * Bitruncated hexateron
 * Small rhombated hexateron
 * Small birhombated dodecateron
 * Great rhombated hexateron
 * Great birhombated dodecateron
 * Small prismated hexateron
 * Prismatotruncated hexateron
 * Prismatorhombated hexateron
 * Great prismated hexateron
 * Small cellated dodecateron
 * Cellirhombated dodecateron
 * Celliprismated hexateron
 * Celligreatorhombated hexateron
 * Celliprismatotruncated dodecateron
 * Great cellated dodecateron
 * Demipenteract
 * Rectified penteract
 * Rectified triacontaditeron
 * Penteractitriacontaditeron
 * Truncated penteract
 * Truncated triacontaditeron
 * Bitruncated penteract
 * Bitruncated triacontaditeron
 * Small rhombated penteract
 * Small rhombated triacontaditeron
 * Small birhombated penteractitriacontaditeron
 * Great rhombated penteract
 * Great rhombated triacontaditeron
 * Great birhombated penteractitriacontaditeron
 * Small prismated penteract
 * Small prismated hexateron
 * Prismatotruncated penteract
 * Prismatotruncated triacontaditeron
 * Prismatorhombated penteract
 * Prismatorhombated triacontaditeron
 * Great prismated penteract
 * Great prismated triacontaditeron
 * Small cellated penteractitriacontaditeron
 * Cellirhombated penteractitriacontaditeron
 * Celliprismated penteract
 * Celliprismated triacontaditeron
 * Celligreatorhombated penteract
 * Celligreatorhombated triacontaditeron
 * Celliprismatotruncated penteractitriacontaditeron
 * Great cellated penteractitriacontaditeron
 * Truncated demipenteract
 * Small rhombated demipenteract
 * Great rhombated demipenteract
 * Small prismated demipenteract
 * Prismatotruncated demipenteract
 * Prismatorhombated demipenteract
 * Great prismated demipenteract
 * Scaliform polytera (14 total, includes some alterprisms based on decachoric, demitesseractic and tetracontoctachoric symmetries)
 * Uniform polychoric prisms (52 total)
 * Scaliform polychoric prisms (4 total)
 * Omnisnubs (7 total)
 * Dodecateric doublings (at least 20 total)
 * Decachoric, demitesseractic and tetracontoctachoric alterprisms (45 total)
 * Bitetrahedral diacositetracontachoric alterprisms (at least 9 total)
 * Edge-snubs (infinite)
 * Biorthosnubs (infinite)
 * Triorthowedges (4 total)
 * Disphenoids (infinite, consists of two identical orthogonal regular polygons with a height between them)
 * Prisms (infinite, consists of all the prisms of the 4D isogonal categories not mentioned above and are not duoprisms themselves)
 * Polyhedral duoprisms (infinite, includes antiprism duoprisms)
 * Duoprismatic prisms (infinite)
 * Duoantiprisms (infinite)
 * Duoantiprismatic antiprisms (infinite)
 * Double antiprismoidal antiprisms (infinite)
 * Truncated tetrahedral duocupoliprisms (infinite, consists of a ring of truncated tetrahedral cupoliprisms, the only scaliform is the digonal-truncated tetrahedral duocupoliprism)
 * Bialternatocupolaic truncated tetrahedral duoprisms (infinite, similar to the truncated tetrahedral duocupoliprisms but with alternating truncated tetrahedral prisms and truncated tetrahedral cupoliprisms)
 * Duoantiwedges (infinite, all members are scaliform except for the digonal duoantiwedge, which is the square disphenoid)
 * Duoprismatic cupoliprisms (infinite, related to the duoexpandoprisms, the only scaliform is the triangular-hexagonal duoprismatic cupoliprism)
 * Duoprismatic alterprisms (infinite, related to the ditetragoltriates)
 * Duoprismatic antialterprisms (infinite, related to the antiditetragoltriates)
 * Duoprismatic truncatocupoliprisms (infinite, related to the duotruncatoprisms)
 * Duoantiprismatic alterprisms (infinite, related to the double antiprismoids)
 * Duoantiprismatic antialterprisms (infinite, related to the double gyroantiprismoids)
 * Prismantiprismatic alterprisms (infinite, related to the double prismantiprismoids)
 * Other duoprismatic alterprisms (infinite, includes alterprisms which are based on duoprismatic polychora not mentioned above)
 * Step prism alterprisms (infinite)
 * Swirlchoron alterprisms (infinite)