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

 * Regular polyhedra (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, the ones with double symmetry have regular symmetry variations, the decachoron has a lower symmetry variant as a 10-3 step prism, the small prismatodecachoron has a lower symmetry variant as an expanded 5-2 step prism, the rectified tesseract has lower symmetry variants as a runcic tesseract or a digonal double prismantiprismoid, the truncated hexadecachoron has a lower symmetry variant as a cantic tesseract, the tesseractihexadecachoron has a lower symmetry variant as a runcicantic tesseract, the rectified icositetrachoron has lower symmetry variants as a small rhombated hexadecachoron or a prismatorhombated demitesseract, the truncated icositetrahedron has lower symmetry variants as a great rhombated hexadecachoron or a great prismated demitesseract, the small rhombated icositetrachoron has a lower symmetry variant as a cantic snub icositetrachoron, the prismatorhombated icositetrachoron has a lower symmetry variant as a runcicantic snub icositetrachoron, the snub disicositetrachoron has lower symmetry variants with 16 or 24 snub tetratetrahedra, and the grand antiprism has a lower symmetry variant as a 2-pentagonal antiprism swirlprism)
 * Scaliform polychora (4 total)
 * Rectified isotoxal decachoric and tetracontoctachoric polychora (4 total)
 * Truncated isotoxal decachoric and tetracontoctachoric polychora (4 total)
 * Decachoric and tetracontoctachoric doublings (43 total, with 1 more variation on the small rhombated pentachoron, small rhombated icositetrachoron, and 4 more variations on the great rhombated pentachoron, great rhombated icositetrachoron prismatorhombated pentachoron, prismatorhombated icositetrachoron, and prismatorhombisnub icositetrachoron)
 * Omnisnubs (7 total, the pyritohedral icosahedral antiprism has a lower symmetry variant as an omnisnub tetrahedral antiprism)
 * Snub tetrahedral tetracontoctachoron and related polytopes (at least 8 total)
 * Snub tetrahedral hecatonicosachoron and related polytopes (at least 9 total)
 * Edge-snubs (infinite, non-duoprismatic variants include the edge-snub hexadecachoron and the edge-snub octahedral hosochoron, the duoprismatic-based cases fall under the prismantiprismoids)
 * Polyhedral prisms (17 total, variations are the same as the polyhedral bases)
 * Antiprismatic prisms (infinite, variations are the same as the antiprism 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 doublings (infinite)
 * Swirlprisms (infinite)
 * Swirlprism doublings (infinite)

Polytera

 * Regular polytera (3 total)
 * Non-regular uniform polytera (55 total)
 * 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)