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 (half symmetry variants exist with two alternating edge lengths)

Polyhedra

 * Regular polyhedra (5 total, the tetrahedron has lower symmetry variants as a tetragonal disphenoid or rhombic disphenoid, both of which are noble, the cube has lower symmetry variants as a square prism, cuboid, or rectangular trapezoprism, the octahedron has a lower symmetry variant as a triangular antiprism, and the icosahedron has lower symmetry variants with pyritohedral or chiral tetrahedral symmetry)
 * Archimedean solids (13 total, the cuboctahedron has a lower symmetry variant as a small rhombitetratetrahedron, the truncated octahedron has a lower symmetry variant as a great rhombitetratetrahedron, and the rhombicuboctahedron has a lower symmetry variant with pyritohedral symmetry)
 * 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, the pentachoron has lower symmetry variants as a 5-2 step prism/5-2 gyrochoron, which is noble, the tesseract has lower symmetry variants as a cubic prism, square duoprism, square-square duoprism, rectangular-square duoprism, or rectangular-rectangular duoprism, the hexadecachoron has lower symmetry variants as a rectangular duopyramid, digonal duoantiprism, tetrahedral antiprism, digonal-digonal duoantiprism, tetragonal disphenoidal antiprism, rhombic disphenoidal antiprism, 1-tetrahedral swirlprism or 8-3 step prism, the icositetrachoron has lower symmetry variants as a 3-tetrahedral swirlprism/3-tetrahedral swirltegum or 1-cubic swirlprism/1-cubic swirltegum, the hecatonicosachoron has a lower symmetry variant as a 1-icosahedral swirlprism, and the hexacosichoron has lower symmetry variants as a 1-dodecahedral swirlprism, 5-cubic swirlprism, or 15-tetrahedral 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 tetracontoctachoron has a lower symmetry variant as a 12-cubic swirlprism, the small rhombated icositetrachoron has a lower symmetry variant as a cantic snub icositetrachoron, the small prismatotetracontoctachoron has a lower symmetry variant with cubic swirlprism symmetry, 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, the rectified hexacosichoron has a lower symmetry variant as a 6-dodecahedral swirlprism, and the grand antiprism has a lower symmetry variant with pentagonal antiprismatic swirlprism symmetry)
 * Scaliform polychora (4 total)
 * Rectified isotoxal decachoric and tetracontoctachoric polychora (4 total)
 * Truncated isotoxal decachoric and tetracontoctachoric polychora (4 total)
 * Decachoric and tetracontoctachoric doublings (33 total, with 1 more variation on the small rhombated pentachoron, small rhombated icositetrachoron, great rhombated pentachoron, great rhombated icositetrachoron, and prismatorhombisnub icositetrachoron, and 3 more variations on the prismatorhombated pentachoron and prismatorhombated icositetrachoron)
 * Omnisnubs (7 total, the pyritohedral icosahedral antiprism has a lower symmetry variant as an omnisnub tetrahedral antiprism)
 * Bialternatosnubs (infinite, non-duoprismatic variants include the bialternatosnub hexadecachoron and the bialternatosnub octahedral hosochoron, the duoprismatic-based cases contain a ring of alternating prisms and antiprisms, lower symmetry variants have twisted antiprisms)
 * 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)
 * 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 gyroantiprismoidss (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)
 * 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 (41 total)
 * Bialternatosnubs (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)