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
1. Regular polygons (half symmetry variants exist with two alternating edge lengths)

Polyhedra
1. 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)

2. 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)

3. 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)

4. 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
1. 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 square duoprism, cubic prism, square-square duoprism, rectangular-square duoprism, or rectangular-rectangular duoprism, with the first being noble, the hexadecachoron has lower symmetry variants as a 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)

2. 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, and the rectified hexacosichoron has a lower symmetry variant as a 6-dodecahedral swirlprism)