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)