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The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory . The pseudorhombicuboctahedron – which is not isogonal – demonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which ...
In geometry, the Rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has a total of 62 faces: 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, with 60 vertices, and 120 edges.
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other).
If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-isotoxal. k-uniform tilings with the same vertex figures can be further identified by their wallpaper group symmetry.
For this reason, convex isohedral polyhedra are the shapes that will make fair dice. [1] Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal.
An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal.The duals of isotoxal polygons are isogonal polygons.Isotoxal -gons are centrally symmetric, thus are also zonogons.
The convex shapes in this family range from the octahedron itself through the regular icosahedron to the cuboctahedron, with its square faces subdivided into two right triangles in a flat plane. Extending the range of the parameter past the proportion that gives the cuboctahedron produces non-convex shapes, including Jessen's icosahedron.
The same shape was depicted by Leonardo da Vinci, in his illustrations for Luca Pacioli's plagiarism of della Francesca's book in 1509. Although Albrecht Dürer omitted this shape from the other Archimedean solids listed in his 1525 book on polyhedra, Underweysung der Messung, a description of it was found in his posthumous papers, published in ...