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The 600-cell has icosahedral cross sections of two sizes, and each of its 120 vertices is an icosahedral pyramid; the icosahedron is the vertex figure of the 600-cell. Another polytope with regular icosahedrons as its cell is the semiregular 4-polytope of snub 24-cell.
A plane containing a cross-section of the solid may be referred to as a cutting plane. The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid. For instance, while all the cross-sections of a ball are disks, [2] the cross-sections of a cube depend on how the cutting plane is related to the ...
In this respect, ideal polyhedra are different from Euclidean polyhedra (and from their Euclidean Klein models): for instance, on a Euclidean cube, any geodesic can cross at most two edges incident to a single vertex consecutively, before crossing a non-incident edge, but geodesics on the ideal cube are not limited in this way. [26]
Nevertheless, some polyhedrons may not possess one or two of those symmetries: A polyhedron with vertex-transitive and edge-transitive is said to be quasiregular, although they have regular faces, and its dual is face-transitive and edge-transitive. A vertex- but not edge-transitive polyhedron with regular polygonal faces is said to be semiregular.
The dual of a non-convex polyhedron is also a non-convex polyhedron. [2] ( By contraposition.) There are ten non-convex isotoxal polyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron: the five (quasiregular) hemipolyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron, and their five (infinite) duals:
(The icosidodecahedron is the equatorial cross-section of the 600-cell, and the decagon is the equatorial cross-section of the icosidodecahedron.) These radially golden polytopes can be constructed, with their radii, from golden triangles which meet at the center, each contributing two radii and an edge.
Other than rhombic triacontahedron, it is one of two Catalan solids that each have the property that their isometry groups are edge-transitive; the other convex polyhedron classes being the five Platonic solids and the other two Archimedean solids: its dual polyhedron and icosidodecahedron. Denoting by a the edge length of a rhombic dodecahedron,
The faces of the icosahedron are eight congruent equilateral triangles with the short side length, and twelve congruent obtuse isosceles triangles with one long edge and two short edges. [8] Jessen's icosahedron is vertex-transitive (or isogonal), meaning that it has symmetries taking any vertex to any other vertex. [9]