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The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of complexity and size at the same radius). [a] It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the 24-cell, [5] as the 24-cell can be deconstructed into three overlapping instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into ...
In analogy with the cross-section of a solid, the cross-section of an n-dimensional body in an n-dimensional space is the non-empty intersection of the body with a hyperplane (an (n − 1)-dimensional subspace). This concept has sometimes been used to help visualize aspects of higher dimensional spaces. [7]
This forms a dihedron, {1,2}, with two hemispherical monogonal faces which share one 360° edge and one vertex. Its dual, a hosohedron , {2,1} has two antipodal vertices at the poles, one 360° lune face, and one edge ( meridian ) between the two vertices.
A toroidal polyhedron with 6 × 4 = 24 quadrilateral faces. Polyhedra with the topological type of a torus are called toroidal polyhedra, and have Euler characteristic V − E + F = 0. For any number of holes, the formula generalizes to V − E + F = 2 − 2N, where N is the number of holes.
In geometry, isotoxal polyhedra and tilings are defined by the property that they have symmetries taking any edge to any other edge. [1] Polyhedra with this property can also be called "edge-transitive", but they should be distinguished from edge-transitive graphs , where the symmetries are combinatorial rather than geometric.
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon).Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube.
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,
When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its ...