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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. The unit-radius 600-cell has tetrahedral cells of edge length 1 φ {\textstyle {\frac {1}{\varphi }}} , 20 of which meet at each vertex to form an icosahedral pyramid (a 4 ...
Construction from the vertices of a truncated octahedron, showing internal rectangles. The Cartesian coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.
The truncated icosahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. [5] It has the same symmetry as the regular icosahedron, the icosahedral symmetry, and it also has the property of vertex-transitivity.
The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio [11]
In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron, one of the Archimedean solids. It has 60 vertices and 120 edges, and is a quartic graph Archimedean graph. [5] Square centered Schlegel diagram
Icosahedral symmetry fundamental domains A soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry. Rotations and reflections form the symmetry group of a great icosahedron. In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron.
The vertices of Jessen's icosahedron may be chosen to have as their coordinates the twelve triplets given by the cyclic permutations of the coordinates (,,). [1] With this coordinate representation, the short edges of the icosahedron (the ones with convex angles) have length 6 {\displaystyle {\sqrt {6}}} , and the long (reflex) edges have ...
The truncation involves cutting away corners; to preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners, and an example can be found in truncated icosahedron constructed by cutting off all the icosahedron's vertices, having the same symmetry as the ...