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A regular icosahedron is topologically identical to a cuboctahedron with its 6 square faces bisected on diagonals with pyritohedral symmetry. The icosahedra with pyritohedral symmetry constitute an infinite family of polyhedra which include the cuboctahedron, regular icosahedron, Jessen's icosahedron, and double cover octahedron. Cyclical ...
An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa. [17] An icosahedron can be inscribed in an octahedron by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two golden sections. Because the golden sections are ...
Three members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid; the icosahedron, a Platonic solid; and the gyroelongated triangular bipyramid if it is made with equilateral triangles, but because it has coplanar faces is not strictly convex.
In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (J 48).As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either J 34 or the icosidodecahedron) by inserting a decagonal antiprism between its two halves.
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 runcitruncated icosahedral honeycomb, t 0,1,3 {3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism, and triangular prism cells, with an isosceles-trapezoidal pyramid vertex figure. The runcicantellated icosahedral honeycomb is equivalent to the runcitruncated icosahedral honeycomb. Viewed from center of triangular prism
If the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron.It can also be considered as a parabidiminished icosahedron, a shape formed by removing two pentagonal pyramids from a regular icosahedron leaving two nonadjacent pentagonal faces; a related shape, the metabidiminished icosahedron (one of the Johnson solids), is likewise form from the icosahedron by ...
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.
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