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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 .
In geometry, the rectified truncated icosahedron is a convex polyhedron. It has 92 faces: 60 isosceles triangles , 12 regular pentagons , and 20 regular hexagons . It is constructed as a rectified , truncated icosahedron , rectification truncating vertices down to mid-edges.
This polyhedron is the truncation of the great icosahedron: . The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, [1] great rhombicosidodecahedron, [2] [3] omnitruncated dodecahedron or omnitruncated icosahedron [4] is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
The pentakis truncated icosahedron is a convex polyhedron constructed as an augmented truncated icosahedron, adding pyramids to the 12 pentagonal faces, creating 60 new triangular faces. It is geometrically similar to the icosahedron where the 20 triangular faces are subdivided with a central hexagon, and 3 corner triangles.
In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U 68. It has 62 faces (30 squares , 20 hexagons , and 12 decagrams ), 180 edges, and 120 vertices. [ 1 ]
The triakis icosahedron is a Catalan solid, the dual polyhedron of the truncated dodecahedron. The truncated dodecahedron is an Archimedean solid , with faces that are regular decagons and equilateral triangles , and with all edges having unit length; its vertices lie on a common sphere, the circumsphere of the truncated decahedron.
A regular icosahedron can be distorted or marked up as a lower pyritohedral symmetry, [2] [3] and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron, and pseudo-icosahedron. [4] This can be seen as an alternated truncated octahedron .