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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 .
Convex regular icosahedron A tensegrity icosahedron. In geometry, an icosahedron (/ ˌ aɪ k ɒ s ə ˈ h iː d r ən,-k ə-,-k oʊ-/ or / aɪ ˌ k ɒ s ə ˈ h iː d r ən / [1]) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) ' twenty ' and ἕδρα (hédra) ' seat '.
It is constructed as a rectified, truncated icosahedron, rectification truncating vertices down to mid-edges. As a near-miss Johnson solid , under icosahedral symmetry , the pentagons are always regular, although the hexagons, while having equal edge lengths, do not have the same edge lengths with the pentagons, having slightly different but ...
The regular icosahedron's dual polyhedron is the regular dodecahedron, and their relation has a historical background on the comparison mensuration. It is analogous to a four-dimensional polytope, the 600-cell. The appearance of regular icosahedron can be found in nature, such as virus with icosahedral-shaped shells and radiolarians.
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.
Net 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 .
Let be the golden ratio.The 12 points given by (,,) and cyclic permutations of these coordinates are the vertices of a regular icosahedron.Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points (,,) together with the points (, /,) and cyclic permutations of these coordinates.
Net A geometric construction of the Tribonacci constant (AC), with compass and marked ruler, according to the method described by Xerardo Neira. 3d model of a pentagonal icositetrahedron In geometry , a pentagonal icositetrahedron or pentagonal icosikaitetrahedron [ 1 ] is a Catalan solid which is the dual of the snub cube .