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The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" ( /- d r ə / ) or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others.
It has icosahedral symmetry (I h) and the same vertex arrangement as a rhombic triacontahedron. This can be seen as the three-dimensional equivalent of the compound of two pentagons ({10/2} "decagram"); this series continues into the fourth dimension as the compound of 120-cell and 600-cell and into higher dimensions as compounds of hyperbolic ...
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 ...
Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools.
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.
For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak. Vertex figure : not itself an element of a polytope, but a diagram showing how the elements meet.
An example can be found in the model of a buckminsterfullerene, a truncated icosahedron-shaped geodesic dome allotrope of elemental carbon discovered in 1985. [17] In other engineering and science applications, its shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both the gadget and ...
In the above coordinates, the first 12 vertices form a regular icosahedron, the next 20 vertices (those with R) form a regular dodecahedron, and the last 30 vertices (those with S) form an icosidodecahedron. Normalizing all vertices to the unit sphere gives a spherical disdyakis triacontahedron, shown in the adjacent figure.