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Similarly, a k-isohedral tiling has k separate symmetry orbits (it may contain m different face shapes, for m = k, or only for some m < k). [ 6 ] ("1-isohedral" is the same as "isohedral".) A monohedral polyhedron or monohedral tiling ( m = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions.
There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, [1] orbifold notation, [2] and order.
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 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 their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 59 such stellations of the regular icosahedron. Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them. Other stellations have more than one face in each plane or form compounds of simpler polyhedra.
This shape is called a plesiohedron. The tiling generated in this way is isohedral, meaning that it not only has a single prototile ("monohedral") but also that any copy of this tile can be taken to any other copy by a symmetry of the tiling. [1] As with any space-filling polyhedron, the Dehn invariant of a plesiohedron is necessarily zero. [3]
Each Catalan solid has constant dihedral angles, meaning the angle between any two adjacent faces is the same. [1] Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other.
[7] Each edge of the triakis icosahedron has endpoints of total degree at least 13. By Kotzig's theorem, this is the most possible for any polyhedron. The same total degree is obtained from the Kleetope of any polyhedron with minimum degree five, but the triakis icosahedron is the simplest example of this construction. [8]