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The body of the tables contain the characters in the respective irreducible representations for each respective symmetry operation, or set of symmetry operations. The symbol i used in the body of the table denotes the imaginary unit: i 2 = −1. Used in a column heading, it denotes the operation of inversion.
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.
Pyritohedral symmetry has the symbol (3*2), [3 +,4], with order 24. Tetrahedral symmetry has the symbol (332), [3,3] + , with order 12. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles .
This list includes these: all 75 nonprismatic uniform polyhedra; a few representatives of the infinite sets of prisms and antiprisms; one degenerate polyhedron, Skilling's figure with overlapping edges. It was proven in Sopov (1970) that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling ...
Chemical symbols are the abbreviations used in chemistry, mainly for chemical elements; but also for functional groups, chemical compounds, and other entities. Element symbols for chemical elements, also known as atomic symbols , normally consist of one or two letters from the Latin alphabet and are written with the first letter capitalised.
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 ...
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.
Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property.