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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.
The dual of a non-convex polyhedron is also a non-convex polyhedron. [2] ( By contraposition.) There are ten non-convex isotoxal polyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron: the five (quasiregular) hemipolyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron, and their five (infinite) duals:
In Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, the 12 isosceles faces are arranged differently so that the figure is non-convex and has right dihedral angles. It is scissors congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.
A rhombic dodecahedron is an isohedral and isotoxal polyhedron A great icosidodecahedron is an isogonal and isotoxal star polyhedron A great rhombic triacontahedron is an isohedral and isotoxal star polyhedron The trihexagonal tiling is an isogonal and isotoxal tiling The rhombille tiling is an isohedral and isotoxal tiling with p6m (*632 ...
Finite spherical symmetry groups are also called point groups in three dimensions.There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.
A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly 12 pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular and that there are always 12 of them. If the vertices are not constrained to a sphere, the polyhedron can be constructed with planar equilateral (but not in general ...
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 ...
(Definition varies among authors; e.g. some exclude solids with dihedral symmetry, or nonconvex solids.) Uniform if every face is a regular polygon, i.e. it is regular, quasiregular or semi-regular. Semi-uniform if its elements are also isogonal. Scaliform if all the edges are the same length. Noble if it is also isohedral (face-transitive).