Isohedral figure

==Classes of isohedra by symmetry==

A polyhedron (or polytope in general) is '''k-isohedral' if it contains k faces within its symmetry fundamental domains. 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''). ("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. An '''m-hedral' polyhedron or tiling has m different face shapes ("dihedral", "trihedral''"... are the same as "2-hedral", "3-hedral"... respectively). Here are some examples of k-isohedral polyhedra and tilings, with their faces colored by their k symmetry positions: ==Related terms==

A cell-transitive or isochoric figure is an n-polytope (n ≥ 4) or n-honeycomb (n ≥ 3) that has its cells congruent and transitive with each others. In 3 dimensions, the catoptric honeycombs, duals to the uniform honeycombs, are isochoric. In 4 dimensions, isochoric polytopes have been enumerated up to 20 cells. A facet-transitive or isotopic figure is an n-dimensional polytope or honeycomb with its facets ((n−1)-faces) congruent and transitive. The dual of an isotope is an isogonal polytope. By definition, this isotopic property is common to the duals of the uniform polytopes. • An isotopic 2-dimensional figure is isotoxal, i.e. edge-transitive. • An isotopic 3-dimensional figure is isohedral, i.e. face-transitive. • An isotopic 4-dimensional figure is isochoric, i.e. cell-transitive. ==See also==