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In geometry, isotoxal polyhedra and tilings are defined by the property that they have symmetries taking any edge to any other edge. [1] Polyhedra with this property can also be called "edge-transitive", but they should be distinguished from edge-transitive graphs, where the symmetries are combinatorial rather than geometric.
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 also 2-isohedral tilings by special cases of type 1, type 2, and type 4 tiles, and 3-isohedral tilings, all edge-to-edge, by special cases of type 1 tiles. There is no upper bound on k for k-isohedral tilings by certain tiles that are both type 1 and type 2, and hence neither on the number of tiles in a primitive unit.
Pages in category "Isohedral tilings" The following 76 pages are in this category, out of 76 total. ... Octagonal tiling; Order-1 digonal tiling; Order-2 apeirogonal ...
The problem of anisohedral tiling has been generalised by saying that the isohedral number of a tile is the lowest number orbits (equivalence classes) of tiles in any tiling of that tile under the action of the symmetry group of that tiling, and that a tile with isohedral number k is k-anisohedral.
The second part of Hilbert's eighteenth problem asked for a single polyhedron tiling Euclidean 3-space, such that no tiling by it is isohedral (an anisohedral tile). The problem as stated was solved by Karl Reinhardt in 1928, but sets of aperiodic tiles have been considered as a natural extension. [7]
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