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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.
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.
The 3.12.12 vertex figure alone generates a truncated hexagonal tiling, while the 3.4.3.12 only exists in this 2-uniform tiling. There are 2 3-uniform tilings that contain both of these vertex figures among one more. It has square symmetry, p4m, [4,4], (*442). It is also called a demiregular tiling by some authors.
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 ...
All four tilings are 2-isohedral. The chiral pairs of tiles are colored in yellow and green for one isohedral set, and two shades of blue for the other set. The pgg symmetry is reduced to p2 when the chiral pairs are considered distinct. The tiling by type 9 tiles is edge-to-edge, but the others are not. Each primitive unit contains eight tiles.
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.
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