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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.
Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive). Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal. The dual of an isotoxal polyhedron is also an isotoxal polyhedron. (See the Dual polyhedron article.)
An example is the sphinx tiling, an aperiodic tiling formed by a pentagonal rep-tile. [20] The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a parallelogram and then tiling the plane by translation of this parallelogram, [ 20 ] a pattern that can be extended to any non-convex pentagon that has two ...
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 ...
§6.2 Isohedral tiling, §6.3 isogonal tiling, §6.4 isotoxal tiling, list of isotoxal tilings, §6.5 striped pattern, §6.6 Evgraf Fedorov, Alexei Vasilievich Shubnikov, planigon, Boris Delone: 7: Classification with respect to symmetries §7.1 Conjugate element, §7.7 arrangement of lines, §7.8 Circle packing: 8: Colored patterns and tilings
If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-isotoxal. k-uniform tilings with the same vertex figures can be further identified by their wallpaper group symmetry.
It is an isohedral (face-transitive) ... The hexagonal trapezohedron also exists as a spherical tiling, ... VRML model <6> Archived 2020-09-15 at the Wayback Machine
The snub square tiling is an Archimedean tiling, and as the dual to an Archimedean tiling this form of the Cairo pentagonal tiling is a Catalan tiling or Laves tiling. [14] It is one of two monohedral pentagonal tilings that, when the tiles have unit area, minimizes the perimeter of the tiles.