Search results
Results from the WOW.Com Content Network
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:
Infinite-order apeirogonal tiling; Infinite-order digonal tiling; ... Media in category "Isohedral tilings" The following 2 files are in this category, out of 2 total.
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.
This file contains additional information, probably added from the digital camera or scanner used to create or digitize it. If the file has been modified from its original state, some details may not fully reflect the modified file.
This nomenclature is used in the diagrams below, where the tiles are also colored by their k-isohedral positions within the symmetry. A primitive unit is a section of the tiling that generates the whole tiling using only translations, and is as small as possible.
This shape is called a plesiohedron. The tiling generated in this way is isohedral, meaning that it not only has a single prototile ("monohedral") but also that any copy of this tile can be taken to any other copy by a symmetry of the tiling. [1] As with any space-filling polyhedron, the Dehn invariant of a plesiohedron is necessarily zero. [3]
In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 14 triangles meeting at each vertex. The image shows a Poincaré disk model projection of the hyperbolic plane.
The dual to this tiling represents the fundamental domains of [∞,5*] symmetry, orbifold notation *∞∞∞∞∞ symmetry, a pentagonal domain with five ideal vertices. The order-5 apeirogonal tiling can be uniformly colored with 5 colored apeirogons around each vertex, and coxeter diagram: , except ultraparallel branches on the diagonals.