Search results
Results from the WOW.Com Content Network
Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. Some tetrahedra that are not regular, including the Schläfli orthoscheme and the Hill tetrahedron, can tessellate.
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
In geometry, a tessellation of dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit.
If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. The Conway criterion is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane. [19]
It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles, as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a ...
Therefore, the second problem is that this nomenclature is not unique for each tessellation. In order to solve those problems, GomJau-Hogg’s notation [ 3 ] is a slightly modified version of the research and notation presented in 2012, [ 2 ] about the generation and nomenclature of tessellations and double-layer grids.
You might be surprised by how many popular movie quotes you're remembering just a bit wrong. 'The Wizard of Oz' Though most people say 'Looks like we're not in Kansas anymore,' or 'Toto, I don't think
A shape that is rep-n or irrep-n is trivially also irrep-(kn − k + n) for any k > 1, by replacing the smallest tile in the rep-n dissection by n even smaller tiles. The order of a shape, whether using rep-tiles or irrep-tiles is the smallest possible number of tiles which will suffice.