Search results
Results from the WOW.Com Content Network
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.
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.
Angles are A = 140°, B = 60°, C = 160°, D = 80°, E = 100°. [ 13 ] [ 14 ] In 2016 it could be shown by Bernhard Klaassen that every discrete rotational symmetry type can be represented by a monohedral pentagonal tiling from the same class of pentagons. [ 15 ]
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]
The {m/2, m} tilings are stellations of the {m, 3} tilings while the {m, m/2} dual tilings are facetings of the {3, m} tilings and greatenings [ii] of the {m, 3} tilings. The patterns { m /2, m } and { m , m /2} continue for odd m < 7 as polyhedra : when m = 5, we obtain the small stellated dodecahedron and great dodecahedron , [ 18 ] and when ...
The Cairo tiling has been described as one of M. C. Escher's "favorite geometric patterns". [7] He used it as the basis for his drawing Shells and Starfish (1941), in the bees-on-flowers segment of his Metamorphosis III (1967–1968), and in several other drawings from 1967–1968.
Cubic honeycomb. In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
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.