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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.
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 ...
Regular Division of the Plane III, woodcut, 1957 - 1958.. Regular Division of the Plane is a series of drawings by the Dutch artist M. C. Escher which began in 1936. These images are based on the principle of tessellation, irregular shapes or combinations of shapes that interlock completely to cover a surface or plane.
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 ]
3D model of a truncated icosahedron. In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. . Intuitively, it may be regarded as footballs (or soccer balls) that are typically patterned with white hexagons and black pentag
The rhombic dodecahedron is a space-filling polyhedron, meaning it can be applied to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane. It is a parallelohedron because it can be space-filling a honeycomb in which all of its copies meet face-to-face. [ 7 ]
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.