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Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal. [5] The first is made of 2-edges, three around every vertex, the second has hexagonal edges, three around every vertex.
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]
Architectonic tessellation Catoptric tessellation Name Coxeter diagram Image Vertex figure Image Cells Name Cell Vertex figures; J 11,15 A 1 W 1 G 22 δ 4 nc [4,3,4] Cubille (Cubic honeycomb) Octahedron, Cubille: Cube, J 12,32 A 15 W 14 G 7 t 1 δ 4 nc [4,3,4] Cuboctahedrille (Rectified cubic honeycomb) Cuboid, Oblate octahedrille: Isosceles ...
The tesseract can make a regular tessellation of 4-dimensional hyperbolic space, with 5 tesseracts around each face, with Schläfli symbol {4,3,3,5}, called an order-5 tesseractic honeycomb. The Ammann–Beenker tiling is an aperiodic tiling in 2 dimensions obtained by cut-and-project on the tesseractic honeycomb along an eightfold rotational ...
It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs.
The snub square tiling, made of two squares and three equilateral triangles around each vertex, has a bilaterally symmetric Cairo tiling as its dual tiling. [13] The Cairo tiling can be formed from the snub square tiling by placing a vertex of the Cairo tiling at the center of each square or triangle of the snub square tiling, and connecting ...
In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is called an einstein, a word play on ein Stein, German for "one stone". [2]
The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal spheres in ordinary space (see Kepler conjecture ).