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Tessellated designs often appear on textiles, whether woven, stitched in, or printed. Tessellation patterns have been used to design interlocking motifs of patch shapes in quilts. [75] [76] Tessellations are also a main genre in origami (paper folding), where pleats are used to connect molecules, such as twist folds, together in a repeating ...
= : Euclidean 4-space tessellation > : hyperbolic 4-space tessellation. Enumeration of these constraints produce 3 convex polytopes, no star polytopes, 3 tessellations of Euclidean 4-space, and 5 tessellations of paracompact hyperbolic 4-space. The only non-convex regular polytopes for ranks 5 and higher are skews.
In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex.
The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3, with a mirrored sphenoid vertex figure.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space.It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive.
These are just the rules for dualising four-dimensional 4-polytopes, except that the usual finite method of reciprocation about a concentric hypersphere can run into problems. The more regular honeycombs dualise neatly: The cubic honeycomb is self-dual. That of octahedra and tetrahedra is dual to that of rhombic dodecahedra.
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space.With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.