Tessellation of space
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A tessellation of space fills space with solids, e.g. polyhedra.
For example, we can have parallel layers, each with prisms according to a tessellation of the plane. In particular, for every parallelepiped, copies can fill space.
A uniform tessellation is one with a symmetry group that matches each vertex to each other vertex. This implies in turn that the neighborhood of each vertex is alike, i.e. that the same arrangement of cells and faces is repeated (possibly rotated) at each vertex. For example, layers of right prisms according to the three regular tessellations in 2D; that with square cuboids is in a way the most regular, especially with cubes, because then it is congruent in three independent directions.
The Andreini tessellations are tilings of three-dimensional space using Platonic and Archimedean solids such that all vertices are identical. They are special case of uniform tessellation.
 rhombic dodecahedral honeycomb |  Bitruncated cubic honeycomb |  rhombo-hexagonal dodecahedra |
 Tetrahedral-octahedral honeycomb |  Internal view of cubic honeycomb |
External links
- [link] Uniform space-filling using only rhombo-hexagonal dodecahedra
- [link] Uniform space-filling using only rhombic dodecahedra
- [link] Uniform space-filling using only truncated octahedra
- [link] Uniform space-filling using triangular, square, and hexagonal prisms
- [link] Five space-filling polyhedra, Guy Inchbald
- [link] The Archimedean honeycomb duals, Guy Inchbald
- [link] "Guy Inchbald: The Archimedean honeycomb duals 213 July 1997 Article"
- The Mathematical Gazette 80, November 1996, p.p. 466-475.
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