In geometry, a honeycomb is a name for a space-filling tessellation, just as a tiling is a tessellation of a plane or 2-dimensional surface. A tessellated plane seen in street pavement. ...

The term tiling is also sometimes used for 3-space tessellation, and honeycomb was defined to more clearly imply a solid tessellation. By generalization, the term is also sometimes used for higher dimensional tessellations as well. For clarity, George Olshevsky advocates limiting the term honeycomb as 3-space tessellations and expanding a systematic terminology for higher dimensions: tetracomb as tessellations of 4-space, and pentacomb as tessellations of 5-space, and so on. George Olshevsky is a freelance editor, writer, publisher, paleontologist, and mathematician living in San Diego, California. ...

A honeycomb as a space-filling tessellation applies to tessellations of hyperbolic space, such as the hyperbolic small dodecahedral honeycomb. The hyperbolic small dodecahedral honeycomb is one of four regular tessellation (or honeycomb) in hyperbolic 3-space. ...

There are 28 convex uniform honeycombs - the set of uniform honeycombs in Euclidean 3-space filled by convex uniform polyhedral cells. A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A cell is a three-dimensional object that is part of the boundary of a higher-dimensional object, such as a polychoron. ...

Other well-known cell-uniform honeycombs include the rhombic dodecahedral honeycomb, and the rhombo-hexagonal dodecahedron honeycomb. The rhombic dodecahedra honeycomb is a tessellation (or honeycomb) in Euclidean 3-space. ... The rhombo-hexagonal dodecahedron is a convex polyhedron with 8 rhombic and 4 equilateral hexagonal faces. ...

The simplest honeycombs can be stacked layers of prisms according to a tessellation of the plane. In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being a specialize regular form. In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... In geometry, a parallelepiped (pronounced ; meaning of parallel planes) or parallelopipedon is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ...

rhombic dodecahedral honeycomb	Bitruncated cubic honeycomb	rhombo-hexagonal dodecahedra
Tetrahedral-octahedral honeycomb	cubic honeycomb	dodecahedral honeycomb in hyperbolic space.

Space tesselation of rhombic dodecahedra. ... The rhombic dodecahedra honeycomb is a tessellation (or honeycomb) in Euclidean 3-space. ... Tesselation of space using truncated octahedra. ... The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ... Image File history File links Download high resolution version (796x612, 33 KB) Summary Rhombo-hexagonal dodecahedron tessellation Licensing I, the creator of this work, hereby release it into the public domain. ... The rhombo-hexagonal dodecahedron is a convex polyhedron with 8 rhombic and 4 equilateral hexagonal faces. ... Image File history File links Tetrahedral-octahedral_honeycomb. ... The tetrahedral-octahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of alternating tetrahedra and octahedra. ... Image File history File links Download high resolution version (902x902, 1221 KB) Summary View of cubic honeycomb generated by software: [1] Curved Spaces v1. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high resolution version (902x902, 1341 KB) Summary View of Hyperbolic small dodecahedral honeycomb generated by software: [1] Curved Spaces v1. ... The hyperbolic small dodecahedral honeycomb is one of four regular tessellation (or honeycomb) in hyperbolic 3-space. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ...

See also

• Regular honeycombs

This page lists the regular polytopes in Euclidean space. ...

External links

• [1] Uniform space-filling using only rhombo-hexagonal dodecahedra
• [2] Uniform space-filling using only rhombic dodecahedra
• [3] Uniform space-filling using only truncated octahedra
• [4] Uniform space-filling using triangular, square, and hexagonal prisms
• [5] Five space-filling polyhedra, Guy Inchbald
• [6] The Archimedean honeycomb duals, Guy Inchbald
• The Mathematical Gazette 80, November 1996, p.p. 466-475.