![]() ![]() Two are quasiregular (made from two types of regular cells): However, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb. Every regular honeycomb is automatically uniform. ![]() There are 28 convex examples in Euclidean 3-space, also called the Archimedean honeycombs.Ī honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Another interesting family is the Hill tetrahedra and their generalizations, which can also tile the space.Ī 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of is transitive on vertices). In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. There are infinitely many honeycombs, which have only been partially classified. However, not all geometers accept such hexagons. Interpreting each brick face as a hexagon having two interior angles of 180 degrees allows the pattern to be considered as a proper tiling. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell. It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles, as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. ![]() Honeycombs are usually constructed in ordinary Euclidean ("flat") space. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. For other uses, see Honeycomb (disambiguation). ![]()
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