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Cubic honeycomb. In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. 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.
The alternated cubic honeycomb is one of 28 space-filling uniform tessellations in Euclidean 3-space, composed of alternating yellow tetrahedra and red octahedra.. In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs , generated as Wythoff constructions , and represented by permutations of rings of the Coxeter diagrams for each family.
A honeycomb is a natural tessellated structure. The honeycomb is a well-known example of tessellation in nature with its hexagonal cells. [82] In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit.
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 ).
A dual alternated omnitruncated cubic honeycomb is a space-filling honeycomb constructed as the dual of the alternated omnitruncated cubic honeycomb. 24 cells fit around a vertex, making a chiral octahedral symmetry that can be stacked in all 3-dimensions:
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of ...
The honeycomb conjecture states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or body-centered cubic lattice) is ...