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A 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 [isometries of 3-space that preserve the tiling] is transitive on vertices). There are 28 convex examples in Euclidean 3-space, [1] also called the Archimedean honeycombs.
The open end of a cell is typically referred to as the top of the cell, while the opposite end is called the bottom. The cells slope slightly upwards, between 9 and 14°, towards the open ends. [citation needed] Two possible explanations exist as to why honeycomb is composed of hexagons rather than any other shape.
In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} around each edge.
The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t 0,2,3 {6,3,3}, has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.
In the geometry of hyperbolic 3-space, the order-7-3 hexagonal honeycomb (or 6,7,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 hexagonal tiling whose vertices lie on a 2-hypercycle , each of which has a limiting circle on the ideal sphere.
The first generates repeated honeycomb, and the last two are nonuniform but included for completeness. The honeycombs from this group are called alternated cubic because the first form can be seen as a cubic honeycomb with alternate vertices removed, reducing cubic cells to tetrahedra and creating octahedron cells in the gaps.
In the geometry of hyperbolic 3-space, the order-6-3 hexagonal honeycomb or 6,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 hexagonal tiling whose vertices lie on a 2-hypercycle , each of which has a limiting circle on the ideal sphere.
In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-simplex facets. The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the ...