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A 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. Every regular honeycomb is automatically uniform. However, there is just one regular honeycomb in Euclidean 3-space, the cubic honeycomb.
A regular hexagonal grid This honeycomb forms a circle packing, with circles centered on each hexagon.. The honeycomb conjecture states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area.
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 hexagonal honeycomb conjecture The most efficient partition of the plane into equal areas is the regular hexagonal tiling. [12] Related to Thue's theorem. Dodecahedral conjecture The volume of the Voronoi polyhedron of a sphere in a packing of equal spheres is at least the volume of a regular dodecahedron with inradius 1.
Elongated alternated cubic honeycomb; Hexagonal prismatic honeycomb; Triangular prismatic honeycomb; Triangular-hexagonal prismatic honeycomb; Truncated hexagonal prismatic honeycomb; Truncated square prismatic honeycomb; Rhombitriangular-hexagonal prismatic honeycomb; Omnitruncated triangular-hexagonal prismatic honeycomb
Aluminum honeycomb structure Honeycomb structure in nature. Honeycomb structures are natural or man-made structures that have the geometry of a honeycomb to allow the minimization of the amount of used material to reach minimal weight and minimal material cost. The geometry of honeycomb structures can vary widely but the common feature of all ...
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 three-dimensional hyperbolic geometry, the alternated order-6 hexagonal tiling honeycomb is a uniform compact space-filling tessellation (or honeycomb).As an alternation, with Schläfli symbol h{4,3,6} and Coxeter-Dynkin diagram or , it can be considered a quasiregular honeycomb, alternating triangular tilings and tetrahedra around each vertex in a trihexagonal tiling vertex figure.