The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.
A geometric honeycomb is a space-filling 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.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
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The triangulartilinghoneycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called paracompact...
geometry, the triangulartiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the...
is one of three regular tilings of the plane. The other two are the triangulartiling and the square tiling. Hexagonal tiling is the densest way to arrange...
entirely of triangular prisms. It is constructed from a triangulartiling extruded into prisms. It is one of 28 convex uniform honeycombs. It consists...
hexagonal tiling cells: The rectified hexagonal tilinghoneycomb, t1{6,3,3}, has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex...
(hexagonal) symmetry form projects into a triangulartiling. A square symmetry projection forms a square tiling. It is related to the regular 4-polytope...
honeycombs with square tiling cells: The rectified square tilinghoneycomb, t1{4,4,3}, has cube and square tiling facets, with a triangular prism vertex figure...
derives from the fact that it combines a regular hexagonal tiling and a regular triangulartiling. Two hexagons and two triangles alternate around each vertex...
In geometry, the elongated triangulartiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex...
There are only 3 unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by...
{3}}}}.} The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. The centers of the hexagons of a honeycomb form a hexagonal...
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