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The rectified order-5 hexagonal tiling honeycomb, t 1 {6,3,5}, has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure. It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.
These 12 order-5 vertices can be truncated such that all edges are equal length. The original 30 rhombic faces become non-regular hexagons, and the truncated vertices become regular pentagons. The hexagon faces can be equilateral but not regular with D 2 symmetry.
In geometry, the truncated order-5 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t 0,1 {6,5}. Related polyhedra and tiling
The cD can more accurately be called a pentatruncated rhombic triacontahedron, because only the (12) order-5 vertices of the rhombic triacontahedron are truncated. The dual of the chamfered dodecahedron is the pentakis icosidodecahedron. The cD is the Goldberg polyhedron GP V (2,0) or {5+,3} 2,0, containing pentagonal and hexagonal faces.
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane.There are 2 dodecagons (12-sides) and one triangle on each vertex.. As the name implies this tiling is constructed by a truncation operation applied to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations.
Hexagonal tiling is the densest way to arrange circles in two dimensions. 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 truncated hexagonal tiling honeycomb, t 0,1 {6,3,3}, has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure. It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling , t{∞,3} with apeirogonal and triangle faces:
Truncated order-5 square tiling: 5.8 2: t{4,5} Truncated order-5 pentagonal tiling: 5.10 2: t{5,5} Truncated order-5 hexagonal tiling: 5.12 2: t{6,5} Pentaapeirogonal ...