Triangular prismatic honeycomb
Triangular prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  {3,6}×{∞} or t_{0,3}{3,6,2,∞} 
Coxeter diagrams  
Space group Coxeter notation 
[6,3,2,∞] [3^{[3]},2,∞] [(3^{[3]})^{+},2,∞] 
Dual  Hexagonal prismatic honeycomb 
Properties  vertextransitive 
The triangular prismatic honeycomb or triangular prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed entirely of triangular prisms.
It is constructed from a triangular tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Contents
 1 Related honeycombs
 1.1 Hexagonal prismatic honeycomb
 1.2 Trihexagonal prismatic honeycomb
 1.3 Truncated hexagonal prismatic honeycomb
 1.4 Rhombitrihexagonal prismatic honeycomb
 1.5 Snub hexagonal prismatic honeycomb
 1.6 Truncated trihexagonal prismatic honeycomb
 1.7 Elongated triangular prismatic honeycomb
 1.8 Gyrated triangular prismatic honeycomb
 1.9 Gyroelongated triangular prismatic honeycomb
 2 References
Related honeycombs[edit]
Hexagonal prismatic honeycomb[edit]
Hexagonal prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbols  {6,3}×{∞} or t_{0,1,3}{6,3,2,∞} 
Coxeter diagrams 

Cell types  4.4.6 
Vertex figure  triangular bipyramid 
Space group Coxeter notation 
[6,3,2,∞] [3^{[3]},2,∞] 
Dual  Triangular prismatic honeycomb 
Properties  vertextransitive 
The hexagonal prismatic honeycomb or hexagonal prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space made up of hexagonal prisms.
It is constructed from a hexagonal tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
This honeycomb can be alternated into the gyrated tetrahedraloctahedral honeycomb, with pairs of tetrahedra existing in the alternated gaps (instead of a triangular bipyramid).
Trihexagonal prismatic honeycomb[edit]
Trihexagonal prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  r{6,3}x{∞} or t_{1,3}{6,3}x{∞} 
Vertex figure  Rectangular bipyramid 
Coxeter diagram  
Space group Coxeter notation 
[6,3,2,∞] 
Dual  Rhombille prismatic honeycomb 
Properties  vertextransitive 
The trihexagonal prismatic honeycomb or trihexagonal prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1:2.
It is constructed from a trihexagonal tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Truncated hexagonal prismatic honeycomb[edit]
Truncated hexagonal prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  t{6,3}×{∞} or t_{0,1,3}{6,3,2,∞} 
Coxeter diagram  
Cell types  4.4.12 3.4.4 
Face types  {3}, {4}, {12} 
Edge figures  Square, Isosceles triangle 
Vertex figure  Triangular bipyramid 
Space group Coxeter notation 
[6,3,2,∞] 
Dual  Triakis triangular prismatic honeycomb 
Properties  vertextransitive 
The truncated hexagonal prismatic honeycomb or tomotrihexagonal prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of dodecagonal prisms, and triangular prisms in a ratio of 1:2.
It is constructed from a truncated hexagonal tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Rhombitrihexagonal prismatic honeycomb[edit]
Rhombitrihexagonal prismatic honeycomb  

Type  Uniform honeycomb 
Vertex figure  Trapezoidal bipyramid 
Schläfli symbol  rr{6,3}×{∞} or t_{0,2,3}{6,3,2,∞} s_{2}{3,6}×{∞} 
Coxeter diagram  
Space group Coxeter notation 
[6,3,2,∞] 
Dual  Deltoidal trihexagonal prismatic honeycomb 
Properties  vertextransitive 
The rhombitrihexagonal prismatic honeycomb or rhombitrihexagonal prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of hexagonal prisms, cubes, and triangular prisms in a ratio of 1:3:2.
It is constructed from a rhombitrihexagonal tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Snub hexagonal prismatic honeycomb[edit]
Snub hexagonal prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  sr{6,3}×{∞} 
Coxeter diagram  
Symmetry  [(6,3)^{+},2,∞] 
Dual  Floret pentagonal prismatic honeycomb 
Properties  vertextransitive 
The snub hexagonal prismatic honeycomb or simotrihexagonal prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1:8.
It is constructed from a snub hexagonal tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Truncated trihexagonal prismatic honeycomb[edit]
Truncated trihexagonal prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  tr{6,3}×{∞} or t_{0,1,2,3}{6,3,2,∞} 
Coxeter diagram  
Space group Coxeter notation 
[6,3,2,∞] 
Vertex figure  irr. triangular bipyramid 
Dual  Kisrhombille prismatic honeycomb 
Properties  vertextransitive 
The truncated trihexagonal prismatic honeycomb or tomotrihexagonal prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of dodecagonal prisms, hexagonal prisms, and cubes in a ratio of 1:2:3.
It is constructed from a truncated trihexagonal tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Elongated triangular prismatic honeycomb[edit]
Elongated triangular prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbols  {3,6}:e×{∞} s{∞}h_{1}{∞}×{∞} 
Coxeter diagrams  
Space group Coxeter notation 
[∞,2^{+},∞,2,∞] [(∞,2)^{+},∞,2,∞] 
Dual  Prismatic pentagonal prismatic honeycomb 
Properties  vertextransitive 
The elongated triangular prismatic honeycomb or elongated antiprismatic prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of cubes and triangular prisms in a ratio of 1:2.
It is constructed from an elongated triangular tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.
Gyrated triangular prismatic honeycomb[edit]
Gyrated triangular prismatic honeycomb  

Type  Convex uniform honeycomb 
Schläfli symbols  {3,6}:g×{∞} {4,4}f{∞} 
Cell types  (3.4.4) 
Face types  {3}, {4} 
Vertex figure  
Space group  [4,(4,2^{+},∞,2^{+})] ? 
Dual  ? 
Properties  vertextransitive 
The gyrated triangular prismatic honeycomb or parasquare fastigial cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space made up of triangular prisms. It is vertexuniform with 12 triangular prisms per vertex.
It can be seen as parallel planes of square tiling with alternating offsets caused by layers of paired triangular prisms. The prisms in each layer are rotated by a right angle to those in the next layer.
It is one of 28 convex uniform honeycombs.
Pairs of triangular prisms can be combined to create gyrobifastigium cells. The resulting honeycomb is closely related but not equivalent: it has the same vertices and edges, but different twodimensional faces and threedimensional cells.
Gyroelongated triangular prismatic honeycomb[edit]
Gyroelongated triangular prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbols  {3,6}:ge×{∞} {4,4}f_{1}{∞} 
Vertex figure  
Space group Coxeter notation 
[4,(4,2^{+},∞,2^{+})] ? 
Dual   
Properties  vertextransitive 
The gyroelongated triangular prismatic honeycomb or elongated parasquare fastigial cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of cubes and triangular prisms in a ratio of 1:2.
It is created by alternating laters of cubes and triangular prisms, with the prisms alternating in orientation by 90 degrees.
It is related to the elongated triangular prismatic honeycomb which has the triangular prisms with the same orientation.
This is related to a spacefilling polyhedron, elongated gyrobifastigium, where cube and two opposite triangular prisms are augmented together as a single polyhedron:
References[edit]
 George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
 Branko Grünbaum, Uniform tilings of 3space. Geombinatorics 4(1994), 49  56.
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (1.9 Uniform spacefillings)
 A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
 Klitzing, Richard. "3D Euclidean Honeycombs tiph".
 Uniform Honeycombs in 3Space VRML models