Triangular prismatic honeycomb

The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms.

Triangular prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbol	{3,6}×{∞} or t0,3{3,6,2,∞}
Coxeter diagrams
Space group Coxeter notation	[6,3,2,∞] [3[3],2,∞] [(3[3])+,2,∞]
Dual	Hexagonal prismatic honeycomb
Properties	vertex-transitive

It is constructed from a triangular tiling extruded into prisms.

It is one of 28 convex uniform honeycombs.

It consists of 1 + 6 + 1 = 8 edges meeting at a vertex, There are 6 triangular prism cells meeting at an edge and faces are shared between 2 cells.

Related honeycombs

Hexagonal prismatic honeycomb

Hexagonal prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbols	{6,3}×{∞} or t0,1,3{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	vertex-transitive

The hexagonal prismatic honeycomb or hexagonal prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space 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 tetrahedral-octahedral honeycomb, with pairs of tetrahedra existing in the alternated gaps (instead of a triangular bipyramid).

There are 1 + 3 + 1 = 5 edges meeting at a vertex, 3 Hexagonal Prism cells meeting at an edge, and faces are shared between 2 cells.

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Trihexagonal prismatic honeycomb

Trihexagonal prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbol	r{6,3}x{∞} or t1,3{6,3}x{∞}
Vertex figure	Rectangular bipyramid
Coxeter diagram
Space group Coxeter notation	[6,3,2,∞]
Dual	Rhombille prismatic honeycomb
Properties	vertex-transitive

The trihexagonal prismatic honeycomb or trihexagonal prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. 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.

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Truncated hexagonal prismatic honeycomb

Truncated hexagonal prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t{6,3}×{∞} or t0,1,3{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	vertex-transitive

The truncated hexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. 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.

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Rhombitrihexagonal prismatic honeycomb

Rhombitrihexagonal prismatic honeycomb
Type	Uniform honeycomb
Vertex figure	Trapezoidal bipyramid
Schläfli symbol	rr{6,3}×{∞} or t0,2,3{6,3,2,∞} s2{3,6}×{∞}
Coxeter diagram
Space group Coxeter notation	[6,3,2,∞]
Dual	Deltoidal trihexagonal prismatic honeycomb
Properties	vertex-transitive

The rhombitrihexagonal prismatic honeycomb or rhombitrihexagonal prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. 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.

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Truncated trihexagonal prismatic honeycomb

Truncated trihexagonal prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbol	tr{6,3}×{∞} or t0,1,2,3{6,3,2,∞}
Coxeter diagram
Space group Coxeter notation	[6,3,2,∞]
Vertex figure	irr. triangular bipyramid
Dual	Kisrhombille prismatic honeycomb
Properties	vertex-transitive

The truncated trihexagonal prismatic honeycomb or tomo-trihexagonal prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. 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.

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Snub trihexagonal prismatic honeycomb

Snub trihexagonal prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbol	sr{6,3}×{∞}
Coxeter diagram
Symmetry	[(6,3)+,2,∞]
Dual	Floret pentagonal prismatic honeycomb
Properties	vertex-transitive

The snub trihexagonal prismatic honeycomb or simo-trihexagonal prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1:8.

It is constructed from a snub trihexagonal tiling extruded into prisms.

It is one of 28 convex uniform honeycombs.

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Snub trihexagonal antiprismatic honeycomb

Snub trihexagonal antiprismatic honeycomb
Type	Convex honeycomb
Schläfli symbol	ht0,1,2,3{6,3,2,∞}
Coxeter-Dynkin diagram
Cells	hexagonal antiprism octahedron tetrahedron
Vertex figure
Symmetry	[6,3,2,∞]+
Properties	vertex-transitive

A snub trihexagonal antiprismatic honeycomb can be constructed by alternation of the truncated trihexagonal prismatic honeycomb, although it can not be made uniform, but it can be given Coxeter diagram: and has symmetry [6,3,2,∞]⁺. It makes hexagonal antiprisms from the dodecagonal prisms, octahedra (as triangular antiprisms) from the hexagonal prisms, tetrahedra (as tetragonal disphenoids) from the cubes, and two tetrahedra from the triangular bipyramids.

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Elongated triangular prismatic honeycomb

Elongated triangular prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbols	{3,6}:e×{∞} s{∞}h1{∞}×{∞}
Coxeter diagrams
Space group Coxeter notation	[∞,2+,∞,2,∞] [(∞,2)+,∞,2,∞]
Dual	Prismatic pentagonal prismatic honeycomb
Properties	vertex-transitive

The elongated triangular prismatic honeycomb or elongated antiprismatic prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. 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.

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Gyrated triangular prismatic honeycomb

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	vertex-transitive

The gyrated triangular prismatic honeycomb or parasquare fastigial cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of triangular prisms. It is vertex-uniform 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 two-dimensional faces and three-dimensional cells.

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Gyroelongated triangular prismatic honeycomb

Gyroelongated triangular prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbols	{3,6}:ge×{∞} {4,4}f1{∞}
Vertex figure
Space group Coxeter notation	[4,(4,2+,∞,2+)] ?
Dual	-
Properties	vertex-transitive

The gyroelongated triangular prismatic honeycomb or elongated parasquare fastigial cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.

It is created by alternating layers 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 space-filling polyhedron, elongated gyrobifastigium, where cube and two opposite triangular prisms are augmented together as a single polyhedron:

References

• Olshevsky, George (2006). "Uniform Panoploid Tetracombs" (PDF). (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Grünbaum, Branko (1994). "Uniform tilings of 3-space". Geombinatorics. 4 (2): 49–56.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 978-0-471-01003-6.
  • Paper 22: Coxeter, H.S.M. (1940). "Regular and Semi-Regular Polytopes I". Mathematische Zeitschrift. 46: 380–407. doi:10.1007/BF01181449. 1.9 Uniform space-fillings
• Andreini, A. (1905). "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): 75–129.
• Klitzing, Richard. "3D Euclidean Honeycombs tiph".
• Uniform Honeycombs in 3-Space VRML models