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] ,2,∞][3] )+ ,2,∞]
Dual
Hexagonal prismatic honeycomb
Properties
vertex-transitive
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 .
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.
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] ,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.
Trihexagonal prismatic honeycomb
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 .
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 .
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,∞}2 {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 .
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 .
Snub trihexagonal prismatic honeycomb
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 .
Snub trihexagonal antiprismatic honeycomb
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 : + . 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 .
Elongated triangular prismatic honeycomb
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 .
Gyrated triangular prismatic honeycomb
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.
Gyroelongated triangular prismatic honeycomb
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) . 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 ISBN 978-0-471-01003-6 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
