# Triangular tiling honeycomb > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Triangular_tiling_honeycomb > Markdown URL: https://mediated.wiki/source/Triangular_tiling_honeycomb.md > Source: https://en.wikipedia.org/wiki/Triangular_tiling_honeycomb > Source revision: 1305161998 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Triangular tiling honeycomb |- |bgcolor=#ffffff align=center colspan=2|320px |- |bgcolor=#e7dcc3|Type||[Hyperbolic regular honeycomb](/source/List_of_regular_polytopes)
[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||{3,6,3}
h{6,3,6}
h{6,3[3]} ↔ {3[3,3]} |- |bgcolor=#e7dcc3|[Coxeter-Dynkin diagram](/source/Coxeter-Dynkin_diagram)s||{{CDD|node_1|3|node|6|node|3|node}}
{{CDD|node_h1|6|node|3|node|6|node}} ↔ {{CDD|branch_10ru|split2|node|6|node}}
{{CDD|node_h1|6|node|split1|branch}} ↔ {{CDD|node_1|splitsplit1|branch4|splitsplit2|node}} ↔ {{CDD|branch_10ru|split2|node|6|node_h0}} |- |bgcolor=#e7dcc3|Cells||[{3,6}](/source/Triangular_tiling) 40px 40px |- |bgcolor=#e7dcc3|Faces||[triangle](/source/triangle) {3} |- |bgcolor=#e7dcc3|[Edge figure](/source/Edge_figure)||[triangle](/source/triangle) {3} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||40px 40px 40px
[hexagonal tiling](/source/hexagonal_tiling) |- |bgcolor=#e7dcc3|[Dual](/source/Dual_polytope)||[Self-dual](/source/Self-dual_polytope) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)s||\overline{Y}_3, [3,6,3]
\overline{VP}_3, [6,3[3]]
\overline{PP}_3, [3[3,3]] |- |bgcolor=#e7dcc3|Properties||Regular |} The '''triangular tiling honeycomb''' is one of 11 paracompact regular space-filling [tessellation](/source/tessellation)s (or [honeycombs](/source/honeycomb_(geometry))) in [hyperbolic 3-space](/source/Hyperbolic_space). It is called ''paracompact'' because it has infinite [cells](/source/Cell_(geometry)) and [vertex figure](/source/vertex_figure)s, with all vertices as [ideal point](/source/ideal_point)s at infinity. It has [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {3,6,3}, being composed of [triangular tiling](/source/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](/source/vertex_figure) is a [hexagonal tiling](/source/hexagonal_tiling). {{Honeycomb}} == Symmetry == [[File:Hyperbolic subgroup tree 363.png|left|thumb|Subgroups of [3,6,3] and [6,3,6]]] It has two lower reflective symmetry constructions, as an [alternated](/source/Alternation_(geometry)) [order-6 hexagonal tiling honeycomb](/source/order-6_hexagonal_tiling_honeycomb), {{CDD|node_h1|6|node|3|node|6|node}} ↔ {{CDD|branch_10ru|split2|node|6|node}}, and as {{CDD|node_1|splitsplit1|branch4|splitsplit2|node}} from {{CDD|node_1|3|node|6|node_g|3sg|node_g}}, which alternates 3 types (colors) of triangular tilings around every edge. In [Coxeter notation](/source/Coxeter_notation), the removal of the 3rd and 4th mirrors, [3,6,3*] creates a new [Coxeter group](/source/Coxeter_group) [3[3,3]], {{CDD|node|splitsplit1|branch4|splitsplit2|node}}, subgroup index 6. The fundamental domain is 6 times larger. By Coxeter diagram there are 3 copies of the first original mirror in the new fundamental domain: {{CDD|node_c2|3|node_c1|6|node|3|node}} ↔ {{CDD|node_c2|splitsplit1|branch4_c1|splitsplit2|node_c1}}. {{Clear}} == Related Tilings == It is similar to the 2D hyperbolic [infinite-order apeirogonal tiling](/source/infinite-order_apeirogonal_tiling), {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface. : 240px == Related honeycombs == The triangular tiling honeycomb is a [regular hyperbolic honeycomb](/source/List_of_regular_polytopes) in 3-space, and one of eleven paracompact honeycombs. {{Regular_paracompact_H3_honeycombs}} There are [nine uniform honeycombs](/source/Paracompact_uniform_honeycombs) in the [3,6,3] [Coxeter group](/source/Coxeter_group) family, including this regular form as well as the [bitruncated](/source/Bitruncation_(geometry)) form, t1,2{3,6,3}, {{CDD|node|3|node_1|6|node_1|3|node}} with all [truncated hexagonal tiling](/source/truncated_hexagonal_tiling) facets. {{363_family}} The honeycomb is also part of a series of [polychora](/source/Regular_4-polytope) and honeycombs with triangular [edge figure](/source/edge_figure)s. {{Symmetric_tessellations}} === Rectified triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Rectified triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||r{3,6,3}
h2{6,3,6} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)||{{CDD|node|3|node_1|6|node|3|node}}
{{CDD|node_h1|6|node|3|node_1|6|node}} ↔ {{CDD|branch_10ru|split2|node_1|6|node}}
{{CDD|node|splitsplit1|branch4_11|splitsplit2|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|6|node_h0}} ↔ {{CDD|node|3|node_1|6|node_g|3sg|node_g}} |- |bgcolor=#e7dcc3|Cells||[r{3,6}](/source/trihexagonal_tiling) 40px
[{6,3}](/source/hexagonal_tiling) 40px |- |bgcolor=#e7dcc3|Faces||[triangle](/source/triangle) {3}
[hexagon](/source/hexagon) {6} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||80px
[triangular prism](/source/triangular_prism) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||\overline{Y}_3, [3,6,3]
\overline{VP}_3, [6,3[3]]
\overline{PP}_3, [3[3,3]] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive |} The '''rectified triangular tiling honeycomb''', {{CDD|node|3|node_1|6|node|3|node}}, has [trihexagonal tiling](/source/trihexagonal_tiling) and [hexagonal tiling](/source/hexagonal_tiling) cells, with a [triangular prism](/source/triangular_prism) vertex figure. ==== Symmetry==== A lower symmetry of this honeycomb can be constructed as a ''cantic order-6 hexagonal tiling honeycomb'', {{CDD|branch_10ru|split2|node_1|6|node}} ↔ {{CDD|node_h1|6|node|3|node_1|6|node}}. A second lower-index construction is {{CDD|node|3|node_1|6|node_g|3sg|node_g}} ↔ {{CDD|node|splitsplit1|branch4_11|splitsplit2|node_1}}. 480px {{Clear}} === Truncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Truncated triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||t{3,6,3} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)||{{CDD|node_1|3|node_1|6|node|3|node}}
{{CDD|node_1|6|node|3|node|3|node}} |- |bgcolor=#e7dcc3|Cells||[t{3,6}](/source/hexagonal_tiling) 40px
[{6,3}](/source/hexagonal_tiling) 40px |- |bgcolor=#e7dcc3|Faces||[hexagon](/source/hexagon) {6} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||80px
[tetrahedron](/source/tetrahedron) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||\overline{Y}_3, [3,6,3]
\overline{V}_3, [3,3,6] |- |bgcolor=#e7dcc3|Properties||Regular |} The '''truncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node|3|node}}, is a lower-symmetry form of the [hexagonal tiling honeycomb](/source/hexagonal_tiling_honeycomb), {{CDD|node_1|6|node|3|node|3|node}}. It contains [hexagonal tiling](/source/hexagonal_tiling) facets with a [tetrahedral](/source/tetrahedron) vertex figure. 480px {{Clear}} === Bitruncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Bitruncated triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||2t{3,6,3} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)||{{CDD|node|3|node_1|6|node_1|3|node}} |- |bgcolor=#e7dcc3|Cells||[t{6,3}](/source/truncated_hexagonal_tiling) 40px |- |bgcolor=#e7dcc3|Faces||[triangle](/source/triangle) {3}
[dodecagon](/source/dodecagon) {12} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||80px
[tetragonal disphenoid](/source/tetragonal_disphenoid) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||2\times\overline{Y}_3, [3,6,3](/source/%3C%2Fnowiki%3E3%2C6%2C3) |- |bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive, cell-transitive |} The '''bitruncated triangular tiling honeycomb''', {{CDD|node|3|node_1|6|node_1|3|node}}, has [truncated hexagonal tiling](/source/truncated_hexagonal_tiling) cells, with a [tetragonal disphenoid](/source/tetragonal_disphenoid) vertex figure. 480px {{Clear}} === Cantellated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Cantellated triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||rr{3,6,3} or t0,2{3,6,3}
s2{3,6,3} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)||{{CDD|node_1|3|node|6|node_1|3|node}}
{{CDD|node_h|3|node_h|6|node_1|3|node}} |- |bgcolor=#e7dcc3|Cells||[rr{6,3}](/source/rhombitrihexagonal_tiling) 40px
[r{6,3}](/source/trihexagonal_tiling) 40px
[{}×{3}](/source/Triangular_prism) 40px |- |bgcolor=#e7dcc3|Faces||[triangle](/source/triangle) {3}
[square](/source/square) {4}
[hexagon](/source/hexagon) {6} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||80px
[wedge](/source/wedge_(geometry)) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||\overline{Y}_3, [3,6,3] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive |} The '''cantellated triangular tiling honeycomb''', {{CDD|node_1|3|node|6|node_1|3|node}}, has [rhombitrihexagonal tiling](/source/rhombitrihexagonal_tiling), [trihexagonal tiling](/source/trihexagonal_tiling), and [triangular prism](/source/triangular_prism) cells, with a [wedge](/source/wedge_(geometry)) vertex figure. ==== Symmetry==== It can also be constructed as a '''cantic snub triangular tiling honeycomb''', {{CDD|node_h|3|node_h|6|node_1|3|node}}, a half-symmetry form with symmetry [3+,6,3]. 480px {{Clear}} === Cantitruncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Cantitruncated triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||tr{3,6,3} or t0,1,2{3,6,3} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)||{{CDD|node_1|3|node_1|6|node_1|3|node}} |- |bgcolor=#e7dcc3|Cells||[tr{6,3}](/source/truncated_trihexagonal_tiling) 40px
[t{6,3}](/source/truncated_hexagonal_tiling) 40px
[{}×{3}](/source/triangular_prism) 40px |- |bgcolor=#e7dcc3|Faces||[triangle](/source/triangle) {3}
[square](/source/square) {4}
[hexagon](/source/hexagon) {6}
[dodecagon](/source/dodecagon) {12} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||80px
[mirrored sphenoid](/source/mirrored_sphenoid) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||\overline{Y}_3, [3,6,3] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive |} The '''cantitruncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node_1|3|node}}, has [truncated trihexagonal tiling](/source/truncated_trihexagonal_tiling), [truncated hexagonal tiling](/source/truncated_hexagonal_tiling), and [triangular prism](/source/triangular_prism) cells, with a [mirrored sphenoid](/source/mirrored_sphenoid) vertex figure. 480px {{Clear}} === Runcinated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Runcinated triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||t0,3{3,6,3} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)||{{CDD|node_1|3|node|6|node|3|node_1}} |- |bgcolor=#e7dcc3|Cells||[{3,6}](/source/triangular_tiling) 40px
[{}×{3}](/source/Triangular_prism) 40px |- |bgcolor=#e7dcc3|Faces||[triangle](/source/triangle) {3}
[square](/source/square) {4} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||80px
[hexagonal antiprism](/source/hexagonal_antiprism) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||2\times\overline{Y}_3, [3,6,3](/source/%3C%2Fnowiki%3E3%2C6%2C3) |- |bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive |} The '''runcinated triangular tiling honeycomb''', {{CDD|node_1|3|node|6|node|3|node_1}}, has [triangular tiling](/source/triangular_tiling) and [triangular prism](/source/triangular_prism) cells, with a [hexagonal antiprism](/source/hexagonal_antiprism) vertex figure. 480px {{Clear}} === Runcitruncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Runcitruncated triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)s||t0,1,3{3,6,3}
s2,3{3,6,3} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)s||{{CDD|node_1|3|node_1|6|node|3|node_1}}
{{CDD|node_h|3|node_h|6|node_1|3|node_1}} |- |bgcolor=#e7dcc3|Cells||[t{3,6}](/source/hexagonal_tiling) 40px
[rr{3,6}](/source/rhombitrihexagonal_tiling) 40px
[{}×{3}](/source/Triangular_prism) 40px
[{}×{6}](/source/hexagonal_prism) 40px |- |bgcolor=#e7dcc3|Faces||[triangle](/source/triangle) {3}
[square](/source/square) {4}
[hexagon](/source/hexagon) {6} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||80px
[isosceles-trapezoidal](/source/isosceles_trapezoid) [pyramid](/source/pyramid_(geometry)) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||\overline{Y}_3, [3,6,3] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive |} The '''runcitruncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node|3|node_1}}, has [hexagonal tiling](/source/hexagonal_tiling), [rhombitrihexagonal tiling](/source/rhombitrihexagonal_tiling), [triangular prism](/source/triangular_prism), and [hexagonal prism](/source/hexagonal_prism) cells, with an [isosceles-trapezoidal](/source/isosceles_trapezoid) [pyramid](/source/pyramid_(geometry)) [vertex figure](/source/vertex_figure). ==== Symmetry==== It can also be constructed as a '''runcicantic snub triangular tiling honeycomb''', {{CDD|node_h|3|node_h|6|node_1|3|node_1}}, a half-symmetry form with symmetry [3+,6,3]. 480px {{Clear}} === Omnitruncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Omnitruncated triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||[Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb) |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||t0,1,2,3{3,6,3} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)||{{CDD|node_1|3|node_1|6|node_1|3|node_1}} |- |bgcolor=#e7dcc3|Cells||[tr{3,6}](/source/truncated_trihexagonal_tiling) 40px
[{}×{6}](/source/hexagonal_prism) 40px |- |bgcolor=#e7dcc3|Faces||[square](/source/square) {4}
[hexagon](/source/hexagon) {6}
[dodecagon](/source/dodecagon) {12} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)||80px
[phyllic disphenoid](/source/phyllic_disphenoid) |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||2\times\overline{Y}_3, [3,6,3](/source/%3C%2Fnowiki%3E3%2C6%2C3) |- |bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive |} The '''omnitruncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node_1|3|node_1}}, has [truncated trihexagonal tiling](/source/truncated_trihexagonal_tiling) and [hexagonal prism](/source/hexagonal_prism) cells, with a [phyllic disphenoid](/source/phyllic_disphenoid) vertex figure. 480px {{Clear}} === Runcisnub triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Runcisnub triangular tiling honeycomb |- |bgcolor=#e7dcc3|Type||Paracompact scaliform honeycomb |- |width=100 bgcolor=#e7dcc3|[Schläfli symbol](/source/Schl%C3%A4fli_symbol)||s3{3,6,3} |- |bgcolor=#e7dcc3|[Coxeter diagram](/source/Coxeter_diagram)||{{CDD|node_h|3|node_h|6|node|3|node_1}} |- |bgcolor=#e7dcc3|Cells|||[r{6,3}](/source/Trihexagonal_tiling) 40px
[{}x{3}](/source/Triangular_prism) 40px
[{3,6}](/source/triangular_tiling) 40px
[tricup](/source/triangular_cupola) 40px |- |bgcolor=#e7dcc3|Faces||[triangle](/source/triangle) {3}
[square](/source/square) {4}
[hexagon](/source/hexagon) {6} |- |bgcolor=#e7dcc3|[Vertex figure](/source/Vertex_figure)|| |- |bgcolor=#e7dcc3|[Coxeter group](/source/Coxeter_group)||\overline{Y}_3, [3+,6,3] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive, non-uniform |} The '''runcisnub triangular tiling honeycomb''', {{CDD|node_h|3|node_h|6|node|3|node_1}}, has [trihexagonal tiling](/source/trihexagonal_tiling), [triangular tiling](/source/triangular_tiling), [triangular prism](/source/triangular_prism), and [triangular cupola](/source/triangular_cupola) cells. It is [vertex-transitive](/source/vertex-transitive), but not uniform, since it contains [Johnson solid](/source/Johnson_solid) [triangular cupola](/source/triangular_cupola) cells. {{Clear}} == See also == * [Convex uniform honeycombs in hyperbolic space](/source/Convex_uniform_honeycombs_in_hyperbolic_space) * [Regular tessellations of hyperbolic 3-space](/source/List_of_regular_polytopes) * [Paracompact uniform honeycomb](/source/Paracompact_uniform_honeycomb)s == References == *[Coxeter](/source/H.S.M._Coxeter), ''[Regular Polytopes](/source/Regular_Polytopes_(book))'', 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296) * ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, {{LCCN|99035678}}, {{isbn|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf Regular Honeycombs in Hyperbolic Space]) Table III * [Jeffrey R. Weeks](/source/Jeffrey_Weeks_(mathematician)) ''The Shape of Space, 2nd edition'' {{isbn|0-8247-0709-5}} (Chapter 16-17: Geometries on Three-manifolds I, II) * [Norman Johnson](/source/Norman_Johnson_(mathematician)) ''Uniform Polytopes'', Manuscript ** [N.W. Johnson](/source/Norman_Johnson_(mathematician)): ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 ** N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 13: Hyperbolic Coxeter groups Category:Regular 3-honeycombs Category:Self-dual tilings Category:Triangular tilings --- Adapted from the Wikipedia article [Triangular tiling honeycomb](https://en.wikipedia.org/wiki/Triangular_tiling_honeycomb) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Triangular_tiling_honeycomb?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.