Alternated hexagonal tiling honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Alternated hexagonal tiling honeycomb |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb<BR>Semiregular honeycomb |- |bgcolor=#e7dcc3|Schläfli symbols||h{6,3,3}<BR>s{3,6,3}<BR>2s{6,3,6}<BR>2s{6,3^[3]}<BR>s{3^[3,3]} |- |bgcolor=#e7dcc3|Coxeter diagrams||{{CDD||node_h1|6|node|3|node|3|node}} ↔ {{CDD|branch_10ru|split2|node|3|node}}<BR>{{CDD||node_h|3|node_h|6|node|3|node}}<BR>{{CDD||node|6|node_h|3|node_h|6|node}}<BR>{{CDD|branch_hh|split2|node_h|6|node}} ↔ {{CDD||node_h0|6|node_h|3|node_h|6|node}}<BR>{{CDD|branch_hh|splitcross|branch_hh}} ↔ {{CDD|branch_hh|split2|node_h|6|node_h0}} ↔ {{CDD|node_h0|6|node_h|3|node_h|6|node_h0}} |- |bgcolor=#e7dcc3|Cells|||{3,3} 40px<br>[[triangular tiling|{3^[3]}]] 40px |- |bgcolor=#e7dcc3|Faces||triangle {3} |- |bgcolor=#e7dcc3|Vertex figure||40px {{CDD|node_1|3|node_1|3|node}}<br>truncated tetrahedron |- |bgcolor=#e7dcc3|Coxeter groups||<math>{\overline{P}}_3</math>, [3,3^[3]]<BR>1/2 <math>{\overline{V}}_3</math>, [6,3,3]<BR>1/2 <math>{\overline{Y}}_3</math>, [3,6,3]<BR>1/2 <math>{\overline{Z}}_3</math>, [6,3,6]<BR>1/2 <math>{\overline{VP}}_3</math>, [6,3^[3]]<BR>1/2 <math>{\overline{PP}}_3</math>, [3^[3,3]] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive, quasiregular |} In three-dimensional hyperbolic geometry, the '''alternated hexagonal tiling honeycomb''', h{6,3,3}, {{CDD||node_h1|6|node|3|node|3|node}} or {{CDD|branch_10ru|split2|node|3|node}}, is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, as an alternation of a hexagonal tiling honeycomb.

{{Honeycomb}}

== Symmetry constructions == [[File:Hyperbolic subgroup tree 336-direct.png|120px|thumb|left|Subgroup relations]] It has five alternated constructions from reflectional Coxeter groups all with four mirrors and only the first being regular: {{CDD|node_c1|6|node|3|node|3|node}} [6,3,3], {{CDD|node_c1|3|node_c1|6|node|3|node}} [3,6,3], {{CDD|node|6|node_c1|3|node_c1|6|node}} [6,3,6], {{CDD|branch_c1|split2|node_c1|6|node}} [6,3^[3]] and [3^[3,3]] {{CDD|branch_c1|splitcross|branch_c1}}, having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)^*] (remove 3 mirrors, index 24 subgroup); [3,6,3^*] or [3^*,6,3] (remove 2 mirrors, index 6 subgroup); [1⁺,6,3,6,1⁺] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3^[3,3]]. The ringed Coxeter diagrams are {{CDD|node_h|6|node|3|node|3|node}}, {{CDD|node_h|3|node_h|6|node|3|node}}, {{CDD|node|6|node_h|3|node_h|6|node}}, {{CDD|branch_hh|split2|node_h|6|node}} and {{CDD|branch_hh|splitcross|branch_hh}}, representing different types (colors) of hexagonal tilings in the Wythoff construction. {{Clear}}

== Related honeycombs== The alternated hexagonal tiling honeycomb has 3 related forms: the cantic hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node_1|3|node}}; the runcic hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node|3|node_1}}; and the runcicantic hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node_1|3|node_1}}.

===Cantic hexagonal tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Cantic hexagonal tiling honeycomb |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |bgcolor=#e7dcc3|Schläfli symbols||h₂{6,3,3} |- |bgcolor=#e7dcc3|Coxeter diagrams||{{CDD||node_h1|6|node|3|node_1|3|node}} ↔ {{CDD|branch_10ru|split2|node_1|3|node}} |- |bgcolor=#e7dcc3|Cells||r{3,3} 40px<br>t{3,3} 40px<br>h₂{6,3} 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>hexagon {6} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>wedge |- |bgcolor=#e7dcc3|Coxeter groups||<math>{\overline{P}}_3</math>, [3,3^[3]] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive |} The '''cantic hexagonal tiling honeycomb''', h₂{6,3,3}, {{CDD||node_h1|6|node|3|node_1|3|node}} or {{CDD|branch_10ru|split2|node_1|3|node}}, is composed of octahedron, truncated tetrahedron, and trihexagonal tiling facets, with a wedge vertex figure.

{{Clear}}

===Runcic hexagonal tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Runcic hexagonal tiling honeycomb |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |bgcolor=#e7dcc3|Schläfli symbols||h₃{6,3,3} |- |bgcolor=#e7dcc3|Coxeter diagrams||{{CDD||node_h1|6|node|3|node|3|node_1}} ↔ {{CDD|branch_10ru|split2|node|3|node_1}} |- |bgcolor=#e7dcc3|Cells||{3,3} 40px<br>{}x{3} 40px<br>rr{3,3} 40px<br>[[triangular tiling|{3^[3]}]] 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>square {4}<BR>hexagon {6} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>triangular cupola |- |bgcolor=#e7dcc3|Coxeter groups||<math>{\overline{P}}_3</math>, [3,3^[3]] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive |} The '''runcic hexagonal tiling honeycomb''', h₃{6,3,3}, {{CDD||node_h1|6|node|3|node|3|node_1}} or {{CDD|branch_10ru|split2|node|3|node_1}}, has tetrahedron, triangular prism, cuboctahedron, and triangular tiling facets, with a triangular cupola vertex figure. {{Clear}}

===Runcicantic hexagonal tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Runcicantic hexagonal tiling honeycomb |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |bgcolor=#e7dcc3|Schläfli symbols||h_2,3{6,3,3} |- |bgcolor=#e7dcc3|Coxeter diagrams||{{CDD||node_h1|6|node|3|node_1|3|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|3|node_1}} |- |bgcolor=#e7dcc3|Cells||t{3,3} 40px<br>{}x{3} 40px<br>tr{3,3} 40px<br>h₂{6,3} 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>square {4}<BR>hexagon {6} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>rectangular pyramid |- |bgcolor=#e7dcc3|Coxeter groups||<math>{\overline{P}}_3</math>, [3,3^[3]] |- |bgcolor=#e7dcc3|Properties||Vertex-transitive |} The '''runcicantic hexagonal tiling honeycomb''', h_2,3{6,3,3}, {{CDD||node_h1|6|node|3|node_1|3|node_1}} or {{CDD|branch_10ru|split2|node_1|3|node_1}}, has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.

{{Clear}}

== See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs * Semiregular honeycomb * Hexagonal tiling honeycomb

== References == {{reflist}} *Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. {{ISBN|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;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] {{Webarchive|url=https://web.archive.org/web/20160610043106/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf |date=2016-06-10 }}) Table III * Jeffrey R. Weeks ''The Shape of Space, 2nd edition'' {{ISBN|0-8247-0709-5}} (Chapters 16–17: Geometries on Three-manifolds I, II) *N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, ''The size of a hyperbolic Coxeter simplex'', Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [https://link.springer.com/article/10.1007%2FBF01238563] [https://web.archive.org/web/20140223225217/http://homeweb1.unifr.ch/kellerha/pub/TGarticle.pdf] * N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, ''Commensurability classes of hyperbolic Coxeter groups'', (2002) H³: p130. [http://www.sciencedirect.com/science/article/pii/S0024379501004773]

Category:Hexagonal tilings Category:3-honeycombs