Hexagonal tiling-triangular tiling honeycomb: Difference between revisions

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|bgcolor=#e7dcc3|[[Schläfli symbol]]||{(3,6,3,6)} or {(6,3,6,3)}

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|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|label6|branch_10r|3ab|branch|label6}} or {{CDD|label6|branch_01r|3ab|branch|label6}} or {{CDD|label6|branch|3ab|branch_10l|label6}} or {{CDD|label6|branch|3ab|branch_01l|label6}}<BR>[[File:CDel K6 636 10.png]] ↔ {{CDD|node_1|6|node_g|3sg|node_g|6|node}}

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|bgcolor=#e7dcc3|Cells||[[Triangular tiling|{3,6}]] [[File:Uniform_tilingUniform tiling 63-t2-red.pngsvg|40px]]<BR>[[Hexagonal tiling|{6,3}]] [[File:Uniform_tiling 63-t0.pngsvg|40px]]<BR>[[Trihexagonal tiling|r{6,3}]] [[File:Uniform_tiling 63-t1.pngsvg|40px]]

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|bgcolor=#e7dcc3|Faces||[[triangular]] {3}<BR>[[square]] {4}<BR>[[hexagon]] {6}

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|bgcolor=#e7dcc3|Vertex figure||[[File:Uniform_tiling_63-t02.pngsvg|80px]]<BR>[[rhombitrihexagonal tiling]]

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|bgcolor=#e7dcc3|[[Coxeter group]]||[(6,3)^[2]]

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|bgcolor=#e7dcc3|Properties||Vertex-uniform, edge-uniform

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''hexagonal tiling-triangular tiling honeycomb''' is a [[paracompact uniform honeycomb]], constructed from [[triangular tiling]], [[hexagonal tiling]], and [[trihexagonal tiling]] cells, in a [[rhombitrihexagonal tiling]] [[vertex figure]]. It has a single-ring Coxeter diagram, {{CDD|label6|branch_10r|3ab|branch|label6}}, and is named by its two regular cells.

{{Honeycomb}}

== Symmetry==

A lower symmetry form, index 6, of this honeycomb can be constructed with [(6,3,6,3^*)] symmetry, represented by a triangular bipyramidal[[cube]] fundamental ___domain, and an octahedral [[Coxeter diagram]] [[File:CDel K6 636 10.png]].

{{Clear}}

== Related honeycombs==

The ''cyclotruncated octahedral-hexagonal tiling honeycomb'', {{CDD|label6|branch_10r|3ab|branch_10l|label6}} has a higher symmetry construction as the [[order-4 hexagonal tiling]].

== See also ==

* [[Convex uniformUniform honeycombs in hyperbolic space]]

* [[List of regular polytopes]]

== References ==

*[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. {{ISBN |0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;294–296)

*[[H.S.M. Coxeter|Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 {{ISBN |0-486-40919-8}} (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)

* [[Jeffrey Weeks (mathematician)|Jeffrey R. Weeks]] ''The Shape of Space, 2nd edition'' {{ISBN |0-8247-0709-5}} (Chapter 16-17: Geometries on Three-manifolds I, II)

* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript

** [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966

** N.W. Johnson: ''Geometries and Transformations'', Manuscript, (20112018) Chapter 13: Hyperbolic Coxeter groups

[[Category:HoneycombsHexagonal (geometry)tilings]]

[[Category:3-honeycombs]]