en>Kmarinas86 at 06:25, 7 December 2014

2014-12-07T06:25:42Z

← Older revision		Revision as of 08:25, 7 December 2014
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	~~!bgcolor=#e7dcc3 colspan=2\|Order-6 tetrahedral honeycomb~~
	\|-
	~~\|bgcolor=#ffffff align=center colspan=2\|[[File:H3 336 CC center~~.~~png\|320px]]~~<BR>~~[[Perspective projection]] view<BR>within [[Poincaré disk model]]~~
	\|-
	~~\|bgcolor=#e7dcc3\|Type\|\|[[List of regular polytopes#Tessellations of hyperbolic 3-space\|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb]]~~
	\|-
	~~\|bgcolor=#e7dcc3\|[[Schläfli symbol]]s\|\|{3,3,6}<BR>{3,3<sup>[3]</sup>}~~
	\|-
	~~\|bgcolor=#e7dcc3\|[[Coxeter diagram]]s\|\|{{CDD\|node_1\|3\|node\|3\|node\|6\|node}}<BR>{{CDD\|node_1\|3\|node\|3\|node\|6\|node_h0}} = {{CDD\|node_1\|3\|node\|split1\|branch}}~~
	\|-
	~~\|bgcolor=#e7dcc3\|Cells\|\|[[Tetrahedron]] {3,3}<BR>[[File:Uniform_polyhedron-33-t0~~.~~png\|40px]]~~
	\|-
	~~\|bgcolor=#e7dcc3\|Faces\|\|[[Triangle]] {3}~~
	\|-
	~~\|bgcolor=#e7dcc3\|Edge figure\|\|[[Hexagon]] {6}~~
	\|-
	~~\|bgcolor=#e7dcc3\|Vertex figure\|\|[[Triangular tiling]] {3,6}~~<BR>~~[[File:Uniform_tiling_63-t2.png\|80px]] [[File:Uniform tiling 333-t1.png\|80px]]~~
	\|-
	~~\|bgcolor=#e7dcc3\|Dual\|\|[[Hexagonal tiling honeycomb]], {6,3,3}~~
	\|-
	~~\|bgcolor=#e7dcc3\|[[Coxeter-Dynkin_diagram#Ranks_4.E2.80~~.~~9310\|Coxeter groups]]\|\|~~<~~math~~>~~{\bar{V}}_3</math>, [6,3,3]<BR><math>{\bar{P}}_3</math>, [3,3<sup>[3]</sup>]~~
	\|-
	~~\|bgcolor=#e7dcc3\|Properties\|\|Regular~~
	\|}
	In the [[geometry]] of [[Hyperbolic space\|hyperbolic 3-space]], the '''order-6 tetrahedral honeycomb''' a regular space-filling [[tessellation]] (or [[honeycomb (geometry)\|honeycomb]]). With [[Schläfli symbol]] {3,3,6}. It has six [[tetrahedra]] {3,3} around each edge. All vertices are [[ideal vertex\|ideal vertices]] with infinitely many tetrahedra existing around each ideal vertex in an [[triangular tiling]] [[vertex arrangement]]. <~~ref~~>~~Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III</ref>~~

	~~== Symmetry constructions ==~~
	~~[[File~~:~~Hyperbolic subgroup tree 336-direct.png\|200px\|thumb\|left\|[[Coxeter_diagram#Subgroup_relations_of_paracompact_hyperbolic_groups\|Subgroup relations]]]]~~
	~~It has~~ a ~~second construction~~ as a ~~uniform honeycomb, [[Schläfli symbol]] {3,3<sup>[3]</sup>}~~, with ~~alternating types or colors of tetrahedral cells.~~

	~~== Related polytopes and honeycombs ==~~

	~~It is [[List of regular polytopes#Tessellations of hyperbolic 3-space\|one of 15 regular hyperbolic honeycombs]]~~ in ~~3-space, 11 of which like~~ this ~~one are paracompact, with infinite cells or vertex figures.~~

	~~It is similar to the 2-dimensional hyperbolic tiling, [[infinite-order triangular tiling]], {3,∞}, for having all ideal vertices made of regular simplices.~~
	~~:[[File:H2checkers iii~~.~~png\|120px]]~~

	~~It is [[Paracompact_uniform_honeycomb#.5B6.2C3.2C3.5D_family\|one of 15 uniform paracompact honeycombs]] in the [6,3,3] Coxeter group, along with its dual [[hexagonal tiling honeycomb]], {6,3,3}.~~

	~~The ''rectified order-6 tetrahedral honeycomb'', t~~<~~sub~~>1<~~/sub~~>~~{3,3,6} has tetrahedron~~ and ~~triangular tiling cells connected in a hexagonal prism vertex figure:~~
	~~: [[File:Hyperbolic rectified order-6 tetrahedral honeycomb.png\|240px]]~~

	~~It a part of a sequence of [[regular polychora]] and honeycombs with [[tetrahedron\|tetrahedral]] [[cell (geometry)\|cells]].~~
	~~{{Tetrahedral cell tessellations}}~~

	~~It a part of a sequence of honeycombs with [[triangular tiling]] [[vertex figure]]s.~~
	~~{{Hexagonal tiling vertex figure tessellations}}~~

	~~== See also ==~~
	* [[Convex uniform honeycombs in hyperbolic space]]
	* [[List of regular polytopes]]

	~~== References ==~~
	~~{{reflist}}~~
	*[[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. 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 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)~~

	~~[[Category:Honeycombs (geometry)]]~~

en>Yobot: Remove invisible unicode control characters + other fixes, added orphan tag using AWB (9871)

2014-01-18T01:15:09Z

Remove invisible unicode control characters + other fixes, added orphan tag using AWB (9871)

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{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Order-6 tetrahedral honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[File:H3 336 CC center.png|320px]] [[Perspective projection]] view within [[Poincaré disk model]]
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]] [[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||{3,3,6} {3,3[3]}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node|3|node|6|node}} {{CDD|node_1|3|node|3|node|6|node_h0}} = {{CDD|node_1|3|node|split1|branch}}
|-
|bgcolor=#e7dcc3|Cells||[[Tetrahedron]] {3,3} [[File:Uniform_polyhedron-33-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle]] {3}
|-
|bgcolor=#e7dcc3|Edge figure||[[Hexagon]] {6}
|-
|bgcolor=#e7dcc3|Vertex figure||[[Triangular tiling]] {3,6} [[File:Uniform_tiling_63-t2.png|80px]] [[File:Uniform tiling 333-t1.png|80px]]
|-
|bgcolor=#e7dcc3|Dual||[[Hexagonal tiling honeycomb]], {6,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin_diagram#Ranks_4.E2.80.9310|Coxeter groups]]||<math>{\bar{V}}_3</math>, [6,3,3] <math>{\bar{P}}_3</math>, [3,3[3]]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-6 tetrahedral honeycomb''' a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). With [[Schläfli symbol]] {3,3,6}. It has six [[tetrahedra]] {3,3} around each edge. All vertices are [[ideal vertex|ideal vertices]] with infinitely many tetrahedra existing around each ideal vertex in an [[triangular tiling]] [[vertex arrangement]]. <ref>Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III</ref>

== Symmetry constructions ==
[[File:Hyperbolic subgroup tree 336-direct.png|200px|thumb|left|[[Coxeter_diagram#Subgroup_relations_of_paracompact_hyperbolic_groups|Subgroup relations]]]]
It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {3,3[3]}, with alternating types or colors of tetrahedral cells.

== Related polytopes and honeycombs ==

It is [[List of regular polytopes#Tessellations of hyperbolic 3-space|one of 15 regular hyperbolic honeycombs]] in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

It is similar to the 2-dimensional hyperbolic tiling, [[infinite-order triangular tiling]], {3,∞}, for having all ideal vertices made of regular simplices.
:[[File:H2checkers iii.png|120px]]

It is [[Paracompact_uniform_honeycomb#.5B6.2C3.2C3.5D_family|one of 15 uniform paracompact honeycombs]] in the [6,3,3] Coxeter group, along with its dual [[hexagonal tiling honeycomb]], {6,3,3}.

The ''rectified order-6 tetrahedral honeycomb'', t1{3,3,6} has tetrahedron and triangular tiling cells connected in a hexagonal prism vertex figure:
: [[File:Hyperbolic rectified order-6 tetrahedral honeycomb.png|240px]]

It a part of a sequence of [[regular polychora]] and honeycombs with [[tetrahedron|tetrahedral]] [[cell (geometry)|cells]].
{{Tetrahedral cell tessellations}}

It a part of a sequence of honeycombs with [[triangular tiling]] [[vertex figure]]s.
{{Hexagonal tiling vertex figure tessellations}}

== See also ==
* [[Convex uniform honeycombs in hyperbolic space]]
* [[List of regular polytopes]]

== References ==
{{reflist}}
*[[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. 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 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)

[[Category:Honeycombs (geometry)]]

Spitzer resistivity - Revision history

en>Kmarinas86 at 06:25, 7 December 2014

en>Yobot: Remove invisible unicode control characters + other fixes, added orphan tag using AWB (9871)