en>CsDix: /* See also */ removed parameter

2013-03-14T16:05:21Z

See also: removed parameter

← Older revision		Revision as of 18:05, 14 March 2013
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	~~Nice to satisfy you~~, ~~my title~~ is ~~Numbers Held although I don't really~~ like ~~becoming known~~ as ~~like that~~. ~~To do aerobics is~~ a ~~factor that I~~'~~m totally addicted~~ to. ~~California is where I've always been residing~~ and ~~I love every day residing right here. Hiring~~ is ~~my profession~~.<br><br>~~my site~~: [http://www.~~youporntime~~.com/~~user~~/~~KBiaggini at home std~~ test]		{\| class="wikitable" align="right" style="margin-left:10px" width="250"
			!bgcolor=#e7dcc3 colspan=2\|Tesseractic honeycomb
			\|-
			\|bgcolor=#ffffff align=center colspan=2\|[[Image:Tesseractic tetracomb.png\|280px]]<br>[[Perspective projection]] of a 3x3x3x3 red-blue chessboard.
			\|-
			\|bgcolor=#e7dcc3\|Type\|\|[[List_of_regular_polytopes#Tessellations_of_Euclidean_4-space\|Regular 4-space honeycomb]]<BR>[[Uniform_polyteron#Regular_and_uniform_honeycombs\|Uniform 4-honeycomb]]
			\|-
			\|bgcolor=#e7dcc3\|Family\|\|[[Hypercubic honeycomb]]
			\|-
			\|bgcolor=#e7dcc3\|[[Schläfli symbol]]s\|\| {4,3,3,4}<BR>t<sub>0,4</sub>{4,3,3,4}<BR>{4,3,3<sup>1,1</sup>}<BR>{4,4}<sup>2</sup><BR>{4,3,4}x{∞}<BR>{4,4}x{∞}<sup>2</sup><BR>{∞}<sup>4</sup>
			\|-
			\|bgcolor=#e7dcc3\|[[Coxeter-Dynkin diagram]]s\|\|{{CDD\|node_1\|4\|node\|3\|node\|3\|node\|4\|node}}<BR>{{CDD\|node_1\|4\|node\|3\|node\|3\|node\|4\|node_1}}<BR>{{CDD\|nodes\|split2\|node\|3\|node\|4\|node_1}}<BR>{{CDD\|node_1\|4\|node\|4\|node\|2\|node_1\|4\|node\|4\|node}}<BR>{{CDD\|node_1\|4\|node\|4\|node\|2\|node_1\|infin\|node\|2\|node_1\|infin\|node}}<BR>{{CDD\|node_1\|infin\|node\|2\|node_1\|infin\|node\|2\|node_1\|infin\|node\|2\|node_1\|infin\|node}}
			\|-
			\|bgcolor=#e7dcc3\|4-face type\|\|[[tesseract\|{4,3,3}]] [[File:Schlegel_wireframe_8-cell.png\|40px]]
			\|-
			\|bgcolor=#e7dcc3\|Cell type\|\|[[cube\|{4,3}]] [[File:Hexahedron.png\|20px]]
			\|-
			\|bgcolor=#e7dcc3\|Face type\|\|[[square (geometry)\|{4}]]
			\|-
			\|bgcolor=#e7dcc3\|Edge figure\|\|8 [[cube\|{4,3}]]<BR>([[octahedron]])
			\|-
			\|bgcolor=#e7dcc3\|Vertex figure\|\|16 [[tesseract\|{4,3,3}]]<BR>([[16-cell]])
			\|-
			\|bgcolor=#e7dcc3\|[[Coxeter group]]s\|\|<math>{\tilde{C}}_4</math>, [4,3,3,4]<BR><math>{\tilde{B}}_4</math>, [4,3,3<sup>1,1</sup>]
			\|-
			\|bgcolor=#e7dcc3\|Dual\|\|[[Self-dual polytope\|self-dual]]
			\|-
			\|bgcolor=#e7dcc3\|Properties\|\|[[vertex-transitive]], [[edge-transitive]], [[face-transitive]], [[cell-transitive]]
			\|}
			In [[Four-dimensional space\|four-dimensional]] [[euclidean geometry]], the '''tesseractic honeycomb''' is one of the three [[List_of_regular_polytopes#Tessellations_of_Euclidean_4-space\|regular]] space-filling [[tessellation]]s (or [[honeycomb (geometry)\|honeycomb]]s), represented by [[Schläfli symbol]] {4,3,3,4}, and constructed by a 4 dimensional packing of [[tesseract]] facets.

			Its [[vertex figure]] is a [[16-cell]]. Two tesseracts meet at each cubic [[Cell (geometry)\|cell]], four meet at each [[Square (geometry)\|square]] [[Face (geometry)\|face]], eight meet on each [[Edge (geometry)\|edge]], and sixteen meet at each [[Vertex (geometry)\|vertex]].

			It is an analog of the [[square tiling]], {4,4}, of the plane and the [[cubic honeycomb]], {4,3,4}, of 3-space. These are all part of the [[hypercubic honeycomb]] family of tessellations of the form {4,3,...,3,4}. Tessellations in this family are [[Self-dual polytope\|Self-dual]].

			==Coordinates ==
			Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).

			== Constructions ==
			There are many different [[Wythoff construction]]s of this honeycomb. The most symmetric form is [[Regular polytope\|regular]], with [[Schläfli symbol]] {4,3,3,4}. Another form has two alternating [[tesseract]] facets (like a checkerboard) with Schläfli symbol {4,3,3<sup>1,1</sup>}. The lowest symmetry Wythoff construction has 16 types of facets around each vertex and a prismatic product Schläfli symbol {∞}<sup>4</sup>. One can be made by [[sterication\|stericating]] another.

			== Related polytopes and tessellations ==
			{{C4_honeycombs}}

			{{B4 honeycombs}}

			The [[24-cell honeycomb]] is similar, but as a [[body centered cubic]], it has vertices positioned at integers (i,j,k,l), and half integers (i+1/2,j+1/2,k+1/2,l+1/2).

			The [[tesseract]] can make a regular tessellation of the [[N-sphere\|4-sphere]], with three tesseracts per face, with Schläfli symbol {4,3,3,3}, called a ''order-3 tesseractic honeycomb''. It is topologically equivalent to the regular polytope [[penteract]] in 5-space.

			The tesseract can make a regular tessellation of 4-dimensional [[hyperbolic space]], with 5 tesseracts around each face, with Schläfli symbol {4,3,3,5}, called an [[List_of_regular_polytopes#Tessellations_of_hyperbolic_4-space\|order-5 tesseractic honeycomb]].

			=== Birectified tesseractic honeycomb ===
			A [[birectified tesseractic honeycomb]], {{CDD\|node_1\|split1\|nodes\|4a4b\|nodes}}, contains all rectified 16-cell ([[24-cell]]) facets and is the [[Voronoi tessellation]] of the [[16-cell honeycomb#D4 lattice\|D<sub>4</sub><sup>*</sup> lattice]]. Facets can be identically colored from a doubled <math>{\tilde{C}}_4</math>×2, <nowiki>[[</nowiki>4,3,3,4]] symmetry, alternately colored from <math>{\tilde{C}}_4</math>, [4,3,3,4] symmetry, three colors from <math>{\tilde{B}}_4</math>, [4,3,3<sup>1,1</sup>] symmetry, and 4 colors from <math>{\tilde{D}}_4</math>, [3<sup>1,1,1,1</sup>] symmetry.

			== See also ==
			Regular and uniform honeycombs in 4-space:
			*[[16-cell honeycomb]]
			*[[24-cell honeycomb]]
			*[[5-cell honeycomb]]
			*[[Truncated 5-cell honeycomb]]
			*[[Omnitruncated 5-cell honeycomb]]

			== References ==
			* [[Coxeter\|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)\|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
			* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
			** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
			* [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)'' - Model 1
			* {{KlitzingPolytopes\|flat.htm\|4D\|Euclidean tesselations}} x∞o x∞o x∞o x∞o, x∞x x∞o x∞o x∞o, x∞x x∞x x∞o x∞o, x∞x x∞x x∞x x∞o,x∞x x∞x x∞x x∞x, x∞o x∞o x4o4o, x∞o x∞o o4x4o, x∞x x∞o x4o4o, x∞x x∞o o4x4o, x∞o x∞o x4o4x, x∞x x∞x x4o4o, x∞x x∞x o4x4o, x∞x x∞o x4o4x, x∞x x∞x x4o4x, x4o4x x4o4x, x4o4x o4x4o, x4o4x x4o4o, o4x4o o4x4o, x4o4o o4x4o, x4o4o x4o4o, x∞x o3o3o d4x, x∞o o3o3o d4x, x∞x x4o3o4x, x∞o x4o3o4x, x∞x x4o3o4o, x∞o x4o3o4o, o3o3o *b3o4x, x4o3o3o4x, x4o3o3o4o - test - O1
			{{Honeycombs}}

			[[Category:Honeycombs (geometry)]]
			[[Category:5-polytopes]]

en>Amalas: /* See also */ adding bore pitch

2012-09-01T13:17:59Z

Bore (engine) - Revision history

en>CsDix: /* See also */ removed parameter

en>Amalas: /* See also */ adding bore pitch