|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {| class="wikitable" align="right" style="margin-left:10px" width="320"
| | Hello and welcome. My title is Irwin and I totally dig that title. I am a meter reader. His spouse doesn't like it [http://tcciaarusha-bic.info/groups/simple-stuff-you-could-do-to-remove-candida/ over the counter std test] way he does but what he really likes doing is to do aerobics and he's been performing it for fairly a whilst. North Dakota is our birth place. |
| !bgcolor=#e7dcc3 colspan=2|{{PAGENAME}}
| |
| |-
| |
| |bgcolor=#ffffff align=center colspan=2|
| |
| [[Image:Cantitruncated cubic tiling.png|190px]] [[File:HC A6-A4-P2.png|110px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Schläfli symbol]]||tr{4,3,4}<BR>t<sub>0,1,2</sub>{4,3,4}
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|4|node_1|3|node_1|4|node}}
| |
| |-
| |
| |bgcolor=#e7dcc3|Vertex figure||[[Image:Cantitruncated cubic honeycomb verf.png|50px]][[File:Omnitruncated alternated cubic honeycomb verf.png|50px]]<BR>(Irreg. [[tetrahedron]])
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter group]]||[4,3,4], <math>{\tilde{C}}_3</math>
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Space group]]<BR>[[Fibrifold notation]]||Pm{{overline|3}}m (221)<BR>4<sup>−</sup>:2
| |
| |-
| |
| |bgcolor=#e7dcc3|Dual||[[triangular pyramidille]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
| |
| |}
| |
| The '''cantitruncated cubic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space, made up of [[truncated cuboctahedron|truncated cuboctahedra]], [[truncated octahedron|truncated octahedra]], and [[cube]]s in a ratio of 1:1:3.
| |
| | |
| [[John Horton Conway]] calls this honeycomb a '''n-tCO-trille''', and its dual [[triangular pyramidille]].
| |
| | |
| == Images==
| |
| | |
| [[Image:Cantitruncated Cubic Honeycomb.svg|250px]]
| |
| | |
| == Symmetry ==
| |
| | |
| Cells can be shown in two different symmetries. The linear [[Coxeter-Dynkin diagram]] form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of [[truncated cuboctahedron]] cells alternating.
| |
| {| class="wikitable" width=320
| |
| !Construction
| |
| !Cantitruncated cubic
| |
| !Omnitruncated alternate cubic
| |
| |- valign=top
| |
| ![[Coxeter group]]
| |
| ![4,3,4], <math>{\tilde{C}}_3</math><BR>=<[4,3<sup>1,1</sup>]>
| |
| ![4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math>
| |
| |-
| |
| ![[Space group]]||Pm{{overline|3}}m (221)||Fm{{overline|3}}m (225)
| |
| |-
| |
| ![[Fibrifold]]||4<sup>−</sup>:2||2<sup>−</sup>:2
| |
| |- align=center
| |
| !Coloring
| |
| |[[Image:Cantitruncated_Cubic_Honeycomb.svg|80px]]
| |
| |[[Image:Cantitruncated_Cubic_Honeycomb2.svg|80px]]
| |
| |-
| |
| ![[Coxeter-Dynkin diagram]]
| |
| !{{CDD|node_1|4|node_1|3|node_1|4|node}}
| |
| !{{CDD|node_1|4|node_1|split1|nodes_11}}
| |
| |-
| |
| ![[Vertex figure]]
| |
| |[[Image:Cantitruncated cubic honeycomb verf.png|80px]]
| |
| |[[File:Omnitruncated alternated cubic honeycomb verf.png|80px]]
| |
| |- align=center
| |
| !Vertex<BR>figure<BR>symmetry
| |
| |[ ]<BR>order 2
| |
| |[ ]<sup>+</sup><BR>order 1
| |
| |}
| |
| | |
| == Related honeycombs==
| |
| The [4,3,4], {{CDD|node|4|node|3|node|4|node}}, [[Coxeter group]] generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The [[Expansion (geometry)|expanded]] cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.
| |
| {{C3 honeycombs}}
| |
| | |
| The [4,3<sup>1,1</sup>], {{CDD|node|4|node|split1|nodes}}, [[Coxeter group]] generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.
| |
| {{B3 honeycombs}}
| |
| | |
| === Alternation ===
| |
| [[File:Alternated_cantitruncated_cubic_honeycomb_verf.png|150px|thumb|Vertex figure for alternated bitruncated cubic honeycomb, with 5 tetrahedral, one icosahedral, and two snub cube cells, but edge-lengths can't be made equal.]]
| |
| {| width=400 class=wikitable
| |
| |[[Image:Alternated cantitruncated cubic honeycomb.png|400px]]<BR>
| |
| This image shows a partial honeycomb of the [[Alternation (geometry)|alternation]] of the ''cantitruncated cubic honeycomb''. It contains three types of cells: [[snub cube]]s, [[Icosahedron|icosahedra]] (snub tetrahedron), and [[Tetrahedron|tetrahedra]]. In addition the gaps created at the alternated vertices form tetrahedral cells.<BR>This honeycomb exists in two mirror image forms. Although it is not uniform, constructionally it can be given as [[Coxeter-Dynkin diagram]]s {{CDD|node_h|4|node_h|split1|nodes_hh}} or {{CDD|node_h|4|node_h|3|node_h|4|node}}.
| |
| |}
| |
| | |
| ==See also==
| |
| {{Commons category|Cantitruncated cubic honeycomb}}
| |
| *[[Architectonic and catoptric tessellation]]
| |
| | |
| == References ==
| |
| * [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, (2008) ''The Symmetries of Things'', ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
| |
| * {{The Geometrical Foundation of Natural Structure (book)|page=197}} Chapter 5 (Polyhedral packing and spacing filling): Fig. 5-13, p.176 shows this honeycomb. Fig. 5-34 shows a partial honeycomb of the alternation with only snub cube cells show.
| |
| * [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
| |
| * [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49 - 56.
| |
| * '''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 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
| |
| * [[Alfredo Andreini|A. Andreini]], ''Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative'' (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
| |
| * {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|x4x3x4o - grich - O18}}
| |
| * [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 06-Grich]
| |
| | |
| [[Category:Honeycombs (geometry)]]
| |
| | |
| {{polychora-stub}}
| |
Hello and welcome. My title is Irwin and I totally dig that title. I am a meter reader. His spouse doesn't like it over the counter std test way he does but what he really likes doing is to do aerobics and he's been performing it for fairly a whilst. North Dakota is our birth place.