|
|
| Line 1: |
Line 1: |
| {| class="wikitable" align="right" style="margin-left:10px" width="320"
| | I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my complete title. What me and my family members adore is bungee leaping but I've been taking on new things recently. Invoicing is my profession. Mississippi is where his home is.<br><br>my homepage - phone psychic readings ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 the original source]) |
| !bgcolor=#e7dcc3 colspan=2|16-cell honeycomb
| |
| |-
| |
| |bgcolor=#ffffff align=center colspan=2|[[Image:Demitesseractic tetra hc.png|280px]]<br>[[Perspective projection]]: the first layer of adjacent 16-cell facets.
| |
| |-
| |
| |bgcolor=#e7dcc3|Type||[[List_of_regular_polytopes#Five_Dimensions_2|Regular 4-space honeycomb]]<BR>[[Uniform_polyteron#Regular_and_uniform_honeycombs|Uniform 4-honeycomb]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Family||[[Alternated hypercube honeycomb]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Schläfli symbol]]||{3,3,4,3}
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|3|node|3|node|4|node|3|node}}
| |
| |-
| |
| |bgcolor=#e7dcc3|4-face type||[[16-cell|{3,3,4}]] [[File:Schlegel_wireframe_16-cell.png|40px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Cell type||[[tetrahedron|{3,3}]] [[File:Tetrahedron.png|20px]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Face type||[[Triangle|{3}]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Edge figure]]||[[cube]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Vertex figure]]||[[File:24-cell_t0_F4.svg|80px]]<BR>[[24-cell]]
| |
| |-
| |
| |bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{F}}_4</math> = [3,3,4,3]
| |
| |-
| |
| |bgcolor=#e7dcc3|Dual||[[Icositetrachoric honeycomb|{3,4,3,3}]]
| |
| |-
| |
| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]], [[edge-transitive]], [[face-transitive]], [[cell-transitive]]
| |
| |}
| |
| In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''16-cell honeycomb''' is the one of three regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 4-space. The other two are the [[tesseractic honeycomb]] and the [[24-cell honeycomb]]. This honeycomb is constructed from [[16-cell]] [[Facet (mathematics)|facet]]s, three around every edge. It has a [[24-cell]] [[vertex figure]].
| |
| | |
| This [[vertex arrangement]] or lattice is called the B<sub>4</sub>, D<sub>4</sub>, or [[F4 (mathematics)#F4_lattice|F<sub>4</sub> lattice]].<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html</ref><ref name="www2.research.att.com">http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html</ref>
| |
| | |
| == Alternate names==
| |
| * Hexadecachoric tetracomb/honeycomb
| |
| * Demitesseractic tetracomb/honeycomb
| |
| | |
| ==Coordinates==
| |
| | |
| As a regular honeycomb, {3,3,4,3}, it has no lower dimensional analogues, but as an [[Alternation (geometry)|alternated]] form (the '''demitesseractic honeycomb''', h{4,3,3,4}) it is related to the [[alternated cubic honeycomb]].
| |
| | |
| Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.
| |
| | |
| == D4 lattice ==
| |
| Its [[vertex arrangement]] is called the [[D4 lattice|D<sub>4</sub> lattice]] or ''F<sub>4</sub> lattice''.<ref name="www2.research.att.com"/> The vertices of this lattice are the centers of the [[3-sphere]]s in the densest possible [[sphere packing|packing]] of equal spheres in 4-space; its [[kissing number]] is 24, which is also the highest possible in 4-space.<ref name="Musin">{{cite journal |author=O. R. Musin |title=The problem of the twenty-five spheres |year=2003 |journal=Russ. Math. Surv. |volume=58 |pages=794–795 |doi=10.1070/RM2003v058n04ABEH000651}}</ref>
| |
| :{{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_1|3|node|3|node|4|node|3|node}}
| |
| | |
| The D{{sup sub|+|4}} lattice (also called D{{sup sub|2|4}}) can be constructed by the union of two 4-demicubic lattices, and is identical to the [[tesseractic honeycomb]]:
| |
| :{{CDD|nodes_10ru|split2|node|split1|nodes}} + {{CDD|nodes_01rd|split2|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|3|node|4|node}}
| |
| | |
| This packing is only a lattice for even dimensions. The kissing number is 2<sup>3</sup>=8, (2<sup>n-1</sup> for n<8, 240 for n=8, and 2n(n-1) for n>8).<ref>Conway (1998), p. 119</ref>
| |
| | |
| The D{{sup sub|*|4}} lattice (also called D{{sup sub|4|4}} and C{{sup sub|2|4}}) can be constructed by the union of all four 5-demicubic honeycombs, but it is identical to the ''D<sub>4</sub> lattice'': It is also the 4-dimensional [[body centered cubic]], the union of two [[4-cube honeycomb]]s in dual positions.
| |
| :{{CDD|nodes_10ru|split2|node|split1|nodes}} + {{CDD|nodes_01rd|split2|node|split1|nodes}} + {{CDD|nodes|split2|node|split1|nodes_10lu}} + {{CDD|nodes|split2|node|split1|nodes_01ld}} = {{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|3|node|4|node}} + {{CDD|node|4|node|3|node|3|node|4|node_1}}. | |
| | |
| The [[kissing number]] of the D{{sup sub|*|4}} lattice (and D<sub>4</sub> lattice) is 24<ref>Conway (1998), p. 120</ref> and its [[Voronoi tessellation]] is a [[24-cell honeycomb]], {{CDD|node_1|split1|nodes|4a4b|nodes}}, containing all rectified 16-cells ([[24-cell]]) [[Voronoi cell]]s, {{CDD|node|4|node|3|node_1|3|node}} or {{CDD|node_1|3|node|4|node|3|node}}.<ref>Conway (1998), p. 466</ref>
| |
| | |
| == Symmetry constructions ==
| |
| | |
| There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored [[16-cell]] facets.
| |
| | |
| {|class='wikitable'
| |
| !Name
| |
| ![[Coxeter group]]
| |
| ![[Schläfli symbol]]
| |
| ![[Coxeter diagram]]
| |
| ![[Vertex figure]]<BR>Symmetry
| |
| ![[Facet (geometry)|Facets]]/verf
| |
| |-
| |
| !16-cell honeycomb
| |
| |<math>{\tilde{F}}_4</math> = [3,3,4,3]||{3,3,4,3}||{{CDD|node_1|3|node|3|node|4|node|3|node}}||{{CDD|node_1|3|node|4|node|3|node}}<BR>[3,4,3], order 1152||24: [[16-cell]]
| |
| |-
| |
| !4-demicube honeycomb
| |
| |<math>{\tilde{B}}_4</math> = [3<sup>1,1</sup>,3,4]||{3<sup>1,1</sup>,3,4}<BR> = h{4,3,3,4}||{{CDD|nodes_10ru|split2|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|4|node}}||{{CDD|node|3|node_1|3|node|4|node}}<BR>[3,3,4], order 384||16+8: [[16-cell]]
| |
| |-
| |
| !
| |
| |<math>{\tilde{D}}_4</math> = [3<sup>1,1,1,1</sup>]||{3<sup>1,1,1,1</sup>}<BR> = h{4,3,3<sup>1,1</sup>}||{{CDD|nodes_10ru|split2|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|split1|nodes}}||{{CDD|node|3|node_1|split1|nodes}}<BR>[3<sup>1,1,1</sup>], order 192||8+8+8: [[16-cell]]
| |
| |}
| |
| | |
| == Related honeycombs==
| |
| {{F4 honeycombs}}
| |
| | |
| {{C4 honeycombs}}
| |
| | |
| {{B4 honeycombs}}
| |
| | |
| {{D4 honeycombs}}
| |
| | |
| == See also ==
| |
| Regular and uniform honeycombs in 4-space:
| |
| *[[Tesseractic honeycomb]]
| |
| *[[24-cell honeycomb]]
| |
| *[[Truncated 24-cell honeycomb]]
| |
| *[[Snub 24-cell honeycomb]]
| |
| * [[5-cell honeycomb]]
| |
| * [[Truncated 5-cell honeycomb]]
| |
| * [[Omnitruncated 5-cell honeycomb]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| == References ==
| |
| * [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
| |
| ** pp. 154–156: Partial truncation or alternation, represented by ''h'' prefix: h{4,4} = {4,4}; h{4,3,4} = {3<sup>1,1</sup>,4}, h{4,3,3,4} = {3,3,4,3}, ...
| |
| * '''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)''
| |
| * {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} x3o3o4o3o - hext - O104
| |
| * {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}}
| |
| {{Honeycombs}}
| |
| | |
| {{DEFAULTSORT:Demitesseractic Honeycomb}}
| |
| [[Category:Honeycombs (geometry)]]
| |
| [[Category:5-polytopes]]
| |
I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my complete title. What me and my family members adore is bungee leaping but I've been taking on new things recently. Invoicing is my profession. Mississippi is where his home is.
my homepage - phone psychic readings (the original source)