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{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Demipenteractic honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||[[Uniform_polypeton#Regular and uniform honeycombs|uniform honeycomb]]
|-
|bgcolor=#e7dcc3|Family||[[Alternated hypercubic honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||
{{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} or {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node}}
<BR>{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} or {{CDD|node_h1|4|node|3|node|3|node|split1|nodes}}
<BR>{{CDD|node_h|4|node|3|node|3|node|3|node|4|node_h}}
<BR>{{CDD|node_h|4|node|3|node|3|node|4|node|2|node_h|infin|node}}
<BR>{{CDD|node_h|4|node|3|node|split1|nodes|2|node_h|infin|node}}
<BR>{{CDD|node_h|4|node|3|node|3|node|4|node_h|2|node_h|infin|node}}
<BR>{{CDD|node_h|4|node|3|node|4|node|2|node_h|infin|node|2|node_h|infin|node}}
<BR>{{CDD|node_h|4|node|split1|nodes|2|node_h|infin|node|2|node_h|infin|node}}
<BR>{{CDD|node_h|4|node|3|node|4|node_h|2|node_h|infin|node|2|node_h|infin|node}}
<BR>{{CDD|node_h|4|node|4|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}
<BR>{{CDD|node_h|4|node|4|node_h|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}
<BR>{{CDD|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}
|-
|bgcolor=#e7dcc3|[[Facet (geometry)|Facets]]||[[5-orthoplex|{3,3,3,4}]] [[File:5-cube t4.svg|25px]]<BR>[[5-demicube|h{4,3,3,3}]] [[File:5-demicube t0 D5.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Rectified 5-orthoplex|t<sub>1</sub>{3,3,3,4}]] [[File:Rectified pentacross.svg|25px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{B}}_5</math> [4,3,3,3<sup>1,1</sup>]<BR><math>{\tilde{D}}_5</math> [3<sup>1,1</sup>,3,3<sup>1,1</sup>]
|}
The '''5-demicube honeycomb''', or '''demipenteractic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 5-space. It is constructed as an [[Alternation (geometry)|alternation]] of the regular [[5-cube honeycomb]].
 
It is the first tessellation in the [[Cubic_honeycomb#Alternated_hypercube_tessellations|demihypercube honeycomb]] family which, with all the next ones, is not regular, being composed of two different types of [[Uniform polytope|uniform]] [[Facet (mathematics)|facet]]s. The [[5-cube]]s become alternated into [[5-demicube]]s h{4,3,3,3} and the alternated vertices create [[5-orthoplex]] {3,3,3,4} facets.
 
== D5 lattice ==
The [[vertex arrangement]] of the '''5-demicubic honeycomb''' is the '''D<sub>5</sub> lattice''' which is the densest known [[sphere packing]] in 5 dimensions.<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.html</ref> The 40 vertices of the [[rectified 5-orthoplex]] [[vertex figure]] of the ''5-demicubic honeycomb'' reflect the [[kissing number]] 40 of this lattice.<ref>''Sphere packings, lattices, and groups'', by [[John Horton Conway]], Neil James Alexander Sloane, Eiichi Bannai
[http://books.google.com/books?id=upYwZ6cQumoC&lpg=PP1&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19#v=onepage&q=&f=false]</ref>
 
The D{{sup sub|+|5}} packing (also called D{{sup sub|2|5}}) can be constructed by the union of two D<sub>5</sub> lattices. The analogous packings form lattices only in even dimensions. The kissing number is 2<sup>4</sup>=16 (2<sup>n-1</sup> for n&lt;8, 240 for n=8, and 2n(n-1) for n&gt;8).<ref>Conway (1998), p. 119</ref>
:{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} + {{CDD|nodes|split2|node|3|node|split1|nodes_10lu}}
 
The D{{sup sub|*|5}}<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html</ref> lattice (also called D{{sup sub|4|5}} and C{{sup sub|2|5}}) can be constructed by the union of all four 5-demicubic lattices:<ref>Conway (1998), p. 120</ref> It is also the 5-dimensional [[body centered cubic]], the union of two 5-cube honeycombs in dual positions.
:{{CDD|nodes_10ru|split2||node|3|node|split1|nodes}} + {{CDD|nodes_01rd|split2|node|3|node|split1|nodes}} + {{CDD|nodes|split2|node|3|node|split1|nodes_10lu}} + {{CDD|nodes|split2|node|3|node|split1|nodes_01ld}} = {{CDD|node_1|4|node|3|node|3|node|3|node|4|node}} + {{CDD|node|4|node|3|node|3|node|3|node|4|node_1}}
 
The [[kissing number]] of the D{{sup sub|*|5}} lattice is 10 (''2n'' for n≥5) and it [[Voronoi tessellation]] is a [[tritruncated 5-cubic honeycomb]], {{CDD|branch_11|3ab|nodes|4a4b|nodes}}, containing all with [[bitruncated 5-orthoplex]], {{CDD|node|4|node|3|node_1|3|node_1|3|node}} [[Voronoi cell]]s.<ref>Conway (1998), p. 466</ref>
 
== Symmetry constructions ==
 
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of differened colors on the 32 [[5-demicube]] facets around each vertex.
 
{|class='wikitable'
![[Coxeter group]]
![[Schläfli symbol]]
![[Coxeter-Dynkin diagram]]
![[Vertex figure]]<BR>Symmetry
![[Facet (geometry)|Facets]]/verf
|-
|<math>{\tilde{B}}_5</math> = [3<sup>1,1</sup>,3,3,4]<BR>= [1<sup>+</sup>,4,3,3,4]||{3<sup>1,1</sup>,3,3,4}<BR> = h{4,3,3,3,4}||{{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node}}||{{CDD|node|3|node_1|3|node|3|node|4|node}}<BR>[3,3,3,4]
||32: [[5-demicube]]<BR>10: [[5-orthoplex]]
|-
|<math>{\tilde{D}}_5</math> = [3<sup>1,1</sup>,3,3<sup>1,1</sup>]<BR>= [1<sup>+</sup>,4,3,3<sup>1,1</sup>]||{3<sup>1,1</sup>,3,3<sup>1,1</sup>}||{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|split1|nodes}}||{{CDD|node|3|node_1|3|node|split1|nodes}}<BR>[3<sup>2,1,1</sup>]
||16+16: [[5-demicube]]<BR>10: [[5-orthoplex]]
|-
|<math>{\tilde{C}}_5</math> = ([[4,3,3,4,2<sup>+</sup>]])||ht<sub>0,4</sub>{4,3,3,4}||{{CDD|node_h|4|node|3|node|3|node|3|node|4|node_h}}||
||16+8+8: [[5-demicube]]<BR>10: [[5-orthoplex]]
|}
 
== Related honeycombs==
{{D5 honeycombs}}
 
== See also ==
*[[Uniform polytope]]
Regular and uniform honeycombs in 5-space:
*[[5-cube honeycomb]]
*[[5-demicube honeycomb]]
* [[5-simplex honeycomb]]
* [[Truncated 5-simplex honeycomb]]
* [[Omnitruncated 5-simplex honeycomb]]
 
== References ==
{{reflist}}
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
** pp.&nbsp;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]
* {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}}
 
== External links ==
*{{GlossaryForHyperspace | anchor=half | title=Half measure polytope }}
 
{{Honeycombs}}
 
{{DEFAULTSORT:Demipenteractic Honeycomb}}
[[Category:Honeycombs (geometry)]]
[[Category:6-polytopes]]

Revision as of 07:25, 3 December 2013

Demipenteractic honeycomb
(No image)
Type uniform honeycomb
Family Alternated hypercubic honeycomb
Schläfli symbol h{4,3,3,3,4}
Coxeter diagram

Template:CDD or Template:CDD
Template:CDD or Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD
Template:CDD

Facets {3,3,3,4}
h{4,3,3,3}
Vertex figure t1{3,3,3,4}
Coxeter group B~5 [4,3,3,31,1]
D~5 [31,1,3,31,1]

The 5-demicube honeycomb, or demipenteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.

It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h{4,3,3,3} and the alternated vertices create 5-orthoplex {3,3,3,4} facets.

D5 lattice

The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions.[1] The 40 vertices of the rectified 5-orthoplex vertex figure of the 5-demicubic honeycomb reflect the kissing number 40 of this lattice.[2]

The DTemplate:Sup sub packing (also called DTemplate:Sup sub) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]

Template:CDD + Template:CDD

The DTemplate:Sup sub[4] lattice (also called DTemplate:Sup sub and CTemplate:Sup sub) can be constructed by the union of all four 5-demicubic lattices:[5] It is also the 5-dimensional body centered cubic, the union of two 5-cube honeycombs in dual positions.

Template:CDD + Template:CDD + Template:CDD + Template:CDD = Template:CDD + Template:CDD

The kissing number of the DTemplate:Sup sub lattice is 10 (2n for n≥5) and it Voronoi tessellation is a tritruncated 5-cubic honeycomb, Template:CDD, containing all with bitruncated 5-orthoplex, Template:CDD Voronoi cells.[6]

Symmetry constructions

There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of differened colors on the 32 5-demicube facets around each vertex.

Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry 		Facets/verf
B~5 = [31,1,3,3,4]
= [1+,4,3,3,4]		 {31,1,3,3,4}
= h{4,3,3,3,4}		 Template:CDD = Template:CDD Template:CDD
[3,3,3,4] 		32: 5-demicube
10: 5-orthoplex
D~5 = [31,1,3,31,1]
= [1+,4,3,31,1]		 {31,1,3,31,1} Template:CDD = Template:CDD Template:CDD
[32,1,1] 		16+16: 5-demicube
10: 5-orthoplex
C~5 = ([[4,3,3,4,2+]]) ht0,4{4,3,3,4} Template:CDD 16+8+8: 5-demicube
10: 5-orthoplex
Cantellated 4-simplex honeycomb
(No image)
Type Uniform 4-honeycomb
Schläfli symbol t0,2{3[5]}
Coxeter diagram Template:CDD
4-face types t1{33}
t0,2{33}
t0,3{33}
Cell types Tetrahedron
Octahedron
Cuboctahedron
Triangular prism
Vertex figure triangular elongated-antiprismatic prism
Symmetry A~4×2, [[3[5]]]
Properties vertex-transitive

In four-dimensional Euclidean geometry, the cantellated 4-simplex honeycomb or cantellated 5-cell honeycomb is a space-filling tessellation honeycomb.

Alternate names

  • small cyclorhombated pentachoric tetracomb
  • small prismatodispentachoric tetracomb

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See also

Regular and uniform honeycombs in 4-space:

Notes

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References

  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • 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 [1]
    • (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)
    • (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 136
  • Template:KlitzingPolytopes x3o3x3o3o3*a - scyropot - O136

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See also

Regular and uniform honeycombs in 5-space:

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  • Coxeter, H.S.M. 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}={31,1,4}, 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 [2]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
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  1. ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.html
  2. ↑ Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai [3]
  3. ↑ Conway (1998), p. 119
  4. ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html
  5. ↑ Conway (1998), p. 120
  6. ↑ Conway (1998), p. 466
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