Arrhenius plot

2013-04-22T12:59:21Z

195.113.35.48:

{| class="wikitable" align="right" style="margin-left:10px" width="360"
!bgcolor=#e7dcc3 colspan=2|8-demicubic honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||[[8-polytope#Regular_and_uniform_honeycombs|Uniform 8-space honeycomb]]
|-
|bgcolor=#e7dcc3|Family||[[Alternated hypercube honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,3,3,3,3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node_h}} {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|[[Facet (geometry)|Facets]]||[[octacross|{3,3,3,3,3,3,4}]] [[demiocteract|h{4,3,3,3,3,3,3}]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Rectified octacross]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{B}}_8</math> [4,3,3,3,3,3,31,1] <math>{\tilde{D}}_8</math> [31,1,3,3,3,3,31,1]
|}
The '''8-demicubic honeycomb''', or '''demiocteractic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 8-space. It is constructed as an [[Alternation (geometry)|alternation]] of the regular [[8-cubic honeycomb]].

It is composed of two different types of [[Facet (mathematics)|facet]]s. The [[8-cube]]s become alternated into [[8-demicube]]s h{4,3,3,3,3,3,3} [[Image:Demiocteract ortho petrie.svg|25px]] and the alternated vertices create [[8-orthoplex]] {3,3,3,3,3,3,4} facets [[Image:Cross graph 8 Nodes highlighted.svg|25px]].

== D8 lattice ==
The [[vertex arrangement]] of the '''8-demicubic honeycomb''' is the '''D8 lattice'''.<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D8.html</ref> The 112 vertices of the [[rectified 8-orthoplex]] [[vertex figure]] of the ''8-demicubic honeycomb'' reflect the [[kissing number]] 112 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 best known is 240, from the [[E8 lattice|E8 lattice]] and the [[5 21 honeycomb|521 honeycomb]].

<math>{\tilde{E}}_8</math> contains <math>{\tilde{D}}_8</math> as a subgroup of index 270.<ref>Johnson (2011) p.177</ref> Both <math>{\tilde{E}}_8</math> and <math>{\tilde{D}}_8</math> can be seen as affine extensions of <math>D_8</math> from different nodes: [[File:Affine D8 E8 relations.png]]

The D{{sup sub|+|8}} lattice (also called D{{sup sub|2|8}}) can be constructed by the union of two D8 lattices. This packing is only a lattice for even dimensions. The kissing number is 240. (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).<ref>Conway (1998), p. 119</ref> It is identical to the [[E8 lattice]]. At 8-dimensions, the 240 contacts contain both the 27=128 from lower dimension contact progression (2n-1), and 16*7=112 from higher dimensions (2n(n-1)).
:{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|split1|nodes}} + {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|split1|nodes_10lu}} = {{CDD|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}.

The D{{sup sub|*|8}} lattice (also called D{{sup sub|4|8}} and C{{sup sub|2|8}}) can be constructed by the union of all four ''D8 lattices'':<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds8.html</ref> It is also the 7-dimensional [[body centered cubic]], the union of two [[7-cube honeycomb]]s in dual positions.
:{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|split1|nodes}} + {{CDD|nodes_01rd|split2|node|3|node|3|node|3|node|3|node|split1|nodes}} + {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|split1|nodes_10lu}} + {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|split1|nodes_01ld}} = {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} + {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node_1}}.

The [[kissing number]] of the D{{sup sub|*|8}} lattice is 16 (''2n'' for n≥5).<ref>Conway (1998), p. 120</ref> and its [[Voronoi tessellation]] is a [[quadrirectified 8-cubic honeycomb]], {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|4a4b|nodes}}, containing all [[trirectified 8-orthoplex]] [[Voronoi cell]], {{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}.<ref>Conway (1998), p. 466</ref>

==See also ==
*[[8-cubic honeycomb]]
*[[Uniform polytope]]

==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}={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 [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]
* [[Norman W. Johnson|N.W. Johnson]]: ''Geometries and Transformations'', Manuscript, (2011)
* {{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}}

[[Category:Honeycombs (geometry)]]
[[Category:9-polytopes]]

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Arrhenius plot