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{| align=right class="wikitable" width=250
| valign=top|[[File:Uniform tiling 44-t1.png|120px]]<BR>An ''alternated square tiling'' or [[checkerboard]] pattern.<BR>{{CDD|node_h1|4|node|4|node}} or {{CDD|nodes|split2-44|node_1}}
| valign=top|[[File:Uniform tiling 44-t02.png|120px]]<BR>An expanded square tiling.<BR>{{CDD|nodes_11|split2-44|node}}
|-
| valign=top|[[File:Tetrahedral-octahedral honeycomb.png|121px]]<BR>A partially filled ''alternated cubic honeycomb'' with tetrahedral and octahedral cells.<BR>{{CDD|node_h1|4|node|3|node|4|node}} or {{CDD|nodes_10ru|split2|node|4|node}}
| valign=top|[[File:Tetrahedral-octahedral honeycomb2.png|121px]]<BR>A subsymmetry colored alternated cubic honeycomb.<BR>{{CDD|node_1|split1|nodes|split2|node}}
|}
 
In [[geometry]], the '''alternated hypercube honeycomb''' (or '''demicubic honeycomb''') is a dimensional infinite series of [[Honeycomb (geometry)|honeycomb]]s, based on the [[hypercube honeycomb]] with an [[Alternation (geometry)|alternation]] operation. It is given a [[Schläfli symbol]] h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of [[Coxeter_diagram#Infinite_Coxeter_groups|Coxeter group]] <math>{\tilde{B}}_{n-1}</math> for n ≥ 4. A lower symmetry form <math>{\tilde{D}}_{n-1}</math> can be created by removing another mirror on a order-4 [[Peak (geometry)|peak]].
 
The alternated hypercube facets become [[demihypercube]]s, and the deleted vertices create new [[orthoplex]] facets. The [[vertex figure]] for honeycombs of this family are [[Rectification (geometry)|rectified]] orthoplexes.
 
These are also named as h&delta;<sub>n</sub> for an (n-1)-dimensional honeycomb.
 
{| class="wikitable"
!rowspan=3|h&delta;<sub>n</sub>
!rowspan=3 width=150|Name
!rowspan=3|[[Schläfli symbol|Schläfli<BR>symbol]]
!colspan=3|[[Coxeter notation|Symmetry family]]
|-
!<math>{\tilde{B}}_{n-1}</math><BR>[4,3<sup>n-4</sup>,3<sup>1,1</sup>]
!<math>{\tilde{D}}_{n-1}</math><BR>[3<sup>1,1</sup>,3<sup>n-5</sup>,3<sup>1,1</sup>]
|-
!colspan=3|[[Coxeter-Dynkin diagram]]s by family
|-
| h&delta;<sub>2</sub>
| ''[[Apeirogon]]''
|{&infin;}
|{{CDD|node_h1|infin|node}}<BR>{{CDD|node_1|infin|node_1}}
|
|-
| h&delta;<sub>3</sub>
| ''Alternated square tiling''<BR>(Same as {4,4})
| h{4,4}=t<sub>1</sub>{4,4}<BR>t<sub>0,2</sub>{4,4}
|{{CDD|node_h1|4|node|4|node}}<BR>{{CDD|nodes_hh|split2-44|node}}<BR>{{CDD|nodes|split2-44|node_1}}
|{{CDD|nodes_11|split2-44|node}}
|-
| h&delta;<sub>4</sub>
| ''[[Alternated cubic honeycomb]]''
| h{4,3,4}<BR>{3<sup>1,1</sup>,4}
|{{CDD|node_h1|4|node|3|node|4|node}}<BR>{{CDD|nodes_hh|4a4b|branch}}<BR>{{CDD|nodes_10ru|split2|node|4|node}}
|{{CDD|node_1|split1|nodes|split2|node}}
|-
| h&delta;<sub>5</sub>
| ''[[16-cell tetracomb]]''<BR>(Same as {3,3,4,3})
| h{4,3<sup>2</sup>,4}<BR>{3<sup>1,1</sup>,3,4}<BR>{3<sup>1,1,1,1</sup>}
|{{CDD|node_h1|4|node|3|node|3|node|4|node}}<BR>{{CDD|nodes_hh|4a4b|nodes|split2|node}}<BR>{{CDD|nodes_10ru|split2|node|3|node|4|node}}
|{{CDD|nodes_10ru|split2|node|split1|nodes}}
|-
| h&delta;<sub>6</sub>
| ''[[5-demicube honeycomb]]''
| h{4,3<sup>3</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>2</sup>,4}<BR>{3<sup>1,1</sup>,3,3<sup>1,1</sup>}
|{{CDD|node_h1|4|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|nodes_hh|4a4b|nodes|3ab|branch}}<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}}
|{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}}
|-
| h&delta;<sub>7</sub>
| ''[[6-demicube honeycomb]]''
| h{4,3<sup>4</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>3</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>2</sup>,3<sup>1,1</sup>}
|{{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|nodes_hh|4a4b|nodes|3ab|nodes|split2|node}}<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|4|node}}
|{{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}}
|-
| h&delta;<sub>8</sub>
| ''[[7-demicube honeycomb]]''
| h{4,3<sup>5</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>4</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>3</sup>,3<sup>1,1</sup>}
|{{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|nodes_hh|4a4b|nodes|3ab|nodes|3ab|branch}}<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|4|node}}
|{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|split1|nodes}}
|-
| h&delta;<sub>9</sub>
| ''[[8-demicube honeycomb]]''
|h{4,3<sup>6</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>5</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>4</sup>,3<sup>1,1</sup>}
|{{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|nodes_hh|4a4b|nodes|3ab|nodes|3ab|nodes|split2|node}}<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|4|node}}
|{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|split1|nodes}}
|-
|colspan=6|&nbsp;
|-
| h&delta;<sub>n</sub>
| ''n-demicubic honeycomb''
|h{4,3<sup>n-3</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>n-4</sup>,4}<BR>{3<sup>1,1</sup>,3<sup>n-5</sup>,3<sup>1,1</sup>}
|colspan=2|...
|}
 
== References ==
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
*# pp. 122-123, 1973. (The lattice of hypercubes &gamma;<sub>n</sub> form the ''cubic honeycombs'', &delta;<sub>n+1</sub>)
*# 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}
*# p. 296, Table II: Regular honeycombs, &delta;<sub>n+1</sub>
 
{{Honeycombs}}
 
[[Category:Honeycombs (geometry)]]
[[Category:Polytopes]]

Latest revision as of 08:50, 5 December 2013


An alternated square tiling or checkerboard pattern.
Template:CDD or Template:CDD
An expanded square tiling.
Template:CDD

A partially filled alternated cubic honeycomb with tetrahedral and octahedral cells.
Template:CDD or Template:CDD
A subsymmetry colored alternated cubic honeycomb.
Template:CDD

In geometry, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation. It is given a Schläfli symbol h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group B~n−1 for n ≥ 4. A lower symmetry form D~n−1 can be created by removing another mirror on a order-4 peak.

The alternated hypercube facets become demihypercubes, and the deleted vertices create new orthoplex facets. The vertex figure for honeycombs of this family are rectified orthoplexes.

These are also named as hδn for an (n-1)-dimensional honeycomb.

hδn Name Schläfli
symbol 		Symmetry family
B~n−1
[4,3n-4,31,1] 		D~n−1
[31,1,3n-5,31,1]
Coxeter-Dynkin diagrams by family
hδ2 Apeirogon {∞} Template:CDD
Template:CDD
hδ3 Alternated square tiling
(Same as {4,4}) 		h{4,4}=t1{4,4}
t0,2{4,4} 		Template:CDD
Template:CDD
Template:CDD 		Template:CDD
hδ4 Alternated cubic honeycomb h{4,3,4}
{31,1,4} 		Template:CDD
Template:CDD
Template:CDD 		Template:CDD
hδ5 16-cell tetracomb
(Same as {3,3,4,3}) 		h{4,32,4}
{31,1,3,4}
{31,1,1,1} 		Template:CDD
Template:CDD
Template:CDD 		Template:CDD
hδ6 5-demicube honeycomb h{4,33,4}
{31,1,32,4}
{31,1,3,31,1} 		Template:CDD
Template:CDD
Template:CDD 		Template:CDD
hδ7 6-demicube honeycomb h{4,34,4}
{31,1,33,4}
{31,1,32,31,1} 		Template:CDD
Template:CDD
Template:CDD 		Template:CDD
hδ8 7-demicube honeycomb h{4,35,4}
{31,1,34,4}
{31,1,33,31,1} 		Template:CDD
Template:CDD
Template:CDD 		Template:CDD
hδ9 8-demicube honeycomb h{4,36,4}
{31,1,35,4}
{31,1,34,31,1} 		Template:CDD
Template:CDD
Template:CDD 		Template:CDD
 
hδn n-demicubic honeycomb h{4,3n-3,4}
{31,1,3n-4,4}
{31,1,3n-5,31,1} 		...

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
    1. pp. 122-123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
    2. 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}
    3. p. 296, Table II: Regular honeycombs, δn+1

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