Fundamental increment lemma: Difference between revisions

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This honeycomb is one of [[Uniform_polypeton#Regular_and_uniform_honeycombs|12 unique uniform honeycombs]]<ref>[http://mathworld.wolfram.com/Necklace.html], [http://oeis.org/A000029 A000029] 13-1 cases, skipping one with zero marks</ref> constructed by the <math>{\tilde{A}}_5</math> [[Coxeter group]]. The extended symmetry of the hexagonal diagram of the <math>{\tilde{A}}_5</math> Coxeter group allows for [[isomorphism]]s that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:


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{| class=wikitable
![[:File:Regular hexagon symmetries.png|Hexagon<BR>symmetry]]
![[Coxeter notation|Extended<BR>symmetry]]
!Extended<BR>diagram
!Extended<BR>order
!Diagrams
|- align=center
!a1
![3<sup>[6]</sup>]
|{{CDD|node|split1|nodes|3ab|nodes|split2|node}}
|×1
|{{CDD|node_1|split1|nodes_10lur|3ab|nodes|split2|node_1}}
|- align=center
!d2
!<[3<sup>[6]</sup>]>
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c3|split2|node_c4}}
|×2
|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}<sub>[[5-simplex honeycomb|1]]</sub>, {{CDD|node|split1|nodes_11|3ab|nodes|split2|node}}, {{CDD|node_1|split1|nodes_11|3ab|nodes|split2|node}}, {{CDD|node_1|split1|nodes_11|3ab|nodes|split2|node_1}}, {{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node}}
|- align=center
!p2
!<nowiki>[[</nowiki>3<sup>[6]</sup>]]
|{{CDD|node_c3|split1|nodeab_c1-2|3ab|nodeab_c1-2|split2|node_c4}}
|×2
|{{CDD|node|split1|nodes_10lur|3ab|nodes_10lru|split2|node}}<sub>[[Truncated 5-simplex honeycomb|2]]</sub>, {{CDD|node_1|split1|nodes_10lur|3ab|nodes_10lru|split2|node_1}}
|- align=center
!i4
![2[3<sup>[6]</sup>]]
|{{CDD|node_c3|split1|nodeab_c1-2|3ab|nodeab_c1-2|split2|node_c3}}
|×4
|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node_1}}, {{CDD|node|split1|nodes_11|3ab|nodes_11|split2|node}}
|- align=center
!d6
![3[3<sup>[6]</sup>]]
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c1|split2|node_c2}}
|×6
|{{CDD|node_1|split1|nodes|3ab|nodes_11|split2|node}}
|- align=center
!r12
![6[3<sup>[6]</sup>]]
|{{CDD|node_c1|split1|nodeab_c1|3ab|nodeab_c1|split2|node_c1}}
|×12
|{{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}}<sub>[[Omnitruncated 5-simplex honeycomb|3]]</sub>
|}<noinclude>
 
==References==
{{reflist}}
 
[[Category:Encyclopedic content templates]]
</noinclude>

Revision as of 18:37, 1 November 2013

This honeycomb is one of 12 unique uniform honeycombs[1] constructed by the Coxeter group. The extended symmetry of the hexagonal diagram of the Coxeter group allows for isomorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:

Hexagon
symmetry 		Extended
symmetry 		Extended
diagram 		Extended
order 		Diagrams
a1 [3[6]] Template:CDD ×1 Template:CDD
d2 <[3[6]]> Template:CDD ×2 Template:CDD1, Template:CDD, Template:CDD, Template:CDD, Template:CDD
p2 [[3[6]]] Template:CDD ×2 Template:CDD2, Template:CDD
i4 [2[3[6]]] Template:CDD ×4 Template:CDD, Template:CDD
d6 [3[3[6]]] Template:CDD ×6 Template:CDD
r12 [6[3[6]]] Template:CDD ×12 Template:CDD3

References

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  1. ↑ [1], A000029 13-1 cases, skipping one with zero marks
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