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Template:A5 honeycombs - Revision history
2026-06-12T20:06:38Z
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alpha>Sarika: Created page with "This honeycomb is one of 12 unique uniform honeycombs<ref>[http://mathworld.wolfram.com/Necklace.html mathworld: Necklace..."
2022-05-19T10:16:33Z
<p>Created page with "This honeycomb is one of <a href="/index.php?title=Uniform_6-polytope&action=edit&redlink=1" class="new" title="Uniform 6-polytope (page does not exist)">12 unique uniform honeycombs</a><ref>[http://mathworld.wolfram.com/Necklace.html mathworld: Necklace..."</p>
<p><b>New page</b></p><div>This honeycomb is one of [[Uniform_6-polytope#Regular_and_uniform_honeycombs|12 unique uniform honeycombs]]<ref>[http://mathworld.wolfram.com/Necklace.html mathworld: Necklace], {{OEIS el|1=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 [[automorphism]]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:<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
!colspan=5| A5 honeycombs<br />
|-<br />
![[:File:Regular hexagon symmetries.png|Hexagon<BR>symmetry]]<br />
![[Coxeter notation|Extended<BR>symmetry]]<br />
!Extended<BR>diagram<br />
!Extended<BR>group<br />
!Honeycomb diagrams<br />
|- style="text-align:center;"<br />
!a1[[File:Hexagon_symmetry_a1.png|40px]]<br />
![3<sup>[6]</sup>]<br />
|{{CDD|node|split1|nodes|3ab|nodes|split2|node}}<br />
|<math>{\tilde{A}}_5</math><br />
|{{CDD|node_1|split1|nodes_10lur|3ab|nodes|split2|node_1}}<br />
|- style="text-align:center;"<br />
!d2[[File:Hexagon_symmetry_d2.png|40px]]<br />
!<[3<sup>[6]</sup>]><br />
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c3|split2|node_c4}}<br />
|<math>{\tilde{A}}_5</math>×2<sub>1</sub><br />
|{{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}}<br />
|- style="text-align:center;"<br />
!p2[[File:Hexagon_symmetry_p2.png|40px]]<br />
!{{Brackets|3<sup>[6]</sup>}}<br />
|{{CDD|node_c3|split1|nodeab_c1-2|3ab|nodeab_c1-2|split2|node_c4}}<br />
|<math>{\tilde{A}}_5</math>×2<sub>2</sub><br />
|{{CDD|node|split1|nodes_10lur|3ab|nodes_10lru|split2|node}}<sub>[[Cyclotruncated 5-simplex honeycomb|2]]</sub>, {{CDD|node_1|split1|nodes_10lur|3ab|nodes_10lru|split2|node_1}}<br />
|- style="text-align:center;"<br />
!i4[[File:Hexagon_symmetry_i4.png|40px]]<br />
![<[3<sup>[6]</sup>]>]<br />
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c2|split2|node_c1}}<br />
|<math>{\tilde{A}}_5</math>×2<sub>1</sub>×2<sub>2</sub><br />
|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node_1}}, {{CDD|node|split1|nodes_11|3ab|nodes_11|split2|node}}<br />
|- style="text-align:center;"<br />
!d6[[File:Hexagon_symmetry_d6.png|40px]]<br />
!<3[3<sup>[6]</sup>]><br />
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c1|split2|node_c2}}<br />
|<math>{\tilde{A}}_5</math>×6<sub>1</sub><br />
|{{CDD|node_1|split1|nodes|3ab|nodes_11|split2|node}}<br />
|- style="text-align:center;"<br />
!r12[[File:Hexagon_symmetry_r12.png|40px]]<br />
![6[3<sup>[6]</sup>]]<br />
|{{CDD|node_c1|split1|nodeab_c1|3ab|nodeab_c1|split2|node_c1}}<br />
|<math>{\tilde{A}}_5</math>×12<br />
|{{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}}<sub>[[Omnitruncated 5-simplex honeycomb|3]]</sub><br />
|}<noinclude><br />
<br />
==References==<br />
{{reflist}}<br />
<br />
[[Category:Article shared content templates]]<br />
</noinclude></div>
alpha>Sarika