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Template:A7 honeycombs - Revision history
2026-06-13T02:23:37Z
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2022-12-15T04:11:51Z
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alpha>Sarika: Created page with "This honeycomb is one of 29 unique uniform honeycombs<ref>{{mathworld | urlname = Necklace | title = Necklace}}, {{OEIS e..."
2022-05-19T10:16:59Z
<p>Created page with "This honeycomb is one of <a href="/index.php?title=Uniform_8-polytope&action=edit&redlink=1" class="new" title="Uniform 8-polytope (page does not exist)">29 unique uniform honeycombs</a><ref>{{mathworld | urlname = Necklace | title = Necklace}}, {{OEIS e..."</p>
<p><b>New page</b></p><div>This honeycomb is one of [[Uniform_8-polytope#Regular_and_uniform_honeycombs|29 unique uniform honeycombs]]<ref>{{mathworld | urlname = Necklace | title = Necklace}}, {{OEIS el|1=A000029}} 30-1 cases, skipping one with zero marks</ref> constructed by <br />
the <math>{\tilde{A}}_7</math> [[Coxeter group]], grouped by their extended symmetry of rings within the [[regular octagon]] diagram:<br />
{|class="wikitable mw-collapsible mw-collapsed"<br />
!colspan=5| A7 honeycombs<br />
|-<br />
![[:File:Regular octagon symmetries.png|Octagon<BR>symmetry]]<br />
![[Coxeter notation|Extended<BR>symmetry]]<br />
!Extended<BR>diagram<br />
!Extended<BR>group<br />
!Honeycombs<br />
|- style="text-align:center;"<br />
!a1[[File:Octagon symmetry a1.png|40px]]<br />
![3<sup>[8]</sup>]<br />
|{{CDD|node|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}<br />
|<math>{\tilde{A}}_7</math><br />
|<br />
{{CDD|node_1|split1|nodes_10lur|3ab|nodes|3ab|nodes_10lru|split2|node}}<br />
{{CDD|node_1|split1|nodes_10lur|3ab|nodes|3ab|nodes|split2|node_1}}<br />
{{CDD|node_1|split1|nodes_10lur|3ab|nodes_10lr|3ab|nodes|split2|node_1}}<br />
{{CDD|node_1|split1|nodes_10lur|3ab|nodes|3ab|nodes_11|split2|node}}<br />
{{CDD|node_1|split1|nodes_10lur|3ab|nodes_10lr|3ab|nodes_11|split2|node}}<br />
|- style="text-align:center;"<br />
!d2[[File:Octagon symmetry d2.png|40px]]<br />
!<[3<sup>[8]</sup>]><br />
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c3|3ab|nodeab_c4|split2|node_c5}}<br />
|<math>{\tilde{A}}_7</math>×2<sub>1</sub><br />
|<br />
{{CDD|node|split1|nodes_11|3ab|nodes|3ab|nodes|split2|node}}<br />
{{CDD|node|split1|nodes|3ab|nodes_11|3ab|nodes|split2|node}}<br />
{{CDD|node|split1|nodes_11|3ab|nodes_11|3ab|nodes|split2|node}}<br />
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node}}<sub>[[7-simplex honeycomb|1]]</sub><br />
{{CDD|node_1|split1|nodes_11|3ab|nodes|3ab|nodes|split2|node}}<br />
{{CDD|node_1|split1|nodes|3ab|nodes_11|3ab|nodes|split2|node}} <br />
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes_11|split2|node}} <br />
<br />
{{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes|split2|node}}<br />
{{CDD|node_1|split1|nodes_11|3ab|nodes|3ab|nodes_11|split2|node}}<br />
{{CDD|node_1|split1|nodes|3ab|nodes_11|3ab|nodes_11|split2|node}}<br />
{{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes_11|split2|node}}<br />
{{CDD|node_1|split1|nodes_11|3ab|nodes|3ab|nodes|split2|node_1}}<br />
{{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes|split2|node_1}}<br />
|- style="text-align:center;"<br />
!p2[[File:Octagon symmetry p2.png|40px]]<br />
!{{Brackets|3<sup>[8]</sup>}}<br />
|{{CDD|branch_c1|3ab|nodeab_c2|3ab|nodeab_c3|3ab|branch_c4}}<br />
|<math>{\tilde{A}}_7</math>×2<sub>2</sub><br />
|<br />
{{CDD|branch_11|3ab|nodes|3ab|nodes|3ab|branch}} <sub>[[Cyclotruncated 7-simplex honeycomb|2]]</sub><br />
{{CDD|branch|3ab|nodes_11|3ab|nodes|3ab|branch}} <br />
{{CDD|branch_11|3ab|nodes_11|3ab|nodes|3ab|branch}} <br />
{{CDD|branch_11|3ab|nodes|3ab|nodes_11|3ab|branch}} <br />
{{CDD|branch_11|3ab|nodes_11|3ab|nodes_11|3ab|branch}} <br />
{{CDD|branch_11|3ab|nodes_11|3ab|nodes|3ab|branch_11}} <br />
|- style="text-align:center;"<br />
!d4[[File:Octagon symmetry d4.png|40px]]<br />
!<2[3<sup>[8]</sup>]><br />
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c3|3ab|nodeab_c2|split2|node_c1}}<br />
|<math>{\tilde{A}}_7</math>×4<sub>1</sub><br />
|<br />
{{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|split2|node_1}}<br />
{{CDD|node|split1|nodes_11|3ab|nodes_11|3ab|nodes_11|split2|node}}<br />
|- style="text-align:center;"<br />
!p4[[File:Octagon symmetry p4.png|40px]]<br />
!<nowiki>[2[</nowiki>3<sup>[8]</sup>]]<br />
|{{CDD|branch_c1|3ab|nodeab_c2|3ab|nodeab_c2|3ab|branch_c1}}<br />
|<math>{\tilde{A}}_7</math>×4<sub>2</sub><br />
|<br />
{{CDD|branch_11|3ab|nodes|3ab|nodes|3ab|branch_11}} <br />
|- style="text-align:center;"<br />
!d8[[File:Octagon symmetry d8.png|40px]]<br />
![4[3<sup>[8]</sup>]]<br />
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c1|3ab|nodeab_c2|split2|node_c1}}<br />
|<math>{\tilde{A}}_7</math>×8<br />
|{{CDD|node_1|split1|nodes|3ab|nodes_11|3ab|nodes|split2|node_1}}<br />
|- style="text-align:center;"<br />
!r16[[File:Octagon symmetry r16.png|40px]]<br />
![8[3<sup>[8]</sup>]]<br />
|{{CDD|node_c1|split1|nodeab_c1|3ab|nodeab_c1|3ab|nodeab_c1|split2|node_c1}}<br />
|<math>{\tilde{A}}_7</math>×16<br />
|{{CDD|node_1|split1|nodes_11|3ab|nodes_11|3ab|nodes_11|split2|node_1}}<sub>[[Omnitruncated 7-simplex honeycomb|3]]</sub><br />
|}<noinclude><br />
==References==<br />
{{reflist}}<br />
<br />
[[Category:Article shared content templates]]<br />
</noinclude></div>
alpha>Sarika