Template:A5 honeycombs: Difference between revisions

Latest revision as of 09:41, 15 December 2022

This honeycomb is one of 12 unique uniform honeycombs^[1] constructed by the ${\tilde {A}}_{5}$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\tilde {A}}_{5}$ Coxeter group allows for automorphisms 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:

A5 honeycombs
Hexagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
a1File:Hexagon symmetry a1.png	[3^[6]]	File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node.png	${\tilde {A}}_{5}$	File:CDel node 1.png File:CDel split1.png File:CDel nodes 10lur.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node 1.png
d2File:Hexagon symmetry d2.png	<[3^[6]]>	File:CDel node c1.png File:CDel split1.png File:CDel nodeab c2.png File:CDel 3ab.png File:CDel nodeab c3.png File:CDel split2.png File:CDel node c4.png	${\tilde {A}}_{5}$ ×2₁	File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node.png₁, File:CDel node.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node.png, File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node.png, File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node 1.png, File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png
p2File:Hexagon symmetry p2.png	[[3^[6]]]	File:CDel node c3.png File:CDel split1.png File:CDel nodeab c1-2.png File:CDel 3ab.png File:CDel nodeab c1-2.png File:CDel split2.png File:CDel node c4.png	${\tilde {A}}_{5}$ ×2₂	File:CDel node.png File:CDel split1.png File:CDel nodes 10lur.png File:CDel 3ab.png File:CDel nodes 10lru.png File:CDel split2.png File:CDel node.png₂, File:CDel node 1.png File:CDel split1.png File:CDel nodes 10lur.png File:CDel 3ab.png File:CDel nodes 10lru.png File:CDel split2.png File:CDel node 1.png
i4File:Hexagon symmetry i4.png	[<[3^[6]]>]	File:CDel node c1.png File:CDel split1.png File:CDel nodeab c2.png File:CDel 3ab.png File:CDel nodeab c2.png File:CDel split2.png File:CDel node c1.png	${\tilde {A}}_{5}$ ×2₁×2₂	File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes.png File:CDel split2.png File:CDel node 1.png, File:CDel node.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png
d6File:Hexagon symmetry d6.png	<3[3^[6]]>	File:CDel node c1.png File:CDel split1.png File:CDel nodeab c2.png File:CDel 3ab.png File:CDel nodeab c1.png File:CDel split2.png File:CDel node c2.png	${\tilde {A}}_{5}$ ×6₁	File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png
r12File:Hexagon symmetry r12.png	[6[3^[6]]]	File:CDel node c1.png File:CDel split1.png File:CDel nodeab c1.png File:CDel 3ab.png File:CDel nodeab c1.png File:CDel split2.png File:CDel node c1.png	${\tilde {A}}_{5}$ ×12	File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png₃

References

↑ mathworld: Necklace, OEIS sequence A000029 13-1 cases, skipping one with zero marks

[1] thworld: Necklace, OEIS sequence A000029 13-1 cases, skipping one with zero marks

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Revision as of 15:46, 19 May 2022 (view source) alpha>Sarika (Created page with "This honeycomb is one of 12 unique uniform honeycombs<ref>[http://mathworld.wolfram.com/Necklace.html mathworld: Necklace...")	Latest revision as of 09:41, 15 December 2022 (view source) Indicwiki (talk \| contribs) m (1 revision imported from alpha:Template:A5_honeycombs)
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