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https://en.formulasearchengine.com/w/index.php?title=Fundamental_increment_lemma&diff=27937
Fundamental increment lemma
2013-11-01T16:37:02Z
<p>190.19.175.174: /* Differentiability in higher dimensions */ The error funciton Phi takes values in R not in R^{n}</p>
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<div>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:<br />
<br />
{| class=wikitable<br />
![[:File:Regular hexagon symmetries.png|Hexagon<BR>symmetry]]<br />
![[Coxeter notation|Extended<BR>symmetry]]<br />
!Extended<BR>diagram<br />
!Extended<BR>order<br />
!Diagrams<br />
|- align=center<br />
!a1<br />
![3<sup>[6]</sup>]<br />
|{{CDD|node|split1|nodes|3ab|nodes|split2|node}}<br />
|×1<br />
|{{CDD|node_1|split1|nodes_10lur|3ab|nodes|split2|node_1}}<br />
|- align=center<br />
!d2<br />
!<[3<sup>[6]</sup>]><br />
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c3|split2|node_c4}}<br />
|×2<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 />
|- align=center<br />
!p2<br />
!<nowiki>[[</nowiki>3<sup>[6]</sup>]]<br />
|{{CDD|node_c3|split1|nodeab_c1-2|3ab|nodeab_c1-2|split2|node_c4}}<br />
|×2<br />
|{{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}}<br />
|- align=center<br />
!i4<br />
![2[3<sup>[6]</sup>]]<br />
|{{CDD|node_c3|split1|nodeab_c1-2|3ab|nodeab_c1-2|split2|node_c3}}<br />
|×4<br />
|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node_1}}, {{CDD|node|split1|nodes_11|3ab|nodes_11|split2|node}}<br />
|- align=center<br />
!d6<br />
![3[3<sup>[6]</sup>]]<br />
|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c1|split2|node_c2}}<br />
|×6<br />
|{{CDD|node_1|split1|nodes|3ab|nodes_11|split2|node}}<br />
|- align=center<br />
!r12<br />
![6[3<sup>[6]</sup>]]<br />
|{{CDD|node_c1|split1|nodeab_c1|3ab|nodeab_c1|split2|node_c1}}<br />
|×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:Encyclopedic content templates]]<br />
</noinclude></div>
190.19.175.174