Fundamental increment lemma

This honeycomb is one of 12 unique uniform honeycombs constructed by the ${\tilde {A}}_{5}$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\tilde {A}}_{5}$ 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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