Template:A4 honeycombs

This honeycomb is one of seven unique uniform honeycombs^[1] constructed by the ${\tilde {A}}_{4}$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

A4 honeycombs
Pentagon symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
a1	[3^[5]]	File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel branch.png	${\tilde {A}}_{4}$	(None)
i2	[[3^[5]]]	File:CDel split1.png File:CDel nodeab c2.png File:CDel 3ab.png File:CDel branch c3.png	${\tilde {A}}_{4}$ ×2	File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel branch.png ₁,File:CDel node.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel branch.png ₂,File:CDel node.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel branch 11.png ₃, File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel branch.png ₄,File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel branch 11.png ₅,File:CDel node.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel branch 11.png ₆
r10	[5[3^[5]]]	File:CDel split1.png File:CDel nodeab c1.png File:CDel 3ab.png	${\tilde {A}}_{4}$ ×10	File:CDel node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel 3ab.png File:CDel branch 11.png ₇

References

↑ mathworld: Necklace, OEIS sequence A000029 8-1 cases, skipping one with zero marks