Template:A4 honeycombs

This honeycomb is one of seven unique uniform honeycombs[http://mathworld.wolfram.com/Necklace.html mathworld: Necklace], {{OEIS el|1=A000029}} 8-1 cases, skipping one with zero marks constructed by the ${\tilde{A}}_4$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

class="wikitable mw-collapsible mw-collapsed" !colspan=5\| A4 honeycombs
Pentagon symmetry !Extended symmetry !Extended diagram !Extended group !Honeycomb diagrams
a1 ![3^[5]] !{{CDD\|node\|split1\|nodes\|3ab\|branch}} \| ${\tilde{A}}_4$ \| (None)
i2 !{{Brackets\|3^{{{Bracket\|5}}}}} !{{CDD\|node_c1\|split1\|nodeab_c2\|3ab\|branch_c3}} \| ${\tilde{A}}_4$ ×2 \|{{CDD\|node_1\|split1\|nodes\|3ab\|branch}} ₁,{{CDD\|node\|split1\|nodes_11\|3ab\|branch}} ₂,{{CDD\|node\|split1\|nodes\|3ab\|branch_11}} ₃, {{CDD\|node_1\|split1\|nodes_11\|3ab\|branch}} ₄,{{CDD\|node_1\|split1\|nodes\|3ab\|branch_11}} ₅,{{CDD\|node\|split1\|nodes_11\|3ab\|branch_11}} ₆
r10 ![5[3^[5]]] !{{CDD\|node_c1\|split1\|nodeab_c1\|3ab\|branch_c1}} \| ${\tilde{A}}_4$ ×10 \|{{CDD\|node_1\|split1\|nodes_11\|3ab\|branch_11}} ₇

class="wikitable mw-collapsible mw-collapsed"

!colspan=5| A4 honeycombs

Pentagon
symmetry

!Extended
symmetry

!Extended
diagram

!Extended
group

!Honeycomb diagrams

![3^[5]]

!{{CDD|node|split1|nodes|3ab|branch}}

| ${\tilde{A}}_4$

| (None)

!{{Brackets|3^{{{Bracket|5}}}}}

!{{CDD|node_c1|split1|nodeab_c2|3ab|branch_c3}}

| ${\tilde{A}}_4$ ×2

|{{CDD|node_1|split1|nodes|3ab|branch}} ₁,{{CDD|node|split1|nodes_11|3ab|branch}} ₂,{{CDD|node|split1|nodes|3ab|branch_11}} ₃,

{{CDD|node_1|split1|nodes_11|3ab|branch}} ₄,{{CDD|node_1|split1|nodes|3ab|branch_11}} ₅,{{CDD|node|split1|nodes_11|3ab|branch_11}} ₆

r10

![5[3^[5]]]

!{{CDD|node_c1|split1|nodeab_c1|3ab|branch_c1}}

| ${\tilde{A}}_4$ ×10

|{{CDD|node_1|split1|nodes_11|3ab|branch_11}} ₇

References

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