Template:A5 honeycombs

This honeycomb is one of 12 unique uniform honeycombs[http://mathworld.wolfram.com/Necklace.html mathworld: Necklace], {{OEIS el|1=A000029}} 13-1 cases, skipping one with zero marks 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:

class="wikitable mw-collapsible mw-collapsed" !colspan=5\| A5 honeycombs
Hexagon symmetry !Extended symmetry !Extended diagram !Extended group !Honeycomb diagrams
style="text-align:center;" !a140px ![3^[6]] \|{{CDD\|node\|split1\|nodes\|3ab\|nodes\|split2\|node}} \| ${\tilde{A}}_5$ \|{{CDD\|node_1\|split1\|nodes_10lur\|3ab\|nodes\|split2\|node_1}}
style="text-align:center;" !d240px !<[3^[6]]> \|{{CDD\|node_c1\|split1\|nodeab_c2\|3ab\|nodeab_c3\|split2\|node_c4}} \| ${\tilde{A}}_5$ ×2₁ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes\|split2\|node}}₁, {{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}}
style="text-align:center;" !p240px !{{Brackets\|3^{{{Bracket\|6}}}}} \|{{CDD\|node_c3\|split1\|nodeab_c1-2\|3ab\|nodeab_c1-2\|split2\|node_c4}} \| ${\tilde{A}}_5$ ×2₂ \|{{CDD\|node\|split1\|nodes_10lur\|3ab\|nodes_10lru\|split2\|node}}₂, {{CDD\|node_1\|split1\|nodes_10lur\|3ab\|nodes_10lru\|split2\|node_1}}
style="text-align:center;" !i440px ![<[3^[6]]>] \|{{CDD\|node_c1\|split1\|nodeab_c2\|3ab\|nodeab_c2\|split2\|node_c1}} \| ${\tilde{A}}_5$ ×2₁×2₂ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes\|split2\|node_1}}, {{CDD\|node\|split1\|nodes_11\|3ab\|nodes_11\|split2\|node}}
style="text-align:center;" !d640px !<3[3^[6]]> \|{{CDD\|node_c1\|split1\|nodeab_c2\|3ab\|nodeab_c1\|split2\|node_c2}} \| ${\tilde{A}}_5$ ×6₁ \|{{CDD\|node_1\|split1\|nodes\|3ab\|nodes_11\|split2\|node}}
style="text-align:center;" !r1240px ![6[3^[6]]] \|{{CDD\|node_c1\|split1\|nodeab_c1\|3ab\|nodeab_c1\|split2\|node_c1}} \| ${\tilde{A}}_5$ ×12 \|{{CDD\|node_1\|split1\|nodes_11\|3ab\|nodes_11\|split2\|node_1}}₃

class="wikitable mw-collapsible mw-collapsed"

!colspan=5| A5 honeycombs

Hexagon
symmetry

!Extended
symmetry

!Extended
diagram

!Extended
group

!Honeycomb diagrams

style="text-align:center;"

!a140px

![3^[6]]

|{{CDD|node|split1|nodes|3ab|nodes|split2|node}}

| ${\tilde{A}}_5$

|{{CDD|node_1|split1|nodes_10lur|3ab|nodes|split2|node_1}}

style="text-align:center;"

!d240px

!<[3^[6]]>

|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c3|split2|node_c4}}

| ${\tilde{A}}_5$ ×2₁

|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}}₁, {{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}}

style="text-align:center;"

!p240px

!{{Brackets|3^{{{Bracket|6}}}}}

|{{CDD|node_c3|split1|nodeab_c1-2|3ab|nodeab_c1-2|split2|node_c4}}

| ${\tilde{A}}_5$ ×2₂

|{{CDD|node|split1|nodes_10lur|3ab|nodes_10lru|split2|node}}₂, {{CDD|node_1|split1|nodes_10lur|3ab|nodes_10lru|split2|node_1}}

style="text-align:center;"

!i440px

![<[3^[6]]>]

|{{CDD|node_c1|split1|nodeab_c2|3ab|nodeab_c2|split2|node_c1}}

| ${\tilde{A}}_5$ ×2₁×2₂

|{{CDD|node_1|split1|nodes|3ab|nodes|split2|node_1}}, {{CDD|node|split1|nodes_11|3ab|nodes_11|split2|node}}