Triangular tiling: Difference between revisions

Revision as of 15:18, 20 September 2018

Revision as of 10:03, 7 October 2018

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:{{CDD|node_1|split1|branch}} + {{CDD|node|split1|branch_10lu}} + {{CDD|node|split1|branch_01ld}} = dual of {{CDD|node_1|split1|branch_11}} = {{CDD|node_1|split1|branch}}

The vertices of the triangular tiling are the centers of the densest possible [[circle packing]].<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, pattern 1</ref> Every circle is in contact with 6 other circles in the packing ([[kissing number]]). The packing density is {{frac|{{pi}}|{{sqrt|12}}}} or 90.69%. Since the union of 3 A₂ lattices is also an A₂ lattice, the circle packing can be given with 3 colors of circles.

The [[voronoi cell]] of a triangular tiling is a [[hexagon]], and so the [[voronoi tessellation]], the hexagonal tiling, has a direct correspondence to the circle packings.

:[[File:1-uniform-11-circlepack.svg|200px]]

{| class=wikitable

!A₂ lattice circle packing

!A{{sup sub|*|2}} lattice circle packing

|-

|[[File:triangular tiling circle packing.png|240px]]

|[[File:Triangular tiling circle packing3.png|240px]]

|}

== Geometric variations ==