Why is it called hexagonal and not triagonal packing?

Question

The densest way to pack circles in a plane is called hexagonal packing (https://en.wikipedia.org/wiki/Circle_packing).

In my understanding, the basic cell of this packing is the displayed hexagon. The tiling can be extended infinitely by "copying" and attaching this basic cell in all directions.

To my question: Would it not be the same thing to have the basic cell as a triangle (of which there are 6 in the hexagon)? This would be more concise. If you define this basic cell of the packing, why not make it as small as possible? It is repeated by definition.

Answer 1

When you draw in the triangles, not all the triangles are oriented in the same direction, so it is impossible to translate (without rotating) some triangles onto each other. Describing the tiling as hexagonal gives you this nice property that each unit cell may be translated onto each other, giving us a better description of the shape of the tiling.

Answer 2

In fact the centers of the red discs define the Delaunay triangulation, i.e. the triangle tiling (trat, x3o6o). Whereas the centers of the white bits inbetween would describe the according Voronoi tiling, which indeed is a hexagonal tiling (hexat, o3o6x), but different from your drawn in hexagon.

The Delaunay tiling generally is the dual of the Voronoi tiling. (Sometimes Delaunay also gets transcribed from russian as Delone.)

--- rk