Iterated Voronoi diagrams of Penrose Tilings

[pcallahan]

Penrose tilings (either rhombs or kites and darts) are angular and in my opinion not that aesthetically pleasing aside from mathematical interest. That's a personal preference, and I know they have been used decoratively to great effect.

By contrast, the Voronoi diagram of the vertices has a natural look. It could be cells in tissue, sunflower seeds, or a cobblestone pavement. After making a Voronoi diagram of a Penrose rhomb tiling, I wondered if the process could be iterated. I.e., take the vertices of the last tessellation and construct its Voronoi diagram.

I don't think the results improve beyond the second case, but I'm posting here anyone is curious. I'll follow up with three more after this. Note that some tessellations have stable oscillations, like a square or hex/triangle mesh. I was hoping for something similar but what I realized is that as the cells grow irregular in shape, we get clusters of internal vertices that are close but not identical.

The original rhomb tiling.

rhomb.png (386.71 KiB) Viewed 190 times

The Voronoi diagram of its vertices.

voronoi1.png (441.12 KiB) Viewed 190 times

The Voronoi diagram of the vertices of the Voronoi diagram.

voronoi2.png (616.79 KiB) Viewed 190 times

[pcallahan]

It gets more complicated in successive iterations and less practical as a tiling. You can see that the cells break up into slivers and the size is increasingly non-uniform. I am actually starting to like these as I look at them, but the first Voronoi is the one I like best. It breaks up the plane into 7 distinct shapes, all of them reasonably compact.

voronoi3.png (704.39 KiB) Viewed 189 times

voronoi4.png (925.33 KiB) Viewed 189 times

voronoi5.png (1.12 MiB) Viewed 189 times