This picture by Greg Egan is more profound than it looks. The area of each white space doesn't change as the wheels roll. But that's not all!
What shape is the worst at densely packing the plane? To make this question interesting we should ask about convex shapes - remember, a shape is convex if whenever two points are in it, so is the line between them. Then the answer seems to be a regular heptagon... though nobody has been able to prove this. The densest packing of regular heptagons fills up about 89.3% of the plane. For comparison, circles fill up 90.7%.
But suppose we also demand that our shape be centrally symmetric - meaning that if you reflect it vertically and then reflect it horizontally, it looks the same. Packings of centrally symmetric convex shapes are a lot easier to understand.
What's the worst centrally symmetric convex shape for densely packing the plane? It seems to be a regular octagon with its corners smoothed in a certain clever way. Nobody has proved this... but Thomas Hales, the guy who proved Kepler's conjecture on densely packing spheres, wants to prove this conjecture too!
The densest packing of these smoothed octagons fills up about 90.2% of the plane.
But there's not just one densest packing! You can turn the smoothed octagons while keeping the density the same! And that's what you see here.
Notice that the centers of the octagons move as we turn them. This makes me a bit sea-sick. But the math is beautiful.
There's a lot more to say about this, and I said it here:
I recommend this paper, too:
• Yoav Kallus, Least efficient packing shapes, http://arxiv.org/abs/1305.0289.
#spnetwork arXiv:1305.0289 #geometry #packing