In my paper Pessimal Packing Shapes, I put forward a conjecture that any convex shape in the plane can be packed more densely than can regular heptagons using a double lattice construction. However, I would be delighted to be proven wrong. I wrote this little javascript program to construct double lattice packing arrangements of convex shapes.
I give you control of 14 polygon vertices. You cannot make an arbitrary convex shape, but you have considerable freedom. (If you want more control points, feel free to download and modify the page source code). A packing arrangement is constructed in real time and the density is displayed at the bottom of the page.
Can you make a convex shape whose best packing is not as dense as the best packing of the regular heptagon? The number to beat is 0.892690686…. If you succeed, copy the coordinates specifying the shape you constructed.
shape area:
unit cell area:
density:
coordinates:
Further reading: A packing pessimization problem, by John Baez.
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