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Reference: https://en.wikipedia.org/wiki/Smoothed_octagon Background: The smoothed octagon is conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex ...

We extend Which convex pentagon gives least packing density? by going from convex pentagons to general ones. Question: Which pentagon gives the least packing density on the Euclidean plane? Note: All ...

Background: Consider trying to cover the largest possible scaled copy of a planar region $C$ with specified shape with n instances of a tile $T$ of specified shape and size. Several families of this ...