What is the limit, as the radius of the disk increases, of the greatest area, in proportion to the area of the disk, of the region covered by regular pentagons of the same fixed size, all lying within a large disk and not overlapping (except perhaps at boundary points)?
2026-03-27 01:44:54.1774575894
How far can the plane be tiled by congruent regular pentagons?
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You posted only two weeks too early!
This question is more concisely phrased as asking for the maximum packing density of the regular pentagon. This had long been conjectured to be $\frac{5-\sqrt{5}}3\approx0.92131\ldots$, but was only resolved in the affirmative by Hales and Kusner on Feb 23, 2016. The optimal packing looks like this: