The thinnest known covering of the plane with congruent regular pentagons is shown in my answer to: Terrible tilers for covering the plane . The covering is of the "double-lattice" type and is known to be the thinnest among double lattice coverings with regular pentagons (to be published). Also, it is conjectured to be the thinnest one among all coverings with congruent regular pentagons. The conjecture is still open.
Wlodek Kuperberg
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