. Is it known what is the thinnest covering of the infinite plane by regular pentagons?Q

By *covering* I mean every point of the plane is covered.
By *thinnest* I mean the proportion of the plane covered more than once is minimal among all coverings.

This seems like it must be known, but I cannot find it, perhaps because I don't know the correct terminology.

This is a natural attempt:

If I've calculated correctly, this covering doubly covers about $38\%$ of the plane: $$\tfrac{1}{2} \left(3-\sqrt{5}\right) \approx 0.382 \;.$$ I am interested because the above covering can be achieved by "rolling" a dodecahedron, and I'd like to know if there is a thinner cover which might not be "rollable."