Let $C$ be a convex shape in the plane. Your task is to cover the plane with copies of $C$, each under any rigid motion. My question is essentially: What is the worst $C$, the shape that forces the most wasteful overlap?

To be more precise, assume $C$ has unit area. Let $n_C(A)$ be the fewest copies of $C$ (under any rigid motions) that suffice to cover a disk of area $A$. I seek the $C$ that maximizes the "waste": $$ w(C) = \lim_{A \to \infty} n_C(A) / A \;.$$ So if $C$ is a perfect tiler of the plane, then $\lim_{A \to \infty} n_C(A) = A$ (because $C$ has area $1$) and $w(C)=1$, i.e., no waste.

Consider a regular pentagon $P$, which cannot tile the plane. Here is one way to cover the plane with regular pentagons:

If I've calculated correctly, this arrangement shows that $w(P) \le 1.510$. So one could cover an area $A=100$ with about $151$ unit-area regular pentagons, a $51$% waste. I doubt this is the best way to cover the plane with copies of $P$ (

**below), but it is one way.**

*Q3*Three questions.

. Is it known that the disk is the worst shape $C$ to cover the plane? My understanding is that L.F.Tóth's paper[1], which I have not accessed, establishes this forQ1latticetilings/coverings. Is it known for arbitrary coverings?

*Q1 Answered*. Thanks to several, and especially Yoav Kallus, for pointing me
in the right direction. Q1 remains an open problem. In [2,p.15], what I call
the waste of a convex body $C$ is called $\theta(C)$. It is about $1.209$ for a disk. The best upperbound is $\theta(C) \le 1.228$ due to Dan Ismailescu,
based on finding special tiling "p-hexagons" in $C$. A p-hexagon has two
opposite, parallel edges of the same length.

. Since every triangle, and every quadrilateral, tiles the plane, the first interesting polygonal shape is pentagons. What is the most wasteful pentagon?Q2

. More specifically, what is the waste $w(P)$ for the regular pentagon?Q3

[1] L. Fejes Tóth, "Lagerungen in der Ebene auf der Kugel und im Raum."

*Die Grundlehren der mathematischen Wissenschaften*Vol. 65. Springer-Verlag, 1972. doi:10.1007/978-3-642-65234-9

[2]
Brass, Peter, William OJ Moser, and János Pach. *Research Problems in Discrete Geometry*. Springer Science & Business Media, 2006.

Gerhard Paseman's recent comments, we get a waste ratio of only $w = (5+\sqrt{80})/11 < 1.2676611$ (if I computed right) for regular pentagons $P$ by finding a pentagon $P' \subset P$ of area $|P|/w$ that has two parallel sides and thus tiles the plane: fix a side $s$ of $P$, and let $P'$ consist of all points of $P$ that project to a point of $s$. $\endgroup$22more comments