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From NPR:

How many shapes are able to "tile the plane" — meaning the shapes can fit together perfectly to cover any flat surface without overlapping or leaving any gaps. Mathematicians have proved that all triangles and quadrilaterals, or shapes with four sides, can tile the plane, and they have documented all of the convex hexagons that can do it.But it gets a lot more complicated when dealing with pentagons — specifically convex, or nonregular pentagons with the angles pointing outward. The number of convex pentagons is infinite — and so is the number that could potentially tile the plane. It's a problem that's almost unsolvable because, as McLoud-Mann put it, it has "infinitely many possibilities."