A tiling of a planar shape is called monohedral if all tiles are congruent to each other. We will investigate the possibility of producing monohedral tilings of the disk. Such tilings are produced on a daily basis by pizza chefs by taking radial cuts distributed evenly around the centre of the pizza. After constructing this tiling, a neighbourhood of the origin has non-trivial intersection with each tile. This brings us to the main question of this article:

Can we construct monohedral tilings of the disk such that a neighbourhood of the origin has trivial intersection with at least one tile?

The answer to this question is yes! The particular solution given in Figure 2 appears as the logo of the Mass Program at Penn State [1]. This website states,

[All] known solutions (except for the one in the logo) have the property that the boundaries of all the pieces consist only of circular arcs of the same radius as the original circle. We do not know whether there is a partition of the disc into congruent pieces other than the logo such that not all pieces contain the center and not all boundaries are circle arcs.

The family of solutions mentioned are displayed in Figure 5 and are called T 2 in our notation. 6n

However, this article presents a new family of solutions to this problem which generalise the solution in Figure 2, and for which some of the edges of the tiles are straight; i.e. they are not circular arcs of the same radius of the circle. Each member of this new family contains uncountably many solutions.

We do not claim that this list is exhaustive and it is still an open problem either to find all such tilings or to prove that all have been found.

A specific case of the problem should be mentioned: to produce a monohedral disk tiling such that a neighbourhood of the origin is contained entirely within the interior of a single tile. This was presented in [2] as an unsolved problem, and may be impossible (e.g. [3]). For all known solutions the centre of the disk intersects tiles only at their vertices. It is unknown whether a tiling of the nature discussed here exists such that the origin intersects a tile on its edge, but not at a vertex.