I have been interested in overlapping tiling recently. I’ve made a few cool designs, but I can’t think of any use for overlapping tiles in 2 dimensions. Using overlapping tiles to create a sphere has a purpose, because the fact that the tiles are overlapping holds the sphere together.
The design
in the link shows my overlapping shell sphere design. This tiling has 8 sphere segments (shells) of the same size and shape (the blue and purple elements), and 6 curved cross shaped elements (grey). The cross shaped elements are created flat from a resilient material, and bent to fit into the gaps between the blue and purple shells.
How/why it works:
The two inside edges of the resilient crosses push against the purple shells outward, but the purple shells are kept in place by the blue shells. The purple shells push against the blue shells, but the blue shells are kept in place by the outside edges of the resilient crosses.
Constructing this would be tricky since it would only be stable once all the shells and resilient crosses are in place. It could be done with scaffolding, wherein all the shells are held in place, then the resilient crosses are inserted.
This could be used as an interesting spherical packaging for toys. Once a single resilient cross is removed, the whole sphere falls to pieces.
This design is based on a truncated octahedron. Interestingly only two other designs could be made (based on the great rhombicuboctahedron or the great rhombicosidodecahedron), using the same kind of overlapping design.