Circles

A circle is a shape defined as being those points an equal distance away from the center. We could use the pythagorean theorm to define distance, however that is boring. Instead, we can use the following distance measurement: (change in x + change in y). This is called the Taxicab metric and this gives us a diamond shape.

If we rotate this shape by 45 degrees, we get the square tiling again, which means I have one less picture to make (at least I found a good image editor for Linux now)

In all seriousness, the circle can only cover 90.7% of the plane and so cannot tile it

Fun fact about circles: They the best shape for rolling stuff