Filling the plane with squares and alternating rhombi

Consider the tilings of the whole plane by a blue square and a white rhombus with a 45° angle, so that whenever two rhombi are either adjacent, or connected by lined up squares, they are in different orientations.

It turns out that all these tilings can be seen as digitization of two-dimensional planes in the four-dimensional space: they are strongly ordered. Actually, they form in the Grassmannian Gr(2,4) a curve parametrized by the proportion of the plane covered by rhombi.

The video shows a walk along this curve, with this proportion ranging from 0% to 50% (after x seconds the proportion is x%): the proportion 50% is the maximum and is reached for the Ammann-Beenker tiling, a celebrated non-periodic tiling discovered in the 70's.

One can show that the closer two tilings are on this curve, the larger is the radius of the smallest pattern that they do not share (these tilings are thus not of finite type).