A visualization of tilings as projections from a higher dimensional integer lattice!
A pattern of tiling can be seen as a projection of a slice of a tiling in a higher dimension. Especially interesting is how aperiodic tilings can be seen as a projection of a slice of a periodic tiling in a higher dimension.
If the slope or gradient of the projection plane is rational, it may be written as a fraction, the rise over the run. The rise over the run, when it is able to be expressed as a relation of integers, ultimately describes the interval at which the plane intersects nodes on the higher dimensional integer lattice. The existence of such an interval implies that tilings generated from rational slopes are periodic because the projected pattern between the cast shadows of every intersecting node repeats for each interval.
If the slope is irrational, however, then it doesn't intersect any node other than the origin, and this means the resulting tiling is aperiodic.
The blue square (the "unit hypercube") is used to isolate the slice of the lattice that is projected. The position of the unit hypercube acts as the seed that determines which (of infinitely many, in the case of aperiodic tilings) variation of possible tilings is cast.
I made this visualizer for the case of a 1-dimensional tiling projected from a 2-dimensional lattice to help understand the basic relationships before considering higher dimensional cases, in particular, the Penrose tilings, which can be seen as 2-dimensional shadows from a 5-dimensional lattice where the slope of the projection plane involves the golden ratio.
I think these things are cooler than they have a right to be. XD
- Left click and drag to move the unit hypercube.
- Middle click and drag to pan the camera.
- Middle click and drag while holding shift to rotate the camera.
- Right click and drag to set the slope of the line.
- Shadows from the Fifth Dimension (This is a short summary of projected Penrose tilings.)
- The Empire Problem in Penrose Tilings (Section 2.2 has a good explanation the sort of thing being visualized here.)