4 ms·
I was about to ask the same question but it appears the repeats are not exactly identical at the edges. Still, can one prove this is aperiodic from geometry al
by jsd1982 4y ago
I was about to ask the same question but it appears the repeats are not exactly identical at the edges.
Still, can one prove this is aperiodic from geometry alone? It seems rather difficult to actually prove that fact. Feels intuitive that there must be a period somewhere, however large it may be, on the infinite 2D plane.
- HelloNurse 4y agoFor most sets of shapes a periodic tiling is possible, but by no means guaranteed. For example, consider rectangles with sides of 1 and 3 units: they can cover the plane periodically (e.g. in a simple rectangular grid), but also aperiodically, because you can form a square grid of square 3 by 3 units "metatiles", each encoding one bit of information in the vertical or horizontal orientation of the narrow rectangles; then it's easy to break symmetry by orienting metatiles so that for all integers m and n some metatile differs from the metatile m rows and n columns away, so the period cannot be m rows and n columns.