5 ms·
>If the tile is allowed to tile the plane both periodically and non-periodically, the solution would be obvious. Maybe you can draw a sketch for those of us wh
by GregBuchholz 9y ago
>If the tile is allowed to tile the plane both periodically and non-periodically, the solution would be obvious.
Maybe you can draw a sketch for those of us who aren't immediately seeing the obvious?
- kevinwang 9y agoSquares, but each row of squares has a random offset
- Terr_ 9y agoOr triangles, with similar "slide along the seam" changes.
- irishsultan 9y agoThat is a periodic tiling (just think about what happens if you follow the tiling along the rows instead of along the columns, you'll periodically see the same pattern).
- kevinwang 9y agoNo, you wouldn't necessarily
- irishsultan 9y agohttps://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_regular_polygons#Tilings_that_are_not_edge-to-edge https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_re... It's listed as a periodic tiling. If you move the whole plane one square to the left or right you'll find that everything fits into place nicely, making this a periodic tiling.
- kevinwang 9y agoHm, I'm not knowledgeable about this subject but this seems like the offset on each row is NOT random, but it's instead repeating. Imagine instead that each time I added a new row, I chose the offset to be, say, a completely new value.
- irishsultan 9y agoDoesn't matter, because your new row will still be offset with the same random value to existing rows after moving one square to the left or right.
- nimih 9y agoIf you allow reflections of the single tile, an easy counterexample is the pinwheel tiling[1], which is a non-periodic tiling composed entirely of isometric triangles. An (imo less satisfying) example which does not require reflection of the tile would be as follows: Take a rectangle with width equal to twice the height. You can use this tile to create squares which are either split vertically or horizontally. Put a single horizontally split square at the origin, then tile the remainder of the plane with vertically split squares: this tiling is (rather trivially) not periodic. [1]https://en.wikipedia.org/wiki/Pinwheel_tiling https://en.wikipedia.org/wiki/Pinwheel_tiling