5 ms·

I dont't get it. In the picture, each tile is composed of 8 identical quadrilaterals. So why don't these quadrilaterals constitute a simpler shape that can ti

• twanvl 4y ago
  This tile forces a pattern that does not repeat. If you use the simpler shape you can tile a wall such that it never repeats, but you can also make a repeating pattern.
• plopilop 4y ago
  The key point (often missed by these articles) is that aperiodic tilings do not have a (infinite) periodic pattern. This means that you cannot draw a shape on these tiles and say: "the tiling is based on infinite repetitions of this shape, and only this shape". Of course individual tiles will repeat, but never in an infinite periodic pattern. Edit: a novelty of this paper is that their shape is "truly" aperiodic, which means no matter how hard you try, you will end up with aperiodic tiling. Existing one-shape aperiodic tilings had to add constraints on how to put two shapes next to each other to ensure aperiodicity.
  • notfed 4y ago
    "Of course individual tiles will repeat" What does this mean? How can an individual tile "repeat"?
    • plopilop 4y ago
      I meant "a given orientation of the tile will be present many times (or infinitely many)". It's very probable (I did not read the paper, only their website page) that the tile only occupies a finite amount of orientations in the tiling and therefore at least (and probably more if not all) one orientation will also be present an infinite amount of times. However this does not imply periodicity of the tiling.
• teawrecks 4y ago
  Someone below mentioned that the important part isn't that it tiles aperiodically, but that it ONLY tiles aperiodically. There are simpler shapes that will tile both aperiodically and periodically.