I love how there's a mix of simple things like why you can't tile things with more than 6 edges and really complex things like what the headline is about.
Unless I'm wrong, the link you gave only speaks about impossibility for regular convex polygons, whereas the original links deals with the more general impossibility for any convex polygon with more than 6 sides.
When it came to the description of "einsteins" (a single tile aperiodic tessellation), I couldn't help but think of the images in the 3rd edition of The Scheme Programming Language:
All three of the Scheme examples can be tiled periodically even though they aren't tiled periodically in the examples. How to tile them periodically is left as an exercise to the reader, but it's not hard.
"Depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, the problem is either open or solved."
The spiral images have tiles that are reflected, not just rotated. You'd otherwise need a third dimension to do another rotation. I think that might be considered two different tiles by the rules laid out in the article.
Me neither - then I googled her and found that she had died just over a week ago (02/07/17). As sad as this is, it seems that she was a remarkably bright woman who lived a long and fruitful life. It's satisfying that in the seemingly ultra-inaccessible world of modern mathematics that (for want of a better term) normal people can still make headway.
I was recently watching some Computerphile videos and was surprised to learn that several geometric problems have fundamental applications outside the physical sciences, such as geometric sphere packing and error correcting codes. Does tiling research have any known applications in CS or information theory?
Wang tiles, mentioned only at the end of the article, encode Turing-equivalent computation in their simple color-matching rule. It is unknown exactly how useful Wang tiles are to theory of computation, other than as a wonderful demonstration of how easily Turing-complete systems arise naturally.
I suspect that tiling may have applications in resource management, particularly when allocating resources like memory which have an underlying topological structure.
Wang tiles made from DNA (using bases to encode the "colours", so matching up base pairs enforces the rules) have been used to perform computation.
AFAIK it's not yet a practical tool, but the choice of Wang tiles was practical for the experimenters since they're known to be Turing complete, and hence make for a more compelling demonstration.
Working with geographic systems, using tiling is a way to perform spatial indexing. If there are novel approaches to nonstandard (rectangular) tiling that better group geographic data for analysis, this could potentially lead to faster geographic computations in an area that is well known for having quite expensive approaches.
Material physics obviously. If can aperiodically pack molecules better than before you got a new super material. With an "Einstein" (single molecul) it would be easiest, of course. Penrose like tilers need two and they are not so easy to arrange.
When they talk about the einstein, I assume they mean a shape that can only tile the plane non-periodically. If the tile is allowed to tile the plane both periodically and non-periodically, the solution would be obvious.
That 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).
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.
Hm, 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.
If 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.
The UI is not perfect, especially the tiling designer (but once you master the keyboard shortcut it works okay). And sin of sins, it's written in Java! But I do think some of the built-in example are nice. I'm biased.
