Tilings are easy to understand and relate to... you can tile your bathroom floor with them for example. Despite that superficial banality, it turns out that it takes a lot of mathematical cleverness to construct and analyze them fully. This particular result is one that people have been chasing for decades and has involved some of the smartest mathamaticians in the world, including Conway and Penrose.
I see thanks. I guess, hearing about other famous hard problems, I am used to seeing comments like: “if problem X could be solved it would unlock lots of other important areas of math.” Wondering if this tiling area draws attention for its own sake alone or if the problems here are similarly vectors to attack larger issues.
Math has problems where everyone suspects the proof will open doors and give valuable insight- I'm mostly plugged into topology where there are a lot of those. There are also problems that aren't interesting except in their difficulty, which drives the creation of new techniques and tools- Fermat's Last Theorem isn't particularly useful, but the effort to prove it created a vast body of spinoff work. But you also see problems that are more passive, waiting for someone to approach them with new firepower. Tilings are more like that- a testing ground for new techniques, and a way for mathematicians to keep their wits sharp.
Also, they do have some inherent beauty. I mean an aperiodic tiling is crazy right? And with one tile?
One big name is the Borel Conjecture- a very productive aspect of topology is developing finer and finer tools to detect differences between spaces. The Borel Conjecture essentially states that for a certain class of spaces, a well used and loved tool is equivalent to an extremely strong tool.
I was thinking more historically though. The development of those tools was driven by specific problems- classifying the behavior of higher dimensional spheres, determining if genus uniquely classifies spaces (not at all, but I believe people were once hopeful), even knot theory is an outgrowth of this kind of research.
Earthquake resistance of buildings. See Incan usage of aperiodic masonry to spread out the frequency response of the construction.
Similar principle with Apple's laptop fan blades.
Same mechanism might be responsible for the Boson peak phenomena in amorphous materials and quasicrystals, where the macro-structure creates extra capacity for absorbing lower-than-lattice-frequency vibrations than what the crystal-structure alone predicts.
Tilings can encode arbitrary computation. For example, any Turing machine can be encoded as a set of Wang tiles. Some shapes can tile the plane; others can't (they inevitably get stuck, with no space to attach new copies without overlap). This is precisely the Halting Problem.
One famous application of this is to encode these shapes using complementary snippets of DNA, to perform massively-parallel computation at the nano-scale: https://www.nature.com/articles/35035038
Not really. Aperiodic tilings, while lacking symmetry in a strict sense, still tend to look quite regular and repetitive, and thus non-natural.[1] Aperiodicity alone doesn't guarantee the "natural" look, which is unsurprising because natural textures are not stitched together from tiles either.
Hmmm... now that we know what shapes these tiles take, maybe we can look for or design molecules with the "same shape" (and yes I know molecules aren't 2D planar objects, but some can be approximated as such, e.g. the benzene ring) and with the right molecule-to-molecule attractions, such that they naturally arrange themselves into these aperiodic tilings.
Aluminum Oxide (aka Sapphire, Ruby) and ALON are already transparent despite being crystals.
Being extremely hard and resistant they are used in applications like watch crystals, windows in grocery-store barcode scanners and armored car windows.
