Aperiodic Tilings
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August 21, 2021
Intro to the How and Why of Aperiodic Tilings<br>Tilings! Also called tessellations, patterns in the Euclidian plane, space filling arrangements of infinitely many copies of geometric shapes, or just considered to be things in photos of interior design magazines... Tilings have a rich history, from the Sumerians in 4000BC, via the Romans' geometric mosaics, through the Girih and Zellij tilings used in Medieval mosques, to the modern theorems and ideas that have lead to new materials, facts about arrangements of things and packings, and interesting YT videos on the quirkier side of 'covering the plane with shapes'...
To anyone who knows me personally, or professionally, it should be no surprise that I get rather very excited when I see videos on YouTube or blog posts or book sections on tilings.
The reason why might not be that well known - but I spent the years on my PhD studying the computability and unsolvability of tilings!! (For the curious, here's my thesis: https://etheses.whiterose.ac.uk/26051/ ) I've found them fascinating for most of my life, and this post aims to share a portion of that.
There is a rich theory and history of results, many of which are surprisingly recent! Granted, 'recent' in pure maths does indeed include 'about 40 years ago', but when you consider that tilings in one shape or form (geddit?) go back over 6,000 years, it is remarkable that some of these ideas, as best we know from analysing old and ancient texts and artefacts, have only come to light in the 20th and 21st centuries.
In this blog post, I want to talk about something that I've not seen explained or covered on YT or in many blogs - I want to show how to answer the following question: What makes a tiling aperiodic?
Essentially, how do we show that a given tiling is 'aperiodic'? How do you prove that a tiling will never ever repeat itself in the plane? It sounds daunting, but it's surprisingly straightforward.
Whilst there are plenty of resources that talk about Penrose tilings, for example, and extoll the virtues of an aperiodic tiling, such as this frankly excellent video from Veritasium; 'The Infinite Pattern that Never Repeats', the fact of 'how and why' tilings are, or can be aperiodic is often glossed over.
The mathematics that makes this fact true of Penrose tilings, and indeed other tilings such as those from Berger in the 60's, is rather beautiful and I think quite intuitive if you touch on all the salient points with some clarity.
But first...<br>What is a tiling?<br>It may seem rather intuitive, but defining a few terms will be useful later on:<br>A tiling is a covering of the plane using selected shapes from a tile set.<br>A tile set is just the set of prototypes for the shapes we are going to use. This is also called a 'palette' in the literature - poetically giving the impression of an artist choosing from the set the most beautiful options with which to cover the plane! 😊<br>A patch is a finite 'block' of tiles covering some finite portion of the plane - think of it as a 'cutting' from the overall tiling of the plane.<br>The idea of a tiling then is that we have a 'finite box' of tiles, each of which we can make infinitely many copies of and arrange them in the plane, usually just using rotations and translations, such that we can cover the plane entirely using just shapes from that initial box.
Now, there are lots and lots of questions and parameters you can apply... for example, in Wang tilings you are not allowed to rotate the square diagonally quadrisected and coloured tiles, whereas in most other tilings you can. In some tilings you might admit that you can 'flip' the tiles - like a pancake - so that they are mirrored across some axis - but we won't consider those for now.
There are also questions about 'what counts as a tiling?' You might be able to tile the plane but you have to admit one or finitely many holes - is that ok? Do you have to use every tile in the original tile set? And use each one infinitely often?
There are actually no hard and fast rules in general - this is in part what makes tilings quite interesting when you play with these options! For ease of use, we'll make the following assumptions - for a given tile set \(S\):<br>You must use every tile \(t\) in \( S \).<br>You must use each tile infinitely often.<br>You are not allowed any holes in your resulting \(S\)-tiling.<br>Rule 1 is fairly redundant, it just maintains that our tile sets will be as small as possible. Rule 2 maintains that our tilings are in some sense 'uniform', and rule 3 means we don't have to worry about any holes - if there's a hole, it's not a tiling! (Isn't pure maths nice, the way you can just 'get rid' of cases you don't want?)
So what is periodicity?<br>Firstly, let's define periodicity - put simply, if a tiling is periodic, then it has a 'period'. This will be some linear shift (translation and/or...