The Telltale Tiles

One of your Maths Masters recently received an invitation to attend a function at RMIT’s Storey Hall on Swanston Street. The fancy invitation card included a photo of Storey Hall’s strikingly tiled walls, together with some mathematical background: 

The dynamic tiles featured throughout the venue were inspired by renowned Oxford Professor of Mathematics, Roger Penrose. The venue reflects his discovery that instead of the 20,426 types of geometric shapes required to cover a continuous surface in a non-periodic pattern, it could be done with just two: a rhombus of 54 degrees and another of 72.

Unfortunately this description, courtesy of the Victorian Department of Education, is hopelessly garbled and almost totally wrong. Plus ça change …

Storey Hall is one of Melbourne’s architectural icons. So, it seems worthwhile clarifying exactly what is mathematically special about its famous façade. 

Take some flat shapes, triangles or squares or what have you. We say that these shapes tile the plane if it is possible to cover the whole plane with copies of the shapes, without overlaps (except on the edges) and without gaps. 

It will probably come as no surprise that the plane can be tiled with equilateral triangles, and also with squares and with regular hexagons.

However, this is just the beginning. Staying with regular shapes, we can tile the planes with combinations, as in the following examples.

The above tilings are all periodic. That is, we can superimpose a parallelogram grid on the tiling so that the inside of each parallelogram looks the same, as pictured below. (Actually, there are subtleties with the precise meaning of “periodic”, but the parallelogram idea suffices for our discussion here).

The focus of this column might suggest that periodic tilings are the boring ones. However, one should not ignore the stunning periodic possibilities, as exhibited, for example, in Islamic art and in the work of M. C. Escher. 

Anyway, what of non-periodic tilings, which the department’s invitation suggests are a big deal? In particular, the implication is that a non-periodic tiling requires at least two different types of tiles. That is not true, and indeed an ordinary square tile suffices: we simply have to shift one row of the previous, periodic tiling.

Pretty boring. Below are two much prettier non-periodic tilings, based upon a single isosceles triangle. And, for another striking example, we direct you to the walls of Federation Square. 

Clearly, non-periodic tilings are easy to come by. So, what’s the big deal?

Notice that if a collection of tiles can be used to tile the plane then there seem to be many different ways to do the tiling. For example the isosceles triangle easily gives a third, periodic tiling:

That brings forth the real question: if we have a non-periodic tiling, can the same tiles also be used to construct a periodic tiling? If the answer is “no”, if the collection of tiles can only be used to tile non-periodically, then the collection is called aperiodic.

For a long time, no one knew whether there existed an aperiodic collection of tiles, and in fact it was suspected that there wasn’t. Then, in 1964, the mathematician Robert Berger constructed an aperiodic collection of 20,426 tiles: that’s the magic number referred to in the department’s invitation.

Berger and others went on to find much smaller collections of aperiodic tiles. In 1973, Roger Penrose found an aperiodic collection consisting of six tiles, and soon after he discovered an aperiodic collection consisting of just two tiles.

Penrose came up with a number of different (though closely related) aperiodic pairs of tiles. One particularly nice pair consists of two rhombuses, embellished with little jigsaw cuts and knobs.

The point of the jigsaw business is to restrict the ways in which rhombus tiles can be placed together. If the cuts weren’t there we could easily make a periodic tiling, just as we did with the isosceles triangle, and that’s exactly what we hope to make impossible.

An attractive alternative to making jigsaw cuts is to add two colours to the tiles:

We then require that in any tiling, the colours must match across the edges. It is these coloured rhombuses that decorate Storey Hall.

There is plenty mysterious about the Penrose tiles. For example, how could Penrose tiles lead to this year’s Nobel prize in chemistry? We hope to write about that soon.

More fundamentally, how do we know that Penrose’s rhombuses can be used to tile the whole plane? And, how do we know that they cannot be used to tile periodically? These are subtle questions, and too deep to go into here: the inquisitive reader is referred to the excellent articles by David Austin.

In any case, we now know what Penrose tiles are, and why they are of interest. It is all fine motivation for applying Roger Penrose’s rhombuses to decorate Storey Hall. 

However, not all is at seems. There is more wrong here than just a confused invitation: next week, we’ll be taking a very close look at Storey Hall.

Puzzle to Ponder: Find all the mistakes in the invitation from the Department of Education.

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator. 

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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