A Very Peculiar Storey

Last week’s column ended with an accusatory teaser: we suggested that there was something mathematically amiss with RMIT’s Storey Hall, one of Melbourne’s architectural icons, It is time to explain.

As we discussed, the tilings on the façade and walls of Storey Hall are based upon Penrose tiles, the two rhombuses pictured below.

The two colours on the tiles indicate the rules for placing them side by side. The Penrose tiles can be used to tile the whole plane while still obeying these colour-matching rules. In fact, it turns out that there are infinitely many ways to do so.

Moreover, and this was the critical point we made last week, any tiling with Penrose tiles that obeys the colour-matching rules will necessarily be non-periodic: the tiling cannot consist of a basic parallelogram region that is repeated over and over. It is the simplicity of the tiles combined with the unavoidable non-periodicity that made Penrose tiles – and their inventor, Roger Penrose – famous.

Let’s now take a look at the Storey Hall tiles:

They are rhombuses of the same proportions and the light-colored regions are in the same locations as the regions on the Penrose tiles. However, the matching regions on the Storey tiles are all one color rather than two. This is a very important distinction. 

With only one color to be matched, many more tilings are now possible. And, as is the case for the example below, some of these new tilings are periodic. 

We reiterate, what is mathematically special about the original Penrose tiles is that the only possible tilings are non-periodic. Now, the reduction from two matching colours to one has completely destroyed this property. 

Why do it? Who knows? 

Still, the Storey Hall tiling is very attractive, and perhaps all is not lost. One can take the tiling and attempt to redo it with proper Penrose tiles. It almost works. 

Below is a reconstruction of the top part of Storey Hall’s façade with Penrose tiles. It works, as far as it goes.

Notice that the façade is missing tiles at spots 1 and 2. This turns out to be unavoidable.

Though the gaps are just the right size to be filled with the fatter rhombuses, this cannot be done so that the colours match, even with single-colour Storey tiles. 

This highlights a central and perplexing difficulty of Penrose tiles: even though we have told you that the whole plane can be tiled, it is not at all clear how to actually do it. Simply laying down Penrose tiles one by one and hoping for the best will likely fail: a configuration with unfillable gaps can easily arise. (For the much fuller story of Penrose tiling, we refer you to the excellent articles by David Austin). 

In particular, as we noted above, the tilings on the Storey Hall façade cannot be extended to fill the gaps at spots 1 and 2. However that does not appear to be an architectural goof. Rather, the gaps seem to have been arranged deliberately, to highlight the peculiar manner in which Penrose tiles work.

However, there are other, seemingly less deliberate, problems. Below is a close-up of the lower left part of the façade.

The matching rules, even Storey Hall’s simplified one-colour rules, are violated at the two corners circled. Perhaps the architects would argue some aesthetic purpose for this as well. However, to us these corners appear to be straight-forward goofs. (Similar problem corners also occur in the tiling inside Storey Hall).

There are other ways in which the Storey Hall tiling is puzzling, though none as glaring as the above issues. Anyway, the next time you walk past Storey Hall, maybe stop a moment to look, and see what does and does not make sense to you.

Notwithstanding all our criticism, we do really like Storey Hall. It is a beautifully designed, very functional building with stunning walls and façade. We simply wish the architects had considered more carefully the nature of Penrose tiles. Or, to the extent the “mis-tiling” was done deliberately, we wish we could understand why.

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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