Federation Forensics

We recently had a careful look at the beautiful tiling of RMIT’s Storey Hall. We were surprised by some of the tiling’s mathematical oddities, and that got us to pondering: what about Melbourne’s other famous geometric façade?

We took a mathematical trip to Federation Square back in 2007. On that occasion we explained the simple rule underlying pinwheel tiling. However, we never actually checked how closely the walls of Federation Square stuck to the rules.

To recall, the pinwheel tiling begins with a right-angled triangle of just the right shape:

We then add four identical triangles to create a super-triangle of the same shape:

Next, in exactly the same way, we add four super-triangles to build a super-super-triangle:

Since each new triangle is the same special shape, we can repeat the process indefinitely, filling the whole plane with successively super-sized triangles. The result is definitely reminiscent of Federation Square:

We’ll now take a closer look at Federation Square, beginning with the ACMI wall, facing Flinders Street. If the tiling on this wall is really part of the pinwheel tiling then we should be able to group the triangular tiles into super-triangles. This is indeed the case: the red triangles outlined below are all super-triangles, each consisting of five basic tiles.

In turn, the red super-triangles can be combined into green super-super-triangles:

However, at the next stage we run into trouble. It partly works, as the purple triangles outlined below are the desired super-super-super-triangles:

However, the green tiles on the left side of the wall cannot be combined in this manner. Close, but no cigar. 

Now, was this carelessness on the architects’ part, or were they forced to cheat a little? We suspect the latter. 

In fact, it is difficult to find rectangular portions of the pinwheel tiling of just the right size. Taking the short side of a basic triangle as our unit, the ACMI wall is 45 x 30 units. We can now hunt for rectangles with these proportions.

Rectangles within the pinwheel tiling are of two types: either the perimeter consists entirely of hypotenuses of the basic tiles, or the perimeter contains no hypotenuses. The green and purple rectangles outlined below are examples of the two types.

For the Federation Square walls, the architects chose to go with no-hypotenuse rectangles. This means that the ACMI wall could have been modelled on the 6 x 4 purple rectangle. Of course, the tiles required to do so would have been massive.

However, we can also consider each triangle in the previous diagram to be a super-super-triangle. In this case the purple rectangle would have dimensions 30 x 20. Or, going to the next possible level, we can regard the purple rectangle as being 150 x 100.

But is there actually a rectangle of size 45 x 30, or even close? We don’t know. So, though the architects could definitely have made the ACMI wall an exact pinwheel tiling, this may well have necessitated basic tiles of a significantly different size. 

Moreover, so far we have only considered one wall: even if tiles were chosen of a size ensuring one wall could be tiled perfectly, those tiles may well fail to work for the next wall. In fact, this seems to have been what happened.

Consider the northern wall of Federation Square, pictured above. It is 20 x 30 units, which means that it definitely can be properly tiled, and indeed it has been. The tiling is exactly the one we indicated above; each purple triangle outlined below is a super-super-triangle of the pinwheel tiling.

We investigated three other walls at Federation Square. The smallest wall followed the pinwheel tiling exactly, and the other two were similar to the ACMI wall: they followed the pinwheel rules up to the super-super-triangle level, but then things broke down. 

Anyway, we now have a clear sense of what does and what doesn’t work in Federation Square. Overall, we’re impressed. True, it would have been terrific to have all the walls follow the pinwheel rules exactly, but to complain would be churlish; the architects have done a beautiful job with a very unwieldy mathematical construct. 

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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