Federation Square’s hidden gem

We're back! And we're back in Federation Square.

This is our third visit to Melbourne's famous façade. In 2007 we wrote about the simple principle behind the intriguing triangular tiling. Then last year we took a closer look at the mathematics of the tiling, to see what the architects got right and what (arguably) they got wrong.

We definitely weren't planning another visit so soon. However, while researching for our column on tennis maths, we had an unexpected encounter with Federation Square's doppelgänger. It appeared in an article by mathematician Roger Nelson, author of the wonderful Proofs Without Words and its sequel.

To explain this appearance, let's first recall the special property of Federation Square's façade. We begin with a big, judiciously chosen portion of the façade, as outlined in the picture above.

This triangular piece is just the right shape so that it can be replaced by five red sub-triangles of the same shape. Then those sub-triangles can be replaced by green sub-sub-triangles, and so on.

We can stop with the orange tiles pictured, giving us part of the Federation Square tiling. However this is when the doppelgänger appears, enticing us to keep going.

Let's begin again with the five red sub-triangles. However, at the next stage we only replace the rightmost red triangle.

Similarly, at the next stage we only replace the rightmost green triangle.

We do the same at the next stage, and the next, and ... we just don't stop. There are infinitely many stages, where at each stage the rightmost miniscule triangle is cut into five micro-miniscule triangles.

It's a cool picture but is there a point to it?

Recalling the mantra "one half base times height", we see that the original brown triangle has area 1. It follows that each of the four red triangles have area 1/5, the green ones have area 1/5 x 1/5, the orange ones have area (1/5)³, and so on. Summing the areas of all the sub-triangles, our diagram shows at a glance that

Or, after dividing both sides by 4,

That is a simple and beautiful summing of an infinite geometric series. It is an infinite analogue of the finite sums we considered in our previous column.

Not surprisingly, other infinite series can be similarly summed, and Roger Nelson uses different right-angled triangles to calculate four such sums. For example, the diagram

demonstrates that

You can probably guess the general formula:

This formula is true for any number N > 1 (and any N < -1), and there is a pretty and well-known algebraic proof of the formula. However, we'll stick to geometry and to whole numbers.

We have taken care of N = 3 and N = 5, so can we find triangles to demonstrate the formula for other whole numbers N? It is not difficult to find right-angled triangles that work for N = 2 and N = 4, but not all numbers are so easy. Indeed, there is no triangle that works for N = 6.

However there is no reason to stick to triangles. The same argument would work for any figure that can be dissected into smaller figures (all of the same size) of exactly the same shape. Such figures were first studied by mathematician Solomon Golomb, who gave them the charmingly apt name of rep-tiles. They were then popularised by Martin Gardner's column. Some rep-tiles are not too difficult to find, but others are clever and beautiful.

Of course, if we're willing to go down a dimension then the simplest rep-tile is a plain old line segment. 

Summing the lengths of the sub-segments, we obtain our first geometric sum (for N = 5), exactly as we did with Federation Square's triangles. Still, Federation Square does it with much more style.

Puzzles to Ponder: A rep-tile that splits into N pieces is called a rep-N. (For example, the Federation Square triangles are rep-5.) Find right-angled triangles that are rep-2 and rep-4. Given any N, find a quadrilateral that is rep-N.

Free Public Lecture, Sunday July 29: Burkard will give a presentation entitled Melbourne Maths. See Burkard at the Melbourne Museum, 10:30-11:30. For more information and to register, please visit the MAV website.

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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