Melbourne’s catenary chaos

You just moved into a new house on a busy road, and it seems that you’ll never get used to the noise of the thundering trucks. Except that you do. 

Melburnians are expert at shutting out the ugliness of city living. (The great food and coffee helps). And there is one form of visual pollution that we all work very hard to ignore: the ludicrous tangle of cables and power lines suspended from the ossified jungle of dead trees. 

The fine product of third world planning, Melbourne’s wiry mess is truly grotesque. So, it is with sincere apologies that we now ask you to focus upon this ugliness.

We want to ponder the shape of the curves made by all those cables. Here is a very slack example, anchored at two level points.

It may look familiar: is Melbourne suffering from parabolic pollution, and is this finally justification for inflicting all that quadratic torture on school students? Nope.

For comparison, over the blue cable we have superimposed a red parabola, which also passes through the two anchor points as well as the bottom point of the cable. The curves pictured look similar but they are not identical.

So, if hanging cables are not parabolas then what are they? Actually, it’s not a “they”, it’s an “it”.

Though hanging cables appear to come in various shapes, there is a sense in which they are all just portions of one master curve, the so-called catenary. (“Catena” is the Latin word for chain). The blue cable above is the bottom part of the catenary. 

How can we identify all hanging cables? First hammer two nails into your classroom wall. (Don’t bother asking for permission from your teacher; she won’t mind). Then hang a thin cable or chain between the nails, forming the blue curve. Next, picture all of the scaled up versions of the blue curve and shift them to pass through the nails, as shown.

What we are claiming is that this picture represents all the shorter ways of hanging a cable between the two nails. (To get the longer, droopier cables, we would have to imagine starting with an even larger blue cable, passing through higher nails, and then scale this larger cable downwards).

How can we see this? To begin, notice that the shape of a hanging cable does not depend upon its weight. For example, having a cable of twice the weight would clearly take the same shape as two of the original cables hanging side by side. 

It’s a little tricky, but this independence from weight actually shows that a scaled image of a hanging cable again represents a hanging cable. (Instead of a uniform cable, imagine a string of pearls; then, simply by looking at the forces acting on each pearl, it is clear that scaling up the string of pearls will hang in exactly the same manner as the original string. The same must then be true for uniform chains.)

It is also the case that any part of a hanging cable is still a hanging cable. That is, if you imagine pinching your fingers around a cable at two points, and let the pieces of chain above your fingers go loose, the cable below your fingers won’t change shape.

Together, these observations prove our claim, that any hanging cable is just a scaled portion of one master curve, our catenary. But exactly what shape is the catenary? 

To determine the precise shape takes work, and either a little calculus or a lot of cleverness. (One approach is to first carefully analyse a pearl necklace, with many small pearls, connected by short, essentially weightless pieces of string).

It turns out that the catenary can be written easily in terms of the exponential function, involving the fundamental mathematical constant e. (e is often erroneously referred to as “Euler’s number”, but that’s another story). University students know of the catenary as the hyperbolic cosine function, though they are seldom told why this “just another function” is at all interesting.

As well as describing hanging cables, the catenary is famous for another striking appearance in nature — the soap film that forms between two parallel circular wire loops. This soap film, which takes the shape of a rotated catenary, is known as a catenoid.

And here is a final eccentric application of the catenary. Imagine constructing a catenary-shaped floor as in the picture below. Then, a square wheel of just the right size can roll smoothly over the floor. And a smooth ride on a square wheeled bicycle is not just a theoretical possibility: it can be an amusing reality. 

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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