What was man's greatest invention? A remote communication device like the telephone or the radio? Maybe you value higher the versatility of the personal computer or the World Wide Web which links them all together? Then again, how about the Saturn V rocket which enabled man to leave this planet and take a few faltering footsteps on our satellite neighbour? Many would agree, however that the greatest achievement was one of the earliest: the wheel.

**On a Roll**

In a nutshell, the wheel facilitates travel because it rolls. Because the centre is always the same distance from the edge, the hub is always the same distance above the ground. If we rig up a vehicle which attaches to the wheel's centre, then we shouldn't see any of that unwanted vertical movement which would throw us around in our seats, or rapidly deposit our goods outside the cart.

These days we use machine tools to produce circular wheels to a high specification, but earlier versions fashioned from wood will have been less circular and more eccentric; fans of *The Flintstones* may remember Fred's buggy with its rough and ready wheels hewn from blocks of stone.

But what about wheels of a different shape? Other things roll - pencils for example. These usually have a hexagonal cross section and they roll with a somewhat bumpy motion. If the wheel's cross section is a polygon with a large number of sides, then it will be rounder and will roll more smoothly. If you have a square, however, then you might think it would make a very poor wheel.

That's not necessarily the case.

**Humps and Bumps**

Of course, we don't always have flat surfaces to travel on. If you drive a vehicle on a bumpy track, you will be grateful for the suspension system which damps the effects of the vertical motion. But, thinking about it, if the bumps and indentations in your bumpy wheel fit exactly with the bumps and indentations on the ground, then it will travel as smoothly as a circular wheel on a flat surface. In fact, there's no reason whatsoever that you can't effortlessly ride a square-wheeled bicycle - you just need the right shaped potholes for the corners to fit into.

Well, you'll be pleased to know that mathematicians have studied this problem and come up with the answer. The road must be made up of a series of bumps of a particular shape - a curve which they call a *catenary*. The word comes from the Latin *catena*, meaning 'chain'. Nothing to do with bicycle chains, it's just that the curve of each bump, if you turned it upside down, is the same curve that you would get if you held a chain at each end and let it hang down under its own weight^{1}.

As the chain is heavy, the tension caused by its weight pulls it into a particular U-shaped curve - one which is fiendishly difficult to describe in terms of simple mathematical functions. The great Galileo himself actually got it wrong, believing it to be a curve known as a parabola. It was only with the development of calculus that mathematicians found it to be a shape related to exponential curves. We'll try to describe these, but if you're not in the mood for the maths, then feel free to skip the next couple of sections.

**Exponential Curves**

Exponential curves have slopes which increase or decrease at a rate which varies with the position along the curve. The most simple exponential function is known as *y = e*^{x}. It's a curve which starts out very flat for negative values of x, and then as x increases, it rapidly slopes upwards, disappearing off the top of your graph for some positive value of x.

Another well-known curve is the reciprocal curve *y = e*^{-x}. This curve starts out very high for negative values of x, and then rapidly drops as x increases past zero, gradually flattening and becoming closer and closer to zero (but never quite getting there). Its graph is a reflection of the graph of *y = e*^{x} in the y-axis. This curve is known as the exponential decay curve. If you have a geiger counter and you're plotting the number of particles given off by a radioactive source, then over time the curve of the graph will decay with this shape.

The symbol *e* is a fundamental mathematical constant, a bit like pi. It's also known as Euler's constant, being introduced by the great mathematician himself. Those of you old enough to remember books of logarithmic tables may remember natural logarithms, which are taken from the base *e*. They're used widely throughout mathematics, but these days we look them up using a calculator. Like pi, *e* is irrational - its value to ten decimal places is 2.7182818284, but it would go on to have an infinite number of decimal places. There are a number of ways of calculating the value of *e* - one is by evaluating the series *e = 1 + 1/1 + 1/(2x1) + 1/(3x2x1) + 1/(4x3x2x1) + ...* This series is accurate to ten decimal places after adding 14 terms.

Our catenary curve is actually the average of the two exponential functions described above, or as mathematicians would write it: *f(x) = (e*^{x} + e^{-x})/2

**Hyperbolics**

The curve also has another name, the *hyperbolic cosine*, also written as cosh(x). You may find this function on a scientific calculator, as it crops up in engineering applications. In fact there's a family of hyperbolic functions, including the hyperbolic sine, sinh(x) and the hyperbolic tangent, tanh(x)^{2}.

One place to see a hyperbolic cosine in all its glory is the St Louis Arch, St Louis, Missouri. This striking catenary-shaped monument completed in 1965 commemorates Thomas Jefferson and the westward expansion of the United States. Of all arch shapes, the catenary is the most stable in supporting its own weight. It becomes less stable however if it's load-bearing, and civil engineers would use other shapes, for example parabolas, in those situations.

**Cycle Path-ic Personality**

So, if you want to ride a square-wheeled bicycle, all you need to do is construct a road of hyperbolic cosine-shaped humps, and you're away. Sounds ridiculous, doesn't it? Maybe, but enter US mathematician Professor Stan Wagon. Wagon has indeed constructed such a road, a feat which earned him an entry in *Ripley's Believe It or Not*. The length of the curve of each hump is the same length as one side of the square wheel. You can see Stan's square-wheeled cycling track on his website, and you can even try it out for yourself if you visit the Science Centre at Macalester College, St Paul, Minnesota.