Monkey Business

The legendary image of an army of monkeys typing letters at random and eventually producing the works of Shakespeare seems to have emerged gradually over a long period of time. In Gulliver’s Travels, written in 1726, Jonathan Swift tells of a mythical Professor of the Grand Academy of Lagado who aims to generate a catalogue of all scientific knowledge by having his students continuously generate random strings of letters by means of a mechanical printing device. The first mechanical typewriter had been patented in 1714. 

After several eighteenth- and nineteenth-century French mathematicians used the example of a great book being composed by a random deluge of letters from a printing works as an example of extreme improbability, the monkeys appear first in 1909, when the French mathematician Émile Borel suggested that randomly typing monkeys would eventually produce every book in France’s Bibliothèque Nationale. Arthur Eddington took up the analogy in his famous book The Nature of the Physical World in 1928, where he anglicised the library: ‘If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum.

’Eventually this oft-repeated example picked the ‘Complete Works of Shakespeare’ as the prime candidate for random recreation. Intriguingly, there was a website that once simulated an ongoing random striking of typewriter keys and then did pattern searches against the ‘Complete Works of Shakespeare’ to identify matching character strings. This simulation of the monkeys’ actions began on 1 July 2003 with 100 monkeys, and the population of monkeys was effectively doubled every few days until recently. In that time they produced more than 1035 pages, each requiring 2,000 keystrokes.

A running record was kept of daily and all-time record strings until the Monkey Shakespeare Simulator Project site stopped updating in 2007. The daily records are fairly stable, around the 18- or 19-character-string range, and the all-time record inches steadily upward.

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A Sense of Balance

Whatever you do in life, there will be times when you feel you are walking a tightrope between success and failure, trying to balance one thing against another or to avoid one activity gobbling up every free moment of your time. But what about the people who really are walking a tightrope. The other day I was watching some old newsreel film of a now familiar sight: a crazy tightrope walker making a death-defying walk high above a ravine and a rushing river. One slip and he would have become just another victim of Newton’s law of gravity.

We have all tried to balance on steps or planks of wood at times, and we know from experience that some things help to keep you balanced and upright: don’t lean away from the center, stand up straight, keep your center of gravity low. All the things they teach you in circus school. But those tightrope walkers always seem to carry very long poles in their hands. Sometimes the poles flop down at the ends because of their weight, sometimes they even have heavy buckets attached. Why do you think the funambulists do that?

The key idea you need to understand why the tightrope walker carries a long pole to aid balance is inertia. The larger your inertia, the slower you move when a force is applied. It has nothing to do with center of gravity. The farther away from the center that mass is distributed, the higher a body’s inertia is, and the harder it is to move it. Take two spheres of different materials that have the same diameter and mass, one solid and one hollow, and it will be the hollow one with all its mass far away at its surface that will be slower to move or to roll down a slope. 

Similarly, carrying the long pole increases the tightrope walker’s inertia by placing mass far away from the body’s center line – inertia has units of mass times distance squared. As a result, any small wobbles about the equilibrium position happen more slowly. They have a longer time period of oscillation, and the walker has more time to respond to the wobbles and restore his balance. Compare how much easier it is to balance a one-metre stick on your finger compared with a 10-centimetre one.

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Pylon of the Month

There are some fascinating websites about, but none was more beguiling than the iconic Pylon of the Month,fn1 once devoted to providing monthly pin-ups of the world’s most exciting and seductive electricity pylons. The ones shown on the website below are from Scotland. Alas, Pylon of the Month now seems to have become a cobweb site, but there is still something to learn from it, since for the mathematician every pylon tells a story. It is about something so prominent and ubiquitous that, like gravity, it goes almost unnoticed.

Next time you go on a train journey, look carefully at the pylons as they pass swiftly by the windows. Each is made of a network of metal struts that make use of a single recurring polygonal shape. That shape is the triangle. There are big triangles and smaller ones nested within them. Even apparent squares and rectangles are merely separate pairs of triangles. The reason forms a small part of an interesting mathematical story that began in the early nineteenth century with the work of the French mathematician Augustin-Louis Cauchy.

Of all the polygonal shapes that we could make by bolting together straight struts of metal, the triangle is special. It is the only one that is rigid. If they were hinged at their corners, all the others can be flexed gradually into a different shape without bending the metal. A square or a rectangular frame provides a simple example: we see that it can be deformed into a parallelogram without any buckling. This is an important consideration if you aim to maintain structural stability in the face of winds and temperature changes. It is why pylons seem to be great totems to the god of all triangles.

If we move on to three-dimensional shapes then the situation is quite different: Cauchy showed that every convex polyhedron (i.e. in which the faces all point outwards) with rigid faces, and hinged along its edges, is rigid. And, in fact, the same is true for convex polyhedra in spaces with four or more dimensions as well.

What about the non-convex polyhedra, where some of the faces can point inwards?They look much more squashable. Here, the question remained open until 1978 when Robert Connelly found an example with non-convex faces that is not rigid and then showed that in all such cases the possible flexible shifts keep the total volume of the polyhedron the same. However, the non-convex polyhedral examples that exist, or that may be found in the future, seem to be of no immediate practical interest to structural engineers because they are special in the sense that they require a perfectly accurate construction, like balancing a needle on its point. 

Any deviation from it at all just gives a rigid example, and so mathematicians say that ‘almost every’ polyhedron is rigid. This all seems to make structural stability easy to achieve – but pylons do buckle and fall down. I’m sure you can see why.

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