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.