Gauss, railways and corrugated iron

I really enjoyed listening to the In Our Time episode on mathematician Carl Friedrich Gauss this morning. As is sometimes the case with these podcasts, I hear about theories I might once have been expected to study and to which I now wish I had been paying more attention. This morning, two theories in particular piqued my inner engineer.

An early internet?

The first concerns long-distance communication. While experimenting with electricity, he and his colleague rigged up a wire across town from the university to their lab and experimented with sending messages to each other. This they did decades before the invention of the telegraph. They even invented their own code language.

Apparently, having got this basic set up working, Gauss then pondered on the idea of sending signals down railway rails between towns. I find this astonishing. Had this been pursued, this could have led very rapidly to a communication network across Germany. An early internet? Unfortunately, it seems Gauss was less interested in convincing business people of his ideas.

Why corrugated iron is so stiff

In the bonus material at the end of the podcast we hear an example to illustrate Gaussian surfaces. As I understand it, Gauss found a way to codify surfaces as follows. Flat surfaces get the number zero. Surfaces that rise to a peak get the number +1 and saddle points get the number -1. He also had a theory that however you manipulate it, a surface this is made with one geometry can’t form into another.

A practical example of this is that it is not possible to fold a piece of paper (geometry 0) up into a sphere (geometry 1).

But the example I like is of corrugated iron and corrugated cardboard. To create a corruagated sheet, you take a flat sheet and put ridges in it in one direction. A ridged sheet is still geometry 0. But if you try to fold the sheet across the ridges you would be changing the geometry into lots of peaks and troughs (-1), which Gaussian theory says is not possible. And we find this to be true: corrugated materials are indeed very stiff. Nice one, Gauss.

(Thinking out loud, I find myself wondering what is the actual physical mechanism – what is going on at molecular level – that makes the material work in this way. And then I realise that this is the wrong way to think of it. These are properties of pure geometry that are true regardless of whether the material is imaginary or real. Maybe someone who knows more about these things could help me out here?)

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