## Friday, 21 October 2011

### Number spotting

I have been travelling around the country by train this week, giving keynote lectures to various conferences for primary school teachers: in Ormskirk, Leeds (twice) and Uttoxeter. The two subjects I get asked to talk about most at present are Creativity in Mathematics and Mathematical Anxiety and its Effect on Learning.

Train travel always involves spending some time on platforms at rail stations. I haven't yet taken up train spotting, but I am inclined to do a bit of number spotting. If you look you'll see that trains and trucks and rail stations have measurements written all over the place. It's fun trying to work out what they are for.

Last week I noticed that on the side of some trucks was written 'minimum curve 41 m' and on some others 'minimum curve 50 m'. These labels struck me as surprising at first, because thinking about 'curvature' in the way its used in mathematics I would have expect the measurement to give the maximum curve, not the minimum. Clearly, what the 41 m sign, for example, must mean is 'the minimum radius of the curve' on any curved stretch of track. Mathematically, the smaller the radius, the greater is the curvature, and vice versa.

The curvature is a measurement of how quickly an arc or curved track is turning. So a curved track which is an arc of a circle with radius 10 m would have much more curvature than one which is an arc of a circle with a radius 100 m. If you ran around these two circles you would get much dizzier on the one with the smaller radius, because you would be turning much more quickly.

We measure the curvature of an arc by calculating what angle you turn through when you travel one unit of length along the arc. So we could measure curvature in degrees per metre, for example. To do this we use this formula for an arc with radius r metres:

curvature = 180/(∏r) degrees per metre

∏ (pi) is 3.1416 to four decimal places. So, for a rail track with radius 41 m, the curvature is 180 ÷ (3.1416 × 41) which is about 1.40 degrees per metre. For a rail track with radius 50 m, the curvature works out as about 1.15 degrees per metre.

So the 'minimum curve 41 m' truck can cope with a bend which turns at a maximum rate of 1.40 degrees per metre. And the 'minimum curve 50 m' truck can cope with a bend which turns at a maximum rate of 1.15 degrees per metre.

At which point my train arrived.