Dr Ron Knott in the Department of Mathematics at Surrey University is not a name I recognised, but reading his resume, I now realise that I have heard him talk a few times about Mathematics on Radio 4, both on Simon Singh’s 5 Numbers series, and in Melvyn Bragg’s In Our Time podcast.

I was looking for some information about exact values of trigonometric ratios, and came across his most informative site. I was extremely pleasantly surprised to discover that for some values the trigonometric functions give exact solutions in terms of phi, the golden ratio, among other information.

For example, did you know that the cosine of 27 degrees is exactly a half of the square root of (two plus the square root of (two subtract phi)). (One day when I finish writing my own equation display movies, I’ll write that out in a prettier way, Dr Knott’s website tries a little harder than I do). I love that the number 27, which clearly wants to be prime so much it tricks generations of children into thinking it is, the square root of two and the golden ratio are connected inextricably through the circle-based cosine function. Fantastic!

The whole page, indeed the whole of his site in general, is steeped in extremely interesting, and relatively accessible mathematics with Fibonacci numbers, Egyptian Fractions and so on and so forth. It’s mostly a site for KS4 and beyond (14 years old +), with most material for the older students. Some of it is not for the faint-hearted. However, it is a valuable resource for mathematicians of all hues, and well worth a look.

I was recently invited to do an IQ test. One of the questions was as follows: “You walk five miles north, five miles west, then five miles south. How far are you from where you started?” The answer that they were looking for was 5 miles.

Perhaps our use of maps convince us of this logic; a logic based on Cartesian 2-dimensional geometry. Unfortunately, we do not live on a Cartesian plane!

How should we understand this IQ test question when we correctly consider that we live on a sphere? A good way to consider this is to think about what happens at the two poles!

The South Pole

Start at the south pole. Travel five miles north. Travelling west is to travel parallel to the equator, so then when travelling five miles west, you get no further from the pole. Then travel five miles south. You arrive where you started, with no distance between where you started and where you are now.

The North Pole

The north pole example is more difficult to imagine, and some may think there’s a trick here. If you start five miles and a bit south of the north pole, move five miles north then five miles west in many tight circles around the pole ending up exactly opposite to where you were when you originally arrived near the pole. Then move five miles south. More or less, you are now ten miles away from where you started.

The trick here is the ‘bit’ which ensures two things: firstly that you can in fact travel west; secondly that having travelled five miles west you finish up exactly opposite where you were when you started moving west. It might be argued that if you ended up exactly at the pole then you would be unable to move west at all. The ‘bit’ ensures that there is a trivially small circle around the pole that you can travel west around. It is also necessary to end up exactly opposite where you started to maximise the resultant difference between the two starting positions. If you imagine the bit as a radius of a circle around the pole, then it can be calculated as any r such that 5 miles = (2n+1)*pi*r where n is a positive integer.

Summary

Thinking about a problem often involves thinking around its extremes or limits. When thinking about compass bearings on a sphere, the poles offer places where their odd relationship to each other are most apparent. The IQ problem assumes that we live somewhere where the relationship between the compass bearings closely resembles the relationship between Cartesian axes. At the poles this similarity breaks down most markedly. By thinking about moving to and and from the poles, it transpires that if you move five miles north, five miles west, and five miles south, you may end up a distance of x miles from where you started, where x is such that: 0 miles <= x < 10 miles.

Now, if the IQ test was testing for this as an answer, I would have been suitably impressed!

Philosophical Footnote

The earliest known argument against the earth being flat comes from Aristotle, who argued that the shadow that the Earth casts onto the moon during lunar eclipses is always circular. The only object which casts a circular shadow irrespective of its orientation is a sphere, and since night and day convince us that the earth does not have a constant orientation with respect to the moon and the sun, the Earth must be spherical. (Aristotle De Caelo, 297b31-298a10)

Our natural instincts about the world are perhaps that it exists on a plane that is looped; the outside of a cylinder with the poles at the top and bottom of the cylinder. This is a practical simplified model of the earth because until you get into the arctic and antarctic more or less, two people moving north are moving more or less parallel to each other, and the consequences of longitude and latitude working on quite separate principles need not be considered.

We could of course change the way that north and south work, and make them akin to east and west. Perhaps the great circle through Grenwich could be the East-West equator, as in some senses it is, and we could therefore define an east and west pole, somewhere on the equator! It is an interesting thought experiment. Would our concepts be more easily understandable if we did this? Why did North and South become defined as it is?