## Use of Mathematics and Hyperscopes

The new “Use of Mathematics” A-level has been hotly debated recently. I would like to start by saying that I agree that things need to be done on this topic. There are some deep issues in the Mathematical culture in Britain and this A-level is aimed at addressing them. A good account of this is given in the open letter sent by ACME to various political figures.

Although changes need to be made, however, we need to be careful about nature of this change.  The proposal at the moment is too much about fitting numbers into equations.  (See Tim Gowers’ analysis). One way to think about the new A-level is that it could play a role similar to “Classical Civilization” when compared to Latin and Greek. This is already slightly troubling as the perception could easily be that this is a light-weight maths.  However there is a good argument for Classical Civilization as interesting history of Ancient Greece and Rome, that informs so much of our culture is made accessible without the language barrier.  Is the same true of mathematics, are there useful mathematical tools that are hidden behind a complex language?  I do not think so, and I will illustrate that with an example. But first some pretty pictures!

Images from a Hyperscope

A hyperscope is a hyperbolic kaleidascope. It has five mirrors arranged in a pentagon. However the mirrors are not flat. Each is bent so that they meet at 90 degrees. Forming this shape:

Right angled pentagon

As the mirrors meet at 90 degrees there are precisely four chambers round each corner, but as they are bent each chamber is a slightly different shape to the last. The result is a glimpse into the negatively curved world of the Hyperbolic plane.

Using one red mirror shows how the extra hyperbolic space is folded away to fit in Euclidean space.

In order to make this I wanted to use standard A4 acrylic mirrors, so I did not need to do any cutting. Each mirror is placed into a groove cut into a piece of MDF, and the mirrors have to fit tightly at the corners. I was therefore faced with a problem. I knew the width of my mirrors but they would be bent, so I needed to make this the distance round a circular arc.  Now let us assume I have successfully completed “Use of Maths” A-level and I recognise this as a mathematical problem. I go onto the world’s best source of equations (wikipedia) to see if I can find anything. In real life I did exactly that, as I am lazy and wanted the answer quick. Unfortunately the ratio between the length of a cord (a line between two points on a circle) and an arc (the curve between two points) is not given.  A couple of google searches later and I gave up.

I gave up as I had a better option. I could just work it out myself. It is not hard, just involving a little trigonometry.  I illustrate with an image. The arc is labelled A, the radius r and the cord C, the angle is $\theta$. An additional line splits C in two and gives two right angled triangles. Which should hint to the answer.

This example is to me “Use of mathematics”. I had a practical problem and wanted to solve it. There was a little trick to realising the tools I needed to solve it, but after that the mathematics was basic. In fact I was lucky enough to have learned all the mathematics I needed here by the time I was 13.  As someone who is perhaps more thoughtful than pratical I have to confess that my perfect calculations failed on “Use of the real world” and the mirrors had 1mm too little space. Luckily such things can be bodged.

If we are going to invest the money in developing a new A-level, therefore, let us play on those practical connections that mathematics has and get people involved in them. Some people, like I did, become engaged in the logic and clarity of maths itself. However for most it is only when they find out how it can solve a problem for them that it becomes interesting. So lets get people building mathematical toys to illustrate trigonomety and geometry. Designing fabric patterns to show symmetry. Working with the basics of google’s pagerank algorithm to show the power of linear algebra. Encoding and decoding messages to learn about factoring prime numbers.  With a little imagination we should be able to cover the whole syllabus.

There is even a model for what we might want to achieve. The Salter’s A-level in Chemistry is a full Chemistry A-level. It is not “Use of Chemistry” as it covers the full criteria (subjects the A-level must cover). However the teaching starts with the applications and moves back to the theory. The theory is therefore seen in a wider context from the start.  Why are we being less ambitious for maths? Is the subject really only accessible to some people? Can’t we find the ways to motivate children to put in the hard work required to gain useful and beautiful insights? We need the changes in the maths syllabus to make a real difference and not just make things look good so the numbers show the problem is getting better.