# Why stop at 3 dimensions?

A friend of mine (who is not a maths teacher) recently sent me this article that Marcus Du Sautoy wrote in 2013 when he was president of the MA. He makes the case for a “Mathematical Literature” GCSE to sit alongside the utilitarian curriculum a bit like English Literature sits alongside English Language GCSE. It should aim to develop a student’s love of mathematics through doing maths for enjoyment to foster an appreciation of our rich and varied mathematical heritage.

At the end of the article Du Sautoy he says he is a mathematician not an educationalist. But I think the example he gives in here is a great one. There is definitely a lesson in it, and not just for higher attaining classes. Once the basics of plotting coordinates in a Cartesian system are secure, I think this would be accessible to any KS2/3 class upwards. But this comes with a health warning: I haven’t tried it yet,  so (as with anything you find on the Internet) proceed at your own risk!

## The lesson plan

Prerequisites:

• Pupils need to know how to plot coordinates in 2D (1st quadrant only)
• They need to know what a square is and be able to reason why the basic square has coordinates (0,0), (0,1), (1,0), (1,1).  You could maybe start with something like this from Don Steward, although this is already maybe more than is required.
• They need a systematic way of finding arrangements of things.  e.g. how many ways are there of arranging the letters ABC? ABCD? etc.

So next, we can introduce the 3D coordinate system.  Visualising things in 3d can be hard; often it is not closely correlated to mathematical attainment. But there are a wealth of options now to show and “spin” a 3D object on a screen. Here is one I created on Geogebra:

I would spend some time on this, looking at systematic ways to make sure we have the coordinates of all 8 vertices.  You might want to take the approach I wrote about here.  I would try to convince students that they could have done this systematically with having had to see the cube in 3D space.  Because what they are going to do next is find the coordinates of an object that they definitely can’t see – a 4D cube.

Although it’s impossible to see in our 3D world, we can use maths to work out the coordinates of each vertex of a 4D cube.  Again, if this is done systematically students will hopefully soon see that the number of vertices doubles each time we add another dimension. Here is 2D, 3D and 4D:

The next part of the article is where it really gets interesting. Because as well as this being an important piece of maths in its own right, it also has an application in computing.  It is used in error correction when sending digital signals. The rules are reasonably straightforward and the article provides an example of a piece of code that contains an error.  Can your students apply the rule to find the incorrect bit?