Brilliant Geometry Exhibition Review
This summer I helped to curate an exhibition on mathematical projections at Summerhall in Edinburgh. Here I’d like to explain a few of the key concepts from the exhibition, illustrated by the exhibits themselves.
Stereographic Projection
We all know that the Earth is (more or less) a sphere. However, if you want a map of the Earth, it isn’t very convenient to carry a miniature sphere around in your pocket. It would be nice to have something that you could fold, something that you could draw on. Maybe something like a piece of paper. But how can we make a round object flat? Unfortunately there is no way to make a perfect map of the sphere on a flat surface. But we can preserve certain characteristics of the original sphere in our flat map.
The map of the world that we are most used to, the Mercator projection, preserves the relative areas of the countries. This projection fits onto a rectangle. However, if we would instead like to preserve the angles between lines, we can use stereographic projection. We can perform this projection in real life. First we need a transparent sphere, with some pattern marked on it, and a wall. Then, by holding a light source at the North Pole of the sphere directly opposite the wall, we see the pattern projected onto the wall. The pattern will be distorted, but all the angles between lines will be exactly the same. If we wanted to make a map of the countries of the world in this way, we would need an infinitely large wall, because the closer the country is to the North Pole, the larger it will appear on the map. In fact, there would be a single point of the sphere which would never make it on to the map. Can you guess which?
At the exhibition we had a 3D-printed model of the Earth which we projected on the wall using a torch. All the countries on the sphere appeared to be the wrong-way round. This is because it is actually the inside of the sphere which is projected onto the wall by stereographic projection.