If you’ve ever given someone a ball or other round object as a gift, you’ve probably run into an ancient mathematical problem – it’s really difficult to neatly cover a sphere in wrapping paper. You might as well stop struggling and put the sticky tape down, because it turns out it’s actually impossible. A sphere is what mathematicians call a developable surface, meaning it can’t be flattened into a sheet without stretching or tearing – and you can’t wrap spherical presents perfectly!
This inescapable fact causes more problems than just a bit of dodgy gift-wrapping. Everyone from geologists to video game designers need ways to translating spheres to flat surfaces. People have come up with all sorts of clever ways to unwrap spheres, but the ancient Greek mathematician Hipparachus discovered one of the oldest methods – stereographic projection.
Around the world in two dimensions
It works like this. Picture a glass sphere resting on a table, with a tiny laser pointer at the north pole. Shine the laser pointer through the sphere and you’ll see two dots; one on the surface of the sphere and another on the table. This effectively means that a particular point on the sphere can be mapped to a particular point on the table without any overlapping – with just one exception. It’s not possible to direct the laser pointer from the north pole to the table without passing through the sphere. Because no overlapping is allowed, the north pole can’t be part of the projection.
What does the full projection look like? It’s easiest to think about what happens to the equator, which is mapped to a circle with a radius twice that of the sphere’s. Everything in the lower half of the sphere (the southern hemisphere) is mapped inside this circle, while everything in the upper half of the sphere (the northern hemisphere) is mapped outside.
This explains why the north pole isn’t part of the projection. Think about what happens to the laser pointer as you direct it higher up the surface of the sphere. The pointer gets closer and closer to a 90° angle and the dot on the table moves further and further away. A dot representing the north pole would have to be infinitely far away in all directions!
The stretching effect that occurs as you move towards the north pole is just a compromise we have to accept, because whatever method you use when projecting the sphere, something’s gotta give. Stereographic projection can’t accurately reflect the distance between two points, but it does preserve angles and shapes. In other words, a triangle on the sphere is still a triangle in the projection – it just isn’t the same size.
Projection: it’s all in how you look at it
This distortion means stereographic projection isn’t always suitable. A stereographic map of the Earth doesn’t tell you how long it takes to fly from London to Washington, because the distance is inaccurate, but it does show the direction to head because the angles are correct.
Mapping the Earth’s surface is just one of many applications for stereographic projection. In science, geologists who study cracks in the Earth’s crust use the projection to represent fault lines, and it lets chemists produce a 2D diagram for analysing the 3D structure of crystals.
Creative artists like stereographic projection too. Video game designers and 3D animators use flat images called textures to cover 3D models with graphical detail, and stereographic projection helps them preserve the shape of their artwork. The technique is also popular with photographers, who use it to create surreal ‘planets’.
Finally, mathematicians find stereographic projection useful for exploring the strange realms of imaginary numbers and higher dimensions. It can be hard to visualise these concepts, but translating from the sphere to the plane and back can help simplify complex problems. Just something to think about as you wrestle with the wrapping paper.