We looked briefly at some medieval maps, especially the Hereford Mappa Mundi. Although people knew very well at that time that the world is round, these depictions tend to be very flat-looking and the roundness of the Earth doesn’t seem to pose any great problem. This was changed somewhat by the discovery of the Americas at the end of the 15th century.
In the 16th century Mercator made a very famous map using a technique that (conceptually, at least) projects the globe onto a cylinder, which can then be cut and laid flat. This method preserves angles, making it very useful for the transatlantic navigation. But it distorts sizes and locations, especially near the top and bottom of the map: if this is the projection you’re used to, the world might look very different from the way you think it does, as this clip from US TV drama The West Wing dramatizes:
In class we made our own maps using a rather different technique, called stereographic projection. It’s awkward to do by hand and produces some very severe distortions, but it has an interesting geometric relationship to the Mercator projection — we discussed that in class, using images from this excellent article.
We’re interested in this for a few reasons. One is that we’ll be using this technique, and others very like it, in later sessions. Stereographic projection is one of those tricks that comes up surprisingly often in maths, and in all kinds of weird contexts, so it’s good to know about.
But since this course is about the interplay of geometry and topology, the problem of transferring designs from spheres to planes is very pertinent anyway. In fact, this coming week we’ll look at the opposite problem — confronted by Islamic artisans in the middle ages — of transferring patterns from flat drawings onto domes and spheres.
The difference between the sphere and the plane is topological, but that’s quite hard to “see”. The geometric effects of this difference, though, are plain to see, and will be even plainer in the next session.