in which we learn about cross-ratio and the complex projective line.
We then looked at the cross-ratio of four distinct points . We know there is a unique Möbius map sending to , to and to . Then the cross-ratio is , i.e. it is the image of under that unique Möbius map. In particular, . The formula we had for the unique Möbius map gives us a formula for the cross-ratio as well: , with the usual special cases for infinity. From this formula it is clear that the double transpositions fix the cross-ratio: . It is also very easy to show from our actual definition (rather than the formula) that Möbius maps preserve cross-ratio: for any Möbius map , we have . This follows from the uniqueness of the fore-mentioned Möbius map: If , then sends to , to and to , so . That immediately shows us that four points in lie on a circle or line if and only if their cross ratio is real.
As a little extra, we looked at the complex projective line. This is the set of one-dimensional subspaces of , which we can represent by their “slopes”: corresponds to , which is a complex number when , or when .
Understanding today’s lecture
Understanding cross-ratios comes best if you look at them from both points of view: the definition via Möbius maps as well as the formula.
After a whole groups lecture course (and of course three other courses as well), you may be able to understand roughly half of the maths puns in this:
Listen to it again after some Analysis, and then after third year, and then… (there is at least one Category Theory joke in it).
Going a little deeper
There is a lot to say about projective spaces, but this blog is too small to contain them 🙂
If you want to test your understanding and memory of our groups lecture course, try this:
Question 7 on Rachel Camina’s third example sheet from last year. We have actually done all this in lectures, but not in the same place, so you’ll have to “pull the threads together”.