# The shape of a theorem

(or Pappus revisited)

(or Pappus revisited)

*Accessibility options are available in the header. In particular, a dark mode is provided.*# 1 Introductions

This essay is all about a very special proof, one that I first learnt around six years ago. Every few months I suddenly remember it, and my mind gets blown all over again. No other proof has had that effect on me and I’ve never seen any attempts to popularise it online. Our journey will take us through a variety of mathematical disciplines such as projective geometry, topology and computer assisted proofs. Like all the best maths, it involves interactions between disparate fields. So what theorem is this proof going to prove? Pappus’.

The fact that a result like Pappus’ theorem has such a great proof is a big surprise (to me at least). You see, I’ve never been one to say things like all of maths is beautiful

. Don’t get me wrong, I love maths. Most of my friends would agree, I get *too* excited about some maths. But Pappus’ theorem… well, let’s take a look at it.

**Pappus’ theorem.**

Let \(\{A,B,C\}\) and \(\{X,Y,Z\}\) be two distinct triples of collinear points in \(\mathbb{R}^2\) such that the lines \(AB\) and \(XY\) don’t coincide. Then the points \(AY \cap BX\), \(AZ \cap CX\) and \(BZ \cap CY\) are also collinear.

*Note:* The brightly coloured dots in this figure (and subsequent ones) are interactive. Try moving them along the lines in the Pappus configuration above; you can also pan and zoom to get a better view if stuff moves off-screen.

Before proceeding, let’s quickly clarify some terminology and notation. Three points are called *collinear* if they all lie on a common line, \(\mathbb{R}^2\) is the set of pairs of real numbers and the \(\cap\)

symbol denotes intersection. From now on when technical terms come up, I’ll give their definition in a footnote like this one.^{1}

Hopefully you can see what I was driving at before: at first glance, Pappus’ theorem is just a bit dry. It falls right in line with Pythagoras and Thales’ theorems as the kind of thing mathematicians are always trying to convince people don’t form the entirety of the subject. Well, our goal today is to uncover a truly fascinating bit of structure underlying this seemingly innocuous result, which will then lead to the promised earth-shaking proof. However, before that, we’re going to have a look at a *different* proof of Pappus’s theorem. And I warn you, this proof is *even less exciting* than the statement. But that’s kind of the point. You’ll see.

**Structure of this essay.**

- Section 2 is a self-contained crash course on the basics of projective geometry.
- Section 3 introduces and analyses a more
classical

proof of Pappus’ theorem. - Section 4 proves Ceva’s theorem and explains how to glue multiple copies of the result together.
- Section 5 presents the
*raison d’être*of this essay: a topological proof of Pappus’s theorem. - Finally, Section 6 discusses more recent progress in extending the ideas of the previous sections.

*Note:* Sections 4 and 5 form the mathematical heart of this essay. If you’re less interested in the projective context, Sections 2 and 3 may be considered optional.

The smallest non-zero natural number.↩︎