What is the Jordan Curve Theorem?
The Jordan Curve Theorem is the sort of thing I adore about mathematics. It states (I paraphrase):
A 2D loop that doesn’t cross itself has an inside and an outside (and the two are separate).
That might strike you as the kind of thing that’s obvious. Indeed, for a long while, the whole mathematical community was on your side: it’s so obvious, it doesn’t need to be proved.
And it is obvious, at least for polygons – the trouble comes when you look at loops made out of something like a Koch snowflake curve or a Weierstrass function that doesn’t behave like the kind of closed curve you’re probably imagining.
Why is it interesting?
I think it’s interesting because it’s an example of something that’s straightforward to state, blooming obvious to the passer-by, yet quite difficult to prove formally. (For a long while, mathematicians didn’t think Jordan’s proof of the result was quite solid; in fact, it just left a few details to the reader.)
It can be generalised to more dimensions (a simple closed surface in 3D has an inside and an outside; a closed volume in 4D has an inside and an outside, too).
It’s a result that’s important because it reminds us that sometimes even the most obvious things still need to be proved.
Who was Camille Jordan?
Marie Ennemond Camille Jordan was born in Lyon, France in 1838. He studied at the École Polytechnique, and worked as an engineer before becoming a teacher.
Among other things, he popularised Galois theory and studied groups generally; he was also renowned for eccentric notation. What’s not to love?
Although many things are named after Camille Jordan, some things that aren’t include Gauss-Jordan elimination (Wilhelm Jordan) and Jordan algebras (Pascual Jordan).
He died in Paris in 1922.
A selection of other posts
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