The Moduli Space of Acute Triangles

Recently Quanta magazine came out with an article explaining modular forms:

• Jordana Cepelewicz, Behold modular forms, the ‘fifth fundamental operation’ of math, Quanta, 23 September 2023.

It does a heroically good job. One big thing it doesn’t do is explain these funny looking ‘fundamental domains’ in the upper half-plane:

That is: where does this picture come from and why is it important?

By sheer coincidence, I just wrote a little article explaining the concept of ‘moduli space’ through an...