### The Quintic, the Icosahedron, and Elliptic Curves

#### Posted by John Baez

Old-timers here will remember the days when Bruce Bartlett and Urs Schreiber were regularly talking about 2-vector spaces and the like. Later I enjoyed conversations with Bruce and Greg Egan on quintics and the icosahedron. And now Bruce has come out with a great article linking those topics to elliptic curves!

- Bruce Bartlett, The quintic, the icosahedron, and ellliptic curves,
*Notices of the American Mathematical Society***71**(April 2024), 447–453.

It’s expository and fun to read.

I can’t do better than quoting the start:

There is a remarkable relationship between the roots of a quintic polynomial, the icosahedron, and elliptic curves. This discovery is principally due to Felix Klein (1878), but Klein’s marvellous book misses a trick or two, and doesn’t tell the whole story. The purpose of this article is to present this relationship in a fresh, engaging, and concise way. We will see that there is a direct correspondence between:

- “evenly ordered” roots $(x_1, \dots, x_5)$ of a Brioschi quintic $x^5 + 10b x^3 + 45b x^2 + b^2 = 0$,
- points on the icosahedron, and
- elliptic curves equipped with a primitive basis for their 5-torsion, up to isomorphism.
Moreover, this correspondence gives us a very efficient direct method to actually calculate the roots of a general quintic! For this, we’ll need some tools both new and old, such as Cremona and Thongjunthug’s complex arithmetic geometric mean, and the Rogers–Ramanujan continued fraction. These tools are not found in Klein’s book, as they had not been invented yet!

If you are impatient, skip to the end to see the algorithm.

If not, join me on a mathematical carpet ride through the mathematics of the last four centuries. Along the way we will marvel at Kepler’s Platonic model of the solar system from 1597, witness Gauss’ excitement in his diary entry from 1799, and experience the atmosphere in Trinity College Hall during the wonderful moment Ramanujan burst onto the scene in 1913.

The prose sizzles with excitement, and the math lives up to this.

## Re: The Quintic, the Icosahedron, and Elliptic Curves

This looks amazing, Bruce. I’m going to have to take my time with this and savor it.

I just wanted to link to this YouTube video that explains, very clearly, from soup to nuts, Arnold’s proof of the unsolvability of the quintic. The visuals are excellent.