johncarlosbaez

The 24-cell is a four-dimensional regular polytope with 24 vertices and 24 octahedral faces. The 600-cell is a four-dimensional regular polytope with 120 vertices and 600 tetrahedral faces. Since 120/24 = 5, you might hope that there’s a way to partition the 600-cell’s vertices into the vertices of five 24-cells.

And there is! A new paper by some high school students shows what we've long believed: there are exactly 10 ways.

In fact, they show there are 25 ways to choose some of the vertices of the 600-cell to be the vertices of a 24-cell. These 25 ways can listed in a 5×5 array, so that each row and each column gives one way to partition the 600-cell’s vertices into the vertices of five 24-cells.

For details and pictures go here:

https://johncarlosbaez.wordpress.com/2020/11/30/the-600-cell-part-4/

A student asked me if any of Ramanujan's formulas remain unproved. I had to ask around. Here's the answer:

https://johncarlosbaez.wordpress.com/2020/11/27/ramanujans-last-formula/