This is a question combining number theory and geometry. I am asking it purely from curiosity, but I think it might be a useful and interesting question.
Start with an equilateral triangle of perimeter 1. Inside of that triangle, inscribe the largest square possible, and add its perimeter to the perimeter of the triangle. Then, inscribe a regular pentagon inside of the square, and add its perimeter. Keep on doing this infinitely, and what would be the sum of the perimeters of all of the polygons?
After some thought, my guess is that it diverges to infinity, very slowly, because even as the polygons get smaller, the rate at which they get smaller decreases. But it might be a convergent series.