Comment by empath75
There's a series of two excellent videos on youtube about this proof:
Detailed description of the problem of constructable regular polygons and a gloss of the proof. https://www.youtube.com/watch?v=EX7U0DGBmbM
A full explanation of the proof: https://www.youtube.com/watch?v=Gdy1u4lsjDw
I learned about the construction of this shape from a numberphile video a couple of years back. https://www.youtube.com/watch?v=87uo2TPrsl8
There's a bit at the end showing the construction used in place of a building number on the front of the Mathematical Sciences Research Institute (MSRI) building at 17 Gauss Way, UC Berkeley.