110 Chapter 9

9.17 For each integer n ≥ 3, let d(n) be the smallest dimension such that there is a regular planar n-

gon realized at points of rational coordinates in dimension d(n), if possible. For which values

of n is d(n) defined, and what is the complete list of values of the function d(n)?

Credits

I learned the main idea of this chapter from a post by Vaughn Climenhaga on MathOver-

flow, Climenhaga (2017); he had presented the hexagonal case, crediting the idea to Gy

¨

orgy

Elekes during an instance of the Conjecture and Proof course in the Budapest Semesters in

Mathematics. The proof of theorem 76 grew out of an exchange among myself, John Baez,

and Yao Liu on Twitter at https://twitter.com/JDHamkins/status/1154865078493208576.

Exercise 9.9 was inspired by math teacher Kate Belin in her sidewalk-math tweet at

https://twitter.com/katebelin/status/1161701927496929282. Exercise 9.17 was suggested

by David Madore at https://twitter.com/gro

tsen/status/1161660140879306754. The chess

adage about looking for a better move appears in annotations of William Wayte (1878).