I almost thought that footnote wouldn’t even be there! This book is that glib. The footnote contains the most interesting fact presented in the entire book.

I must compulsively check all these constructions to see if they are “real” they probably are— but still.

I wonder how the heptagon approximation was created— I get the desire to just know how to make a serviceable regular solid— but to talk of heptagons but not unconstructability ? heresy I say!

I wonder how the heptagon approximation was created

💯 I'm wondering too. Answering this would probably require to extend approximation theory, i.e. continued fractions, to arbitrary algebraic fields. Not sure if that should be done ... (or has been done).

@helge @futurebird as a computer-programming-poisoned mind, I'd probably approximate anything trickier than a triangle by repeated bisection of angles using the binary representation of the fraction. You can get that as close as you like if you're patient.

That includes the pentagon, which has an exact construction but I can never remember how to do it.

Nooooooooooooo! That's inefficient.

If my memory serves me right, it's probably one of the worst possible ways to do it. All numbers of the form n/2^k can be constructed ... Now, one needs to dig through the approximation theory, on why this converges slowly in the number of steps.