# gjm comments on On exact mathematical formulae

• One thing that isn’t made, aha, ex­plicit in this ex­cel­lent ar­ti­cle, and that maybe should be, is that con­sid­er­ing ruler-and-com­passes con­struc­tions to be ex­plicit turns out to be ex­actly the same thing as con­sid­er­ing tak­ing square roots to be ex­plicit be­cause (once you have the idea of co­or­di­nate ge­om­e­try, which for some rea­son no one thought of un­til Descartes) it pretty eas­ily tran­spires that the nu­mer­i­cal op­er­a­tions you can do on co­or­di­nates us­ing ruler-and-com­passes con­struc­tions are ex­actly ar­ith­metic plus square roots. (Hand­wav­ily, the square roots come from the fact that the equa­tion of a cir­cle in­volves squar­ing the co­or­di­nates.)

And then (this uses, in some sense, “half” of Galois the­ory, so it’s eas­ier than the in­sol­u­bil­ity of the quin­tic) you can use this to prove that in­deed you can’t “du­pli­cate the cube” us­ing ruler and com­passes (it means tak­ing cube roots, and it turns out that no quan­tity of ar­ith­metic-plus-square-roots will let you do that), and that an­other fa­mous an­cient prob­lem—tri­sect­ing an­gles—can’t be done with ruler and com­passes for es­sen­tially the ex­act same rea­son, and (this is a bit more difficult; it was one of Gauss’s first im­por­tant the­o­rems) that the reg­u­lar poly­gons you can con­struct with ruler and com­passes are those whose num­ber of sides is a power of 2 times some num­ber of dis­tinct primes of form 2^2^n+1. So you can do 3 and 5 and 17, but not 9 (two 3s: not al­lowed) and not 11 (wrong sort of prime).

Of course, for all the rea­sons daoza­ich gives here, one shouldn’t care too much about what can be done with ruler and com­passes, as such. But these are still re­ally cool the­o­rems.

• You’re right, I should have made that clearer, thanks!