From: ken@straton.demon.co.uk (Ken Starks) Newsgroups: sci.math Subject: Re: Trisection of angle Date: Thu, 24 Sep 1998 18:34:20 BST graham_fyffe@hotmail.com wrote: > AFAIK, the original problem of trisecting the angle does not allow any > kind of constructions to help you do it. This is kind of vague, I know. > Anyway, some guy whose name has slipped my mind had come up with a way > to construct a bunch of curves on a sheet of paper, using the allowed > tools, which enabled him to trisect arbitrary angles. The math > community said "that doesn't count, since you had to construct that big > ugly thing first." > So, it all depends on how you define "construct", and where you draw the > line (pardon the pun) Perhaps you should try a book by George E Martin: "Geometric Constructions" ( Springer-Verlag ISBN 0-387-98276-0) This covers, in modern notation, many kinds of construction including the trisection of an angle using a 'tomohawk' The chapter headings will give you some idea. Euclidean Constructions The Ruler and compass The Compass and the Mohr-Mascheroni Theorem The ruler The ruler and dividers The Poncelot-steiner Theorem and double rulers The ruler and rusty compass Sticks The marked ruler Paperfolding The result you need for the smallest constructable angle ( with straight edge and compass ) as an integral number of degrees is: The Gauss-Wantzel Theorem. -------------------------- A regular n-gon is constructable with ruler and compass iff n is an integer greater than 2 such that the greatest odd factor of n is either 1 or a product of distinct Fermat primes. A 'Fermat prime' is one of the form ((2^(2^n))-1), and there are 4 known: 3, 5, 17, 257, 65537 For a whole number of degrees in your n-gon you therefore need a to factorise 360 into one of n x 3 x E n x 5 x E or n x 15 x E where E is a power of two. The Largest n is obviously 120, giving a three degree angle. By the way, Gauss proved that a 17-gon is construcable when he was 19. As well as a construction, you also get a formula: 16 cos(360/17)° = -1 + root17 + root(34 - 2root17) +2root( 17 + 3root17 - root(34 - 2root17) - 2root(34+2root17) )