Lalalalal

“YOU CAN’T GET THERE FROM HERE”: WHY YOU CAN’T TRISECT AN ANGLE, DOUBLE THE CUBE, OR SQUARE THE CIRCLE (WITH STRAIGHTEDGE AND COMPASS)
RAVI VAKIL

The three impossibilities: (i) You can’t trisectan angle with straightedge and compass. (ii) You can’t double the cube with straightedge and compass. (iii) You can’t square the circle with straightedge and compass. 1. (Archimedes’ trick) Show thatyou can trisect an angle using a “marked straightedge” (such as a ruler). Hint: First show that in the following figure, angle B is one third angle A.

B

A

2. Show that cos 20 is a root of (1)8x3 − 6x − 1 = 0. (Hint: Work out the formula for cos 3θ in terms of cos θ, and use cos 60 = 1/2.) Show that (1) has no rational roots (so cos 20 is irrational). Constructible numbers are the realnumbers that you can write down using rational numbers, the four arithmetic operations, and square roots. (All intermediate calculations must be real as well.) 3. Let S be the set of points you couldthen construct using straightedge and compass, starting with the points (0, 0) and (1, 0). Show that S is precisely the set of points (x, y) where x and y are constructible. (This has many parts, butnone of them are technically difficult. For example, show that if you have two line segments of length a, and b = 0, and a √ rational number r, then you can construct line segments of lengths a ± b, ab,a/b, a, and ra.) A real field F is a subset of R containing 0 and 1, closed under the four operations. In other words, if you add, subtract, multiply, or divide two elements of F , you’ll √ get anotherelement of F . Suppose z > 0 is an element of F such that z isn’t
Date: Friday, December 8.
1

√ in F . Then the elements of the form a + b z (where a, b ∈ F ) also√ form a real field, and this iscalled a quadratic extension of F , and is denoted F ( z). Define √ √ the conjugate of a + b z to be a − b z. Hence constructable numbers are those numbers that are elements of towers of quadratic...