Impossible Constructions

Date: 01/14/98 at 17:39:05
From: Coconut
Subject: Euclidean Geometry

Using only a compass and straightedge, what are the three ancient 
impossible construction problems of Euclidean geometry? Under what 
restrictions are they impossible?

Date: 01/14/98 at 20:12:18
From: Doctor Wilkinson
Subject: Re: Euclidean Geometry

(1) Trisecting an angle: given an angle construct an angle one third 
    as large. The problem has to be solved for an arbitrary angle.  
    Some particular angles such as 90 degrees can be trisected easily. 

(2) Duplicating the cube: given the side of a cube, construct the side 
    of a cube with twice the volume.

(3) Squaring the circle: given the radius of a circle, construct the 
    side of a square of the same area.

All three problems are impossible if you adhere strictly to the rules, 
using only a compass and an unmarked straightedge. If you are allowed 
to make marks on the straightedge or to cheat in various other ways, 
you can trisect the angle, for example.

The first two problems were proved to be impossible by Pierre Laurent
Wantzel in 1837, though this was already known to Gauss around 1800.

The third problem was proved to be impossible by Lindemann in 1882.

The impossibility proofs depend on the fact that the only quantities 
you can get by doing straightedge-and-compass constructions are those 
you can get from the given quantities by addition, subtraction, 
multiplication, division, and taking square roots. The first two 
problems require in effect taking a cube root. The third problem 
requires constructing pi, and what Lindemann showed was that pi is a 
so-called transcendental number, which means that it is not the root 
of an algebraic equation with integer coefficients.

I hope this helps clear things up a little.

-Doctor Wilkinson,  The Math Forum
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