Associated Topics || Dr. Math Home || Search Dr. Math

### 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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Constructions
High School Geometry
High School History/Biography
Middle School Geometry
Middle School History/Biography

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search