Impossible ConstructionsDate: 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/ |
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