classical problems of constructibility


There are at least three classical problems of constructibility:

  1. 1.

    Trisecting the angle: Can an arbitrary angle be trisected?

  2. 2.

    Doubling the cube: Given an arbitrary cube, can a cube with double the volume be constructed?

  3. 3.

    Squaring the circle: Given a circle (http://planetmath.org/Circle), can a square with the same area as the circle be constructed?

The ancient Greeks knew that these constructions were possible using various tools:

Since the ancient Greeks were interested in performing constructions with the minimalPlanetmathPlanetmath amount of tools, they wanted to know whether these constructions were possible using only compass and straightedge. With this constraint in place, answers were elusive until the advent of abstract algebra. The problem was that, working in geometryMathworldPlanetmathPlanetmath alone, there is really no way to prove that a construction is impossible. By using abstract algebra, people could finally prove that certain compass and straightedge constructions were impossible.

The discovery that trisecting the angle using only compass and straightedge is impossible is attributed to Pierre Wantzel. He actually proved a sharper result from which the result about trisecting the angle immediately follows.

Theorem 1 (Wantzel).

It is impossible to trisect a 60 angle using only compass and straightedge.

Proof.

It should first be noted that 60 is a constructible angle (http://planetmath.org/Constructible2) since cos60=12 is a constructible number. (See the theorem on constructible angles for more details.) Thus, we are working in the field of constructible numbers.

Suppose that 20 is a constructible angle. Then cos20 is also a constructible number. Using the triple angle formulasMathworldPlanetmathPlanetmath (http://planetmath.org/TrigonometricIdentities), we have that cos60=4cos320-3cos20. Thus, 12=4cos320-3cos20. Therefore, 8cos320-6cos20=1. Hence, (2cos20)3-3(2cos20)-1=0.

Let α=2cos20. Then α is a constructible number and α3-3α-1=0. Since 1 and -1 are not roots of x3-3x-1, the polynomialPlanetmathPlanetmath is irreducible (http://planetmath.org/IrreduciblePolynomial) over by the rational root theorem. Thus, x3-3x-1 is the minimal polynomial for α over . Hence, [(α):]=3, contradicting the theorem on constructible numbers. The result follows. ∎

The discovery that doubling the cube using only compass and straightedge is impossible is also attributed to Pierre Wantzel.

Theorem 2 (Wantzel).

Doubling the cube is impossible using only compass and straightedge.

Proof.

By scalingMathworldPlanetmath so that the sides of the original cube are of length 1, the possibility of this construction is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) to 23 being a constructible number. Since [(23):]=3, the theorem on constructible numbers yields that 23 is not a constructible number. ∎

The discovery that squaring the circle is impossible is attributed to Ferdinand von Lindemann.

Theorem 3 (Lindemann).

Squaring the circle is impossible using only compass and straightedge.

Proof.

By scaling so that the radius (http://planetmath.org/Radius2) of the circle is of length 1, the possibility of this construction is equivalent to π being a constructible number. Note that π is transcendental. (See this result (http://planetmath.org/ProofOfLindemannWeierstrassTheoremAndThatEAndPiAreTranscendental2) for more details.) Thus, π is also transcendental. Therefore, [(π):] is not even finite, let alone a power of 2. The theorem on constructible numbers yields that π is not a constructible number. ∎

References

  • 1 Rotman, Joseph J. A First Course in Abstract Algebra. Upper Saddle River, NJ: Prentice-Hall, 1996.
Title classical problems of constructibility
Canonical name ClassicalProblemsOfConstructibility
Date of creation 2013-03-22 17:18:37
Last modified on 2013-03-22 17:18:37
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 14
Author Wkbj79 (1863)
Entry type Topic
Classification msc 01A20
Classification msc 51M15
Classification msc 12D15
Related topic TheoremOnConstructibleNumbers
Related topic TheoremOnConstructibleAngles
Related topic CompassAndStraightedgeConstruction
Related topic ConstructibleAnglesWithIntegerValuesInDegrees
Defines trisecting the angle
Defines doubling the cube
Defines squaring the circle