classical problems of constructibility
There are at least three classical problems of constructibility:
Trisecting the angle: Can an arbitrary angle be trisected?
Doubling the cube: Given an arbitrary cube, can a cube with double the volume be constructed?
The ancient Greeks knew that these constructions were possible using various tools:
A way of squaring the circle is discussed in the entry variants on compass and straightedge constructions.
Since the ancient Greeks were interested in performing constructions with the minimal 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 geometry 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 angle using only compass and straightedge.
It should first be noted that is a constructible angle (http://planetmath.org/Constructible2) since 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 is a constructible angle. Then is also a constructible number. Using the triple angle formulas (http://planetmath.org/TrigonometricIdentities), we have that . Thus, . Therefore, . Hence, .
Let . Then is a constructible number and . Since and are not roots of , the polynomial is irreducible (http://planetmath.org/IrreduciblePolynomial) over by the rational root theorem. Thus, is the minimal polynomial for over . Hence, , 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.
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.
By scaling so that the radius (http://planetmath.org/Radius2) of the circle is of length , the possibility of this construction is equivalent to being a constructible number. Note that is transcendental. (See this result (http://planetmath.org/ProofOfLindemannWeierstrassTheoremAndThatEAndPiAreTranscendental2) for more details.) Thus, is also transcendental. Therefore, is not even finite, let alone a power of . The theorem on constructible numbers yields that is not a constructible number. ∎
- 1 Rotman, Joseph J. A First Course in Abstract Algebra. Upper Saddle River, NJ: Prentice-Hall, 1996.
|Title||classical problems of constructibility|
|Date of creation||2013-03-22 17:18:37|
|Last modified on||2013-03-22 17:18:37|
|Last modified by||Wkbj79 (1863)|
|Defines||trisecting the angle|
|Defines||doubling the cube|
|Defines||squaring the circle|