| • Science | • People | • Locations | • Timeline |
The most famous ruler-and-compass problems have been proven impossible, in several cases by the results of Galois theory. In spite of these impossibility proofs, some mathematical novices persist in trying to solve these problems. Many of them fail to understand that many of these problems are trivially solvable provided that other geometric transformations are allowed: for example, squaring the circle is possible using geometric constructions, but not possible using ruler and compass alone.
Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books.
The "ruler" and "compass" of ruler-and-compass constructions is an idealization of rulers and compasses in the real world:
Each construction must be exact. "Eyeballing" it(essentially looking at the construction and guessing at its accuracy, or using some form of measurement, such as the units of measure on a ruler) and getting close does not count as a solution.
Stated this way, ruler and compass constructions are a parlor game, rather than a serious practical problem. Figuring out how to do any particular construction is an interesting puzzle, but the persistent interest in the problem derived from what you can’t do this way.
The three classical unsolved construction problems were:
For 2000 years people tried to find constructions within the limits set above, and failed. The reason? Because all three are impossible.
The straight edge and compass give you the ability to produce ratios which are solutions to quadratic equations, but doubling the cube and trisecting the angle require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio. Curiously, origami (i.e. paper folding without any equipment) is more powerful and can be used to solve cubic equations, and thus solve two of the classical problems.
How do you prove something impossible? There are many different ways, but this particular problem we carefully demarcate the limit of the possible, and show that to solve these problems you must transgress that limit.
Using a ruler and compass, you can impose coordinates on the plane. Draw two points, and draw the line through them. Call that the x-axis, and define the length between the two points to be one. One construction that you can do is draw perpendiculars, so draw a perpendicular to your x-axis, and call it your y-axis. We now have a Cartesian coordinate system on the plane.
You can identify a point (x,y) in the Euclidean plane with the complex numberThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , wher x + y i. In ruler and compass construction, one starts with a line segment of length one. If one can construct a given point on the complex plane, then one says that the point is constructible. By standard constructions of Euclidean geometryEuclidean geometry In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. Euclidean geometry someti one can construct the complex numbers in the form x.+ yi with x and y rational numberIn mathematics, a rational number (or informally fraction is a ratio of two integers, usually written as the vulgar fraction a ''b where b is not zero. The set of all rational numbers is denoted by Q or in blackboard bold. Using the set-builder notation is. More generally, using the same constructions, one can, given complex numbers a and b, construct a + b, a - b, a * b, and a / b. This shows that the constructible points form a fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil, which one treats as a subfield of the complex numbers. Moreover, one can show that the given a constructible length one can construct its complex conjugateIn mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number z a + ib (where a and b are real numbers) is defined to be z a − ib''. It is also often denoted and square rootIn mathematics, the square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is. For example, since. This example suggests how square roots can arise whe.
The only way to construct points is as the intersection of two lines, of a line and a circle, or of two circles. Using the equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extensionField theory Algebraic number theory In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have L ''K n and L is generated over K by a root of a polynomial X n − a with a in K and K contains n di of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form x + y √ k, where x, y, and k are in F.
Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for a constructible point (and therefore of any constructible length) has degree a power of 2. In particular, any constructible point (or length) is an algebraic numberAbstract algebra Algebra In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form : a x n + a x n minus;1 + ··· + a x + a 0 where n is a posit.