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Ruler-and-compass construction

From Academic Kids

A number of ancient problems in plane geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass.

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.

Contents

Ruler and compass

The "ruler" and "compass" of ruler-and-compass constructions is an idealization of rulers and compasses in the real world:

  • The ruler is infinitely long, but it has no markings on it and has only one edge (thus making it a straightedge instead of what we usually think of as a ruler). The only thing you can use it for is to draw a line segment between two points, or to extend an existing line.
  • The compass can be opened arbitrarily wide, but (unlike most real compasses) it also has no markings on it. It can only be opened to widths you have already constructed.

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 appear to be 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 is in the problems 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. All three have now been proven under mathematical rules to be impossible.

The straightedge 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.

Constructible points and lengths

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 number 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 geometry one can construct the complex numbers in the form x+yi with x and y rational numbers. 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 field, a subfield of the complex numbers. Moreover, one can show that the given a constructible length one can construct its complex conjugate and square root.

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 extension 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+yk, 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 number.

Impossible constructions

Squaring the circle

The most famous of these problems, "squaring the circle", involves constructing a square with the same area as a given circle using only ruler and compass.

Squaring the circle has been proved impossible, as it involves generating a transcendental ratio, namely 1:√π. Only algebraic ratios can be constructed with ruler and compass alone. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason.

Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and has been solved many times in antiquity.

Doubling the cube

Doubling the cube: using only ruler and compass, construct the side of a cube that has twice the volume of a cube with a given side. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its minimal polynomial over the rationals has degree 3.

Angle trisection

Angle trisection: using only ruler and compass, construct an angle that is one-third of a given arbitrary angle. This requires taking the cube root of an arbitrary complex number with absolute value 1, and is likewise impossible.

Specifically, one can show that the angle of 60° cannot be trisected. If it could be trisected, then the minimal polynomial of cos 20° would have to have a power-of-two degree.

Using the trigonometric identity cos(3α) = 4cos³(α) - 3cos(α), one sees that, letting cos 20° = y, that 8y³ - 6y - 1 = 0, so, substituting x = 2y, x³ - 3x - 1 = 0. The minimal polynomial for x is a factor of this, but if it were not irreducible, then it would have a rational root which, by the rational root theorem, must be 1 or -1, which are clearly not roots. Therefore the degree for the minimal polynomial for cos 20° is of degree 3, so cos 20° is not constructible and 60° cannot be trisected.

Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the more powerful (but physically easy) operations of paper folding, or origami. Huzita's axioms (types of folding operations) can construct cubic extensions (cube roots) of given lengths, whereas ruler-and-compass can construct only quadratic extensions (square roots). See Mathematics of origami.

Constructing regular polygons

Some regular polygons (e.g. a pentagon) are easy to construct with ruler and compass; others are not. This led to the question: Is it possible to construct all regular polygons with ruler and compass?

Carl Friedrich Gauss in 1796 showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes. Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was proved by Pierre Wantzel in (1836). See constructible polygon.

Constructing with only ruler or only compass

It is possible (according to the Mohr-Mascheroni theorem) to construct anything with just a compass that can be constructed with ruler and compass. It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but (by the Poncelet-Steiner theorem) given a single circle and its center, they can be constructed. A further generalisation of this theorem (due to K. Venkatachala Iyengar) gives constructions using only a ruler given a point and five distinct points equidistant from it.

Recent research

Simon Plouffe has written a paper showing how ruler and compass can be used as a simple computer with unexpected power to compute binary digits of certain numbers.

Reference

  • Simon Plouffe. "The Computation of Certain Numbers Using a Ruler and Compass." Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.3.

See also

External links

fr:Construction la rgle et au compas it:Trisezione di un angolo ja:定規とコンパスによる作図 ru:Построение с помощью циркуля и линейки zh:尺规作图

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