| Compass And Straightedge |
Article Index for Compass |
Website Links For Compass |
Information AboutCompass And Straightedge |
|
Compass and straightedge or '''ruler-and-compass construction''' is the construction of lengths or Angle s using only an idealized Ruler and Compass . The ruler to be used has no markings on it and only one edge, and is known as a Straightedge . A number of ancient problems in Plane Geometry involve this technique. The most famous ruler-and-compass problems have been proven impossible in several cases by 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. RULER AND COMPASS 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 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 one ''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 BASIC CONSTRUCTIONS All compass and straightedge constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are:
For example, starting with the minimal state of a drawing, with just two distinct points, we can create a line or either of two circles. From the two circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle. CONSTRUCTIBLE POINTS AND LENGTHS There are many different ways to prove something is impossible. In this particular problem we carefully demarcate the limit of the possible, and show that to solve these problems one must transgress that limit. Using a ruler and compass, one 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 one ''can'' do is draw perpendiculars, so draw a perpendicular to the ''x''-axis, and call it the ''y''-axis. This results in a Cartesian coordinate system on the plane. One 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 Number s. 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'' + ''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 Number . CONSTRUCTIBLE ANGLES There is a Bijection between the angles that are constructible and the points that are constructible on any constructible circle. The angles that are constructible form an Abelian Group under addition modulo (which corresponds to multiplication of the points on the unit circle viewed as complex numbers). The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example the regular Heptadecagon is constructible because : as discovered by Gauss {Link without Title} . The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots). The only angles of finite order that may be constructed starting with two points are those whose order is a product of a power of two and a set of distinct Fermat Primes . In addition there is a dense set of constructible angles of infinite order. COMPASS AND STRAIGHTEDGE CONSTRUCTIONS AS COMPLEX ARITHMETIC Given a set of points in the Euclidean Plane , selecting any one of them to call 0 and another to be called '''1''', together with an arbitrary choice of Orientation allows us to consider the points as a set of Complex Number s. Given any such interpretation of of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest Field containing the original set of points and closed under the Complex Conjugate and Square Root operations (to avoid ambiguity, we can specify the square root with Complex Argument less than ). The elements of this field are precisely those that may be expressed as a formula in the original points using only the operations of Addition , Subtraction , Multiplication , Division , Complex Conjugate , and Square Root , which is easily seen to be a countable dense subset of the plane. Each of these six operations corresponding to a simple compass and straightedge construction. From such a formula it is straightforward to produce a construction of the corresponding point by combining the constructions for each of the arithmetic operations. More efficient constructions of a particular set of points correspond to shortcuts in such calculations. Equivalently (and with no need to arbitrarily choose two points) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points. The set of ratios constructible using compass and straightedge from such a set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots. For example the real part, imaginary part and modulus of a point or ratio (taking one of the two viewpoints above) are constructible as these may be expressed as : : |
|
|