Lecture Notes 5
The philosophy of constructions
Constructions using compass and straightedge have a long history in Euclidean geometry. Their use reflects the basic axioms of this system. However, the stipulation that these be the only tools used in a construction is artificial and only has meaning if one views the process of construction as an application of logic. In other words, this is not a practical subject, if one is interested in constructing a geometrical object there is no reason to limit oneself as to which tools to use. The value of studying these constructions lies in the rich supply of problems that can be posed in this way. It is important that one be able to analyze a construction to see why it works. It is not important to gain the manual dexterity needed to carry out a careful construction.
Compass vs Dividers
The ancient greek tool used to construct circles is not the modern day compass. Rather, they used a device known as a divider. Dividers consist of just two arms with a central pivot. Should you pick up a divider, the arms will collapse, so it is impossible to use them to transfer lengths from one area to another. Modern compasses remain open when picked up, so such transfers are possible. Given the difference in the two tools, it appears that the modern compass is a more powerful instrument, capable of doing more things. However, this is not true. The ancient dividers can do everything that modern compasses can. Of course, this means that how certain constructions were done by the ancient Greeks are quite different from the way we would do them today. This underscores the statement above; technique is not as important as understanding why it works.
The basic constructions are:
Given a segment which represents the number 1 (a unit segment), the segments which can be constructed from this one by use of compass and straightedge represent numbers called Constructible Numbers. Note that the restrictions imply that the constructible numbers are limited to lying in certain quadratic extensions of the rationals.
Given two constructible numbers one can with straightedge and compass construct their:
- Square Root
Constructions using basic operations.
- Example: Construct a triangle, given the length of one side of the triangle, and the lengths of the altitude and median to that side.
As the third vertex is determined by the intersection of one of two parallel lines with a circle, there are three possibilities for the number of solutions. If b is less than c, there will be no intersection, so no solutions. If b equals c, the lines will be tangent to the circle and we would get two solutions. Finally, if b is greater than c (the situation drawn above) then there will be four points of intersection.
- Example: Construct a triangle, given one angle, the length of the side opposite this angle, and the length of the altitude to that side.
As the position of vertex A is determined by the intersection of a single line with a circle, there are three possibilities for the number of solutions. If the parallel does not intersect the circle, there is no solution. If the parallel is tangent to the circle there is one solution, and finally, if the parallel intersects the circle twice, there are two solutions (as indicated in the situation drawn above).
- Example: Construct a triangle, given the circumcenter O, the center of the nine-point circle N, and the midpoint of one side A'.
This construction always gives a unique triangle provided one exists. If N = A' there will be no nine-point circle, but N could equal O, or A' could equal O and the construction will still work. The points could also be collinear.
Constructions and Impossibility Proofs
An algebraic analysis of the fields of constructible numbers shows the following:
Theorem: If a constructible number is a root of a cubic equation with rational coefficients, then the equation must have at least one rational root.
The three famous problems of antiquity:
- The Delian problem - duplicating the cube. The problem is to construct a cube that has twice the volume of a given cube. A particular instance of this problem would be to construct a cube whose volume is twice that of the unit cube. This entails constructing a side of the larger cube, and in this case that means constructing a length equal to the cube root of 2. This length is a root of the equation x3 - 2 = 0, but this cubic equation with rational coefficients has no rational root.
- Trisection of an Angle - The problem is to find the angle trisectors for an arbitrary angle. The general problem can not be done because it can't be done for some specific angles, for instance an angle of 60°. (Construction of 20 degree angle leads to the cubic equation 8x3 -6x - 1 = 0,
and this does not have roots of the required type). (Wankel)
- Squaring the Circle - The problem is to construct a square that has the same area as the unit circle, i.e. . If this can be done, then the square root of would be constructible. And if that is true, then would also be constructible. But is a transcendental number (Lindemann, 1882), and such numbers are not constructible.
Angle Trisection using a marked straightedge (Archimedes, 287-212 B.C.) and the Conchoid
of Nichomedes (240 B.C.).
Trisectors - are still with us. Readings from Underwood Dudley's, A Budget of Trisections.
Construction of regular polygons (Gauss). These are only possible when the number of sides, n, is of the form
n = 2ap1p2...pk
where the pi are distinct Fermat primes, i.e. prime numbers of the form 22x + 1.
The first few Fermat primes are: p1= 3 = 220 + 1 , p2= 5 = 221 + 1 , p3 = 17 = 222 + 1. Thus, it is possible to construct regular polygons of n sides when n is:
| 3 || 203
| 4|| 22|
| 5|| 205|
| 6 || 213|
| 8 || 23|
| 10|| 215|
| 12|| 223|
| 15 || 203(5)|
| 16 || 24|
| 17 || 2017
The factor of a power of 2 comes from the fact that given any regular n-gon, you can always construct a regular 2n-gon. This is done by inscribing the n-gon in a circle and then constructing the perpendicular bisectors of each of the sides. Extend these to the circle and these points together with the original vertices of the n-gon, form the vertices of a regular 2n-gon. Repeating this will give the higher powers of 2.
It is not possible to construct, with straightedge and compass alone, regular polygons of sides n = 7, 9, 11, 13, 14, 18, 19, ....
Other types of Construction
It can be shown that any construction that can be made with straightedge and compass can be made with compass alone (Mascheroni, 1797). Of course one must understand that a straight line is given as soon as two points on it are determined, since one can't actually draw a straight line with only a compass.
It can also be shown that any construction that can be made with straightedge and compass can be made with straightedge alone, as long as there is a single circle with its center given (Steiner, 18??).
Some constructions by paper folding:
Geometer's Sketchpad demonstration.
- Other constructions are possible, including trisecting angles. Here is a folding of the regular nonagon (9 sided regular polygon) which is impossible to do with straightedge and compass.
(from T.S. Row, Geometric Exercises in Paper Folding).