Ruler and compass constructions

Summary: Ruler and compass constructions form the basis of geometry and have challenged mathematicians for thousands of years.

Ruler and compass constructions have long been important in mathematics. In geometry, a ruler and compass construction refers to a geometric construction that uses only an unmarked ruler and a compass. The ancient construction problems of squaring the circle, duplicating the cube, and trisecting the angle were unsolved until they were proved impossible by algebraic techniques. Early tile makers and architects were also interested in these constructions. Aside from historical considerations, limiting constructions to these two tools is important because the restrictions generate a variety of rich problems. In the twenty-first century, dynamic geometry software programs allow students, teachers, and researchers to explore, save, and share constructions.

Euclid

The most significant early compendium of ruler and compass constructions is Euclid’sElements written c. 300 b.c.e. In fact, Euclid’s book organizes everything around these constructions in an attempt to build as much geometry as possible starting with the most basic tools. Drawing a line using a ruler and a circle using a compass are seen as elementary in Euclid’s tradition—hence, the title Elements—and it is preferred to reduce as much of geometry as possible to these elementary tools. Elements begins with five common notions and five “self evident” postulates. The first three postulates specify the rules for geometric constructions:

  • A straight line segment can be drawn joining any two points.
  • Any straight line segment can be extended indefinitely in a straight line.
  • Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

The final two postulates of Euclid are

  • All right angles are congruent.
  • If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

The last one is the famous fifth postulate and is equivalent to the more common parallel postulate: from a given point not on a given line, one can draw exactly one line parallel to the given line. Euclid based the whole edifice of rigorous geometry on these axioms, hence ruler and compass constructions are at the center of Euclidean geometry.

The Three Classical Problems

Three ancient construction problems captured the imagination of mathematicians for many centuries: doubling a cube, trisecting an angle, and squaring a circle.

  • Doubling a cube: Given the side of a cube, can one construct, using an unmarked ruler and a compass, the side of another cube whose volume is twice the first one?
  • Trisecting an angle: Given an arbitrary angle, can one draw a line, using an unmarked ruler and a compass, that trisects the angle?
  • Squaring a circle: Given a line segment that is the radius of a circle, can one construct, using an unmarked ruler and a compass, the side of a square that has the same area as the original circle?

None of these constructions are possible, but surprisingly, despite more than 2000 years of effort, a satisfactory answer to these three questions was given only in the nineteenth century.

Each of these classical problems has a long history. For example, the problem of doubling a cube was known to the Egyptians, Greeks, and Indians. In one version of the Greek legend, the citizens of Athens consulted the oracle of Apollo at Delos to put a stop to a plague in Athens. The oracle prescribed that the Athenians double the size of their altar. Efforts to find a way of doubling the volume of the cube failed, and it is claimed that Plato (427–347 b.c.e.) had remarked that the oracle really meant to “shame the Greeks for their neglect of mathematics and for their contempt of geometry.” The original legend did not specify the tools to be used, and, in fact, solutions using a number of tools were found. However, a construction using the elementary tools of an unmarked ruler and a compass remained elusive.

Tool Variations

Variations on the tools are possible. For example, if one were allowed to make two marks on the ruler, then with the use of this marked ruler and a compass, one can trisect an arbitrary angle.

An interesting variation arose in the work of Abu’l Wafa Buzjani (940–997 c.e.). Abu’l Wafa in a work aimed at artisans (such as tile makers, designers of intricate patterns, and architects) limited the geometric tools to an unmarked ruler and a “rusty” compass. In other words, he wanted to only use a compass that had a fixed opening and could not be adjusted to draw different sized circles. He believed that working with such a fixed compass would be more accurate, less error-prone, and more useful for artisans. Abu’l Wafa constructs, among other polygons, regular pentagons, octagons, and decagons using a rusty compass. Since the opening of the compass used in Euclid’s Elements could vary, Abu’l Wafa could not rely on the constructions in Elements. Hence, he constructed anew, using the rusty compass, all the needed basic results.

In Europe, the Danish mathematician Georg Mohr (1640–1697) showed, rather surprisingly, that all ruler and compass constructions can be done with a compass alone. In such constructions, one cannot draw a line segment, and a line segment is considered constructed as long as its two endpoints are found. This result is now known as the Mohr–Mascheroni theorem. The Italian Lorenzo Mascheroni (1750–1800) had independently found the same result. Georg Mohr also proved that all ruler and compass constructions can be done with a ruler and a rusty compass. Finally, the German mathematician Jacob Steiner (1796–1863) and the French mathematician Jean-Victor Poncelet (1788–1867) proved that all constructions using a ruler and a compass can be made with a ruler and only one use of the compass.

Proofs

Going back to the classical problems, the first rigorous proof of the impossibility of doubling the cube and trisecting an arbitrary angle using a ruler and a compass was given by the French mathematician Pierre Laurent Wantzel (1814–1848). In 1882, the German mathematician Ferdinand Lindemann (1852–1939) proved that π is transcendental. From this, it followed that one cannot square a circle using a ruler and a compass. In general, using only these tools, it is possible to construct line segments of any rational length as well as line segments whose length is the square root of the length of any already constructed segment. However, one can prove that it is impossible to construct other lengths using the theory of fields that was developed with the help of Niels Henrik Abel and Évariste Galois on the solvability of equations. The proof essentially boils down to the fact that, using a ruler and a compass, one can draw only straight lines and circles, and the only new points are the intersections of these lines and circles. Since lines have linear equations and circles have quadratic equations, finding the points of intersection of these shapes is the same as equating their equations and finding the solutions. These all can be achieved using the quadratic formula, which involves only square roots.

Polygons

Constructing regular polygons with a straightedge and compass is also an interesting ruler and compass construction problem. An n-gon is a regular polygon with n sides. Ancient Greeks could construct regular n-gons for n = 3, 4, 5, and 15 (triangles, squares, regular pentagons, and regular pentadecagons). They also knew that if one can construct a regular n-gon with a straightedge and compass, then one can also construct a regular 2n-gon. Carl Friedrich Gauss (1777–1855) added to this knowledge, by constructing, when he was 19 years old, a regular heptadegon (a 17-gon).

A Fermat prime is a prime number of the form 22k+1, where k is a non-negative integer. The only Fermat primes known are 3, 5, 17, 257, and 65537. It is not known whether there are any other Fermat primes or not. In any case, Gauss stated, and Wantzel gave a proof, that a regular n-gon is constructible with ruler and compass if and only if n is an integer greater than two such that the greatest odd factor of n is either one or a product of distinct Fermat primes.

Bibliography

Hadlock, Charles Robert. “Field Theory and Its Classical Problems.” Carus Mathematical Monographs, 19 (1978).

Katz, Victor, ed. The Mathematics of Egypt, Mesopotamia, China, India, and Islam. A Sourcebook. Princeton, NJ: Princeton University Press, 2007.

Martin, George E. Geometric Constructions. New York: Springer-Verlag, 1998.

Sutton, Andrew. Ruler and Compass: Practical Geometric Constructions. New York: Walker & Co., 2009.