Alphabet of Lines: Geometric Construction

Topics: Carl Friedrich Gauss, Compass and straightedge constructions, Constructible polygon Pages: 3 (810 words) Published: January 27, 2013
In antiquity, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compass (or in Plato's case, a compass only; a technique now called a Mascheroni construction). Although the term "ruler" is sometimes used instead of "straightedge," the Greek prescription prohibited markings that could be used to make measurements. Furthermore, the "compass" could not even be used to mark off distances by setting it and then "walking" it along, so the compass had to be considered to automatically collapse when not in the process of drawing a circle.

Because of the prominent place Greek geometric constructions held in Euclid's Elements, these constructions are sometimes also known as Euclidean constructions. Such constructions lay at the heart of the geometric problems of antiquity of circle squaring, cube duplication, and angle trisection. The Greeks were unable to solve these problems, but it was not until hundreds of years later that the problems were proved to be actually impossible under the limitations imposed. In 1796, Gauss proved that the number of sides of constructible polygons had to be of a certain form involving Fermat primes, corresponding to the so-called Trigonometry Angles.

Although constructions for the regular triangle, square, pentagon, and their derivatives had been given by Euclid, constructions based on the Fermat primes were unknown to the ancients. The first explicit construction of a heptadecagon (17-gon) was given by Erchinger in about 1800. Richelot and Schwendenwein found constructions for the 257-gon in 1832, and Hermes spent 10 years on the construction of the 65537-gon at Göttingen around 1900 (Coxeter 1969). Constructions for the equilateral triangle and square are trivial (top figures below). Elegant constructions for the pentagon and heptadecagon are due to Richmond (1893) (bottom figures below).

Given a point, a circle may be constructed of any desired radius, and a diameter drawn...