Famous Problems of Geometry and How to Solve Them by Benjamin Bold

Author:Benjamin Bold
Language: eng
Format: epub
Publisher: Dover Publications
Published: 1969-06-26T16:00:00+00:00

PROBLEM SET VII–A

1. Construct regular polygons of 2m sides where m equals (a) 2. (b) 3. (c) 4.

2. Construct a regular polygon of 3·2m sides where m equals (a) 0. (b) 1. (c) 2.

3. Assuming that regular polygons of 3 sides and 5 sides can be constructed, show how to construct a regular polygon of 15 sides by finding two integers k and l such that 3k + 5l = 1. How would you then construct a regular polygon of 30 sides?

4. According to what you have learned thus far, is it possible

(a) to construct a regular polygon of 3·3 or 9 sides?

(b) to construct a regular polygon of a + b sides if regular polygons of a and b sides can be constructed, and a and b are relatively prime? Try to prove your conjecture.

For more than 2,000 years the problem of dividing a circle into equal parts remained as left by the ancient mathematicians. Despite the fact that such eminent mathematicians as Fermat and Euler worked on the problem, no further progress was made until the end of the eighteenth century, when Gauss solved the problem completely in 1796.

E. T. Bell in his Development of Mathematics states “The occasion for Gauss’ making mathematics his life work was his spectacular discovery at the age of nineteen concerning the construction of regular polygons by means of straight edge and compasses alone.” Prior to his discovery Gauss had been considering a career in Philology. D. E. Smith in his History of Mathematics quotes Gauss’ own record of his discovery.

“The day was March 29, 1796, and chance had nothing to do with it. Before this, indeed, during the winter of 1796 (my first semester at Gottingen), I had already discovered everything relating to the separation of the roots of the equation, (xp − 1)/(x − 1) = 0 into two groups. After intensive consideration of the relation of all the roots to one another on arithmetical grounds, I succeeded during a holiday at Braunschweig, on the morning of the day alluded to (before I got out of bed,), in viewing the relation in the clearest way, so that I could immediately make application to the 17 sides and to the numerical verifications.”

Courant and Robbins in their book What Is Mathematics? state “He [Gauss] always looked back on the first of his great feats with particular pride. After his death a bronze statue of him was erected in Göttingen; and no more fitting honor could be devised than to shape the pedestal in the form of a regular 17-gon.” Why a regular polygon of 17 sides will become clear later.

Let us now examine Gauss’ remarkable achievement. At the beginning of the chapter we indicated that the Greeks could construct a regular polygon of n sides, if n is of the form , when P1 = 3 and P2 = 5 and r = 0 or 1. In Problem Set VII–A, you were asked to construct a regular polygon of 3 sides, but not of 5.

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