Approximate Construction of Regular Polygons: Two Renaissance Artists

Editor's note: This article was published in March 2006.

Regular polygons and polyhedra have interested mathematicians at least since Euclid (c. 300 BCE), who dedicated Books IV and XIII of his Elements  to them. (In fact, much of the knowledge contained in Book XIII had come down to him from the disciples of Pythagoras (c. 569-470 BCE).)

The interest of these figures lies in their beauty and in the challenge posed by their construction; it is not easy to construct a polygon of, say, five sides, all of them equal! Well, since the sides of such a polygon are all equal, they may be viewed as chords of a circle, all equidistant from its center. This circle will, then, be circumscribed about the polygon. We may, if we wish, take as our initial datum either the side of the n-gon (polygon of n sides), l_n, or the radius R of the circumscribed circle, for there is a unique relation between them:

l_n/2 = R sin((2p/n)/2)              (1)

here, q_n = 2p/n is the central angle subtended by one side l_n, obviously equal to one n -th of the complete circle.

But the sine in Eq.(1) makes things hard if you want to construct l_n with ruler and compass (besides, trigonometric functions were not invented until much later). So the Greek mathematicians relied on ordinary geometry for their constructions. They did well; they found how to construct regular polygons of 3, 4, 5, 6, 8, 10 and 15 sides.      

As is well known, Greek mathematics was forgotten in Europe during the Middle Ages, but was rediscovered in the Renaissance (1400-1600 CE). At this time, artists with a mathematical ability became interested in problems of perspective, in drafting, and in the amazing, quasi-mystical, properties of the golden section --- which appears prominently in connection with the regular pentagon.     For these reasons, some artists looked for procedures to construct regular polygons. It did not matter whether these procedures were old or new; it did not matter either if they were not exact, but only approximate.  They were intended for use by painters, architects and draftsmen, not for contemplation by pure geometers.