# Approximate Construction of Regular Polygons: Two Renaissance Artists

## Introduction

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), ln, or the radius R of the circumscribed circle, for there is a unique relation between them:

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

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

But the sine in Eq.(1) makes things hard if you want to construct ln 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.

## Leonardo da Vinci

Leonardo da Vinci (1452-1519) was many things: painter, physicist, engineer, anatomist... and amateur mathematician. He was not a methodic writer, but now and then he would note down a construction procedure for some regular polygon. (Some of them he later considered bad enough to label them “falso”.)

We will review here his construction of the regular pentagon . (It is approximate; an exact one was given by Euclid.)

Let the given side of the regular pentagon be DA=a. Drawing circles of radius a, centered at D and at A, we obtain points P and R on the perpendicular bisector of DA.  We then divide DA into eight equal parts. We now draw PG ||AD, with PG = AD/8; AG and PS intersect at O, center of the circle circumscribed about the sought pentagon. From here, we can proceed easily: it is only necessary to copy angle AOD = 2p/5 four more times.

How accurate is this procedure? To find out, we need only note that since triangles SAO and PGO  are similar, it follows that AS/PG = SO/OP = 4/1.  And since triangle  DAP is equilateral of side a, SP, the altitude of this triangle, has length a √3/2. Then, the tangent of angle SOA is equal to the tangent of angle GOP.  This tangent is easily calculated to be

PG/OP = ( a/8)/(a√3/10) = (5/12)√3.

Therefore, the sine of angle SOA is (5/73)√73 = 0.585.

This is, in fact, a good approximation, as it makes sin 36° = 0.585 instead of 0.587. [1]  Of course, 36° = (1/2)(360°/5).

## Albrecht Durer

Albrecht Dürer (1471-1528), considered the father of modern German painting, was also a great mathematical amateur. He wrote a book titled Unterweysung der Messung..., which deals with all sorts of geometrical problems. In this book he gives an exact construction of the regular pentagon, but he also gives an approximate construction which is quick to execute in practical drawing. (His book was aimed at artisans, stonemasons, etc., who cared more for simple procedures than for geometrical accuracy.)

This construction is shown below:

Take a fixed opening of the compass: AB = a. Draw circles of radius a, centered at A and B; let these circles intersect at C and D.  Then AB = AD = BD, as we know. Draw the circle of center D and radius DA. This circle will pass through B; let E be the point where it intersects CD, and let F and G be the points where it intersects the circles centered at A and B, respectively. Produce FE until it intersects at H the circle centered at B; produce GE  until it intersects at I the circle centered at A. Then the intersection K of the circles of radius a, centered at H and I, gives the fifth vertex of the pentagon.

How regular is this pentagon?  If it were exactly regular, all sides should subtend an angle of 360°/5 = 72° at the center of the pentagon.  Therefore, all angles at the vertices of the pentagon should be equal to 180°-72°= 108°(since each of them is the sum of two equal angles of [180°-72°]/2). In fact, we will see that angle ABH= 108°21’58” --- a little more than 108°, so that some other angle must be a little less than 108°. First note that  angle FBG = 90°; then, since  FG = 2a and GB = a, we have FB = a√3 Also,  angle DFE = 45°and angle DFC = angle DFA = 60° (F, A, C are collinear, as well as C, B, G and F, D, G ) , and therefore angle AFE = 15°. Since angle BFE subtends an arc of circle FABG equal to that subtended by angle AFE, we also have angle BFE= 15°.

We now use the law of sines for triangle FBH; this gives sin(angleBHF) =  √3 sin15° (observe that since angle BHF < 90°, there is no ambiguity). From here angle BHF can be found and then angle HBF; then, subtracting angle ABF = 30° from angle HBF, we find that angle ABH is approximately 108°22’, a value close to that of Pedoe [2].

Since angle BAI is equal to angle ABH, it too is a little greater than 108°. Also,  each of angles  BHK and AIK is a little greater than 107°, while angle HKI is a little greater than 109°... and still, this would be barely noticeable in our drawing.

Dürer does not warn the reader of his book that  this construction is approximate. (It is, also, executed with a “rusty” compass, i.e., a compass with a fixed opening.) In fact, he also quotes, as we said, the exact construction of the pentagon.

“Dürer’s interest in the construction of regular polygons is explained by the applications of geometry during the Middle Ages in Islamic and Gothic decorative design and, after the invention of guns, in the building of fortified towns. (It is curious that very few buildings  in history have been built based on the pentagonal shape. The Pentagon, near Washington, DC, is a notable exception.)” [2]

Summarizing this survey of Renaissance polygon constructions, we may say that both Leonardo da Vinci and Albrecht Dürer were great mathematical amateurs. “Leonardo wrote a lot about polygons; but it was Dürer, more than Leonardo, who transmitted to us the popular medieval constructions.”  [2]

## References

[1] Coolidge, J.L., The mathematics of great amateurs, Oxford, University Press, 1950.

[2] Pedoe, Dan, Geometry and the liberal arts, London, Penguin, 1976.

Acknowledgement: The author is thankful to an anonymous referee for comments leading to a simplification in the study of Dürer’s construction.