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).