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Mathematics may be likened to a large rock whose interior composition we wish to examine. The older mathematicians appear as persevering stone cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well placed internal charges.

In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

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Approximate Construction of Regular Polygons: Two Renaissance Artists

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).