Constructible Polygon - Compass and Straightedge Constructions

Compass and Straightedge Constructions

Compass and straightedge constructions are known for all constructible polygons. If n = p·q with p = 2 or p and q coprime, an n-gon can be constructed from a p-gon and a q-gon.

• If p = 2, draw a q-gon and bisect one of its central angles. From this, a 2q-gon can be constructed.
• If p > 2, inscribe a p-gon and a q-gon in the same circle in such a way that they share a vertex. Because p and q are relatively prime, there exists integers a,b such that ap + bq = 1. Then 2aπ/q + 2bπ/p = 2π/pq. From this, a p·q-gon can be constructed.

Thus one only has to find a compass and straightedge construction for n-gons where n is a Fermat prime.

• The construction for an equilateral triangle is simple and has been known since Antiquity. See equilateral triangle.
• Constructions for the regular pentagon were described both by Euclid (Elements, ca 300 BC), and by Ptolemy (Almagest, ca AD 150). See pentagon.
• Although Gauss proved that the regular 17-gon is constructible, he did not actually show how to do it. The first construction is due to Erchinger, a few years after Gauss' work. See heptadecagon.
• The first explicit construction of a regular 257-gon was given by Friedrich Julius Richelot (1832).
• A construction for a regular 65537-gon was first given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript. (Conway has cast doubt on the validity of Hermes' construction, however.)