Constructing the
H E P T A D E C A G O N
Young Gauss discovered that the only regular polygons constructable with a prime number of sides, were polygons with a number of sides equal to a Fermat Prime. A Fermat prime is a prime number of the form Fp=22n+1. There are only a handful of these primes in existence. Here is a list of the first few.
Out of all those, only the first 5 are prime. 4,294,967,297 is divisible by 641.
Believe it or not, there are actually constructions for the 257-gon and the 65,537-gon. But I wouldn't waste my time drawing those.
- Draw circle O and points A and P so that OA is perpendicular to OP.
- Find point B on line OA so that OB is 1/4 of OA (Bisect it twice).
- Draw a line through BP. Quadrasect angle OBP so that angle OBC is 1/4 of angle OBP.
- Find point D on line OP extended, so that angle DBC is 45°.
- Find the midpoint H of segment DP.
- Draw a circle on line OP with diameter DP. Circle DP intersects line OA at F.
- Draw a circle at C with radius CF. Circle C intersects line OP at point G and H.
- Draw a perpendicular line from points G and H that intersects Circle O at points P1 and P2.
- Points P, P1, and P2 are three points of the Heptadecagon.
© 2002 Robin Hu