Constructing Polygons

[Sparky]

How do you construct an n-sided polygon using only a straight-edge and a pair of compasses? 3, 4 and 6 -sided are easy enough... Should I remember this from school?

[gavinclarkeu]

(Taken from http://home.wanadoo.nl/zefdamen/Constructions/Constructions_ en.htm) "Not all patterns can be constructed by (strict) ruler-and-compass rules. For instance, a regular heptagon � 7-sided polygon � can not be constructed in this way. From a mathematical law, it has been known, that a regular n-sided polygon can be constructed by ruler-and-compass, if and only if n is a finite product of different numbers from the set of powers of 2 (2, 4, 8, 16, ...) and primes of Fermat (3, 5, 17, 257, 65537, ...). According to this law, regular n-sided polygons are constructible for n = 3 (equilateral triangle), 4 (square), 5 (pentagon), 6 (hexagon), 8 (octagon), 10 (decagon), 12 (dodecagon), 15, 16, 17, 20, ...; however, n-sided polygons with n not present in this series, are not constructible (n = 7 (heptagon), 9 (nonagon), 11 (undecagon), 13, 14, 18, 19, ...)." And no, you shouldn't really remember anything more complex than 3, 4 and 6 sided shapes from school. The last thing I studied on my Maths degree course was why you can't make certain ruler and compass constructions, and it involved using pretty much every thing I had learnt over my entire degreee.

[Sparky]

Thanks for that; very interesting. What prompted the question was finding myself unable to layout a large pentagon, and finding none of my colleagues could do it either! As a 5-sided figure is possible according to your rules, do you have any idea how to do it?

[Sparky]

Found it! It's a golden ratio thing. Easy when you know how!