Date: Wed, 14 May 97 17:43:52 CDT From: rusin (Dave Rusin) To: pshapiro@oz.sunflower.org Subject: Re: Constructing a Pentagon Newsgroups: sci.math In article <862974232.16174@dejanews.com> you write: >Does anyone know how to construct a pentagon using a compass and >straight-edge? Please e-mail me if you do. The proof that the regular n-gon is constructible by compass and straightedge only for certain n is due to Gauss; presumably there's a simple construction at least that old. All you need is a way to construct a 72-degree angle. This will do if you haven't received any other ideas: Start with any line segment OB. (A few cm in the middle of a sheet of notebook paper works out OK). (O will be a vertex of the pentagon). Drop a perpendicular bisector CAC' through OB so that AC and AC' both have the same length as OB. (A is the midpoint of OB) Draw the circle S at C passing thru O. (This will be the circumscribed circle for the pentagon.) Now get that angle: Draw enough of the circle at O with radius AO so you can find the intersection D with the segment OC. Draw a long line L perpendicular to CD at D. Draw enough of the circle at C with radius CC' so you can find its intersections E, F with L. Both angles OCE and OCF are now 72 degrees. The intersections E' and F' of S with CE and CF respectively give two more vertices of the pentagon. You can get the last two vertices G,H by intersecting S with arcs at E' and F' of length OE' (=OF'). As a check, the distance GH should also equal OE'. Of course, it's easier to get a pentagon simply by tying an overhand knot into a thin strip of paper and then pressing the whole thing flat! dave