Compass Ø Straightedge Construction

To help with a personal project I needed to learn how to draw regular polygons with only a compass and straightedge. On 2010 August 15 I started an extensive search via the internet only to find out that there is no general or straightforward way to create regular polygons of any number of sides. Although there are several differing methods for creating three, four, five, six and seven sided regular polygons and a few others, there is no apparent process that will address the creation of any regular polygon. Also some regular polygons are deemed impossible to draw with only a compass and straightedge.

It is obvious that a circle is in essence an infinitely sided regular polygon, which therefore should lend itself to creating any regular polygon. So I set out on a personal quest to figure out an all-encompassing, perfect way to create any regular polygon with only a compass and a straightedge (and of course a good surface to write on).

On the morning of 2010 August 21 I was trying yet another permutation of the many circle and line combinations I'd been toying with over the previous days. When suddenly I made a simple and generally perfect discovery that, by drawing two circles double the diameter of the main circle then placing them along the lower left portion and extending a line through the intersection to a predetermined segment point then on through the upper right portion of the original circle, I could get a good measurement for the side of the target polygon. And with every regular polygon I've tried this method I've been able to get a fairly precise polygon. My goal was reached for I have found an all-encompassing, perfect technique that can, in fact, be used for every single regular polygon. Therefore I came up with the perfect name for this technique-Perfectagøn™. Although Perfectagøn won't replace CAD, it is perfect for creating fairly precise regular polygons when physically drawing with only a compass and straightedge.

An important aspect
of Perfectagøn is to use the compass and straightedge to
equally divide a line into the number of segments equaling the
number of sides that you wish your regular polygon to be. For
instance divide a line into five equal segments for creating a
pentagon. Dividing a line into equal segments can be a rather
tedious step but it is an essential aspect of Perfectagøn.
An alternative to creating a segmented line is to use graph paper
as a way to test the effectiveness of Perfectagøn. However
the specific goal is to be able to create any regular polygon
with only a compass and straightedge, therefore being able to
divide a line into equal segments is an imperative step using
Perfectagøn. The following pages present some compass and
straightedge standards that are required for using Perfectagøn
effectively, then the subsequent and necessary steps for utilizing
Perfectagøn to draw regular polygons with 3, 4, 5, 6, 7,
8, 9, and 13 sides and ultimately any regular polygon.