Circles Inside, Circles OutsideA regular polygon is a 2-dimensional closed shape, consisting only of straight lines, that is both equilateral and equiangular. This means all the sides have the same length, and all angles have the same measure. A regular polygon can either be convex, with all sides “bulging outward,” or shaped like a star (with a mixture of concave and convex sides). The most common examples of convex polygons are the triangle and square, and the most common examples of star polygons are the pentagram and hexagram.
Every regular polygon can be both inscribed and circumscribed by a circle. The circle that is tangent to all sides of the polygon is said to inscribe the polygon. This unique circle is called the incircle, and its radius is referred to as the inradius (r) or apothem of the polygon.
|Incircle with inradius “r”|
|Inradius r and Circumradius R|
Dividing both sides of the equation by r gives us a generalization for the ratio of circumradius/inradius:
Table 1 – Linear and Square Polytonic Ratios
|Table 2 - The First 12 Encapsulated Polygons|
A Look Ahead...As you might guess, many of the above rings compose key elements in the design of the crop circles. Rings can be combined with each other, producing a multiplicative effect of the polytonic ratios which resembles the way musical diatonic ratios are combined.There is much to be explored here, but for now I will leave you with a polytonic representation of the perfect fifth formed with two hexagons and a heptagon. It's actually not a perfect fifth ratio (1.50), but it's close (1.48)
|A Polytonic Perfect Fifth|