The following is a summary of the starting point for my crop circle research. I realized early into my research in 2010 that someone else had already noticed the patterns I was seeing in the crop circle designs. His name was Professor Gerald Hawkins, and he was a pioneer in the study of the geometric properties that appeared in these designs. As an astronomer, however, he should have left the theoretical mathematics to mathematicians, and his "crop circle theorems" are nothing more than the application of a well known formula I will explain below. That being said, I am very grateful for the work of Professor Hawkins, because not many credible scientists even take the subject seriously.
Circles Inside, Circles Outside
A regular polygon is a 2dimensional 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” 
Likewise, a circle can be drawn around the polygon, passing
through all of its vertices, and this circle is said to circumscribe the
polygon. This is referred to as the circumcircle, with a radius called the
circumradius (R).
The formula for the
circumradius (R) of any polygon (“ngon”) can be expressed in terms of its
inradius (r) and number of sides (n) as follows
Suppose we want to compare the size of the circle that nestles the interior of the shape with the circle that caresses the outside.
Dividing both sides of the equation by r gives us a generalization for the ratio of circumradius/inradius:
Inradius r and Circumradius R 
Dividing both sides of the equation by r gives us a generalization for the ratio of circumradius/inradius:
This
formula tells us the ratio of circumradius/inradius is constant for each regular polygon, and
based on a periodic function (cosine) that is used to describe light and sound
waves. Equivalent ratios are produced by looking at circumference or diameter,
because they are both proportional to the radius, which means they grow at the
same rate.
The
ratio of areas of the circumcircle
and incircle form a second set of ratios that is the square of the first. The formula to calculate these values is the square of the circumradius/inradius formula above.
I have named the
ratios defined by these two formulas as the polytonic ratios, and polygons surrounded by circles as encapsulated polygons. Each shape has
two interval ratios that are unique to that polygon and independent of scale. The
values for both the linear and the square polytonic ratios are given in the
table below. The red bold ratios are the ones referenced in Gerald Hawkins’ Theorems.
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 
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