Geometry
It has been claimed that there are features in the design and distribution of crop formations that cannot be purely random.  For example, by drawing lines on a map between the location of formations, many of the subtended angles so formed are identical - too many to be a coincidence.   Fortunately, there is a simple formula that we can use to test this theory.

Subtended Angles
In the left hand diagram below, I have represented seven randomly distributed geographical locations as red dots and labelled then A to G.  There is no particular reason for choosing seven.  All of the following discussion applies equally to any number of points.  You will see that the lines from B and C subtended an angle at location A.  This is the subtended angle at A from points B&C and we identify this angle as BAC.  Lines from all the remaining points create a range of subtended angles.

We now need to derive an expression for the total number of these subtended angles at A.   If we use 'n' to represent the number of geographical locations (7 in this example), we can construct a table n-1 high and n-2 wide containing all of the possible angles that can be subtended at A from the remaining points (B to G).   You can see that there are as many filled cells in the table as there are unfilled.   By representing the results in this way, it is possible to derive the maximum number of subtended angles at A by simply dividing the number of cells in the table by 2.   In other words, no matter how many points there are, the formula for the number of possible subtended angles at any one location will be half the number of cells in the corresponding table:
                        ½ _* (n-1) _* (n-2)
However, this formula only gives the number of possible subtended angles at any one location.  By multiplying it by n, we get the general formula for the total number of possible subtended angles from all 'n' locations.

    The total number of possible subtended angles = ½ _* n _* (n-1) _* ( n-2)

Now, figures vary for each year but, assuming that 2004 was typical, in July there were at least 20 crop formations in Wiltshire alone (from the Crop Circle Connector web site).  Over the whole season, there were at least 36.   Substituting these figures into our formula we get:

For 20 formations, there were: ½ _* 20 _* 19 _* 18  = 3420 possible subtended angles.

For 36 formations, there were: ½ _* 36 _* 35 _* 34 = 21,420 possible subtended angles.

As you can see, there would have been no difficulty in finding identical angles simply because there were so many to choose from.  The subtended angles considered were alway less than 180 degrees.  So, for just twenty formations, there would have been an average of 19 identical angles for each of the possible 180 degrees (3420 divided by 180) and, for all 36 formations, there would have been an average of 119 identical angles for each of the possible 180 degrees (21420 divided by 180).

Some pundits used to become quite excited when they managed to find half a dozen similar - not even identical - angles.  Yet, as you can see, there were obviously a great many more identical angles that they could have found - had they looked more closely.  

Complex Geometry
Over the years, the complexity of formations has increased and, inevitably, so has the time taken to create them.   One way of achieving greater complexity within the timescale is to adopt designs that can be created by a step-and-repeat process.   At the turn of the millennium, numerous very complex designs appeared that were clearly taken from a book entitled "Mosaic and Tessellated Patterns" by John Willson. 

These are just some of the designs from the book that were
recreated as crop formations.

These designs manage to create striking optical illusions by simple repetitive steps - for example, drawing equally spaced radial lines or triangles and alternately filling in the resulting shapes.  

None of these patterns have any mathematical or religious significance.   On the cover of Willson's book, they are described as "op art" designs.   Yet, some avid croppies would brook no suggestion that they were anything less than mystical symbols, despite the fact that the publication from which they were taken was described as a colouring book.