Around 1976, I became enthralled by Eschers patterns. I purchased many books on his art, hoping to find his method, but as far as I could find, there weren't any published. I eventually developed a system on my own by which I could create patterns at will. I have since created about thirty different patterns of my own which include elephants, eagles, angels, bisons, marine reptiles, leaves, gargoyles, dragons, etc. This system I developed, I believe, is the same, or very similar to what Escher employed, because I have since seen books that show the underlying gridwork, and graduation from the basic square grid, to the final curves of the finished pattern. This obviously doesn't require a computer, just some graph paper, a pencil with eraser, and some basic understanding of patterns along with a minor amount of artistic creativity. The graph paper can be a basic square grid, or even triangular or hexagonal, or any other regular gridwork. In a nutshell, simplify your animal or plant shape to its very basic shape, using the grid to block it out, and using common dimensions as much as possible, make a pattern of that block shape. By understanding rotation point s-curves,etc, the block can be gradually transformed into curves. The entire system is too lengthy to go into here, but you may see some of my results on my Alfredman/facebook page which is open to the public. The page is Identified as comic strip and creative writings. I have included some of my patterns on that page,and if any interest is shown, I could go into greater detail on the subject.As far as the circular design works, a grid can be formed using a compass and a straightedge, and a previously designed pattern can be applied to the resulting grid. A basic example of this is also included on my page.
In addition, as to the question of whether this fits as a mathematical query, it does. Patterns are a form of math. Not all math directly concerns numerical computations, geometry is math, as is music. Math can be intuitive, and creative. Check out Eschers book called Fantasy and Symmentry, by Caroline H. Macgillavry, Abrams press. This book contains the periodic drawings of M C Escher, and is used to illustrate how crystals form, basically a crystalography textbook. It explains some various elements of pattern formation, translation, rotation, mirroring etc. In MY patterns, as in music, a form of Jazz can be incorporated, by including a bit of randomizing. By this I mean that while the shapes have a definite structure and regularity, The colors do not have to conform to a regular pattern, although they certainly can. Escher also does this by way of morphing his shapes within a pattern to cause change. Much like visual music.