September 07, 2005
Knotwork Theory
As a professional illustrator I have specialized for many years in Celtic design. One of my favorite aspects of this intricate style is the interwoven bands of knotwork. It took lots of practice before I became comfortable with the underlying pattern for constructing the more complicated knots, but eventually I became familiar enough (through much repetition) with the geometric principles that aid in creating good designs. These knotwork patterns all begin with a system of grids, so Moleskine notebooks with gridded pages have always been a favorite place of mine to work out the details of a knotwork design. Below is the frontispiece to my Knot Theory notebook, a large size Moleskine Volant with squared pages.
It was only quite recently that I discovered this special branch of mathematics devoted to the study of knots, called, surprisingly enough, Knot Theory. It is a subsection of mathematical discipline of Topology which brings us such novelties as the Klein Bottle. In Topology mathematicians study the properties that do not change through deformations like stretching and twisting, but cutting and tearing are strictly not allowed. To a Topologist a doughnut is the same as a coffee cup, a cube equal to a sphere. This would make for an interesting parallel universe if the laws of matter adhered to these topological principles, and additionally a great plot structure for a possible episode of Farscape (also in a parallel universe, one where this series wasn't canned!).
Knot Theory is a relatively new mathematical study, just a little over 100 years old, and a place where new discoveries and proofs are still able to be found, which apparently can't be said about many areas of mathematical study. Imagine my surprise when I realized that the drawings and designs I spent so much time creating might actually be representations of cutting edge mathematical discovery! Okay, so maybe this isn't as thrilling to you as it is to me, but I think it is worth mentioning anyway. Any time I find something that is both mathematical and beautiful I experience a particular kind of thrill. I am not musical, but I imagine that it might compare to the experience of a musical composer - the connection between the mathematical qualities of the musical notation on paper to the full emotional experience of hearing the music performed.

Pages from my Knot Theory notebook - click for larger image.
What Knot Theory offers me is a new way of understanding these designs that I have been creating for years - a contemporary filter through which to look at a centuries old creative process. The academic community involved in the study of Celtic art agrees unanimously that there is no evidence that certain types of knots were ever used as particular symbols. The absence of archaeological proof does not mean that particular symbolisms were not applied, only that we can't know what they were exactly. Through the mathematical mindset of Knot Theory it is possible to see parallels between the ideas present in number theories and symbolisms and the particular qualities possessed by certain knots. From what little I have learned about the topological approach to understanding knots I find already that it offers a compelling way to begin thinking about the original symbolic significance these knots may have been intended to express - even if it means that I have to see a coffee cup when what I am really looking at is a doughnut.
September 07, 2005 in MOLESKINE, Mathematics | Permalink
