Ninth Wave Designs

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.

Pascal's Triangle

Pascal's Triangle (aka the Yanghui triangle in China) is one of those interesting bits of number theory that seems simple on the surface, but under examination proves to be quite complex.  The triangular grid begins with 1 with the other squares filled in with the sum of the two numbers above it. In the example below the green square with the number 2 is the sum of the yellow and blue squares above it both having the number 1 in them.  In this way the entire triangle can be filled in with numbers.

I constructed this example of a Pascal's triangle in the pages of my Moleskine Volant notebook with squared pages.  I use this Moleskine as a place to collect interesting notes and figures on numbers and number theory.

Click on above image for larger version.

The arrangement of numbers in the Pascal's triangle creates many fascinating relationships between numbers and patterns of numbers, the mathematical explanations of which are beyond my scope.  What interests me is the way that such a simple construct can result in so many compelling relationships. Among other mathematical gems, the Fibonacci sequence is revealed in the Pascal triangle, and the coloring pattern I used in this example is one that illustrates it.  If you add the diagonal according to color, first red, then yellow, then blue, and then green, the sums of these color-coded numbers results in the numbers in the summation series of the Fibonacci sequence.  Here's what you get:

• Red: 1
• Yellow: 1
• Blue: 1 + 1 = 2
• Green: 2 + 1 = 3
• Red: 1 + 3 + 1 = 5
• Yellow: 3 + 4 + 1 = 8
• Blue: 1 + 6 + 5 + 1 = 13
• And so on . . .

This adding of numbers along the shallow diagonal results in the numbers 1, 1, 2, 3, 5, 8, 13, . . . which is the number sequence made famous by Fibonacci (aka Leonardo of Pisa), a 12th century Italian mathematician.  This number sequence is a summation series, which means that you add the previous two numbers in the series together to get the next number in the sequence.  The next Fibonacci number after 13 is 21 (8+13) and so on from there.  The relationship between the numbers in this series is closely related to growth patterns in nature, especially in the arrangement of leaves on a stem or petals of flowers.  Again and again the numbers in the Fibonacci sequence show up in nature, in sunflowers, daisies, pine cones and even breeding rabbits.  It is a ratio of growth, where the next step forward is determined by the two previous steps, changing while maintaining a sense of sameness.  The ratio of the numbers in this sequence (i.e., 13/8) approaches the number Phi (1.6180339887...), also known as the Golden Ratio.  Phi is an irrational number, so it cannot be expressed as a fraction.  The ratios generated by neighboring Fibonnaci numbers then cannot be equated exactly with Phi, but they do hover interestingly close to the irrational, and serve at least as a tangible symbol to express something otherwise inexpressible.

All this and much more is behind this simple triangle of numbers.  If you are interested you can read articles from MathWorld on Pascal's Triangle, Fibonacci Numbers and The Golden Ratio.