Artist Statement: The Mathematics of Irish Dance

In first grade, I was jumping around waiting for my teacher to check my work, and she told me I should look into Irish dance. The K-8 school had several dancers that competed internationally—like with actual Irish citizens—and some even won. While I never got past advanced beginner (because I didn’t like to go to competitions), Irish dance was so important to me in a lasting way.

Along with the dance lessons, we also learned about the Celtic culture we were practicing from Breda Yeates, an Irish immigrant. I’m a McElligott on my mom’s side which is very obviously an Irish name but I didn’t feel any connection to the culture in my own right; I never considered until now that my great-great-great-so on relatives might have practiced this dance that Yeates taught us. The dances developed when the Irish were enslaved and not allowed to celebrate their culture. The stiffness of the dancers made their movements unrecognizable as dance (or so I was told). The dancers’ hallmark look includes tight curls, a stiff paneled dress with elaborate embroidered decorations, and poodle socks. The Celtic knots are thought to have been popularized by the spread of Christianity but the designs that predate Christianity are hard to trace if they don’t depict an animal or recognizable symbol.

Celtic knotwork is often one continuous looping ribbon with distinct repeating patters. The circular and square knots are usually centered on a quadrant system and are exactly the same in each quadrant rotated. The unorthodox shaped ap knots are reflections over an axis. On the dresses, the skirt is paneled with the most elaborate design on the front of the dress. For the sake of design, most dresses use knots with 3 or more colorful ribbons. The adornments add a uniqueness to the stage. No two girls (from different schools) wear the same dress in the competitive stage.

For my creative project, I wanted to design a pattern that could be found on a typical dress. The way that Celtic knots curl on themselves and cross over and under in a specific pattern. By far, the hardest part of the design process was making the crossings evenly numbered; often, I would find myself at an impasse to determine what to do next because if I went one way I would put two “overs” or “unders” in a row which throws off the whole design. The design aspect is so intrinsically tied into mathematics, that there are computer programs specifically for designing Celtic knot patterns. Unfortunately, many links to the academic work with these programs are broken or inaccessible to the public but one free program, Knots3D is available on Windows. Unfortunately, I have a Mac and couldn’t run the program but from test pictures supplied on the download site, I was intrigued by the intricacy of the knots and their weaving. The nature of computer generation, of turning a 2D image into a 3D image, and furthermore, of a repeating loop image requires knowledge of topological spaces and geometry.

In my initial development of the knot, I focused too much on the drawing of the physical ribbon instead of controlling or containing the negative space. I was unsatisfied with the design but then, read up on the geometric patterns in Celtic knot work. Each knot has a “skeleton” with vertices in the negative spaces. Then, the midpoint of the side lengths is where the ribbon crosses over or under. The instructor describes the rule for crossing over or under akin to the right hand rule: if you were to physically pick up the ribbon to cross the lines, you would always cross left over right as you follow the path from am arbitrary point. The instructor pointed out that, ignorer to make a large knot, you first have to find a small knot pattern you like and tile it out through the entire area.

Regarding some basic knot theory from introductory documents from class, the definition of a knot is a (closed) tangled string. While the art form exists in a 2D place, the assumption for knots is that they are three dimensional. Without a loss of generality, we can view the 2D image as if it were 3D in order to apply the knot theory terms. Already, I have mentioned that Celtic knots are alternating knots. We can see that the classic Celtic trefoil is a knot and is the simplest nontrivial knot. At first glance, I would have assumed most Celtic knots are trivial but after revisiting some of the knot theory documents, I think that the knot in the outer ring of my design is nontrivial. There are sixteen figure-eight knots (thank you girl scouts) which would not pull apart. As for the inner design, I have no intuition about whether or not it would pull apart by looking at it alone. I presume that the majority of celtic knots are nontrivial because they develop so much upon themselves. The trefoil knot is the simplest knot topologically as mentioned previously, and the Celtic motif of the trefoil appears within much of the art that decorates dresses.

This project was a lot of fun; the nostalgia for me was so inspiring. I hope that in a few years, I can restart and join an academy. The class was asked what we would have changed about the project if we had chosen our topics at the end of the class instead of the beginning, and I will say that while I was inspired by the images in the knot theory papers we read for my topic, I would have decided to focus on the music and dance aspect of Irish culture. The difference between Reels, Slip Jigs, Light Jigs, Treble Jigss, and the Blackbird are so subtle but so profoundly tied into math. I think back to the only dance I remember in full:

O’er-2-3, o’er-2-3, o’er-2-3, o’er-2-3, hop-2-3-4-5-6-7, hop-step-step, o’er-2-3. Repeat. Hop- step-step, hop-step-step, hop-2-3-4-5-6-7, o’er-2-3-4-5-6-7, hop-step-step-and-a-point-hop-back. Repeat.

There absolutely has to be some fun mathematical connection in there!

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