Braid Theory-The wondrous life of corn rows

by Bernat Guillén Pegueroles

This semester we had the great pleasure of giving a workshop on braid theory to a group of W.E.B. DuBois scholars. Our goal from the beginning was to introduce a mathematical concept, in this case braid theory and group theory, in a culturally relevant way.

As we saw in a previous blog post, braids are, in a simplified, abstract way, mathematical objects that we can concatenate and invert and therefore they form a noncommutative group. However, in the real world, braids have a history and a meaning beyond “simply” weaving hair strands. They mean tradition, parental/fraternal relationships, in some cases like cornrows they mean a fight against oppression. How do we explain that, and how do we go from the real world to the abstract world of mathematics? 

During the first part of the workshop, we discussed the history of cornrows as a way to express identity and a way to escape slavery. We started looking at different patterns of cornrows and how, despite them meaning something different to each of us, we could find some commonalities, among them:

  1. The tesselation of the scalp formed by grabbing the different strands of hair.
  2. The fact that our heads are spherical and therefore a cornrow pattern needs to be done on a sphere.
  3. The repetition of the braid with less and less hair, forming something like a fractal.
  4. The very act of braiding, which can be summarized in a few instructions being repeated over and over.

Even in a purely technical approach, these four aspects aren’t the end all be all of braiding (for example, a dutch braid and cornrow have the same instructions but the way hair is added at each step is different). However each of them can lead to deep discussion and exploration of mathematical concepts. In our workshop, we chose to focus on the instructions for braiding and the relation to group theory.

To aid in the exploration of these concepts we used several tools: First of all, we have our friendly mannequin head that some expert students used. We also provided yarn threads for an easier work, and finally, we made a small javascript webapp for trying different braids.

With 3 strands we tried to braid the following (try them in the webapp!) combinations of instructions:

  • 1 over 2 ([0, 1]), then 3 over 2 ([2, 1]), then repeat.
  • 1 over 2, 1 over 2, 2 under 3, 1 under 2, 2 under 3, 1 over 2, then repeat.

In the webapp these two look pretty different, however the astute reader (and anyone who actually tries to braid these with hair or yarn) will notice that they are the same. Indeed, the trio of instructions “2 under 3, 1 under 2, 2 under 3” is equivalent to “1 under 2, 2 under 3, 1 under 2”, and “1 over 2, 1 under 2” is the same as doing nothing. But what do we mean by “equivalent”? In our case, the most natural way to think about it is, well, that if we braid them physically we will get the same braid (although it will be impossible to get a tight braid using the second set of instructions). In an abstract sense, we could imagine “combing” the second braid and the first braid until all we have is a big mess at the end and getting the same mathematical object. That is called the “Artin form” of a braid, and computationally it is doable and lets us compare two braids and see if they are the same. In general, in group theory, knowing if two objects are the same is not trivial (it is known as the “word problem”). 

After seeing these equivalences of instructions with 3 strands, we tried to find non-equivalent braids with 5 strands. Finding 5-strand braids that were not equivalent and at the same time were doable by a single person was a hard task, but we were able to find two different 5-strand braids. 

  1. 2 over 3, 1 over 2, 4 over 3, 5 over 4.
  2. 3 over 2, 1 over 2, 3 over 4, 5 over 4.

Can you find another one? (use the webapp or yarn, we do not recommend trying it on your hair!)

To end the workshop, we had a very productive discussion on the importance of culturally relevant mathematics. Is it a form of approaching a subject and make it more relatable? Are we erasing aspects of our culture when we abstract them and dissect them? See the presentation here.

Braid Theory/Tesselations

Tesselations are another interesting mathematical concept, related to how well can we pack objects in an area, for example, squares and triangles. Another typical tesselation that happens naturally are hexagons. In certain braids where we pick bunches of hair a hexagonal pattern would appear naturally. Mathematicians are still now trying to figure out possible tesselations for different shapes, and maybe some of those could find their way onto a hairstyle!

Braid Theory spherical geometry

Our possible hairstyles would change a lot if our head was flat! Cornrows (attached to the head) need to follow certain rules of spherical geometry. For example, there is a mathematical theorem known as “hairy ball theorem” (n.a. The theorem;s name has sexual innuendo but that’s how it’s called…) that says that you can’t comb the hair on a sphere without getting a “cowlick” or a “whirl”. Or course our heads aren’t spheres full of hair, the face in particular is (usually) hairless, so we can do cornrows like this.