Where does mathematics live?

A Scientific American article brought mathematical knitting to my attention once again, and within the article was a link to Bridges, an organization which oversees the annual Bridges conference that explores the connections between art and mathematics.  Following the link to their 2013 Conference, I found their Short Movie Festival. I’ve watched a number of these short films whose actions explore things like topological objects, fractals, geometric symmetries, and groups. These ideas are presented in visual works, movement performances, as well as in nature.  I thoroughly enjoyed all of them and I found myself asking the question, “So where does mathematics live, really?”  For example, associated with the short movie Dancing Braids by Ester Dalvit is the following note:

Braids can be described as configuration space of points in a disc. These can be visualized as dances: the positions of each dancer are translated into a strand of the braid, the time into a spatial dimension.
This movie is a small part of a long video about braid theory which is available here.

Or with Susan Gerofsky’s film, The Geometry of Longsword Locks, is this:

In traditional English longsword dancing, a team of dancers makes intricate moves while joined together by their wooden or metal ‘swords’. An impressive element of the dance is the variety of traditional geometric, symmetrical sword locks (often stars) created through the movements of all the dancers. The film showcases a longsword dance and the locks created by the physical algorithms of the conjoined dancers’ movement. After showing the dance, questions are offered to spark mathematical explorations by secondary or post-secondary students. These questions include topological and geometric ideas about crossings, angles and edges, and logic-related questions about categorizing lock types and discovering whether new locks could be created through analysis of the physical algorithms that create them. Slow-motion and repeated views help learners explore this rich source of geometry.

On the Simons Foundation website is yet another short video on Change Ringing.

The art or “exercise” of change ringing is a kind of mathematical team sport dating from the 1600s. It originated in England but now is found all over the world. A band of ringers plays long sequences of permutations on a set of peal bells. Understanding the patterns so they can be played quickly from memory is an exact mental exercise which takes months for ringers to perfect. Composers of new sequences must understand the combinatorics of permutations, the physical constraints of heavy bells, and the long history of the art and its specialized vocabulary. Change ringing is a little-known but surprisingly rich and beautiful acoustical application of mathematics.

According to The North American Guild of Change Ringers,

the earliest record we have of these is from 1668:Tintinnalogia: or, The Art of Ringing. Wherein Is laid down plain and easie Rules for Ringing all sorts of Plain Changes. Together with Directions for Pricking and Ringing all Cross Peals; with a full Discovery of the Mystery and Grounds of each Peal.

Perhaps we can ignore the effect of the subject tabs we learned to put in our notebooks when we were young and ask some new questions.  Do these visual and musical experiences represent mathematical concepts or are mathematical concepts actually exploring the elements of these visual and musical experiences?  I lean in the direction of the latter.  In fact, I would argue that one of the major functions of the brain is to integrate experience. The dances shown in two of the short films are, in some sense, an impulsive integration of the things we hear, see and hold, that become shapes within the inherent unity of our experience.  It can be said that mathematics ‘picks up’ on this impulse, and further explores that unity by investigating the paths that are born of these more impulsive harmonies.  Mathematics is then distinguished by its symbolic representation of the flow of patterns created by our living – by the visual, and audio structures that the senses build, as well as the cognitive structures that develop with them. Braids and knots are two of the oldest human impulses to create new experience, and they are two of newest objects investigated by mathematics, which then further integrates them into what we know of number and quantity and symmetry.

There is one more thing, not so much related to the theme of this post, but worth a look.  One of the short movies in the Bridges short movie festival is a poetic approach to the words real and complex that I think is really nicely done.  You can go to it directly here.