Visual artist Owen Schul and mathematician Satyan L. Devados collaborated on a project with the goal to intertwine their disciplines in a direct manner. They decided to work with tree spaces, a mathematical concept related to the geometric relation between different branching configurations. The concept appear in in numerous areas of mathematics, including algebraic topology, enumerative combinatorics, geometric group theory, and biological statistics.
The result of the collaboration is a work titled Cartography of Tree Space, involving acrylic, watercolor, and graphite on 108cm x 108cm wood panels. One of the pieces, titled “Underground”, is shown below
Sketch, computer illustration
In the end, the mathematician had more to say about the art, and the artist had more to inquire about the mathematics. Three visual pieces were produced, forming at triptych, with the following open mathematical questions:
1. What is the least number of associahedral duals needed to cover the tree space?
2. If we fix a caterpillar tree and relabel the leaves, we obtain the permutohedral dual polytopes. What happens when we fix another tree types and relabel its leaves?
3. The braid arrangement naturally appears as a scaffolding for tree space with rooted trees with five leaves. Does this naturally extend to higher dimensions?
4. Expanding on the notion of a map, what can be said about distances between two points in tree space under certain conditions (such as walks restricted to tree types or alternate notions of discrete metrics)?
“The particular object of our study is a configuration space of phylogenetic trees, originally made famous by the work of Billera, Holmes, and Vogtmann. Each point in our space corresponds to a specific geometric, rooted tree with five leaves, where the internal edges of the tree are specified to be nonnegative numbers. From a global perspective, this “tree space” is made of 105 triangles glued together along their edges, where three triangles glue along each edge. This results in 105 distinct edges, and 25 distinct corners.” – Prof. Devados