In our enlightened world, the work of the mathematician and the visual artist are not only viewed as incompatible, but held in tension. The goal of this collaboration is to engage mathematics and the visual arts in a direct manner, with concrete outputs, that does not insult either field. In other words, new mathematical questions need to be formulated and new artworks need to be produced for the success of this venture. Our work is a true collaboration, with the mathematician involved in the drawings and the artist involved in the mathematics.
The particular object of our study is a configuration space of phylogenetic trees. 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 are glued along each edge. This results in 105 distinct edges and 25 distinct corners. This space of trees appears in numerous areas of mathematics, including algebraic topology, enumerative combinatorics, geometric group theory and biological statistics.
Although the space is only two-dimensional (made of numerous triangles), the natural world for this tree space to inhabit is in four dimensions, where the full symmetry of its structure will be made transparent. Our goal was not to describe the space in mathematical terms. Instead, we wanted to describe what it feels like to live in this tree space, to inhabit it as a world like any other world. To this end, we use the world of cartography and map making to invite the viewer to understand tree space.
The collaboration took place during an 18-month timeframe, from September 2013 until February 2015. Roughly, the first six months were spent in understanding the goals of the project and choosing a point of collaboration. The second six months were spent at coffee shops and studios, we would meet and go through both mathematics and sketches. The final six months were focused on crafting extreme details and formulating a unifying vision to the project.
In all of this, the process behind the mathematics and the art was quite similar. Ideas were conjectured, tested and evaluated, both visually and analytically. And there was a sense of incredible freedom to explore these worlds, with a strong instinct guiding the collaborators as to the right road to pursue.
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)?
Professor John Littlewood, a renowned mathematician during the early part of the 20th century, expressed the following duality between images and theory:
A heavy warning used to be given that pictures are not rigorous. This has never had its bluff called and has permanently frightened its victims.
The exhibition is on view at Satellite Berlin
through 26 April, 2015.