# Cartography of Tree Space

Over six years ago, I wrote a post on ways in which mathematics (at  a research level) and art (at a gallery level) can intersect today.  The problem is that in our enlightened world, the work of the mathematician and the visual artist are not only viewed as incompatible, but held in tension. In my post on danger, I alluded to my desire (and fear) in pursuing this venture.

During my sabbatical at Stanford last year, I spent a part of my time having coffee with Owen Schuh, an artist in the SF bay area. The goal of this collaboration was 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 should be a true collaboration, with the mathematician involved in the drawings and the artist involved in the mathematics.

In the end, we created a triptych of works, titled Cartography of Tree Space, involving acrylic, watercolor, and graphite on 108cm x 108cm wood panels.  Our work was picked up for an inaugural gallery showing in Germany by Satellite Berlin.  One of the pieces, titled “Underground”, is shown below.

The particular object of our study is a configuration space of phylogenetic trees, originally made famous by the work of Billera, Holmes, and Vogtmann.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.  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, where Owen and I 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.