# Chris McCarthy

This artwork came out of my dissertation which involved proving theorems related to the Hilbert Metric. The dissertation, "The Hilbert Projective Metric, Multi-type Branching Processes and Mathematical Biology: a Model of the Evolution of Resistance," is about the mathematics underlying some of the models that are helping us to understand the rate at which diseases become resistant to treatment. To better understand the Hilbert Metric, I applied the metric to the 2 simplex (i.e., to a triangle) and wrote a computer program to draw lines uniformly far apart w.r.t. the metric. The resulting grid tiled the simplex with Hilbert Metric equilateral triangles and circles (the hexagons). I added colors. The results are my artwork.

The Hilbert metric is a special way to define the distance between points. The Hilbert metric applied

to the interiors of these two triangles results in a hyperbolic non-Euclidean geometry.

With respect to this hyperbolic geometry the lines coming out of these vertices are parallel (non-intersecting). To our eyes, those lines do not look parallel, unless perhaps we see them as being parallel lines going off into the distance, drawn in perspective.

As we look at the various patterns formed by the lines and the coloring, our mind tries to make sense of the complexity; our perception shifts between seeing hexagons (which are circles with respect to the Hilbert metric), parallel lines, and the occasional parallelepiped.