# Robert Franzosa

Professor of Mathematics
University of Maine
Orono, Maine, USA

I have long had a fascination with the "rubber-sheet geometry" that is the field of topology. The visual and

multidimensional nature of the subject stretches the imagination to see and comprehend the otherwise unseen.

I enjoy working with wood as a medium for both artistic creativity and practical construction. With barks, burls, veneers, and boards, I create works where the natural patterns of the grain and structure of the wood combine to produce satisfying form and function.

It is natural that the topology of my imagination would combine with the topology of wood to create a work of mathematical art.

The Maximum Genus of the Complete Graph on Ten Vertices is Eighteen
45 x 45 x 2 cm
Birds-eye maple veneer, stained and finished with tung oil
2017

The flowing birds-eye maple grain, in contrast to the rigid, symmetric, laser-cut depiction of the polygon and vertices, hints at the topological freedom permitted in deforming the polygon and gluing the edges to obtain the embedding of a graph in a compact surface.

Gluing the 90-gon's vertices and edges according to the numbering yields the complete graph on 10 vertices. Including the polygon interior, the result is an embedding in a compact orientable surface with 10 vertices, 45 edges, and 1 face. The Euler characteristic and genus relationship V-E+F = 2-2g yields g = 18. Thus the embedding is in an 18-hole torus. Since the graph's complement is the single open-disk polygon interior, this is a maximal 2-cell embedding.