A Rabbit Hole between Topology and Geometry

Topology and geometry should be very closely related mathematical subjects dealing with space. However, they deal with different aspects, the first with properties preserved under deformations, and the second with more linear or rigid aspects, properties invariant under translations, rotations, or projections. The present paper shows a way to go between them in an unexpected way that uses graphs on orientable surfaces, which already have widespread applications. In this way infinitely many geometrical properties are found, starting with the most basic such as the bundle and Pappus theorems. An interesting philosophical consequence is that the most general geometry over noncommutative skewfields such as Hamilton's quaternions corresponds to planar graphs, while graphs on surfaces of higher genus are related to geometry over commutative fields such as the real or complex numbers. 1. Introduction The British/Canadian mathematician H.S.M. Coxeter (1907–2003) was one of most influential geometers of the 20th century. He learnt philosophy of mathematics from L. Wittgenstein at Cambridge, inspired M.C. Escher with his drawings, and influenced the architect R. Buckminster Fuller. See [1]. When one looks at the cover of his book “Introduction to Geometry” [2], there is the depiction of the complete graph on five vertices. It might surprise some people that such a discrete object as a graph could be deemed important in geometry. However, Desargues -point -line theorem in the projective plane is in fact equivalent to the graph : in mathematical terms the cycle matroid of is the Desargues configuration in three-dimensional space, and a projection from a general point gives the configurational theorem in the plane. Desargues theorem has long been recognised (by Hilbert, Coxeter, Russell, and so on) as one of the foundational theorems in projective geometry. However, there is an unexplained gap left in their philosophies: why does the graph give a theorem in space? Certainly, the matroids of almost all graphs are not theorems. The only other example known to the author of a geometrical theorem coming directly from a graphic matroid is the complete bipartite graph , which gives the -point -plane theorem in three-dimensional space; see [3]. It is interesting that both and are minimal nonplanar (toroidal) graphs and both lead to configurational theorems in the same manner. In this paper, we explain how virtually all basic linear properties of projective space can be derived from graphs and topology. We show that any map (induced by a graph of vertices and edges) on an