Hindawi Publishing Corporation ISRN Geometry Volume 2013, Article ID 379074, 9 pages http://dx.doi.org/10.1155/2013/379074 Research Article A Rabbit Hole between Topology and Geometry David G. Glynn CSEM, Flinders University, P.O. Box 2100, Adelaide, SA 5001, Australia Correspondence should be addressed to David G. Glynn; david.glynn@linders.edu.au Received 10 July 2013; Accepted 13 August 2013 Academic Editors: A. Ferrandez, J. Keesling, E. Previato, M. Przanowski, and H. J. Van Maldeghem Copyright © 2013 David G. Glynn. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Topology and geometry should be very closely related mathematical subjects dealing with space. However, they deal with diferent aspects, the irst with properties preserved under deformations, and the second with more linear or rigid aspects, properties invariant under translations, rotations, or projections. he 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 ininitely 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 skewields such as Hamilton’s quaternions corresponds to planar graphs, while graphs on surfaces of higher genus are related to geometry over commutative ields such as the real or complex numbers. 1. Introduction he British/Canadian mathematician H.S.M. Coxeter (1907– 2003) was one of most inluential geometers of the 20th century. He learnt philosophy of mathematics from L. Wittgenstein at Cambridge, inspired M.C. Escher with his drawings, and inluenced 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  5 on ive vertices. It might surprise some people that such a discrete object as a graph could be deemed important in geometry. However, Desargues 10-point 10-line theorem in the projective plane is in fact equivalent to the graph  5 : in mathematical terms the cycle matroid of  5 is the Desargues coniguration in three-dimensional space, and a projection from a general point gives the conigurational 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 let in their philosophies: why does the graph give a theorem in space? Certainly, the matroids of almost all graphs are not theorems. he only other example known to the author of a geometrical theorem coming directly from a graphic matroid is the complete bipartite graph  3,3 , which gives the 9-point 9-plane theorem in three-dimensional space; see [3]. It is interesting that both  5 and  3,3 are minimal nonplanar (toroidal) graphs and both lead to conigurational 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 orientable surface of genus , having V vertices,  edges, and  faces, where V −+= 2−2, is equivalent to a linear property of projective space of dimension V −1, coordinatized by a general commutative ield. his property is characterized by a coniguration having V + points and  hyperplanes. his leads to the philosophical deduction that topology and geometry are closely related, via graph theory. If =0 (and the graph is planar), the linear property is also valid for the most general projective spaces, which are over skewields that in general have noncommuta- tive multiplication. his is a powerful connection between the topology of orientable surfaces and discrete conigurational properties of the most general projective spaces. here are various “fundamental” theorems that pro- vide pathways between diferent areas of mathematics. For example, the fundamental theorem of projective geometry (FTPG) describes the group of automorphisms of projective geometries over ields or skewields (all those of dimensions greater than two) as a group of nonsingular semilinear transformations. his most importantly allows the choice of coordinate systems in well-deined ways. Hence, the FTPG is