The invention of the 'dual resonance model' N-point functions B_N motivated the development of current string theory. The simplest of these models, the four-point function B₄, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulae such as the contour-integral representation obtained by Pochhammer in 1890. However, the precise features of the expected multiple-complex-variable generalizations to B_N have not been systematically studied. Here we explore the geometry underlying the dual five-point function B₅, the simplest generalization of the Euler Beta function. The original integrand defining B₅ leads to a polyhedral structure for the five-crosscap surface, embedded in , that has 12 pentagonal faces and a symmetry group of order 120 in . We find a Pochhammer-like representation for B₅ that is a contour integral along a surface of genus 5 in . The symmetric embedding of the five-crosscap surface in is doubly covered by a corresponding symmetric embedding of the surface of genus 4 in that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in . These symmetries enable the construction of elegant visualizations of these surfaces. The key idea of this paper is to realize that the compactification of the set of five-point cross-ratios forms a smooth real algebraic subvariety that is the five-crosscap surface in . It is in the complexification of this surface that we construct the contour integral representation for B₅. Our methods are generalizable in principle to higher dimensions, and therefore should be of interest for further study.