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The invention of the "dual resonance model" N-point functions BN motivatedthe development of current string theory. The simplest of these models, thefour-point function B4, is the classical Euler Beta function. Many standardmethods of complex analysis in a single variable have been applied to elucidatethe properties of the Euler Beta function, leading, for example, to analyticcontinuation formulas such as the contour-integral representation obtained byPochhammer in 1890. Here we explore the geometry underlying the dual five-pointfunction B5, the simplest generalization of the Euler Beta function. Analyzingthe B5 integrand leads to a polyhedral structure for the five-crosscap surface,embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120in PGL(6). We find a Pochhammer-like representation for B5 that is a contourintegral along a surface of genus five. The symmetric embedding of thefive-crosscap surface in RP5 is doubly covered by a symmetric embedding of thesurface of genus four in R6 that has a polyhedral structure with 24 pentagonalfaces and a symmetry group of order 240 in O(6). The methods appeargeneralizable to all N, and the resulting structures seem to be related toassociahedra in arbitrary dimensions.Comment: 43 pages and 44 figure

A contour integral representation for the dual five-point function and a symmetry of the genus-4 surface in