International Journal of Contemporary Mathematical Sciences Vol. 10, 2015, no. 8, 349 - 358 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2015.5838 The Petrie Lengths of All Regular Maps of Genus 2 to 15 Serhan Ulusan Department of Mathematics, Faculty of Arts and Sciences Adnan Menderes University, 09010, Aydın, Turkey Copyright c  2015 Serhan Ulusan. This article is distributed under the Creative Com- mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we calculate the Petrie numbers and the lengths of the Petrie polygons of all reflexible regular maps of genus 2 to 15. Mathematics Subject Classification: 30F10, 05C10 Keywords: Regular map, Petrie polygon, Petrie automorphism, Petrie number, Petrie length 1 Introduction A map M is an embedding of a finite graph G in a Riemann surface X such that the components of X −G , which are called the faces of M, are each homeomorphic to an open disc. M is said to be of type {m, n} if every vertex and face of M has valency m and n respectively. The genus of M is the genus of the underlying surface X . In our maps we require G to be connected and every edge of G to have two vertices. We also require X to be orientable, compact, connected and without boundary. An automorphism of M is an automorphism of X that leaves M invariant and preserves incidence. All automorphisms of M form a group under composition of maps and we will denote it by Aut ± M and the subgroup consisting of conformal automorphisms by Aut + M. M is said to be regular if Aut + M is transitive on the directed edges. If M admits an involution R that fixes the mid-point of an edge and interchanges the two darts without interchanging the two neighboring faces, then M is called reflexible and R is called a reflection of M.