Hexagonal Extensions of Toroidal Maps and Hypermaps Maria Elisa Fernandes, Dimitri Leemans and Asia Ivi´ c Weiss Dedicated to Egon Schulte on the occasion of his 60th birthday Abstract The rank 3 concept of a hypermap has recently been generalized to a higher rank structure in which hypermaps can be seen as “hyperfaces” but very few examples can be found in literature. We study finite rank 4 structures obtained by hexagonal extensions of toroidal hypermaps. Many new examples are produced that are regular or chiral, even when the extensions are polytopal. We also construct a new infinite family of finite nonlinear hexagonal extensions of the tetrahedron. Keywords Regularity · Chirality · Thin geometries · Hypermaps · Abstract polytopes 2000 Math Subj. Class: 52B11 · 20D06 M. E. Fernandes Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal e-mail: maria.elisa@ua.pt D. Leemans (B ) Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand e-mail: dleemans@ulb.ac.be D. Leemans Département de Mathématique, C.P. 216 - Algèbre et Combinatoire, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Brussels, Belgium A. I. Weiss Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada e-mail: weiss@mathstat.yorku.ca © Springer International Publishing AG, part of Springer Nature 2018 M. D. E. Conder et al. (eds.), Discrete Geometry and Symmetry, Springer Proceedings in Mathematics & Statistics 234, https://doi.org/10.1007/978-3-319-78434-2_8 147