Symmetry: Culture and Science Vol. 22, Nos.3-4, 435-458, 2011 *This research was supported by the Slovenian — Hungarian intergovernmental cooperation programme 2008-2009. 3-SIMPLEX TILINGS, SPLITTING ORBIFOLDS AND MANIFOLDS* Emil Molnár, Jenő Szirmai, Jeffrey R. Weeks Address: Budapest University of Technology and Economics, Institute of Mathematics, Department of Geometry, H-1521, Budapest, Egry J. utca 1. H. II. 22. 88 Stars st. Canton NY 13617 USA E-mail: emolnar@math.bme.hu . , szirmai@math.bme.hu ., temp0000@geometrygames.org Homepage: www.math.bme.hu/~emolnar , www.math.bme.hu/~szirmai , www.geometrygames.org Fields of interest: Thurston’s homogeneous geometries, manifolds, orbifolds. Abstract. In Fig.1.1 there are depicted two generalized orbifold series with rotation orders a, b as parameters, each by a simplex T with face pairing identifications. Or, equivalently consider a corresponding combinatorial simplex tiling (T, ) under the fundamental group  generated by the face pairings. Generalized orbifold means that for each -equivalence class of vertices we allow either finite stabilizer or stabilizer, equivariant to Euclidean ( 2 E ) plane (crystallographic) group or hyperbolic ( 2 H ) plane group as well. Then, by cutting regular neighbourhood around the vertices in the latter two cases, we get a usual orbifold with boundary. We are going to show (on the base of the computer software SnapPea by the third author) that our each orbifold splits along a Euclidean 2-suborbifold if a=1. The remaining orbifold with boundary can be endowed with R L ~ S 2 metric if b=1, and 3 H metric if b  2. We construct also splitting manifolds in the case a=b=1. These results are summarized at the end of the paper in three theorems. The general machinery has been developed