In Adler et al [Convex dynamics and applications. Ergod. Th. & Dynam. Sys.25 (2005), 321–352] certain piecewise linear maps were defined in terms of a convex polytope. When the convex polytope is a simplex, the resulting map has a dual nature. On one hand it is defined on ℝN and acts as a piecewise translation. On the other it can be viewed as a translation on the N-torus. What relates its two roles? A natural answer would be that there exists an invariant fundamental set into which all orbits under piecewise translation eventually enter. We prove this for N=1 and for acute and right triangles—i.e. non-obtuse triangles. We leave open the case of obtuse triangles and higher-dimensional simplices. Another question not treated is the connectivity of the invariant fundamental sets which arise.