  The Periodic Event Scheduling Problem (PESP) is the standard mathematical
tool for optimizing periodic timetabling problems in public transport. A
solution to PESP consists of three parts: a periodic timetable, a periodic
tension, and integer periodic offset values. While the space of periodic
tension has received much attention in the past, we explore geometric
properties of the other two components, establishing novel connections between
periodic timetabling and discrete geometry. Firstly, we study the space of
feasible periodic timetables, and decompose it into polytropes, i.e., polytopes
that are convex both classically and in the sense of tropical geometry. We then
study this decomposition and use it to outline a new heuristic for PESP, based
on the tropical neighbourhood of the polytropes. Secondly, we recognize that
the space of fractional cycle offsets is in fact a zonotope. We relate its
zonotopal tilings back to the hyperrectangle of fractional periodic tensions
and to the tropical neighbourhood of the periodic timetable space. To conclude
we also use this new understanding to give tight lower bounds on the minimum
width of an integral cycle basis.
