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The Periodic Event Scheduling Problem (PESP) is the standard mathematicaltool for optimizing periodic timetabling problems in public transport. Asolution to PESP consists of three parts: a periodic timetable, a periodictension, and integer periodic offset values. While the space of periodictension has received much attention in the past, we explore geometricproperties of the other two components, establishing novel connections betweenperiodic timetabling and discrete geometry. Firstly, we study the space offeasible periodic timetables, and decompose it into polytropes, i.e., polytopesthat are convex both classically and in the sense of tropical geometry. We thenstudy this decomposition and use it to outline a new heuristic for PESP, basedon the tropical neighbourhood of the polytropes. Secondly, we recognize thatthe space of fractional cycle offsets is in fact a zonotope. We relate itszonotopal tilings back to the hyperrectangle of fractional periodic tensionsand to the tropical neighbourhood of the periodic timetable space. To concludewe also use this new understanding to give tight lower bounds on the minimumwidth of an integral cycle basis.

The tropical and zonotopal geometry of periodic timetables