A centriole, a notion introduced by B.-Y. Chen and T. Nagano in the 1980s, is a connected component of the set of midpoints between two 'antipodal' points in a compact symmetric space. In our talk we discuss some beautiful geometric properties of centrioles with a particular emphasis on centrioles consisting of midpoints of shortest geodesic arcs, and we describe centrioles in terms of the root system of the ambient symmetric space. Applications to periodicity phenomena in geometry and topology are sketched. This talk is partially based on joint work with A.-L. Mare and on joint work with M. S. Tanaka.