Numerical exploration of a hexagonal string billiard

► The string construction yields a family of billiard tables which are globally C2. ► The boundary of our billiard has strictly positive curvature. ► The hexagonal string billiard has three types of trajectories: focal, inner, and outer. ► Poincaré's SOS is filled solely with periodic points and invariant curves. ► We have gathered numerical evidence against the Birkhoff-Poritsky conjecture.