When a fluid layer is subjected to vertical oscillation,subharmonic standing waves form at its surface; this is the famous instability described by Faraday in 1831.The patterns first observed were classic: stripes,squares and hexagons.The discovery in the 1990s of more complicated spatial structures such as quasicrystalline patterns,superlattices and oscillons led to a resurgence of theoretical interest.Surprisingly,the Faraday instability has been little studied numerically,with the first 2D simulation carried out in 2000 by Chen.The first 3D simulation,carried out by our group using a front-tracking/immersed-boundary method,reproduced the experimental hexagonal standing waves; the variation of the interface height over one oscillation period is illustrated above.These waves are succeeded,however,by long-time recurrent alternation between new patterns we have called quasi-hexagonal and beaded stripes,interconnected by new spatio-temporal symmetries.