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The familiar axisymmetric zones and belts that characterize Jupiter's weather system at lower latitudes give way to pervasive cyclonic activity at higher latitudes. Two-dimensional turbulence in combination with the Coriolis β-effect (that is, the large meridionally varying Coriolis force on the giant planets of the Solar System) produces alternating zonal flows. The zonal flows weaken with rising latitude so that a transition between equatorial jets and polar turbulence on Jupiter can occur. Simulations with shallow-water models of giant planets support this transition by producing both alternating flows near the equator and circumpolar cyclones near the poles. Jovian polar regions are not visible from Earth owing to Jupiter's low axial tilt, and were poorly characterized by previous missions because the trajectories of these missions did not venture far from Jupiter's equatorial plane. Here we report that visible and infrared images obtained from above each pole by the Juno spacecraft during its first five orbits reveal persistent polygonal patterns of large cyclones. In the north, eight circumpolar cyclones are observed about a single polar cyclone; in the south, one polar cyclone is encircled by five circumpolar cyclones. Cyclonic circulation is established via time-lapse imagery obtained over intervals ranging from 20 minutes to 4 hours. Although migration of cyclones towards the pole might be expected as a consequence of the Coriolis β-effect, by which cyclonic vortices naturally drift towards the rotational pole, the configuration of the cyclones is without precedent on other planets (including Saturn's polar hexagonal features). The manner in which the cyclones persist without merging and the process by which they evolve to their current configuration are unknown.