We report for the first time on the formation of spirals like vegetation patterns in isotropic and uniform environmental conditions. The vegetation spirals are not waves and they do not rotate. They belong to the class of dissipative structures found out of equilibrium. Isolated or interacting spirals and arcs observed in South America (Bolivia) and North Africa (Morocco) are interpreted as a result of curvature instability that affects the circular shape of localized patches. The biomass exhibits a dynamical behaviour with arcs that transform into spirals. Interpretation of observations and of the predictions provided by the theory is illustrated by recent measurements of peculiar plant morphology (the alfa plant, or Stipa tenacissima L.) originated from northwestern Africa and the southern part of the Iberian Peninsula.This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2).