Although amphitheatre-shaped valley heads can be cut by groundwater flows emerging from springs, recent geological evidence suggests that other processes may also produce similar features, thus confounding the interpretations of such valley heads on Earth and Mars. To better understand the origin of this topographic form, we combine field observations, laboratory experiments, analysis of a high-resolution topographic map and mathematical theory to quantitatively characterize a class of physical phenomena that produce amphitheatre-shaped heads. The resulting geometric growth equation accurately predicts the shape of decimetre-wide channels in laboratory experiments, 100 m-wide valleys in Florida and Idaho, and kilometre-wide valleys on Mars. We find that, whenever the processes shaping a landscape favour the growth of sharply protruding features, channels develop amphitheatre-shaped heads with an aspect ratio of π.
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