  Although amphitheater-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
amphitheater-shaped heads. The resulting geometric growth equation accurately
predicts the shape of decimeter-wide channels in laboratory experiments,
100-meter wide valleys in Florida and Idaho, and kilometer wide valleys on
Mars. We find that whenever the processes shaping a landscape favor the growth
of sharply protruding features, channels develop amphitheater-shaped heads with
an aspect ratio of pi.
