Shapes of ideal stalagmites

Stalagmites are isolated columns of calcium carbonate growing on a cave floor; their growth is driven by the constant dripping of supersaturated solutions from the roof of the cave. In this paper, we derive a closed-form expression for the shape of a steadily growing stalagmite. Our analysis gives rise to three distinct shapes, all of them observable in nature, with the shape characterized by a single dimensionless parameter. Transitions between different shapes occur at a specific value of this parameter, with additional selection rules determining the shape and size of stalagmites evolving under specific cave conditions. Our theory shows that the stalagmite shape influences the [Formula: see text] isotope shifts, which are an important source of paleoclimatic information.