On the ideal shapes of stalagmites

Stalagmites are column-like formations that rise from the floor of  caves. They are formed by the buildup of minerals deposited from water  dripping from the ceiling. The water dissolves minerals, such as calcium  carbonate, from the rock above. As the water drips down, it loses carbon  dioxide to the cave air. This causes the minerals to come out of  solution and precipitate onto the cave floor, slowly building up the  stalagmite.

Nearly sixty years ago, Franke formulated a mathematical model for the  growth of stalagmites. In this model, the local growth rate of a  stalagmite is proportional to the oversaturation of calcium ions in the  solution dripping down the stalagmite's surface. Franke postulated that - provided the physical conditions in the cave remain constant - after a  sufficiently long period, the stalagmite will assume an ideal shape,  which in later stages of growth will only move upwards without further change in its form. These conclusions were later confirmed in computer  simulations yet the mathematical form of this ideal shape was not  discovered.

As we will show, Franke's model for stalagmite growth can be solved  analytically, finding invariant, Platonic forms of stalagmites that  could be observed in an "ideal cave", under constant physical conditions  and with a constant flow of water dripping from an associated stalactite. Interestingly, it turns out that the shape numerically found in previous numerical studies is just one of a whole family of solutions. These new solutions describe stalagmites with a flat area at  their peak of a certain fixed diameter, and conical stalagmites, with sharply pointed tops. All of these forms are observed in caves.