Seminarium KMMF "Teoria Dwoistości"

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.