# The Calculus of Caves

Deep within a limestone cavern, a fairy tale landscape glistens. Soda straws, draperies, popcorn, and flowstone mimic the everyday objects they're named for, while stalagmites grow upwards from the cave floor and stalactites hang from the ceiling like the fangs of some troglodytic monster. Now, a group of spelunking researchers has come up with a mathematical equation that explains how stalagmites grow.

Cave formations materialize in limestone caverns over thousands of years as water oozes out of the rock and deposits solid calcium carbonate in its path. The diverse shapes of these objects depend on the geometry of the rock and the amount of flowing water. Until recently, no one had been able to quantitatively explain why the deposited calcium carbonate forms graceful shapes instead of big globs of white stuff.

In the 14 January Physical Review Letters, a team of physicists takes the first step. According to graduate student Martin Short and colleagues at the University of Arizona and Kartchner Caverns State Park, the speed of stalagmite growth depends on the thickness of the water flowing over it. The water layer tends to be thickest at the tip of the stalagmite, where the drop of water initially lands, and thins as the droplet spreads down to the wide base. The thicker the water layer, the more calcium carbonate deposited, and the faster the stalagmite grows in that direction. Curiously, the math the physicists came up with predicts stalagmites and stalactites should have an "ideal form", a cone shape that varies only in size, never in the angle of the cone's slope. To verify this, the researchers went to Arizona's Kartchner Caverns, photographed stalagmites, and compared them to virtual stalagmites generated from the equation. The results matched beautifully.

The group "elegantly" explains a phenomenon familiar to many but poorly understood, says Howard Stone, a physicist at Harvard University. Meanwhile, the Arizona group is pushing on to explain other, more complex cave shapes.

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