# Mathematical Digest

### Short Summaries of Articles about Mathematicsin the Popular Press

"Leaves, Flowers and Garbage Bags: Making Waves," by Eran Sharon, MichaelMarder and Harry L. Swinney. American Scientist, May-June 2004, pages 254-261.

If you've ever noticed that fractals appear in the curving edges of some leavesand flowers, you're not alone. Sharon, Marder, and Swinney explored the causesof this phenomenon by seeing what happens when one tears a plastic sheetimprinted with a grid of dots. They noted that the fractal pattern along thetear is the result of "spontaneous symmetry breaking," partly a result of thestretching near the tear. The authors write that "the distances between thedots on the surface after tearing cannot be met if the sheet is flat. Then, toavoid the expensive compression energy, the sheet happily pays cheap bendingenergy, as it buckles out of the plane, while trying to generate saddle pointseverywhere [due to negative Gaussian curvature]."

They also performed experiments with leaves, tubes made of synthetic materialthat would expand in a chemical solution, and computer models. They noted thatwhile many complex biological systems have complex causes, the reasons behindthese fractal patterns are simple. The growth along leaf and flower edgesresults from a uniform growth law, but---as with the plastic sheets---thegeometric limitations of space and the elasticity of the material caused theedges to take on a wavy shape. They conclude, "physics and biology meet at therippled edges of leaves and flowers to provide one of these rare tractableproblems."

--- Claudia Clark