Conformal Model in Leaf Growth

Conformal Model in Leaf Growth
Conformal Model in Leaf Growth - A. Dai and M. Ben Amar

Scientists in the field of physics have demonstrated that a mathematical transformation, the conformal map, is used to predict the growth of leaves. D'Arcy Thompson pioneered the application of mathematics to biology in his 1917 book On Growth and Form.

Now, physicists Martine Ben Amar and Anna Dai of the École Normale Supérieure in Paris have borrowed a page from this centuries-old text. Dai and Ben Amar brought the math of conformal maps to the topic of leaf growth after observing that many of Thompson's diagrams of plant and animal growth resemble conformal maps—angle-preserving transformations. They show how the idea of ​​physical energy minimization provides a powerful motivation for mathematical technique.

In biology, just as in any other physical system, there are various growth mechanisms that show how organisms try to use as little energy as possible. This includes reducing internal elastic stresses in growing organisms.

However, this view of energy minimization growth seems to have little to do with the mathematical view of growth outlined by Dai and Ben Amar in Thompson's book. Imagine drawing a grid over a young leaf and watching it droop as it develops. The lines will stretch and bend as the leaf ages properly, but the angles at which the grid lines meet will not change. This simple mathematical change initially seems to have little effect on the physical forces that contribute to the leaf's growth.

However, Dai and Ben Amar discovered that this map captures key elements of complex physics at work. Using leaves of the Monstera deliciosa (or "Swiss cheese") plant, they revealed that conformal maps replicate leaf growth while reducing elastic stress, demonstrating the physical justification for the mathematical transformation. The analysis has been done in 2D up to this point, and the researchers now want to see if conformal maps can be used to describe the evolution of leaves in 3D.


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