Conformal growth of Arabidopsis leaves

•The growth of Arabidopsis leaves follows a conformal map.•A general mechanism that ensures at growth in plant leaves.•An underlying linear gradient of growth rate in Arabidopsis leaves.•Using complex numbers to model leaf growth.

I show that Arabidopsis leaf growth can be described with good precision by a conformal map, where expansion is locally isotropic (the same in all directions) but the amount of expansion can vary with position. Data obtained by tracking leaf growth over time can be reproduced with almost 90% accuracy by such a map. The growth follows a Möbius transformation, which is a type of conformal map that would arise if there were an underlying linear gradient of growth rate. From the data one can derive the parameters that describe this linear gradient and show how it changes over time. Such a rule has the property of maintaining the flatness of a leaf.