A measure of leaf complexity and probably an instance of  the application of morphometrics. Varying phenotypes in arabidopsis may be due to genetics. In some cases, measuring the perimeter to area ratios may  yield insights into underlying mathematical beauty as patterns in terms of the golden ratio begin to emerge.

http://prometheuswiki.publish.csiro.au/tiki-index.php?%20page=Measuring+leaf+perimeter+and+leaf+area

http://en.m.wikipedia.org/wiki/Morphometrics

https://books.google.com.my/books?id=Gw-lqvE6fNgC&pg=PT127&lpg=PT127&dq=perimeter+to+area+golden+ratio&source=bl&ots=FvG4E2aOXV&sig=rzusl1btfRV5i-nVH6VdOLQ5DPY&hl=en&sa=X&ei=7XIZVcfUL4zn8AXGi4DoDw&ved=0CCcQ6AEwBA

https://www.google.com.my/url?sa=t&source=web&rct=j&ei=DnUZVZ-fOYro8AXCwYHgAg&url=http://onlinelibrary.wiley.com/doi/10.1111/j.1365-313X.2011.04674.x/pdf&ved=0CCYQFjAB&usg=AFQjCNEz9Y_JhLH9p3txHdPGtn6xypLT1A&sig2=0H_b6bmpP-NFkF5kgLAoRw

Perimeter to area ratios measure the complexity of the shape of leaves.The higher the value the more complex the shape. The approach is an instance of the application of morphometrics. Differences in shape complexity as well as varying leave  curvatures of the leaves have been used to tag phenotypes in Arabidopsis. Mathematically, the closer the shape metrics approach the golden ratio, the closer it gets to mathematical perfection/beauty.

https://www.google.com.my/url?sa=t&source=web&rct=j&ei=DnUZVZ-fOYro8AXCwYHgAg&url=http://onlinelibrary.wiley.com/doi/10.1111/j.1365-313X.2011.04674.x/pdf&ved=0CCYQFjAB&usg=AFQjCNEz9Y_JhLH9p3txHdPGtn6xypLT1A&sig2=0H_b6bmpP-NFkF5kgLAoRw

https://www.google.com.my/url?sa=t&source=web&rct=j&ei=DnUZVZ-fOYro8AXCwYHgAg&url=http://www.ncbi.nlm.nih.gov/pubmed/25711708&ved=0CCIQFjAA&usg=AFQjCNF-wm8jbJVwHywLFbVH8xoisILumg&sig2=dstV6KvMuMNurv8SSjbV_Q

https://books.google.com.my/books?id=Gw-lqvE6fNgC&pg=PT127&lpg=PT127&dq=perimeter+to+area+golden+ratio&source=bl&ots=FvG4E2aOXV&sig=rzusl1btfRV5i-nVH6VdOLQ5DPY&hl=en&sa=X&ei=7XIZVcfUL4zn8AXGi4DoDw&ved=0CCcQ6AEwBA

Hi!

I am using several definitions of compactness to study plants as well.

Sometimes what you use as Compactness Area/Perimetes is called Shape Factor or Compactness

Here some references;

http://homepages.inf.ed.ac.uk/cgi/rbf/CVONLINE/entries.pl?TAG157

https://en.wikipedia.org/wiki/Shape_factor_%28image_analysis_and_microscopy%29

http://www.m-hikari.com/imf-password2009/25-28-2009/bribiescaIMF25-28-2009.pdf

This definition of P/A help you to calculate the departure from a circle and is good for leaves. Other measurement I use is the Area/ConvexHullArea that give you a kind of percentage of the surface covered by the leave and the percentage that not. In a leave without petioles this would correspond to the Area should be filled if the leave would not be rugged.

Have also a look to

http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0096889

Regards. Odin

Thanks a lot you all!

Odín, I am just reading the interesting paper you suggest, thanks a lot!!

Some plants show large changes in leaf lobe and/or tooth formation during development - or in relation to environmental variables - and area/perimeter relationships have commonly been used to assess such changes.  A reference from my own work is attached.

Article Elevated CO2 and leaf shape: are dandelions getting toothier

Thank Dr. Sean C. Thomas very much! I am also doing such a work. Your paper is of great value for me. :-)