Geometric entropy of plant leaves: A measure of morphological complexity.

Shape is an objective characteristic of an object. A boundary separates a physical object from its surroundings. It defines the shape and regulates energy flux into and from an object. Visual perception of a definite shape (geometry) of physical objects is an abstraction. While the perceived geometry at an object's sharp interface (macro) creates a Euclidian illusion of actual shape, the notion of diffuse interfaces (micro) allows an understanding of the realistic form of objects. Here, we formulate a dimensionless geometric entropy of plant leaves (SL) by a 2-D description of a phase-field function. We applied this method to 112 tropical plant leaf images. SL was estimated from the leaf perimeter (P) and leaf area (A). It correlates positively with a fractal dimensional measure of leaf complexity, viz., segmental fractal complexity. Leaves with a higher P: A ratio have higher SL and possess complex morphology. The univariate cluster analysis of SL reveals the taxonomic relationship among the leaf shapes at the genus level. An increase in SL of plant leaves could be an evolutionary strategy. The results of morphological complexity presented in this paper will trigger discussion on the causal links between leaf adaptive stability/efficiency and complexity. We present SL as a derived plant trait to describe plant leaf complexity and adaptive stability. Integrating SL into other leaf physiological measures will help to understand the dynamics of energy flow between plants and their environment.