David A. Tanzer, December 25, 2020

Leaf shape: an optimization in nature

On this Christmas day, let us take a moment to think of the tree, and wonder how it actually grows. Slowly move your attention to the leaves, and ponder how they grow. Next, forget about this type of leaf, for a conifer, with parallel veins — as our true subject is the growth of leaves with branching veins.

The growth of a tree leaf is an algorithm of nature which leads to beautiful forms. And with a bit of nice math, some of the main principles of this algorithm can be understood! Here is beauty and reason together.

A paper was published in 2007, by Qinglan Xia, called the formation of a tree leaf, which lays out this math. Strikingly, the paper says:

Based on this model, we also provide some computer visualization of tree leaves, which resemble many known leaves including the maple and mulberry leaf.

In these posts I will chip away at explaining some of the key ideas of this model, doing my best to boil it down, using terms that are as simple as possible – but not so simple that the idea gets lost.

In a word, the leaf consists of cells and a transport system of pipes — the veins — for getting fluid from the root to the cells. The “algorithm” leads to a branching pattern for the veins which minimizes the aggregate cost of supply all the cells with fluid. This is an optimization, which is here performed by nature.

Now let’s begin with the construction of the model proper.

The action takes place on a grid of squares in the plane. A ‘leaf’ is modeled as set of these squares. Each square in the leaf is called a cell, with all cells having the same size.

There is one special cell, called the root, which is centered right above above the origin. Every leaf is required to contain the root cell.

In fact, the growth process is modeled by a sequence of leaves in time. At each moment, there is a “current leaf.” To get from the current leaf to the next leaf, new cells are added, just outside of its fringes, according to specific rules for selecting new cells.

To kick the process off, the initial leaf in the sequence consists of just the root cell.

Well, I’m going to quit while I’m ahead, and stop here. Hopefully you will have gotten some orientation to how the algorithm works. And there’s still time to post this, at the tail end of Christmas Day.

Next time, we will look into the ‘piping system’ within the leaf, for supplying each of the cells with the fluids need for its metabolism.

Reference: Qinglan Xia, The formation of a tree leaf, ESAIM 2007.

Copyright © 2020, David A. Tanzer. All Rights Reserved.