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# Leaf modeling initiative

edited February 27

This paper presents a simplified, quasi-physical model of how a leaf actually grows:

Strikingly, the authors claim:

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

Since this is science, let's reproduce their results!

This will involve:

• Using this study group to go through the paper in detail
• Consolidate the gist of the model into a mathematical specification
• Develop this further into a functional specification for a leaf-plotting program
• Once developed, use the program to test the model
• Can we really make leaves that are close to maple, mulberry etc.?
• Make an "art gallery" of images, to include simulated and actual leaves

• Options
1.
edited February 27

Here is a digest of the paper:

In The Formation of a Tree Leaf by Qinglan Xia, we see a possible key to Nature’s algorithm for the growth of leaf veins. The vein system, which is a transport network for nutrients and other substances, is modeled by Xia as a directed graph with nodes for cells and edges for the “pipes” that connect the cells. Each cell gives a revenue of energy, and incurs a cost for transporting substances to and from it.

The total transport cost depends on the network structure. There are costs for each of the pipes, and costs for turning the fluid around the bends. For each pipe, the cost is proportional to the product of its length, its cross-sectional area raised to a power α, and the number of leaf cells that it feeds. The exponent α captures the savings from using a thicker pipe to transport materials together. Another parameter β expresses the turning cost.

Development proceeds through cycles of growth and network optimization. During growth, a layer of cells gets added, containing each potential cell with a revenue that would exceed its cost. During optimization, the graph is adjusted to find a local cost minimum. Remarkably, by varying α and β, simulations yield leaves resembling those of specific plants, such as maple or mulberry.

A growing network

Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish.

From:

Comment Source:Here is a digest of the paper: > In The Formation of a Tree Leaf by Qinglan Xia, we see a possible key to Nature’s algorithm for the growth of leaf veins. The vein system, which is a transport network for nutrients and other substances, is modeled by Xia as a directed graph with nodes for cells and edges for the “pipes” that connect the cells. Each cell gives a revenue of energy, and incurs a cost for transporting substances to and from it. > The total transport cost depends on the network structure. There are costs for each of the pipes, and costs for turning the fluid around the bends. For each pipe, the cost is proportional to the product of its length, its cross-sectional area raised to a power α, and the number of leaf cells that it feeds. The exponent α captures the savings from using a thicker pipe to transport materials together. Another parameter β expresses the turning cost. > Development proceeds through cycles of growth and network optimization. During growth, a layer of cells gets added, containing each potential cell with a revenue that would exceed its cost. During optimization, the graph is adjusted to find a local cost minimum. Remarkably, by varying α and β, simulations yield leaves resembling those of specific plants, such as maple or mulberry. > A growing network > Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish. From: * [Prospects for a Green Mathematics](https://johncarlosbaez.wordpress.com/2013/02/15/prospects-for-a-green-mathematics/), John Baez and David Tanzer, Azimuth Blog. Originally published in Mathematics of Planet Earth Blog, February 2013.