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# Leaf modeling initiative

edited December 2020

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

Strikingly, the author claims:

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

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1.
edited February 2020

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. 
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2.

Tree leaves are not just optimized as vascular transport networks and solar collectors, but also as cantilever structures to handle gravity and wind loads. Furthermore, some leaves are optimized to gather rain and dew moisture, and others to shed moisture, and there are also morphic optimizations against drought and freezing. These higher-order factors are not part of Qinglan's fine analysis of the transport and collector requirements.

Let me stick my neck out as a novice Category Theory student: For the purposes of top-down Leaf Morphology Category Theory, Qinglan's Transport and Solar Collector Categories can be Monoidalized and Braided into a Leaf Morphology Groupoid with the Structural and other Monoidal Function Categories.

Thanks for any correction or suggestion.

Comment Source:Tree leaves are not just optimized as vascular transport networks and solar collectors, but also as cantilever structures to handle gravity and wind loads. Furthermore, some leaves are optimized to gather rain and dew moisture, and others to shed moisture, and there are also morphic optimizations against drought and freezing. These higher-order factors are not part of Qinglan's fine analysis of the transport and collector requirements. Let me stick my neck out as a novice Category Theory student: For the purposes of top-down Leaf Morphology Category Theory, Qinglan's Transport and Solar Collector Categories can be Monoidalized and Braided into a Leaf Morphology Groupoid with the Structural and other Monoidal Function Categories. Thanks for any correction or suggestion.
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3.
edited December 2020

Interesting comments about tree leaves, thanks. Do you know of any papers on tree leaves with mathematical models of higher order factors such as the ones that you describe?

Comment Source:Interesting comments about tree leaves, thanks. Do you know of any papers on tree leaves with mathematical models of higher order factors such as the ones that you describe?
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4.
edited December 2020

Qinglan's references provide one bread-crumb trail into applicable Botany literature. Fortunately, heuristic application of structural engineering and aerodynamics first-principles will take us far. Aeroelasticity is the engineering-science intersection of structural aerodynamic loading of tree leaves.

By the time we have mapped the Tree Leaf Morphology meta-category space, we'll be ready for more specialist literature. Translating Qinglan's low-level abstractions into Category formalisms without a global ontology would be tedious and possibly bake-in gross first-order omissions.

Papers that treat leave and stems structurally and dynamically, from Materials Science to Aerodynamics perspectives, are easily found, and their references will lead to even more applicable science-

https://jeb.biologists.org/content/jexbio/202/23/3281.full.pdf

https://www.witpress.com/Secure/ejournals/papers/D&NE080202f.pdf

Then there are purely mathematical quasi-leaves, like the Barnsley Fern, which are indeed "computer visualization(s) which resemble many known leaves"-

https://en.wikipedia.org/wiki/Barnsley_fern

Comment Source:Qinglan's references provide one bread-crumb trail into applicable Botany literature. Fortunately, heuristic application of structural engineering and aerodynamics first-principles will take us far. Aeroelasticity is the engineering-science intersection of structural aerodynamic loading of tree leaves. By the time we have mapped the Tree Leaf Morphology meta-category space, we'll be ready for more specialist literature. Translating Qinglan's low-level abstractions into Category formalisms without a global ontology would be tedious and possibly bake-in gross first-order omissions. Papers that treat leave and stems structurally and dynamically, from Materials Science to Aerodynamics perspectives, are easily found, and their references will lead to even more applicable science- https://jeb.biologists.org/content/jexbio/202/23/3281.full.pdf https://www.witpress.com/Secure/ejournals/papers/D&NE080202f.pdf Then there are purely mathematical quasi-leaves, like the Barnsley Fern, which are indeed "computer visualization(s) which resemble many known leaves"- https://en.wikipedia.org/wiki/Barnsley_fern
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5.
edited December 2020

Here are Plant Biomechanics Researchers whose portfolio spans vascular to bulk-structure morphology of leaves-

The Group-

https://www.botanischer-garten.uni-freiburg.de/mitarbeiter/pbg

The Prof-

http://www.bio.uni-freiburg.de/groups/speck-en

What makes this Math "Green" is direct applicability to more sustainable (more efficient) engineered structures, logically made of Polymer, rather than steel and concrete. Thus the Architectural Domain becomes a Carbon Sink rather than a Greenhouse Gas Source. This will tie into Airborne Wind Energy (AWE) and Aerotecture green math, which will naturally replicate optimal botanical morphologies.

Comment Source:Here are Plant Biomechanics Researchers whose portfolio spans vascular to bulk-structure morphology of leaves- The Group- https://www.botanischer-garten.uni-freiburg.de/mitarbeiter/pbg The Prof- http://www.bio.uni-freiburg.de/groups/speck-en What makes this Math "Green" is direct applicability to more sustainable (more efficient) engineered structures, logically made of Polymer, rather than steel and concrete. Thus the Architectural Domain becomes a Carbon Sink rather than a Greenhouse Gas Source. This will tie into Airborne Wind Energy (AWE) and Aerotecture green math, which will naturally replicate optimal botanical morphologies.
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6.

Modeling Leaf aerodynamics is another advanced aspect of leaf morphology. Because most leaves are flattened to optimize solar-collection to surface-area, they happen also to act as wings, which is counterproductive in high wind.

Leaves embody various aerodynamic tricks to dissipate excess energy. A maple leaf will curl into a streamlined tube in high wind. Many leaves flutter to dissipate energy. Leaves and branches collectively bend downwind to reduce and buffer peak wind storing some of it as elastic energy. Leaves are lattice spring-mass elements in a damped "tree" network. Standard mathematical Aerodynamics and Aeroelasticity apply.

The Aerodynamic Model of a Leaf on its stalk has six degrees-of-freedom (DOF)- Pitch, Roll, Yaw, Heave, Surge, and Sway. The first three DOF dominate, interacting in a "Dutch Roll" motion, and last three DOF more constrained, but more expressive of the bulk motion of a branched network.

Its wonderful how simple generative equations and genetics produce fairly optimal dynamic structural aero-forms. How might these mathematical marvels be applied? By coincidence, leaf networks (plants) are dynamical analogues for Airborne Wind Energy Kite Networks in the Planetary Wind topic, with Kites and Leaves mathematically similar.

Comment Source:Modeling Leaf aerodynamics is another advanced aspect of leaf morphology. Because most leaves are flattened to optimize solar-collection to surface-area, they happen also to act as wings, which is counterproductive in high wind. Leaves embody various aerodynamic tricks to dissipate excess energy. A maple leaf will curl into a streamlined tube in high wind. Many leaves flutter to dissipate energy. Leaves and branches collectively bend downwind to reduce and buffer peak wind storing some of it as elastic energy. Leaves are lattice spring-mass elements in a damped "tree" network. Standard mathematical Aerodynamics and Aeroelasticity apply. The Aerodynamic Model of a Leaf on its stalk has six degrees-of-freedom (DOF)- Pitch, Roll, Yaw, Heave, Surge, and Sway. The first three DOF dominate, interacting in a "Dutch Roll" motion, and last three DOF more constrained, but more expressive of the bulk motion of a branched network. Its wonderful how simple generative equations and genetics produce fairly optimal dynamic structural aero-forms. How might these mathematical marvels be applied? By coincidence, leaf networks (plants) are dynamical analogues for Airborne Wind Energy Kite Networks in the Planetary Wind topic, with Kites and Leaves mathematically similar.
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7.
edited December 2020

The classic plant morphology text by Goethe, who declared, "all is leaf"-

https://maypoleofwisdom.files.wordpress.com/2020/03/the-metamorphosis-of-plants.pdf

Goethe intuits a common mathematical basis for forms, "No wonder that the earth expresses itself outwardly in leaves, it so labors with the idea inwardly. The atoms have already learned this law, and are pregnant by it."

As Blake put it, "To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour." Therefore, as many plant parts are evolutionarily modified leaves, understand the Leaf to see "Heaven in a Wildflower". Further, "Eternity in an hour" is Einstein's Relativity.

Nice article on vascularization in animals-

https://www.mitchmedical.us/cardiovascular-system/speculation-on-the-assembly-of-av-networks.html

Precisely the same mathematical optimality forces in play. There are deep interrelations between plumbing and structural stiffness. Their shared features are tapered branched tubing networks with internal hydrostatic pressure. The base pressure is Gibbs zero-point energy, and circulation is Gibbs free-energy component.

Comment Source:The classic plant morphology text by Goethe, who declared, "all is leaf"- https://maypoleofwisdom.files.wordpress.com/2020/03/the-metamorphosis-of-plants.pdf Goethe intuits a common mathematical basis for forms, "No wonder that the earth expresses itself outwardly in leaves, it so labors with the idea inwardly. The atoms have already learned this law, and are pregnant by it." As Blake put it, "To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour." Therefore, as many plant parts are evolutionarily modified leaves, understand the Leaf to see "Heaven in a Wildflower". Further, "Eternity in an hour" is Einstein's Relativity. Nice article on vascularization in animals- https://www.mitchmedical.us/cardiovascular-system/speculation-on-the-assembly-of-av-networks.html Precisely the same mathematical optimality forces in play. There are deep interrelations between plumbing and structural stiffness. Their shared features are tapered branched tubing networks with internal hydrostatic pressure. The base pressure is Gibbs zero-point energy, and circulation is Gibbs free-energy component. 
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8.

Sidenote: I really appreciate this infusion of content into the discussion!

Comment Source:Sidenote: I really appreciate this infusion of content into the discussion!
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9.
edited December 2020

Thank You, David.

Considering next the Network Topology of a Leaf, its aptly a Tree Network, no pun intended, which is a hierarchy of Star Networks on Buses (or Trunks, if you will). These are Fractal Dimensions and we have already seen Leaf forms generated by Fractal Formulas.

Here are two Fractal Pages with formulas that generate various plant forms-

http://paulbourke.net/fractals/fracintro/

https://fractalformulas.wordpress.com/

The claim is therefore well validated, that elegantly simple mathematics generate realistic plant forms, and that real plants in fact embody such math, on a principle of Occam's Razor or Least Action Shannon Information-Energy Equivalence.

Comment Source:Thank You, David. Considering next the Network Topology of a Leaf, its aptly a Tree Network, no pun intended, which is a hierarchy of Star Networks on Buses (or Trunks, if you will). These are Fractal Dimensions and we have already seen Leaf forms generated by Fractal Formulas. Here are two Fractal Pages with formulas that generate various plant forms- http://paulbourke.net/fractals/fracintro/ https://fractalformulas.wordpress.com/ The claim is therefore well validated, that elegantly simple mathematics generate realistic plant forms, and that real plants in fact embody such math, on a principle of Occam's Razor or Least Action Shannon Information-Energy Equivalence. 
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10.

Yes, those are nice, thanks. Though it's a different kettle of fish, as those are purely mathematical leaves, without a basis in a physically plausible model.

I just started a separate thread for digging into Xia's paper in specific.

We can keep this one as a general place for talking about any aspects of leaf modeling.

Comment Source:Yes, those are nice, thanks. Though it's a different kettle of fish, as those are purely mathematical leaves, without a basis in a physically plausible model. I just started a separate thread for digging into Xia's paper in specific. We can keep this one as a general place for talking about any aspects of leaf modeling. 
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11.
edited December 2020

There is a noble Physics tradition, dating back to the Pythagoreans at least, that all of reality is "number" (math). In our time, the idea is seriously expressed that we may live within a simulation. Therefore, a biological leaf could be nothing but a "purely mathematical" leaf.

The opposed view is some "quintessence" of life that math does not do. This is not a settled debate.

Comment Source:There is a noble Physics tradition, dating back to the Pythagoreans at least, that all of reality is "number" (math). In our time, the idea is seriously expressed that we may live within a simulation. Therefore, a biological leaf could be nothing but a "purely mathematical" leaf. The opposed view is some "quintessence" of life that math does not do. This is not a settled debate.
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12.
edited December 2020

Mathematically modeling a Leaf is a multi-physics problem. Static and dynamic structure in relation to vascular transport, insolation, aerodynamics, aeroelasticity, allometry, and so on, all figure in morphological expression.

Biological leaves perform Transpiration and Guttation via their Vascular Transport Networks. Typically >97% of moisture transported to the leaf passes out of its Stomatal Pores. Modeling a leaf realistically as vascular transport network therefore needs to account for stomatal valve function.

Leaf transport networks are therefore substantially a one-way transport of bulk flow, with a small return flow of sugar syrup and other solutes. This reduces "turning cost" of two-way transport accordingly. Return flow is more viscous, as a further transport "cost" factor to introduce in modeling.

https://en.wikipedia.org/wiki/Transpiration

A heat stressed leaf needs to transpire to cool off, with sufficient surface area. A thick cactus can buffer midday heat by its thermal inertial bulk. These are morphological factors.

Further regarding Allometric scaling, tropical rain forest and wetland plants in non-windy locations can sustain the largest leaves, and hot-desert and alpine-polar zone plants the smallest leaves.

Mass Scaling Exponents relate to the form of leaves. We recall that stems and leaves evolved from common ancestral segments. Thin stems and pine-needles would scale somewhat linearly, were it not for non-dimensional transport distance; broad-leaves scale nearly quadratically; and thick bodies, more-or-less cubically. Scaling exponents constrain how big a plant and its parts may grow. Kelp's buoyancy gives it enhanced vertical aspect ratio. Some vines exceed 1km long (ie. Entada phaseoloides), freed from a self-support requirement.

Comment Source:Mathematically modeling a Leaf is a multi-physics problem. Static and dynamic structure in relation to vascular transport, insolation, aerodynamics, aeroelasticity, allometry, and so on, all figure in morphological expression. Biological leaves perform Transpiration and Guttation via their Vascular Transport Networks. Typically >97% of moisture transported to the leaf passes out of its Stomatal Pores. Modeling a leaf realistically as vascular transport network therefore needs to account for stomatal valve function. Leaf transport networks are therefore substantially a one-way transport of bulk flow, with a small return flow of sugar syrup and other solutes. This reduces "turning cost" of two-way transport accordingly. Return flow is more viscous, as a further transport "cost" factor to introduce in modeling. https://en.wikipedia.org/wiki/Transpiration A heat stressed leaf needs to transpire to cool off, with sufficient surface area. A thick cactus can buffer midday heat by its thermal inertial bulk. These are morphological factors. Further regarding Allometric scaling, tropical rain forest and wetland plants in non-windy locations can sustain the largest leaves, and hot-desert and alpine-polar zone plants the smallest leaves. Mass Scaling Exponents relate to the form of leaves. We recall that stems and leaves evolved from common ancestral segments. Thin stems and pine-needles would scale somewhat linearly, were it not for non-dimensional transport distance; broad-leaves scale nearly quadratically; and thick bodies, more-or-less cubically. Scaling exponents constrain how big a plant and its parts may grow. Kelp's buoyancy gives it enhanced vertical aspect ratio. Some vines exceed 1km long (ie. Entada phaseoloides), freed from a self-support requirement.