Structural Emergence: The Fractal Geometry of Nature #SoME3
Mystery of Fractals
There is a mystery that lies at the heart of nature. It lies in plain sight, visible through the eyes of an artist, but with a secret structure we have only recently developed the language to describe.
Santiago Ramón y Cajal, the founder of modern neuroscience, had the vision to articulate this secret structure. In 1894, he wrote:
“The cerebral cortex is similar to a garden filled with innumerable trees, the pyramidal cells, which can multiply their branches thanks to intelligent cultivation, send their roots deeper, and produce more exquisite flowers and fruits every day.”
What might mathematics say about such structures? Also, why, and how, does the brain grow this garden of branching trees?
Also, why, and how, do such branching structures appear at the scale of the universe?
I hope exploring this mystery will change how you see the brain, life, and the universe.
Define Fractals
To answer this mystery, first we must gain the language to describe what these structures have in common. Specifically, what common structure do neurons, trees, and the universe share? In what sense are these similar?
To answer, zoom in on a branch of a neuron,
a tree,
or a part of the universe;
still the shared structure remains as we zoom in; that is, the structure looks the same at different sizes. This property of objects that look the same at different scales is called scale invariance. Objects which display this property of scale invariance are called fractals.
Efficiency of Fractal Design
Our mystery of why neurons, trees, and the universe look similar can then be restated as why should neurons, trees, and glaxy superclusters have a fractal design? How do these designs arise in such diverse contexts? Why are such structures also found in living things, like circulatory systems, lungs, root systems, and non-living things, like river systems?
To begin to understand these questions, let’s look first at why trees adopted this fractal design. To begin to answer this, we’ll start with a simpler question: what problem are trees designed to solve?
Put simply, trees are trying to spread their leaves over as much surface area as they can, while limiting the volume of cells they must construct. To understand why, think about how trees build themselves; they capture the power of the sun and use it to transform carbon dioxide in the air into more of their cells. By maximizing the ratio of their surface area to their volume, they give each of their cells access to as much sunlight as possible, enabling them to thrive.
Fractal geometry, which studies objects with a fractal design, shows that the fractal designs trees have evolved are powerful solutions to the problem of maximizing the ratio of surface area to volume, the problem trees are trying to solve. This answers why trees would “want” to adopt a fractal design; fractal shapes have a very high ratio of surface area to volume.
To illustrate why the ratio of surface area to volume is high for a fractal shape, one classical question we can consider is: How long is the coastline of Britain? While your initial instinct might tell you that this is a simple question of measurement, in fact, you’ll find that the answer depends on how long of a ruler you use in calculating the answer, as illustrated in the picture below:
Fractals are continually broken up into progressively smaller pieces, so they keep zigzagging as you zoom in. This means that as you measure with smaller and smaller units, the coastline keeps expanding.
The jagged pieces of a coastline that continue breaking down as you zoom in mean the length of the coast is much greater compared to the surface area contained within it.
If we scale this up from two to three dimensions, we can see why fractals are a great design for trees; they can increase the size of the perimeter without increasing the size of the interior.
Evolutionary Plausibility
At this point, you may be asking “Ok, maybe fractals are a good design for solving the problem of covering as much area with as little material as possible, but nature doesn’t always find the best solution, does it?”
This is a reasonable objection; there are many structures, such as wheels, that might be efficient designs for nature to adopt. However, evolution only allows for the development of complex structures which can plausibly evolve from simpler structures. We don’t have kangaroos with roller skates, presumably because wheels have no plausible biologically advantageous precursor which could then evolve into wheels.
How, then, did trees develop their branching structure? More specifically, how can trees guide the growth of such seemingly complex branching structures with only simple growth rules that could have plausibly evolved? In other words, how could the genetic code for such complex structures be written compactly? Such branching rules must be reasonably simple in order for them to be evolutionarily plausible.
To answer this question, we return to the idea of scale invariance; what problem is each individual branch of a tree solving? It solves the same problem as the larger tree: maximize the ratio of surface area to volume to provide its cells with as rich an environment as possible. Roughly speaking, each branch is merely a smaller version of the broader tree.
Recursion
This pattern of breaking large problems into smaller instances of the same problem is a powerful design pattern commonly used in computer programming, a paradigm known as recursion. It enables the solution of complex problems to be described very succinctly, which is critical for an approach that can evolve to be represented compactly in the genetic code of an organism.
For instance, trees are able to grow their complex branching structure by recursively solving the problem of maximizing access to sunlight while minimizing volume. By recursive, we mean that each branch is merely a smaller version of the same tree, trying to solve the same problem as the larger tree. Each branch implements this same branching strategy, growing new offshoots when enough sunlight is available.
Guided Growth
Recalling that trees are trying to maximize access to sunlight, it’s important to note that trees don’t build their branching structure at random; they focus growth to areas with the richest access to sunlight. Viewed this way, we can come to a deeper understanding of how trees can direct their growth using only localized growth signals, eliminating the need for a centralized system to direct growth; if a branch has rich access to sunlight, it releases a stronger local growth signal, which enables it to draw more resources and grow faster, while branches in shade do not grow and are more likely to be pruned.
If this process is not clear, consider a branch at the very tip of the tree. If it collects a lot of sunlight, it can build up a chemical signal to send to its parent branch requesting more resources to enable growth. Alternatively, if a branch has little access to sunlight, it can only send a weak growth signal, and so does not grow. The next branch closer to the trunk adds up the growth signals of its sub-branches, and so on for each branch until we reach the trunk. At each branching point, the tree then knows how much resources to allocate to each branch, prioritizing the more promising one.
In this way, the branch structure of the tree comes to direct growth towards the most advantageous sources of energy, avoiding areas with little access to resources. Moreover, it need only rely on localized, recursive growth patterns which can be easily coded for genetically, as they are repeated throughout the tree.
Emergent Shape
Returning to the broader mystery of fractal growth, we now have a tentative solution to why and how trees follow this fractal growth pattern; it is highly efficient at solving the problem of maximizing access to resources for each part of the tree, while relying on a simple growth strategy that can be genetically coded quite simply.
To visualize how localized growth strategies give rise to shapes that are highly efficient designs for particular environments, let’s examine three common tree shapes that arise in different contexts, and analyze how branching out in search of sunlight is a powerful solution despite its simplicity.
First, let’s look at the branching structure of trees in a dense forest. As seen in the image below, trees in this environment typically have few branches near the ground, preferring instead to focus their resources on stretching their long trunks up towards the canopy, where they have more direct sun access.
Next, let’s look at trees growing at the edge of forests, such as trees that grow near river banks. In such cases, the branches of trees will be lopsided, favoring growth towards the empty space where there is less competition for sunlight.
This effect can sometimes be extended to form quite remarkable shapes that reach far out across waterways, where there is little competition for sunlight
Finally, let’s look at solitary trees. These trees do not have competition for sunlight in any direction, so they continue branching out, forming an expansive crown
In each of these cases, we see that the branching structure trees develop is an elegant solution to the sunlight access problem they encounter in the environment. Trees are not born knowing the branching structure that will be most efficient for them to adopt; rather this structure is “learned” from the experience of the branches over the lifecycle of the tree. In this way, a simple recursive growth pattern to expand towards areas with rich access to resources provides a compact means of solving the complex problem of what shape the tree should adopt to best match its environment.
When you look at trees now, I hope you’ll have a deeper appreciation for how their structure is a product of their environment.
Across Nature
We have seen how the fractal design of trees is an elegant solution to the problem of maximizing access to sunlight while minimizing the material required to acquire it. If we assume form follows function, could we perhaps generalize this argument to other fractal structures in living systems?
For circulatory systems, they “want” to maximize surface area available to exchange oxygen throughout the body, while minimizing the volume of the body taken up by delivering blood.
Similarly, our lungs “want” to maximize surface area available to exchange oxygen from the air we breathe, while keeping the lungs as small as possible.
What about neurons? We know miniaturization is important so the brain fits inside the skull, so minimizing volume is important, but could it be that neurons seek to maximize their area to enable the greatest information exchange? I suspect so, see here if you want a deep dive on how this process may operate within the brain. In any case, from the diverse shapes of neurons, if they are shaped by similar forces as those we’ve explored, those forces operate differently in different areas of the brain, as seen in the diversity of their shapes.
Non-living Systems
We still have not addressed the mystery of how these fractal structures occur in non-living things, like river systems or galaxy clusters. To do so, we need to invert our growth picture we described for trees; rather than thinking about the problem of outward growth toward resources, we instead should focus on where resources are drawn when they fall inwards.
In the case of river systems, when water first falls, it will naturally flow downhill to the nearest rivulet. These rivulets will be attracted to lower places yet, combining at the lowest points. As these streams grow in size, flowing down to the lowest available point, they will continue combining, with larger streams naturally attracting more water due to likely being lower down. Preferential attachment is the name given to this tendency for the growth of fractal structures to be proportionate to their size, and is a key feature that gives rise to scale invariance.
This same preferential attachment also explains why fractals occur at the scale of galaxy superclusters; larger clusters will tend to attract surrounding matter more strongly, and so the larger a structure grows, the faster it will grow.
This symmetry between living things growing out versus non-living things falling in is no mistake. In mathematics, many problems have what’s called a dual description, an equivalent problem that is phrased differently. For instance, rather than maximizing a function, you could simply minimize its opposite. Dual descriptions can often clarify the nature of the mathematical structure you wish to study.
In the case of fractals, the idea of preferential attachment provides a bridge between views of living and non-living things. You can look at this as resources falling inwards from a state of high potential to low potential energy, or you can take the dual view of living things growing to reach out along the channels where resources fall, seeking out the points where these resources are attracted most.
Why Not Fractals All the Way Down?
Finally, you may reasonably ask: if these fractal structures occur at the scale of galaxy supercluster, and at the scale of things like river systems, why is it not fractals all the way down? Why do we have isolated structures like spiral galaxies with unique structures all their own?
To answer, recall again the defining characteristic of fractals: scale invariance. They form because the forces shaping them operate the same at small scales as at larger scales. In the case of galaxy clusters, it meant more massive objects attracting more objects to them. For living things, it meant branches following the same growth pattern at all scales. We can rephrase our question, then, from why don’t galaxies have a fractal structure, to instead: why don’t galaxies operate under the same rules at all scales?
Here, we can give a surprisingly simple answer: black holes. Place too much matter in too small a place, and the space itself can no longer support any internal structure. Rather, space itself collapses, with no evident internal structure that can influence the surrounding matter. The reason there are not fractals all the way down, then, is that there is a discontinuity caused by a difference in the forces shaping the evolution of structure at different scales.
Reflect
While we’ve covered a lot of ground, I hope you’ve come to a deeper appreciation for the role fractals play in many of the structures we see in the natural world, and how those structures emerge. In particular, we’ve seen:
- How scale invariance is one of the defining characteristics of fractals
- How fractal design helps maximize the surface for collecting or dispensing resources while minimizing the volume required to construct it
- How recursion enables the representation of fractal design in a compact form, enabling such structures to evolve in living things
- How living things guide their growth to the most promising areas, and the emergent structures that result
- How widespread the fractal design is in nature to solving the problem of maximizing a surface to enable resource exchange while minimizing volume of construction
- How the dual description of preferential attachment of material falling inwards can clarify why such structures occur in non-living systems as well
- Why the lack of internal structure for black holes interrupts the fractal structure of the universe at larger scales, giving rise to galaxies
