The algebra of the earth that reveals its multifunctional genius

The idea of multifunctionality is a wonderful concept. Its key to understanding how a system functions integrally.

A tree has many different roles. For instance, it plays a role in both the carbon cycle and the water cycle. It breathes in carbon dioxide, turns it into complex carbohydrates, and then it decays and that carbon gets used in soil. It also takes in water, uses water in photosynthesis which helps with making the complex carbohydrates, it transpires water, which plays a role in clouds, transpiration cooling, greenhouse gas, and it pushes down water during wet season and draws it up during dry season.

Each of these roles supports the others. The carbon the tree captures gets built into biomass using water. That biomass becomes leaves that shade the soil, keeping it moist. The moist soil hosts bacteria that help the tree access nutrients. The tree’s transpiration helps create the rain that refills the soil moisture. It’s a web where every function makes every other function work better.

A floodplain likewise plays many roles. It slows water, where the soil and plants can cleanse the water. That water can filter down into aquifers, which can be brought up by trees during dry season to hydrate the landscape. It creates wetlands which support insects and fish. And here again, each role reinforces the others: the cleaner water creates better habitat, the insects support the plants that slow the next flood, the recharged aquifer keeps the system resilient through droughts.

Soil is likewise multifunctional. It’s a sponge for rainwater, a filter for toxins, a habitat for rich microbial and insect life, a pharmacy for antibiotics, a nutrient provider for plants. The microbes create the sponge structure, the sponge structure enables the filtration, the filtration protects the microbes. Each function depends on and strengthens the others.

Permaculture has made multifunctionality one of its key tenets. It looks for solutions that have multiple purposes, e.g., placing logs or branches in the landscape in a process called hugelkultur, a process which can work to slow water and which also decays over a long time, providing a habitat for fungi and decomposing to provide carbon to enrich the soil. One intervention, multiple benefits, each benefit supporting the next.

This kind of design, where one element does many jobs and the jobs reinforce each other, seems to be nature’s signature approach. It’s how healthy systems work. And increasingly, it’s how the most successful ecological restoration works. When you restore a degraded landscape by reintroducing the right elements, trees, soil bacteria, water flows, you’re not just fixing one problem. You’re restoring multifunctionality, and that multifunctionality creates cascading benefits that spread through the system.

Rewilding efforts demonstrate multifunctionality. When wolves were reintroduced to Yellowstone, they reduced deer populations that had been overgrazing young trees. With more trees able to grow, beavers had the materials to build natural dams. Those dams created wetlands that recharged aquifers and changed the course of rivers, providing more water for the whole forest system. The wolves weren’t just controlling deer, they were part of a chain of operations affecting trees, water, soil, and the entire landscape. These animals and plants are playing multiple roles in the functioning of the Earth system.

Which raises an interesting question: is there a way to develop a more formal language for this multifunctionality? Is there a way to mathematicize it so we can understand it more precisely, predict it, and design with it intentionally?

The reason a mathematical framework is helpful isn’t necessarily just academic. It’s that our minds have trouble tracking multiple simultaneous effects. We can easily think “the tree captures carbon” or “the tree moves water.” But thinking about how the tree simultaneously affects carbon AND water AND soil AND climate, with each effect influencing the others, that’s much harder to hold in your head.

And when we can’t see all the connections at once, we make decisions as if they don’t exist. We need a way to see the whole web of effects at once.

Mathematics gives us that. Not because math is better than intuition, but because it forces us to write everything down. It makes the invisible visible. And once you can see all the connections on paper, you can start to design solutions that work with them instead of against them.

Representing more formally

Here’s one way to think about it. First, we need to represent the state of a system. I’ll start with a simplified example.

Imagine we want to track the carbon cycle. We could break it into components: the carbon dioxide in the air, the carbon in trees, and the carbon in the soil. Let’s call these three components our “carbon values.” Similarly, for the water cycle, we might track: water in the air, water in trees, water in soil, and water in underground aquifers. These are our “water values.”

Now imagine a row of mail slots in an office. In the first slot, we put the current amount of atmospheric carbon. In the second slot, the carbon in trees. In the third, the carbon in soil. In the fourth slot, water in the air. Fifth slot, water in trees. Sixth, water in soil. Seventh, water in aquifers.

This row of slots with all their values, that’s what mathematicians call a vector. It’s just a way of representing the state of a system at any given moment. All the important values, lined up so you can see them at once.

Now think about what happens when a tree grows. It doesn’t just change one slot, it changes multiple slots at once.

The tree pulls carbon dioxide from the air, so the atmospheric carbon slot decreases. The carbon in the tree slot increases as it builds biomass. When leaves fall and decay, the soil carbon slot increases. At the same time, during dry season, the tree draws water up from the aquifer and soil, so those slots decrease while the tree water slot increases. Then the tree transpires, so the atmospheric water slot increases.

The tree is operating on the entire row of slots simultaneously. It’s transforming the whole state of the system.

We can represent this transformation as a set of rules, what mathematicians call a matrix. The matrix is just a systematic way of writing down: “When a tree acts on the system, here’s what happens to each slot.” It’s the recipe for how the tree transforms the state.

This might seem like a complicated way to describe something simple. But here’s why it’s powerful: once you write it down this way, you can see something crucial. The tree doesn’t just affect carbon and water separately. It couples them together. You can’t change the carbon slots without changing the water slots, because photosynthesis uses both. You can’t change the water slots without changing the carbon slots, because the tree needs its carbon-built structure to move water.

The mathematical framework makes this coupling visible. It shows you that the carbon cycle and water cycle aren’t separate systems. They’re interlocked, and the tree is the mechanism that interlocks them.

We can see this coupling principle at work in other contexts too. Consider what happens when you slow water using earthworks like swales or check dams. Fast water rushing across the landscape operates primarily on the water slots in our vector: it moves water from soil to rivers to ocean, but it leaves the system before much else can happen. But when you slow that water down, the operation changes. Now the slow-water matrix operates on both water and carbon slots simultaneously. The water has time to infiltrate, so the soil water slot increases. That water is now available for plants and soil microbes to use in photosynthesis and biological processes, so the tree carbon slot and soil carbon slot increase. The slow water is enabling the coupling between water and carbon that fast water prevents. It’s creating the conditions for the two cycles to work together. This is matrix formulation makes these couplings explicit. You can see that the slow-water operation isn’t just moving water around, it’s changing how water and carbon interact with each other.

Now we can use this same framework to look at human interventions. Let’s take desalinization and aqueducts as an example.

Usually, we think about these technologies only in terms of water. Desalination takes ocean water and produces fresh water. Aqueducts move that water to where it’s needed. We consider them as operating only on the water cycle.

But that’s not the full picture. Desalinization requires enormous amounts of energy. That energy comes from burning fuel, which means releasing carbon into the atmosphere. Aqueducts require massive pumps running continuously to move water uphill over mountain ranges. In California, 20% of the state’s energy goes to cleaning and moving water around. That’s a lot of carbon emissions. Those emissions can lead to global warming, which in turn causes more droughts in some areas.

So when we write down what desalinization actually does to our system, when we look at all the slots, not just the water ones, we see it’s also increasing the atmospheric carbon slot. It’s operating on both the water cycle and the carbon cycle simultaneously.

This is what we mean by looking at the “full vector space” that a solution operates on. Instead of only asking “what does this do to water?” we ask “what does this do to water and carbon and energy?” We’re forced to see the whole transformation, not just the part we’re focused on.

But there’s something even more important the mathematical framework reveals: ripple effects through time.

Let’s say we extract groundwater. That decreases the aquifer slot in our system state. Now we have a new state, one where the aquifer is lower.

When the tree operates on this new state, something different happens. There’s not enough water for the tree to draw up during dry season, so the tree water slot stays lower. The tree becomes stressed.

Now lightning strikes, something that happens naturally all the time. Lightning might hit dry grass or a dead branch, starting a fire that spreads through the landscape. When fire encounters a healthy, well-hydrated forest, it might burn slowly as a small fire that clears underbrush. But when fire encounters stressed, dried-out trees in a degraded landscape, you get a major conflagration. The fire operation now decreases the tree carbon slot dramatically as the forest burns.

We can keep tracing forward: each operation creates a new state, and that new state determines what the next operation will do. This is what we mean by a chain of operations. The groundwater extraction operation acts on the original state. The tree operation acts on that result. The lightning operation acts on that result. The fire operation acts on that result. Each operation in the sequence acts on the state created by the previous operation. This is how ripple effects propagate through a system.

The mathematical framework can help clarify: one might think groundwater extraction only affects the water cycle. But when you trace through the chain of operations, you see it ultimately affects the carbon cycle too. The connections aren’t always direct, but they’re there, and the math helps you trace them.

The Deep Structure: Groups and Ripple Effects

These operations, tree, bacteria, rain, fire, lightning, they don’t just act in sequences. They have relationships with each other. They form what mathematicians call a group structure.

To understand what a group is, think about the seasonal water cycle. In wet season, rain pushes water down into the soil and aquifers. In dry season, trees pull that water back up from the aquifers and release it into the air through transpiration. The wet season operation and the dry season operation are inverses of each other, they undo what the other does. If you go through four seasons (wet, dry, wet, dry), the water has cycled through the system and returned to a similar state as where it began. These seasonal transformations, the set of operations that cycle the water through the system, form a group.

Different ecosystems have different group structures. A tropical rainforest has different seasonal water operations than a desert. A forest that burns every 50 years has a different fire-succession group than grassland that burns every 3 years. The structure of possible transformations is different.

Now, the operations in an ecosystem also form a group structure. But instead of being just about cycles that return to the same state, it’s about the structure of how transformations can combine and propagate.

When you combine bacteria operation with tree operation, you get soil-building. When you combine tree operation with rain operation, you get hydraulic redistribution. These combinations follow rules. Some operations can be reversed: deforestation undoes what forest-building does. Some operations, when combined in different orders, lead to the same final state. Some operations amplify each other, some cancel each other out.

The group structure captures all of these relationships. It’s the deep mathematical pattern underlying how these operations relate to each other.

And here’s why the group structure matters for understanding Earth systems: it tells you how far ripples can spread.

When you cut down a forest in one watershed, you’re applying a deforestation operation. That operation can combine with a drought operation (less water because no trees to cycle it). The drought operation can combine with a fire operation (dry landscape burns). The fire operation can combine with an erosion operation (nothing holding soil). The erosion operation can combine with a sedimentation operation (rivers clog with soil). The sedimentation operation affects river temperatures, which affects fish populations, which affects ocean ecosystems.

Each operation creates a state that the next operation acts upon. The effects cascade through the system, potentially spreading across continents and oceans.

The group structure is the mathematical description of all possible cascades. It tells you: given these basic operations (tree, bacteria, rain, fire, drought, etc.), what are all the possible ways they can combine? What are the full set of states you can reach by chaining operations together? How far can a local change propagate through the planetary system?

This is what we mean by “the structure of ripple effects.” The group is the pattern of how transformations propagate through Earth systems, the web of possible consequences.

Understanding this structure helps us see that local actions and planetary processes aren’t separate. They’re connected through chains of operations. A decision about land use in one place combines with climate operations, with ocean operations, with atmospheric operations, cascading across the whole Earth system according to the rules of the group structure.

Here’s what makes the group structure perspective particularly illuminating for geoengineering: it reveals why these interventions so often create problems even when they “work.”

Natural operations form what mathematicians call a closed group. You can combine them (Tree · Bacteria · Rain) and they create states that other natural operations can work with effectively. The operations compose well together. But geoengineering operations often don’t fit cleanly into this structure. Not only do levees create more flooding down stream as it causes rivers to speed up, but a levee blocks floodplains from spreading water across the landscape. This stops the aquifer from being recharged during floods. The levee creates a degraded state with depleted aquifer and poor soil that’s incompatible with what natural operations expect. When rain then falls on this degraded state, it can’t recharge aquifers properly because the floodplain pathway is blocked. When trees operate on this state, they can’t access enough water and become stressed. The natural operators are designed to work with floodplains; remove the floodplain and the whole chain breaks down.

This is why geoengineering solutions often seem to solve one problem while creating several others. They’re introducing operations that don’t compose well with the natural group structure. They create states that make subsequent natural operations (drought, fire, rain) more destructive rather than regenerative. Understanding the group structure helps us see this incompatibility mathematically, rather than discovering it through costly trial and error.

Ecological restoration that works with multifunctionality takes the opposite approach. When you restore a floodplain, you’re not just solving flooding, you’re reintroducing an operation that composes well with the natural group structure. The floodplain operation creates a state (recharged aquifer, enriched soil, thriving biodiversity) that makes subsequent natural operations more effective. Rain on a restored floodplain recharges aquifers. Trees on a restored floodplain access deep water during droughts. The restoration creates beneficial cascades rather than destructive ones. This is why ecological restoration often produces multiple benefits that weren’t explicitly planned for, the restored operations naturally compose with other operations in the system to create healthy states that propagate forward through time.

So I believe these ideas, vectors to represent states, matrices to represent operations, chains to represent ripple effects, and groups to represent the deep structure of how operations relate, give us the beginnings of a way to more formally talk about multifunctionality.

But we can also use these concepts informally, without needing to do the actual mathematics. We can ask: Are we looking at the full state space that a solution operates on? Are we considering all the slots it affects, not just the one we’re focused on? Are we thinking about the chain of operations that will follow? Are we asking how this operation will combine with other operations in the system?

These questions come naturally within this framework. It helps us see that multifunctional solutions aren’t just doing more things. They’re operating in ways that fit the group structure of natural systems. They’re creating states that make subsequent operations more beneficial rather than more destructive. They’re working with the deep patterns of how transformations propagate, rather than against them.

I think these group structure ideas, combined with coupled system concepts (introduced two essays back) and other frameworks from complex systems theory (to be discussed in future issues), can help us develop a more integral and holistic way of understanding the Earth system and how better to restore it.

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I started thinking about these group structure ideas earlier this year, and found out during the summer, when we both published something on it within hours of each other, that Ali Bin Shahid had also been thinking about these group ideas applied to nature and our planetary systems.