Complexity Science

Complexity science is the study of how large systems made up of many interacting parts give rise to emergent behaviors — patterns, structures, or dynamics that cannot be understood by analyzing the parts in isolation. It seeks to understand how order, adaptation, and organization emerge in natural, social, and technological systems — from ant colonies and ecosystems to economies, cities, and human societies. Complexity science studies systems that are more than the sum of their parts.

How Living Things Stay “Themselves” (Even When Everything Inside Changes)

December 14, 2025December 14, 2025 ~ Manuel Castañón-Puga

You know that weird moment when you’ve had a rough week, skipped sleep, eaten random snacks, and still wake up feeling like… you’re still you? The paper by Varela, Maturana, and Uribe digs into that everyday mystery from a very grounded angle: not “what are living things made of?”, but “what kind of ongoing pattern makes something a living unit at all?” Their starting point is that focusing only on parts can miss what actually matters: the organization that makes the whole hold together as one distinct “someone” or “something,” whether or not it’s reproducing.

Their key idea is called autopoiesis, which basically means “self-making.” A living system, in their description, is a network of processes that continually produces the very components that sustain the network. At the same time, it builds and maintains a boundary that makes it a recognizable unit in its space. They use the cell as the easy example: a vast web of chemical reactions keeps making molecules that keep those reactions possible, and together those molecules keep the cell as a physical, separate “thing,” even though the actual matter inside is constantly being replaced. In that picture, what makes something alive is not a specific ingredient, but a looping, self-maintaining organization. That’s also why they contrast living systems with allopoietic ones: many machines produce something other than themselves, while an autopoietic system’s “output” is basically its own continued existence as that same kind of unity.

This also changes the way we think about reproduction. The authors argue that reproduction and evolution are essential. Still, they aren’t the basic definition of being alive, because you can’t reproduce unless you already have a living unity to reproduce in the first place. In their view, reproduction happens as a special case of this self-maintaining organization: the unit can split so that the same kind of self-producing network continues in two fragments. To make the idea less abstract, they present a minimal computer model in a simple grid-world: elements bump around randomly, a “catalyst” helps create “links,” and links bond into chains. Sometimes a chain closes into a loop that traps the catalyst inside. Once that happens, new links formed inside can replace boundary links that fall apart. Hence, the boundary stays intact even though parts keep turning over—like fixing a fence plank-by-plank without ever letting the yard stop being enclosed. They even give a practical “checklist” style key (six points) for deciding if something counts as autopoietic: can you find a boundary, identify components, see a rule-governed system, confirm the boundary is produced by interactions of elements, and confirm the components (including boundary ones) are continually made and participate in creating others.

The everyday takeaway is surprisingly useful: it’s a reminder to look for the pattern that keeps something going, not just the content itself. Your body, habits, relationships, even a group project, can “feel alive” or “fall apart” depending on whether the ongoing loop that sustains it is still running—and whether there’s a boundary that protects that loop from getting wrecked by every outside bump. In the paper, when the network of production breaks, the unity disintegrates; when it can compensate for disturbances, it stays autonomous. That’s a simple lens you can apply day-to-day: if you want a system (you, a routine, a shared apartment, a club) to stay stable while everything changes, focus on the repeating actions that rebuild the structure and the boundary conditions that make those actions possible. The authors even point toward how this thinking could guide attempts to build “life-like” systems in chemistry, like imagining a bubble-like structure whose membrane components are produced or modified by reactions that happen within the special conditions created by the membrane itself—because what matters most is not the material, but the self-maintaining loop that makes a unit a unit.

Reference:
Varela, F. G., Maturana, H. R., & Uribe, R. (1974). Autopoiesis: The organization of living systems, its characterization and a model. Biosystems, 5(4), 187–196. https://doi.org/10.1016/0303-2647(74)90031-8

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When Small Changes Create Big Patterns: The Everyday Magic of Synergetics

December 13, 2025December 13, 2025 ~ Manuel Castañón-Puga

You know that moment when a group chat is quiet. Then one tiny thing happens—someone posts a plan, a meme, or a rumor—and suddenly everyone is talking, making decisions, and the whole “vibe” shifts? Haken says science often sees this kind of jump too: lots of small parts (atoms, cells, people, machines) can start “working together” in a way that creates a clear, large-scale pattern, and that pattern can change suddenly when you tweak just one outside condition. Synergetics is his term for studying these shared “pattern-jump” moments across multiple fields, rather than keeping each subject in its own separate box.

To illustrate this, Haken points to systems that appear totally unrelated but behave in surprisingly similar ways when they cross a tipping point. A laser, with low power, acts like a regular lamp with messy, short light bursts. Still, past a specific input, it suddenly produces a long, steady beam because the tiny “oscillators” inside fall into step together. A fluid heated from below can transition from calm heat flow to organized motion, characterized by neat rolls or hexagons. Some chemical reactions can flip into rhythmic color changes or ring-like patterns. Even a loaded structure, like a thin shell, can go from smooth to a buckled pattern with repeating cells. The point isn’t the details of each example; it’s that “order” can appear because the parts begin cooperating, and the switch can be dramatic.

So, how do you discuss “order” without tracking every single part? Haken’s answer is the idea of an order parameter: a small set of numbers that captures the big pattern you actually notice (like “how magnetized” something is, or “how strong” a large-scale wave is). In contrast, the countless minor details fade into the background. He explains a simple yet powerful trick: often, the slow, significant changes (the order parameter) end up steering the faster, less important ones, so the small parts quickly “fall in line” with the larger pattern. This also helps explain why different outcomes can compete. Sometimes several possible patterns are “almost ready” at the same time, and the final result depends on how those options cooperate or fight each other, plus the system’s starting point and random little pushes from fluctuations. In everyday terms, it’s like a team project where a few key decisions set the direction, and everyone else adjusts. Yet, a small bit of randomness (who speaks first, a sudden deadline, a surprise idea) can decide which plan wins.

Once a new pattern exists, Haken says it isn’t the end of the story. If you continually change the outside conditions, the latest pattern can also become unstable, leading to a whole chain of changes: more complex patterns, pulses, and sometimes irregular, chaotic behavior that appears in very different systems. And randomness isn’t just “noise” to ignore—it can help a finite system explore different stable options, which matters for things like reliability, adaptability, and switching (basically, whether something can change modes without breaking). The big everyday takeaway from Haken’s synergetics is this: when you’re looking at a complicated situation—your habits, a friend group, a crowded campus, a workplace—don’t assume you must understand every tiny detail to understand the outcome. Often, a few “big knobs” (the effective order parameters) and a few tipping points explain why patterns form, why they suddenly shift, and why a slight push at the right moment can change everything.

Reference:
Haken, H. (1984). Synergetics. Physica B+C, 127(1–3), 26–36. https://doi.org/10.1016/S0378-4363(84)80007-8

Why Big Changes Often Happen “All at Once” (and What That Means for You)

December 12, 2025December 12, 2025 ~ Manuel Castañón-Puga

You know that feeling when nothing seems to change for ages—same routine, same habits, same you—and then suddenly a lot shifts at once? Perhaps a new friend group forms quickly, you switch majors, you move to a new city, or one tough week forces you to get serious about sleep and financial management. Gould and Eldredge argue that the history of life often follows this pattern: long periods where a species remains essentially unchanged, with only minor fluctuations, followed by brief periods during which a new species emerges and change is concentrated.

In their “punctuated equilibria” view, the main action isn’t a slow, steady makeover happening little by little inside one significant, stable population. Instead, the most noticeable change is tied to speciation—when one line splits and a new species forms—often in small, isolated groups, and on the fossil record clock, which can appear almost instantaneous. The practical takeaway is surprisingly personal: if you expect your life to improve in a perfectly smooth line, you’ll feel like you’re failing whenever progress comes in jumps. Tracking your habits or journaling can reveal weeks of stability, which are just as crucial as the leaps, helping you understand what’s steady, fragile, and triggers change.

They also warn that people can get fooled by their own expectations. Paleontologists, they argue, spent a long time hunting only for slow, steady change and treating gaps and standstills as mistakes or “nothing to see,” which made the “always gradual” story feel more common than it really was. This highlights a bias: if you only look for one type of evidence, you’ll confirm what you already believe. To make more intelligent decisions about fitness, studying, or relationships, try shifting your focus-notice the quiet periods, not just the dramatic moments. Recognizing that significant leaps often follow specific conditions, such as a new environment or constraint, can help you understand your own patterns more effectively.

Finally, Gould and Eldredge say that significant trends can emerge even if individual species don’t slowly transform step by step. A trend can emerge from which species thrive and spread, while others decline. So, the “direction” you see later is more like the result of winners piling up than a single, continuous self-improvement story. They even point to human evolution as an example where they report no detected gradual change within a given hominid group. At the same time, a long-term pattern, such as the development of larger brains, could result from the success of different, mostly stable groups over time. If you bring that down to daily life, it suggests a kinder way to read your own “trend.” You don’t have to force yourself to become a brand-new person through nonstop tiny upgrades. Sometimes the real change comes from choosing a new “branch”: a new circle, a new routine, a new project that fits better—and then letting the stable parts of you do their job until the next meaningful jump.

Reference:
Gould, S. J., & Eldredge, N. (1977). Punctuated equilibria: the tempo and mode of evolution reconsidered. Paleobiology, 3, 115–151. https://www.jstor.org/stable/2400177

Why One Number Is Not Enough to Describe a Messy World

November 26, 2025November 28, 2025 ~ Manuel Castañón-Puga

Imagine a group chat with dozens of friends trying to pick a place to go on Friday night. One person wants to watch a movie, another insists on karaoke, a few only care that it’s affordable, and someone else wants good photos for social media. The more people join, the more tangled the conversation becomes. If you tried to “summarize” the whole chat with just one number that says how close everyone is to agreeing, it would feel a bit ridiculous. The situation is messy, with mini-groups, shifting alliances, and half-made plans. That, in a strange way, is close to what physicists face when they study certain messy materials called spin glasses, and it is precisely the kind of problem Parisi dives into.

Spin glasses are special kinds of magnets where tiny atomic “arrows” don’t all want to point in the same direction. Some pairs prefer to align, others prefer to oppose each other. Hence, the material ends up frustrated and disordered, more like your chaotic group chat than a neat row of soldiers. Earlier, scientists such as Edwards, Anderson, Sherrington, and Kirkpatrick attempted to describe this complex system with a single “order parameter,” essentially a single number that indicates the degree of order in the system. But when they pushed their formulas, they ran into absurd results, like predicting a negative entropy at very low temperature—roughly like saying a playlist has less than zero possible song orders. It was a sign that the description was too simple. Parisi’s key move is to show that, in a careful mathematical treatment called the replica approach, you don’t just need one parameter to describe a spin glass—you actually need an infinite number of them. Instead of one rating, you get a whole curve. This function indicates the similarity between different possible internal arrangements of the material.

Parisi builds this step by step. He starts with the old “one-number” picture, then lets the system be described by more and more parameters—first 1, then 3, then 5, and so on—constantly checking how the predicted energy and other properties behave near the critical temperature where the spin glass appears. With only a few parameters, the description already improves significantly, aligning well with more detailed calculations and computer simulations, and the previously observed negative entropy almost vanishes. As the number of parameters grows, the description approaches a limit where the internal structure of the spin glass is encoded in that function of one variable, defined between 0 and 1. In everyday terms, it’s like going from rating a movie with a single score out of 10 to having a whole profile: acting, soundtrack, story, cinematography, and then even more fine-grained sub-scores. The material is not captured by one label but by an entire landscape of overlapping “moods.”

The interesting part is what this hints at beyond the realm of physics. The function Parisi introduces contains all the information needed to compute physical quantities, such as how strongly the material responds to a magnetic field or how much energy it stores; however, its deeper meaning is still not fully clear in the paper. That uncertainty is a reminder that, in real life, too, complex systems—from your social circle to your mental health or even an online community—are rarely described well by a single number, such as a score, rank, or average. We often try anyway: GPA, follower count, likes, and a single “happiness” scale. Parisi’s work quietly suggests another attitude: when reality is messy and conflicted, expect it to need many parameters, perhaps even a whole continuous spectrum, to be described fairly. Instead of asking “What’s the one number that sums this up?”, we can ask “What is the shape of the whole picture?” Learning to think that way can make us more careful with statistics, more skeptical of oversimplified rankings, and more understanding of materials in a lab, and of people in our lives who, like spin glasses, don’t fit neatly into a single box.

Reference:
Parisi, G. (1979). Infinite Number of Order Parameters for Spin-Glasses. Physical Review Letters, 43(23), 1754–1756. https://doi.org/10.1103/PhysRevLett.43.1754

Why Helping Others (and Expecting It Back) Works Better Than You Think

November 24, 2025November 24, 2025 ~ Manuel Castañón-Puga

Imagine you’re working on a group project. Everyone promises to do their part, but you’ve been burned before—someone slacks off, and suddenly you’re carrying the whole load. Still, once in a while, you meet someone who matches your effort. You help them, and they help you; suddenly, the whole project feels smoother, even fun. That simple loop—I help you, you help me—is more potent than it looks. According to Axelrod and Hamilton, cooperation can flourish even in a world where everyone is trying to get ahead, as long as the same individuals meet repeatedly. They compare this to a game where you choose between helping (cooperating) or taking advantage (defecting). A strategy called “tit for tat”—start by cooperating, then copy whatever the other person did last time—turned out surprisingly effective in their simulations. It wasn’t fancy; it was just friendly, firm, and forgiving, and that was enough to thrive among many different types of players.

Think of a simple example: two students who see each other daily at school. If one person helps another with notes today, the other is more likely to return the favor tomorrow. But if someone takes advantage—say, copying homework and giving nothing back—they’ll quickly face the consequences when the other person withdraws support. Axelrod and Hamilton demonstrate that cooperation is most effective when future interactions are likely. The more you expect to see someone again—friends, classmates, teammates—the more valuable it becomes to treat them fairly. It’s the same reason long-term friendships or stable online communities tend to be kinder: people know their actions will come back to them.

Authors also explain that cooperation often begins within small groups. Even if most people around you act selfishly, a tight-knit circle that consistently helps each other can influence the wider environment. This is why friend groups, clubs, or study teams can create pockets of trust even in competitive settings. Over time, the benefits of mutual support become evident, encouraging more cooperation. Recognizing one another also plays a key role-just as animals rely on scent or territory, humans use faces, names, and digital identities. Once you know who treated you well, you can return kindness to the right person—and avoid rewarding those who didn’t.

In everyday life, this theory encourages long-term thinking and planning by showing how cooperation builds lasting relationships. A small act of generosity can initiate a chain of positive responses, while taking advantage of someone might lead to a quick gain but can damage future opportunities. The work of Axelrod and Hamilton reminds us that cooperation is not naïve; it’s strategic. Being helpful, responding firmly to unfairness, and being willing to forgive are not just moral choices; they are effective ways to strengthen bonds over time. Whether you are working on school projects, dealing with roommates, or navigating social circles, choosing to cooperate first—and maintaining a fair approach afterward—can make life smoother, more productive, and much more satisfying.

Reference:
Axelrod, R., & Hamilton, W. D. (1981). The Evolution of Cooperation. Science, 211(4489), 1390–1396. https://doi.org/10.1126/science.7466396