The S-Curve of Everything

The Lily Pad Puzzle

Here's a riddle that has been humbling smart people at cocktail parties for decades.

If you said day 15, congratulations — you're in excellent company and completely wrong. The answer is day 29. One day before total coverage. Because if the lily pad doubles every day, and it fills the whole pond on day 30, then the day before it must have been exactly half.

This is the puzzle that launched a thousand blog posts about exponential growth, and it's a genuinely useful exercise in mathematical thinking. But here's what those blog posts almost never tell you: the lily pad would never actually cover the whole pond.

Not because the math is wrong — the math is perfectly fine. But because nothing in the real world grows exponentially forever. At some point the lily pad runs out of nutrients, or sunlight, or space. It slows down. It plateaus. It traces not a soaring exponential curve but something much more interesting: an S-curve.

The exponential is a mathematical fantasy. The S-curve is what actually happens.

Verhulst's Brilliant Correction

In 1798, Thomas Malthus published his famous Essay on the Principle of Population, arguing that human population grows exponentially while food supply grows only linearly. The conclusion was grim: famine, disease, and war were mathematically inevitable.1

Malthus wasn't stupid. His exponential model was a reasonable first approximation — and to his credit, it made a specific, testable prediction about the future. But forty years later, a Belgian mathematician named Pierre-François Verhulst looked at the same problem and asked a better question: what happens when the population starts to run out of room?2

Verhulst's insight was elegant. Instead of assuming growth rate stays constant, he made it shrink as the population approached some ceiling — the carrying capacity, which he called K. The result was the logistic equation:

P

Current population

r

Intrinsic growth rate (how fast it grows when unconstrained)

K

Carrying capacity (the ceiling)

1 − P/K

The braking factor: near 1 when P is small, near 0 when P approaches K

That little term (1 − P/K) is doing all the interesting work. When the population is tiny compared to K, this factor is essentially 1, and you get plain exponential growth — Malthus's nightmare scenario. But as P climbs toward K, the brake tightens. Growth slows. And the population asymptotically approaches K without ever quite reaching it.

Think of it like a nightclub. When the first few people arrive, there's space everywhere — you can dance, grab drinks, find your friends. Growth feels unlimited. But as the room fills up, every new arrival makes things a little worse for everyone. The music hasn't changed, the venue hasn't shrunk, but the effective experience has degraded. Eventually, people stop coming. Not because they don't want to — because they can't get through the door.

The result is the S-curve (or sigmoid) — slow start, explosive middle, gentle plateau. It's one of the most important shapes in all of applied mathematics, and once you learn to see it, you'll see it everywhere.

The Pandemic S-Curve

In March 2020, when COVID-19 case counts were doubling every three days, well-meaning data scientists projected horrifying exponential curves into the future. Some models predicted tens of millions of U.S. cases by April.3

They weren't wrong about the math. They were wrong about the model. Pure exponential growth assumes an infinite supply of susceptible people, which is — to put it technically — not how people work. Every epidemic eventually runs into its carrying capacity: the point where enough people have been infected (or vaccinated, or socially distanced) that new infections slow down.

The SIR model that epidemiologists actually use is a close cousin of the logistic equation. It produces the same S-shaped curve: explosive early growth that bends over and plateaus.4 The crucial policy question was never "will this S-curve flatten?" — it always does — but "how high is K?" Interventions like lockdowns and masks didn't change the shape of the curve. They changed the carrying capacity.

This is a point worth lingering on, because it was widely misunderstood. "Flatten the curve" didn't mean "make the exponential stop being exponential." It meant "lower the effective ceiling." Push K down so that hospitals could handle the peak. The S-curve was always going to be an S-curve. The question was whether the inflection point would be above or below the capacity of the ICU system.

This is the deep problem with exponential extrapolation, and it applies far beyond pandemics. When you're on the steep part of the S-curve, looking backward, all you see is acceleration. The flattening hasn't happened yet. And so you project the trend forward in a straight line — or worse, an exponentially curving line — into absurdity.

Try It Yourself: The Logistic Growth Simulator

Play with the parameters and watch how they reshape the S-curve. Toggle the exponential overlay to see when — and how dramatically — the two curves diverge.

Diffusion of Everything

In 1962, sociologist Everett Rogers published Diffusion of Innovations, a book that would become one of the most cited works in social science.5 Rogers studied how new technologies spread through populations — from hybrid corn seeds among Iowa farmers to antibiotics in medical practice — and found the same pattern every time: an S-curve.

His famous adopter categories — innovators, early adopters, early majority, late majority, laggards — are just labels for different positions on the logistic curve. The innovators are the tiny initial population. The early majority hits around the inflection point, when growth is fastest. And the laggards are the people who finally buy a smartphone in 2024.

The pattern is strikingly consistent. Radio took about 38 years to reach 50 million users. Television did it in 13. The internet in 4. Facebook in 3.5. Pokémon Go in 19 days.6 The curves are getting steeper — the growth rate r is increasing — but they're all still S-curves. Every single one plateaus.

This matters because when you're a startup founder in the steep part of the curve, the world feels infinite. Your growth chart looks like a hockey stick. Venture capitalists are throwing money at you. And the temptation is overwhelming to extrapolate: if we grew 20% month-over-month for the last twelve months, surely we'll keep growing 20% month-over-month for the next twelve.

The most dangerous question in business: "Are we early-exponential, or mid-logistic?"

If you're early-exponential — if you're the lily pad on day 10 — then the best is yet to come, exponentially so. But if you're mid-logistic — if you're approaching the inflection point — then growth is about to slow, no matter how brilliant your product or how many engineers you hire. The data, maddeningly, looks the same in both cases.

Moore's Law is perhaps the grandest example. For half a century, the number of transistors on a chip doubled roughly every two years — an exponential run so long and so reliable that people started treating it as a law of physics rather than an empirical observation about an industry on the steep part of its S-curve. Physicists always knew there was a floor: you can't make a transistor smaller than an atom. The question was never whether Moore's Law would saturate, but when. And now, as chip manufacturers struggle with quantum tunneling effects at the 3-nanometer scale, the answer seems to be: right about now.9

Can You Spot the Plateau?

This game tests your intuition about S-curves. You'll see the early part of a growth curve — the part that looks exponential — and try to guess where it will plateau. Spoiler: you'll probably guess too high.

Overshoot and Collapse

The logistic equation assumes organisms are polite. They sense the carrying capacity approaching and gently ease off the accelerator. But real organisms — deer, bacteria, humans building suburbs — don't always get the memo.

When a population blows past K, something worse than a plateau happens: overshoot and collapse. The population exceeds what the environment can sustain, degrades the environment in the process (eating all the food, polluting the water, strip-mining the soil), and then crashes — often to well below where it started.7

The classic case is the reindeer of St. Matthew Island. In 1944, the U.S. Coast Guard introduced 29 reindeer to the remote Bering Sea island. With abundant lichen and no predators, the population exploded to around 6,000 by 1963. Two years later, it was 42. The reindeer had overshot their carrying capacity so dramatically that they destroyed their own food supply.8

Mathematically, you can model this by adding a delay to the logistic equation — letting the braking term respond not to the current population, but to the population some time ago. The result is no longer a smooth S-curve. Depending on the delay, you get oscillations (boom and bust cycles) or outright collapse. Ecologists see this pattern everywhere: lemming cycles, locust plagues, algal blooms that choke a lake and then vanish. The feedback is real, but it's not instantaneous, and that lag is where the chaos lives.

This is what Malthus got wrong — not the direction of his argument, but the shape. He predicted linear food vs. exponential people, leading to perpetual famine. Verhulst saw that the feedback loop would bend the exponential. But the deepest lesson is that even Verhulst's model is optimistic. The real world often overshoots before it corrects, and sometimes the correction is catastrophic.

So the next time someone shows you an exponential growth chart — whether it's a pandemic, a startup's revenue, a cryptocurrency's price, or a population curve — remember the lily pad puzzle. But then remember something the puzzle doesn't teach you: the pond has a size, the lily pad will find it, and if it's growing too fast to notice, things might get ugly before they get stable.

The exponential is the dream. The S-curve is the reality. And overshoot is the nightmare that happens when you confuse the two for too long.