Growth, from folding paper to AI that improves itself
Short scenes, each with something to press or a guess to make. They start with paper and pond weed, move through chain reactions and epidemics, and end at the equation people use for AI improving AI. The colors mean the same thing all the way through, here and on the other pages.
Adding vs. doubling
Fold a sheet of paper.
Paper is about a tenth of a millimeter thick. Fold it in half and the stack is twice as thick. Fold it again and it doubles again.
Guess firstHow many folds until the stack reaches the Moon?
Forty-two folds. Each fold multiplies the stack by two, so 42 folds multiply it by about 4.4 trillion. If each fold had added a tenth of a millimeter instead, the stack would be 4.3 millimeters tall.
Adding grows by the same amount each step. Doubling grows by the same factor each step. Growth by a fixed factor is called exponential.
Real paper gives up after about seven folds, and record attempts with paper kilometers long stop around a dozen. Limits come back later.
Doubling time
Every growth rate has a doubling time.
Something growing 7% a year doesn’t sound fast, but it doubles every 10 years. There’s a shortcut: divide 70 by the yearly percent. At 7% that’s 10 years, and at 2% it’s 35.
Guess firstA city grows 1% a year. About how long until it’s twice as big?
Tap the things under the chart to compare their doubling times.
About 70 years. Growing 1% a year for 70 years multiplies a thing by almost exactly 2, which is where the shortcut comes from.
For fast growth the shortcut breaks, so the chart uses exact numbers. Training compute for the biggest AI models has grown about 4 to 5 times a year since 2010, according to Epoch AI. That’s a doubling about every five and a half months.
Exponential growth
A lily pad doubles every day.
One pad makes two, two make four. Each day the lilies grow by as much as there already is. On day 30 they cover the whole pond.
Guess firstOn what day was the pond half covered?
Day 29. The cover doubles every day, so the day before it’s full, it’s half full. On day 25 the lilies cover 3% of the pond. Anyone looking at it would see open water and think nothing much was happening.
That’s the rule behind exponential growth. How much it grows depends on how much is already there, so the growth feeds itself. The bars under the pond show it: nearly all the change happens in the last five days.
Log scales
A ruler where every step is ×10.
On an ordinary chart, steady doubling hugs the floor for a long time and then shoots up the wall, like the bars under the pond. A log scale fixes that. Each gridline is ten times the one below. Most charts about AI growth use one.
Guess firstThese dots are transistor counts on computer chips from 1971 to 2023. On a log scale, what shape will they make?
A straight line. On a log scale, a straight line means a steady doubling time, and a steeper line means faster doubling. Transistor counts doubled about every two years for fifty years, from 2,300 on Intel’s 4004 in 1971 to 134 billion on Apple’s M2 Ultra in 2023. That’s Moore’s law.
Moore’s law is also a feedback loop. Every generation of chips runs the software used to design the next one. That loop gave steady doubling for half a century rather than an explosion, because each step also needed new factories, new physics, and more money.
Hitting a ceiling
Bacteria double until the food runs out.
This dish has food for about four thousand cells. Each cell splits in two every 20 minutes. So far the colony has doubled six times in a row.
Guess firstWill it keep doubling?
You can’t tell yet. The dashed line keeps doubling forever. The solid line slows as the food runs out and stops near four thousand. The two lines nearly match for the first eight doublings, so the early numbers can’t tell you which one you’re on.
Every real exponential meets a limit sooner or later, and the curve bends into an S. This shape has names: the S-curve, the logistic curve, or a sigmoid. Early on it looks exactly like doubling. Halfway up it is growing fastest. Near the top it flattens out. The question is always how far away the top is.
Stacked S-curves
One technology levels off. The trend can keep going.
Computers have been built five different ways: punch-card machines, relays, vacuum tubes, transistors, and chips. Each one improved fast, then hit its own limit and flattened into an S.
Guess firstIf every one of them leveled off, can computing as a whole keep growing exponentially?
Yes, as long as a new S-curve starts rising before the old one flattens. Each technology on its own is an S. Stacked, their tops trace one long line that keeps climbing. On a log scale, that line is close to straight, which means steady exponential growth for over a century.
Ray Kurzweil built his forecasts on this picture. He calls it the law of accelerating returns, and it’s how he got human-level AI by 2029. The weak point is the word “if”. Nothing guarantees the next curve shows up on time.
AI has stacked S-curves too: bigger neural networks, then transformers, then scaling up training, then reasoning models trained to think step by step. Each looked close to leveling off before the next one took over. So when someone says AI progress is flattening, ask two things: is it one S-curve, or the whole stack? And is a new one already rising underneath?
Critical mass
Add uranium until it goes critical.
When a uranium atom splits, it throws out two or three neutrons. If one of them hits another uranium atom, that atom splits too. In a small lump most neutrons fly out the sides before they hit anything, and the chain dies. A bigger lump catches more of them.
The number that matters is k, the average number of new splits each split causes.
Guess firstWhat happens when k is exactly 1?
It holds steady. Each split causes one more on average, so the reaction keeps going without growing. That’s what critical means, and it’s where a power reactor runs.
Even above 1, a chain can die out early by bad luck if its first few neutrons all escape. Fire again and one will catch.
Enrico Fermi’s team built the first reactor this way in December 1942, stacking uranium and graphite blocks under the stands of a football field in Chicago. Cadmium control rods soaked up neutrons, and they pulled them out slowly until the pile went critical.
Reactors can be controlled because about 0.65% of the neutrons come out seconds late instead of instantly. That delay gives the control rods time to work. A bomb pushes k far above 1, and its chain reaction ends when it blows itself apart.
Epidemics
One sick person, and a number called R.
A disease spreads like a chain reaction. R is how many people each sick person infects. Early COVID-19 had an R around 3. While R stays above 1, cases grow exponentially.
Everyone who recovers is immune for a while. As immunity builds, each sick person finds fewer people left to infect, so R falls on its own.
Guess firstIf R starts at 3, what share of people need to have had it before R drops below 1?
About two thirds. With R = 3, once two out of three people are immune, each sick person can only reach a third as many people who can still catch it, so R drops to 1. That share, 1 − 1/R, is called the herd immunity threshold.
The outbreak doesn’t stop there. Plenty of people are still sick, so it keeps going and ends with almost the whole town infected. Halving contacts early cuts R from 3 to 1.5, which flattens the curve and spares more than a quarter of the town.
Measles has an R of about 12 to 18, so roughly 95% of people need to be immune to stop it.
R for a disease and k for a reactor are the same number: new cases per case. The AI loop has one too.
Faster than doubling
Each doubling takes half as long as the last.
The first doubling takes one minute. The next takes 30 seconds, then 15, then 7½, and so on.
Guess firstHow long until it has doubled infinitely many times?
Two minutes. Add up the times: 1 + ½ + ¼ + ⅛ + … never gets past 2. Infinitely many doublings fit before the two-minute mark, so the size shoots to infinity there. Mathematicians call this a finite-time singularity.
On the chart each step up is one doubling, so steady doubling is a straight line. Halving times bend upward into a wall.
This happens when growing makes the grower better at growing. In 1960 Heinz von Foerster fit this kind of curve to two thousand years of world population. It reached infinity on Friday, 13 November 2026. The fit broke in the 1960s, when birth rates started to fall.
The longest chart
The world economy keeps changing gears.
For most of history, growth was too slow for anyone to notice in a lifetime. Farming sped it up, and industry sped it up again. Each era had its own steady doubling time, and each was much shorter than the last.
Guess firstFrom farming to industry, doubling got about 60 times faster. If the next change were that big, how often would the economy double?
Every few months: 15 years divided by 60 is three months. Using the bigger jump, from foraging to farming, it would be about three weeks. The economist Robin Hanson, who pointed out this pattern, put the next era’s doubling time somewhere between a week and a month.
Michael Kremer explained the long speedup with a loop. More people come up with more ideas, and better ideas let the land feed more people. That’s the same shape as AI improving AI, with people as the researchers. The loop weakened when families started having fewer children.
David Roodman fit one faster-than-exponential curve to 12,000 years of world output. At the median it reaches infinity around 2047. His point is how strong the long-run speedup has been, not that output will literally become infinite.
Recursive self-improvement
Each hammer forges a better hammer.
With a crude hammer, a blacksmith makes a slightly better one. The better hammer makes the next one faster and better again. The tool is improving the thing that makes tools.
Recursive self-improvement is the same loop with AI. The AI does research on AI, the research finds an improvement, and the improvement becomes a better AI, which does the next round.
In 1965 the statistician I. J. Good, who had broken codes with Alan Turing during the war, argued that a machine better than people at designing machines could design a better one still. He called what would follow an intelligence explosion.
Loops like this already exist. Each one ran into its own kind of limit.
The loop, as a race
A runner on a track that gets longer every lap.
Here is the whole loop as one picture. Each lap of the track is one round of AI research, and every finished lap makes the AI twice as good.
After each lap, two things change at once.
Which one grows faster decides everything.
Guess firstThe runner gets 1.5× faster each lap, and the track gets 1.3× longer. What happens to the lap times?
The rule is simple. Divide how much longer the track gets by how much faster the runner gets: 2 ÷ 1.5, 1.5 ÷ 1.5, and 1.05 ÷ 1.5. That’s how each lap compares with the one before it. Above 1 the laps stretch out. Below 1 they shrink toward nothing, and infinitely many fit in a few months, like the halving clock earlier.
That’s the toy version of an intelligence explosion. Nothing real goes to infinity, because something else runs out first. The brakes part is about what runs out.
The same race, in symbols
One symbol at a time.
Economists write the race as one equation. Every symbol in it is one part of the picture you just watched.
Tap a symbol to open its short lesson, or step through them with the arrow.
One number decides it
Speed-up against slow-down.
Every time the AI doubles, its research gets 2λ times faster. With λ = 0.6 that’s 1.5×. That’s the runner speeding up.
And the next doubling needs 2β times as much research as the last one. With β = 0.6 that’s also 1.5×. That’s the track getting longer.
So the laps shrink exactly when λ is bigger than β.
Guess firstλ = 0.6 and β = 0.4. What does the AI do?
Researchers fold the comparison into one number, r = λ ÷ β, called the returns to research. It answers one question: when research effort doubles, how many times does the AI double? If r is above 1, the runner wins. It plays the same role as k in a reactor and R in an epidemic.
Bottlenecks
The slowest step sets the pace.
A round of AI research is more than thinking. Ideas need experiments, experiments need chips, a new model needs months of training, and people check the results.
Here is one round, 10 months long. You get three upgrades. Each one makes a step ten times faster. Tap a step to upgrade it, and see how fast you can make the round.
Tap a step to make it 10× faster.
The best you can do with three upgrades is 5.3×, by upgrading the three biggest steps once each. Upgrading one step twice barely helps, because the other steps are still there.
Now let AI make thinking a thousand times faster and leave everything else alone. The round drops from 10 months to 6, which is 1.7× faster. A thousand-times-faster thinker, and the whole round speeds up by less than double.
Engineers call this Amdahl’s law. In AI, the steps that don’t speed up are the brakes: compute for experiments, training time, human review, and behind them data, power, chips, and money. Whether the loop explodes depends on whether those can grow fast enough too.
Every runaway stops
Every loop on this page hit a brake.
Each runaway here slowed or ended for a reason you can name. Most of the brakes are negative feedback: the growth itself uses up what it needs. A few were chosen by people. The question for AI is which brake bites first, and how hard.
The evidence
So is r above 1?
Researchers estimate r by looking back at how much better software got each time the research effort behind it doubled.
Tap the line to place your guess for AI research. Then look at the estimates.
The estimates straddle 1. Eth & Davidson put current AI software at roughly 0.5 to 2 once hardware is held fixed. Davidson & Houlden’s median is 1.2. Tillman’s October 2026 revisit lands near 1.0.
Other kinds of software range from 0.8 for chess programs to 3.5 for SAT solvers. A small change in r flips the toy model from slowing down to exploding, which is a big part of why careful people disagree about takeoff.
Cybernetics
Every loop here is a feedback loop.
In 1948 the mathematician Norbert Wiener named the study of feedback in animals and machines cybernetics. Its central idea is the sign of the loop. Negative feedback pushes against a change, like a thermostat switching the heat off when the room gets warm. Positive feedback pushes along with it, like a microphone held near its own speaker.
Nudge both and watch what happens. Then give the thermostat a delay.
The thermostat pulls the room back to where it was. The squeal runs away until the amplifier can’t get any louder, which is the ceiling from the bacteria dish.
With a delay, the thermostat overshoots and swings back and forth, like a shower where the hot water takes a few seconds to arrive. AI research has delays too, since you only learn whether an idea worked after an experiment or a training run finishes.
Every runaway on this page is positive feedback. Every brake is a negative feedback loop or a hard limit.
Related math
Where this math comes from.
Each idea on this page has a field behind it. These are good places to keep going.
Four shapes
Every curve on this page was one of four.
Brakes can turn one shape into another. A loop can explode until compute runs short, settle into steady doubling, and then level off at a ceiling.
The Scenarios page runs sixteen futures through one model with every brake built in, from “the bubble pops” to a hard takeoff. Each has its own page where you can stop the clock at any month and see which brake is holding.