One becomes two
Why doubling starts slowly and then gets very big very fast.
What you are looking at
At the start there is one dot. It splits into two dots. Those two split into four, then eight, and so on, until there are over sixteen thousand dots. Each time a dot splits, you see a small flash.
The idea
Say you tell a secret to two friends today. Tomorrow, each of them tells two new friends. The day after, each of those tells two more. How many people know the secret after ten days? More than a thousand. After fourteen days? More than sixteen thousand.
This is called doubling. At the start it feels slow: 1, 2, 4, 8. But every step adds more than all the steps before it put together, so the numbers get big very fast. That is why the last few splits in the animation fill the screen so quickly.
Real doubling always runs into a wall. The friends run out of new friends, or the dots run out of room.
Where else you see it
- Bacteria in good conditions can split about every twenty minutes. Left alone with plenty of food, one cell could become billions in half a day.
- A sheet of paper folded in half again and again gets thick very quickly (and you cannot fold it more than about seven times by hand).
- Anything that spreads by passing to two or more others: a rumor, a video, a sickness.
Go deeper: the proper names and the math
After n rounds of doubling there are 2n particles. Here n = 14, so 214 = 16,384. The growth rate is proportional to the current size, dN/dt = kN, and the solution is N(t) = N₀ekt. If the doubling time is T, then k = ln 2 / T.
Real populations are limited by resources, so growth curves bend into an S shape: the logistic curve. That same equation shows up again in the bifurcation stage, where it behaves in a surprising way.