A single bacterium is dropped into a petri dish at 6 am. Every hour it divides in two: one becomes two, two become four, four become eight. The dish can hold at most a billion bacteria before it is completely full.

Here is the puzzle. When does the dish get half full?

Your gut says: around the middle of the day. The actual answer is: one hour before it is completely full. If the dish fills at hour 30, then at hour 29 it is only half full — and at hour 25 it is only one-thirty-second full. For the first 25 hours, you would look at the dish and think nothing is happening. Then, in the last five hours, it goes from near-empty to overflowing.

The simulator

Drag the hour slider below to walk through the growth of the colony. The number shown is the population size, which doubles every step. The horizontal line is the capacity of the dish — one billion.

A draggable population curve of bacteria doubling every hour for thirty hours A graph whose horizontal axis is hours from zero to thirty, and whose vertical axis is population from zero to about a billion. A curve starts at one bacterium at hour zero, rises imperceptibly for the first twenty hours, then explodes upward in the last five hours to reach a billion at hour thirty. A horizontal capacity line marks the carrying capacity of the dish at one billion. A draggable red point shows the current hour and population. capacity (1 billion) 0 h 10 h 20 h 30 h 0 ~500M ~1B ↔ drag hour from 0 to 30
Bacteria count over $30$ hours of hourly doubling in a dish that holds $2^{30} \approx 10^9$ organisms. The curve is indistinguishable from zero for the first twenty hours — even though the population is genuinely doubling every hour, the values are so small compared to the dish's capacity that they look like zero. Then, in the last five hours, the population goes from $2^{25}$ (about $33$ million, still only $3\%$ of capacity) to $2^{30}$ (a full billion, the entire dish).

The last-five-hours surprise

Here is the population at every hour, written out. Look at when the colony stops being a rounding error and starts filling the dish.

Hour Population Fraction of dish
0 1 10^{-9} (negligible)
10 1{,}024 10^{-6} (negligible)
20 1{,}048{,}576 0.1\%
25 33{,}554{,}432 3\%
26 67{,}108{,}864 6\%
27 134{,}217{,}728 13\%
28 268{,}435{,}456 25\%
29 536{,}870{,}912 50\%
30 1{,}073{,}741{,}824 100\%

Why the last five hours are so dramatic: each hour doubles the population, so between hour 25 (three percent) and hour 30 (one hundred percent), the dish's content multiplies by 2^5 = 32, taking it from "not even noticeable" to "completely packed." The dramatic-looking jump is not a speedup — the doubling rate is constant. It only looks sudden because the earlier hours are invisibly small on the same scale.

The deceptive look of "nothing is happening"

If you were a microbiologist watching the dish grow, you would see nothing for the first 20 hours. The dish would look empty. At hour 25 it would look 3\% occupied — still mostly empty. You would shrug.

Then hour 28 arrives and the dish is one-quarter full. You blink; there is visible cloudy mass for the first time. An hour later the dish is half full. An hour after that it is completely full.

In four hours, you went from "nothing worth reporting" to "overflow." That is what an exponential process looks like from the inside: for most of its run, nothing perceptible happens; and then, at the very end, everything happens.

The technical reason is that the growth rate is proportional to the current size. When the size is small, the absolute increase is small and invisible. When the size is large, the absolute increase per hour is also large — and since the size is doubling, the absolute increase is also doubling every hour. The biggest jumps in absolute numbers happen at the very end.

This pattern is everywhere

The bacteria-in-a-dish story is the template for a long list of real processes:

Every one of these is described by P(t) = P_0 \cdot 2^{t/T}, where T is the doubling time. For bacteria in this article, T is one hour. For money at 7\% interest, T is about 10 years. For a rumour, T might be twenty minutes. The doubling time changes; the shape of the curve does not.

The moral

When you catch yourself thinking "this is growing slowly, I have time," and the growth is exponential (a constant percentage per unit time), re-check. The early hours of exponential growth are slow in absolute terms but fast in percentage terms — and it is the percentage that controls when the wall arrives.

A population at 3\% of capacity is not "nowhere near the wall." If it is doubling every hour, it is five hours from the wall. A disk at 3\% capacity, growing by a constant 5\% per day, is about 70 days from the wall — dangerously close by software-operations standards.

The petri dish is just a vivid way of feeling this. After you have dragged the slider, the reflex "three percent means five doublings away from full" should stay with you. That reflex is what exponents teach you.

Related: Exponents and Powers · A Tower of 2s: Watch Doubling Explode Off the Screen by Step 30 · Why a⁰ = 1: The Halving Staircase That Forces the Answer · Tile-View Proof of the Three Core Exponent Laws