The rice and chessboard problem: why exponential growth fools your brain
Why exponential growth always stops: the S-curve
If one bacterium kept doubling every 20 minutes, its family would weigh more than the Earth in less than two days. That never happens. See what stops every doubling in the real world, and the shape that growth takes when it slows down.
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1
Rono thinks of the pond, his savings, his debt and the outbreak. Then he asks Miva the big question: can something keep doubling for ever?
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Miva answers with bacteria. In a lab, with food and warmth, a common bacterium called E. coli can split in two about every 20 minutes. Then each half splits again.
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If nothing stopped them, the family of one bacterium would weigh more than the whole Earth in less than two days.
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It never happens. The more bacteria there are, the faster they eat their food. Soon the dish has no food and no space left, and new bacteria stop coming.
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Draw the number of bacteria over time, and the curve has the shape of an S. It starts like doubling, then it bends, and at last it lies flat at the limit. This is called an S-curve.
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The lily pads stop when the pond is full. An outbreak slows when fewer people are left who can catch it. Every real doubling meets its limit, but at the start, an S-curve looks just like doubling.
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7
Rono laughs. At least, he says, his brain is only fooled by doubling. Miva smiles. This year, two of the 23 pupils in her class have the same birthday. Rono says that is just luck: a year has 365 days.
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Miva says it is not luck at all. Here too, something grows faster than your brain thinks: not the pupils, but the pairs of pupils who could share a day. How can 23 people be enough? Find out in "The birthday paradox: why 23 people are enough to share a birthday".
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Quick check
A rumour in a school.
Show the answer
No. It doubles at first, then it slows as fewer pupils are left who have not heard it, and it stops at 500. It is an S-curve.
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Go further
Two names for the S and its limit.