The one curve our brains simply cannot feel coming.
If lilies on a pond double every day and fill the pond on day 30, on what day was it half full?
The one curve for everything that changes in proportion to its current size — the exponential function.
Experiment
Hands-on experiment
Predict first — lilies double daily and fill the pond on day 30. On what day was it half full?
Fill the pond yourself (starting from day 24)
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Bacteria divide in proportion to how many there are now, interest accrues on the current balance, and an epidemic spreads in proportion to today's infected count. The common structure: 'the change is proportional to the current amount'.
This kind of growth can never be predicted with straight-line intuition. It looks quiet until yesterday, then suddenly explodes today — when in fact it was growing at the same ratio from the very start.
The function that holds this structure is y = aˣ, the exponential. As the lily puzzle shows — the answer is day 29, not day 30! — in the exponential world, the final step equals everything that came before.
Insight
Insights from the video
“What makes the exponential special: its rate of change looks like itself.”
An exponential grows faster the bigger it is — because its growth speed is proportional to its current size. This is where the only function that is its own derivative (eˣ) is born.
“The lesson of day 29: in the exponential world, 'only half' means 'almost done'.”
Half the pond fills in the final single day. Whether it's the good kind (compound assets) or the bad kind (epidemics, debt), by the time an exponential phenomenon is noticeable, it's often already late.
Misconception
Common misconceptions
Doubling daily means about 60x after 30 days.
It's 2³⁰ ≈ 1 billion times. Estimate repeated multiplication with an additive gut feeling and you'll always underestimate spectacularly. Human intuition is linear; the world's growth is exponential.
The gentle early part of the graph isn't 'exponential' yet.
The gentle beginning is the same ratio of growth. 1→2→4 doesn't look like much, but it's already doubling every step. This is the mathematical reason early response matters in an epidemic.
Formula
Writing it as math
Let's write the explosion and decay from the simulation as formulas.
The exponential function
If a>1 it explodes (growth); if 0<a<1 it vanishes (decay). The growth-rate slider in the experiment was exactly a.
The structure of growth
The formula for 'the change is proportional to the current amount'. Repeat this one line n times and you get (1+r)ⁿ — the same formula from the Compound Interest lab.
Half-life and doubling time
Anything that changes at a constant ratio has a constant 'time to double' and 'time to halve'. The moment you ask for that time, logarithms enter.
In Real Life
Where you meet it in real life
Epidemic spread
If one infected person infects an average of R others, each generation multiplies by R — whether R was above or below 1 was the whole story of pandemic control.
Compound interest and debt
Assets and debts both grow exponentially. The snowball curve from the Compound Interest lab is exactly this function.
Radioactive decay and dating
Carbon-14 halves every 5,730 years. Measure the remaining fraction and you can compute an artifact's age — decaying exponentials at work.
The forgetting curve
Memory fades exponentially too. That's why review timing can be designed mathematically (a learning lab is planned).
Connection
Concepts connect
Previous concept
Exponents
Experience the explosion of repeated multiplication first, and the exponential function is simply that explosion joined into a continuous curve.
← Exponents labLeads to next
Logarithmic Functions
Reading an exploding curve with human eyes takes a compressing lens. The function that appears when you hold the exponential up to a mirror — the logarithmic function.
Go to the Logarithmic Functions lab →