The Cantor Set

Dust with no length, and uncountably many points

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Removing middle thirds forever leaves the Cantor set: total length zero.
• Its points are base-3 expansions with only 0s and 2s, and they are uncountable.
• Its fractal dimension is $\ln 2/\ln 3 \approx 0.63$.

Steps

1. Remove the middle thirds
2. Vanishing length
3. Which points survive?
4. How many points?
5. Is ½ in the Cantor set?
6. A fractional dimension