Big Maths Ideas · Beta

The Cantor set

Remove middle thirds and discover why zero length does not mean no points.

Build one stage at a time. Select an interval to read its endpoints.

N_n=2^n,\quad L_n=\left(\frac23\right)^n

Stage 2 retains 4 closed intervals.

Remaining intervals: 4
Length of each interval: $3^{-2}$
Total remaining length: 0.44444
Selected interval (approx.): 0 ≤ x ≤ 0.11111

Every displayed stage has positive length. In the limit the total length is zero, yet the Cantor set has uncountably many points. Endpoints such as 0, 1/3 and 1 are never removed.

Predict

If the Cantor set has zero length, must it contain no points?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

If the Cantor set has zero length, must it contain no points?

$L_3=\left(\frac23\right)^3$

Hint

Each step keeps two thirds of the previous total length.

Reveal answer

The removed intervals are open. Endpoints survive, and the limiting set has uncountably many points.

$\frac8{27}$

Build an explanation

Hint 1

Each step keeps two thirds of the previous total length.

Hint 2

The removed intervals are open. Endpoints survive, and the limiting set has uncountably many points.

Hint 3

Compare the result with the model, then explain why it occurs.

Worked example

This example uses fixed values, separate from the diagram controls.

Find total remaining length after three stages.

Start with length 1.
Multiply by 2/3 three times.
Calculate (2/3)³.

Answer: 8/27.

Connect this idea

Watch out: The removed intervals are open. Endpoints survive, and the limiting set has uncountably many points.