Tool

Cantor set playground

Pick how much of each interval to remove at every step. The standard Cantor set removes the middle third; varying the fraction gives a whole family of generalised Cantor sets with continuously varying Hausdorff dimension.

Iteration depth: 8

Removed fraction: 0.333 (standard middle thirds)

Construction rule

step 0:  [0, 1]
step n:  for each surviving interval [a, b]:
           let len = b − a, removed = 0.333 · len  (centered)
           keep [a, a + (1 − r) · len / 2] and [b − (1 − r) · len / 2, b]

IFS form:
  f_1(x) = 0.333 · x
  f_2(x) = 0.333 · x + 0.667

At r = 1/3 you recover the standard middle-thirds Cantor set.

At this setting

Intervals at depth 8: 2⁸ = 256

Each interval length: 1.52e-4 of original

Total length: 3.90e-2

Hausdorff dimension

d = log 2 / log(2 / (1 − r)) = 0.6309

At r = 1/3 you recover Cantor’s log 2 / log 3 ≈ 0.6309.

Background reading at /fractals/cantor-set.

FAQ