Build the classic middle-thirds fractal step by step. Adjust depth, colors, thickness, then save a PNG or share a live link with your exact setup.
The Cantor set is a fractal you build by cutting the middle third out of every remaining segment. Use this generator to watch the process unfold, measure what is left, then share the result.
Georg Cantor used this set in 1883 to show a bounded set on the real line could be uncountable yet carry zero length. Every point in the set matches a base-3 expansion with digits 0 or 2, giving a direct bijection with the real interval [0, 1].
Students use the set to learn limits, compactness, self-similarity, measure zero, and ternary representations. Engineers borrow the pattern for fractal antennas, lossless compression schemes, porous material models, and strange attractors in nonlinear systems. Designers use the rule for generative art because its 2:1 scaling keeps detail balanced at every zoom level.
Start with the interval [0, 1]. Remove the open middle third from (1/3, 2/3). Two bars remain. Repeat the cut inside every surviving bar. Keep going forever, and the limit is the Cantor set.
Each step keeps two-thirds of the previous total. After n steps you have (2/3)^n, which tends to zero. Points still survive though: any number whose base-3 digits are only 0 or 2 stays in the set forever.
A self-similar set of N copies scaled by factor r has dimension log N / log(1/r). The Cantor set has N = 2 copies at scale r = 1/3, so its Hausdorff dimension is log 2 / log 3, about 0.6309.
Write a number from [0, 1] in base 3. The number is in the Cantor set if its expansion uses only the digits 0 and 2. For example 1/3 = 0.0222… belongs to the set, while 1/2 = 0.1111… does not.
Press Share. The URL copied to your clipboard encodes iterations, colors, and spacing. Opening the link restores the same view on any device.
Yes. Choose the Mono palette for black-and-white print or the Depth palette for color decks, toggle Dark stage for dark-themed talks, then press PNG to save a high-resolution render.
