🔍 /

Asymmetric Cantor Set Generator

Generate and visualize asymmetric Cantor sets with custom left/right ratios. Compute Hausdorff dimension, export PNG, explore fractal geometry.

User Rating 0.0 ★★★★★
Total Usage 0 times
Category Science & Math
Presets:
Is this tool helpful?

Your feedback helps us improve.

★ ★ ★ ★ ★

About

The classical Cantor set removes the open middle third of every interval at each iteration, producing a self-similar fractal with Hausdorff dimension log 2log 3 ≈ 0.6309. That symmetry is a special case. In the general construction, the left child inherits a fraction rL of the parent interval and the right child inherits rR, where rL + rR < 1. When rL ≠ rR, the resulting set is no longer self-similar in the strict sense. Its Hausdorff dimension d satisfies the implicit equation rLd + rRd = 1, which this tool solves numerically via bisection to 10−12 precision.

Getting the ratios wrong collapses the fractal into trivial dust or a solid segment. If rL + rR ≥ 1, children overlap and the complement has zero measure. This tool enforces the open set condition and computes exact gap sizes at every level. Note: the Hausdorff dimension formula assumes the open set condition holds, which requires the gap 1 − rL − rR > 0.

cantor set fractal generator hausdorff dimension asymmetric fractal iterated function system fractal geometry math visualization

Formulas

The asymmetric Cantor set is constructed by an Iterated Function System (IFS) with two contractions applied to each interval [a, b]:

f1(x) = a + rL ⋅ (x − a)
f2(x) = b − rR ⋅ (b − x)

The left child interval becomes [a, a + rL(b − a)] and the right child becomes [b − rR(b − a), b]. The open gap has length (1 − rL − rR)(b − a).

The Hausdorff dimension d is the unique solution to Moran's equation:

rLd + rRd = 1

This is solved by bisection on d ∈ (0, 1). The total number of intervals at depth n is 2n. The total measure (Lebesgue) of the set at level n is:

Ln = n∏k=1 (rL + rR) = (rL + rR)n

Where rL = left contraction ratio (0 < rL < 0.5), rR = right contraction ratio (0 < rR < 0.5), d = Hausdorff dimension, n = iteration depth, Ln = total remaining measure at level n.

Reference Data

ConfigurationrLrRGap RatioHausdorff Dim dNotes
Classic Cantor1/31/31/30.6309Standard middle-third removal
Fat Cantor (Smith - Volterra)3/83/81/40.7056Positive Lebesgue measure variant
Thin Cantor1/51/53/50.4307Large gaps, sparse dust
Heavy Left0.400.100.500.4380Asymmetric, left-biased
Heavy Right0.100.400.500.4380Mirror of Heavy Left
Extreme Asymmetry0.450.050.500.3979Near-degenerate right branch
Golden Ratio0.3820.2360.3820.5549rL = 1−φ−1
Quarter-Quarter1/41/41/20.5000Dimension exactly 0.5
Near Overlap0.450.450.100.8677Tiny gap, dense fractal
Mild Asymmetry0.300.350.350.6107Slight right bias
Dyadic1/41/21/40.6942Power-of-two scaling
Binary Dust1/41/85/80.3569Very sparse
Balanced Wide0.200.200.600.4307Wide central gap each level
Nearly Full0.480.480.040.9436Approaches dimension 1
Minimal Left0.050.300.650.3222Left branch nearly vanishes

Frequently Asked Questions

The condition rL + rR < 1 guarantees a positive gap between child intervals at every level. If the sum equals or exceeds 1, the children overlap or touch, violating the Open Set Condition required for Moran's equation to yield the correct Hausdorff dimension. The resulting object would be a full interval segment, not a fractal.
For a self-similar IFS satisfying the Open Set Condition, the Hausdorff dimension d solves rLd + rRd = 1. This tool uses bisection on the interval (0, 1) with tolerance 10−12. The function g(d) = rLd + rRd is strictly decreasing in d, guaranteeing a unique root.
As rL → 0, the left branch collapses to a single point. The set degenerates into a one-sided Cantor dust determined solely by rR. The Hausdorff dimension approaches 0 because a single contraction ratio below 1 raised to power d can only equal 1 at d = 0. In practice, ratios below 0.01 produce visually indistinguishable dust.
No. The Hausdorff dimension is a property of the limit set at infinite iterations. It depends only on rL and rR. Increasing depth refines the visual approximation but does not alter the computed dimension. At depth 8, you have 256 intervals; at depth 15, you have 32768. Beyond depth 12, pixel-level detail on standard screens is exhausted.
Assign probability p to the left branch and 1 − p to the right. The resulting self-similar measure on the asymmetric set has a multifractal spectrum. The information dimension differs from the Hausdorff dimension when rL ≠ rR and p ≠ 0.5. This tool computes the geometric (unweighted) Hausdorff dimension only.
The Lebesgue measure at level n equals (rL + rR)n. Since rL + rR < 1, this converges to 0. So the limit set always has zero Lebesgue measure under this construction. For positive-measure Cantor-like sets (Smith - Volterra - Cantor), the gap sizes must decrease faster than geometrically, which requires a different construction not covered here.