About

Cantor Dust is a multi-dimensional generalization of the Cantor Set, constructed by recursively removing the middle portion of line segments (1D) or interior sub-squares (2D). At iteration n, the 1D set contains 2ⁿ segments each of length (1⁄3)ⁿ, yielding a Hausdorff dimension of ln 2ln 3 ≈ 0.6309. The 2D variant retains 4 corner squares from a 3×3 grid at each step, producing a dust with Hausdorff dimension ln 4ln 3 ≈ 1.2619. The resulting set has Lebesgue measure zero and is totally disconnected - no two points share a connected path. Miscounting iterations or misunderstanding the removal pattern leads to incorrect fractal dimension estimates and flawed geometric analysis. This tool computes the exact recursive geometry and renders it at pixel precision.

Formulas

The 1D Cantor Set construction removes the open middle third at each iteration. The number of segments and their lengths at step n:

N_1D(n) = 2ⁿ , L(n) = 13ⁿ

Total remaining length converges to zero:

M_1D(n) = 2ⁿ3ⁿ = (23)ⁿ

For 2D Cantor Dust, a unit square is divided into a 3×3 grid and only the 4 corner sub-squares are retained:

N_2D(n) = 4ⁿ , S(n) = 13ⁿ

The Hausdorff dimension d_H is computed via the self-similarity relation:

d_H = ln Nln s

Where N = number of self-similar copies (2 for 1D, 4 for 2D) and s = scaling factor (3). This yields d_H ≈ 0.6309 (1D) and d_H ≈ 1.2619 (2D).

Reference Data

Iteration n	1D Segments	1D Segment Length	1D Total Length	2D Squares	2D Square Side	2D Total Area
0	1	1	1.000000	1	1	1.000000
1	2	1/3	0.666667	4	1/3	0.444444
2	4	1/9	0.444444	16	1/9	0.197531
3	8	1/27	0.296296	64	1/27	0.087791
4	16	1/81	0.197531	256	1/81	0.039018
5	32	1/243	0.131687	1024	1/243	0.017341
6	64	1/729	0.087791	4096	1/729	0.007707
7	128	1/2187	0.058528	16384	1/2187	0.003425
8	256	1/6561	0.039018	65536	1/6561	0.001522
9	512	1/19683	0.026012	262144	1/19683	0.000677
10	1024	1/59049	0.017341	1048576	1/59049	0.000301
n → ∞	∞	0	0	∞	0	0

Frequently Asked Questions

At each iteration n, the total measure (length in 1D, area in 2D) equals (2⁄3)ⁿ for 1D or (4⁄9)ⁿ for 2D. Both ratios are less than 1, so as n → ∞, the measure converges to 0. However, the set can be put in bijection with the real interval [0,1] via ternary expansion (using only digits 0 and 2), proving it is uncountable. Measure and cardinality are independent concepts in set theory.

The 2D mode is capped at 8 iterations, producing 4⁸ = 65,536 rectangles. The 1D mode allows up to 17 iterations (2¹⁷ = 131,072 segments). Beyond these limits, individual elements become sub-pixel and the computational cost grows exponentially with no visual benefit. The rendering time scales as O(Nⁿ) where N is the branching factor.

The 2D Cantor Dust is the Cartesian product of two independent 1D Cantor Sets: C_2D = C × C. This product structure means the Hausdorff dimension doubles: 2 × 0.6309 ≈ 1.2619. The same logic extends to 3D (d_H ≈ 1.8928) using 8 corner sub-cubes from a 3×3×3 grid. Each dimension added multiplies the number of retained pieces by 2.

Yes. The standard Cantor Set removes 1 of 3 parts, but you can remove different fractions. Removing the middle fifth (keeping 2 pieces at scale 1/5) yields d_H = ln(2)⁄ln(5) ≈ 0.4307. For 2D, keeping different sub-squares (e.g., 5 of 9 instead of 4) produces the Vicsek fractal with d_H ≈ 1.4650. This generator uses the classical construction for mathematical accuracy.

At iteration n, each element has side length 1⁄3ⁿ relative to the canvas. On a 800px canvas, iteration 6 produces elements of ~1.1px, and iteration 7 yields ~0.37px elements that get anti-aliased into faint smudges. The fractal's self-similar structure means the overall pattern stabilizes visually around iteration 5 - 6 for typical screen resolutions.