Fractal starting points

1. Mandelbrot set: definition reference · Hausdorff dimension 2 ≈ 2 (boundary) · Escape-time approximation: 250 iterations per sample, 2×2 samples per pixel, bailout radius 256, a square window of side 2.8248 centered at (-0.7642, 0); samples that do not escape within 250 iterations are ink.
2. Julia set (c = −0.8 + 0.156i): definition reference · Escape-time approximation: 250 iterations per sample, 2×2 samples per pixel, bailout radius 256, a square window of side 3.4857 centered at (0, 0); samples that do not escape within 250 iterations are ink. c lies just outside the Mandelbrot set (its critical orbit escapes after 252 iterations), so the Julia set is disconnected (a Cantor dust); the dark centers mark slow escape at the 250-iteration cap, not an interior.
3. Koch snowflake: definition reference · Hausdorff dimension log 4 / log 3 ≈ 1.2619 (boundary curve) · Curve approximation at iteration 5 of the L-system (3,072 segments, turn 60°), filled and fitted to a 448px square.
4. Sierpiński triangle: definition reference · Hausdorff dimension log 3 / log 2 ≈ 1.585 (set) · Depth-5 approximation (243 cells), filled and fitted to a 448px square.
5. Sierpiński carpet: definition reference · Hausdorff dimension log 8 / log 3 ≈ 1.8928 (set) · Depth-4 approximation (586 cells), filled and fitted to a 448px square.
6. Barnsley fern: definition reference · Point-cloud approximation of the attractor by the chaos game: 1,500,000 points from fixed seed 19, a square window of side 11.4266 centered at (0.2369, 4.9992).
7. Heighway dragon: definition reference · Hausdorff dimension 2 ≈ 2 (limit set (space-filling)) · Curve approximation at iteration 14 of the L-system (16,384 segments, turn 90°), stroked at 1.5px and fitted to a 448px square.
8. Hilbert curve: definition reference · Hausdorff dimension 2 ≈ 2 (limit curve (space-filling)) · Curve approximation at iteration 5 of the L-system (1,023 segments, turn 90°), stroked at 6px and fitted to a 448px square.
9. Pythagoras tree: definition reference · Hausdorff dimension 2 ≈ 2 (set (45° tree)) · Depth-10 approximation with 45° branches (2,047 cells), filled and fitted to a 448px square.
10. Lévy C curve: definition reference · Hausdorff dimension 2 ≈ 2 (limit set (contains open sets)) · Curve approximation at iteration 14 of the L-system (16,384 segments, turn 45°), stroked at 1.5px and fitted to a 448px square.
11. Gosper curve: definition reference · Hausdorff dimension 2 ≈ 2 (limit curve (space-filling)) · Curve approximation at iteration 4 of the L-system (2,401 segments, turn 60°), stroked at 3.5px and fitted to a 448px square.
12. Peano curve: definition reference · Hausdorff dimension 2 ≈ 2 (limit curve (space-filling)) · Curve approximation at iteration 3 of the L-system (728 segments, turn 90°), stroked at 7px and fitted to a 448px square.
13. Cantor set: definition reference · Hausdorff dimension log 2 / log 3 ≈ 0.6309 (set) · Levels 0–5 stacked top to bottom (63 cells), filled and fitted to a 448px square.
14. Vicsek fractal: definition reference · Hausdorff dimension log 5 / log 3 ≈ 1.465 (set) · Depth-4 approximation (625 cells), filled and fitted to a 448px square.

Fractal images are illustrations rendered by Fav Tings from each construction’s published definition; a finite image only approximates the infinite object. References are English Wikipedia articles, used as factual cross-references; nothing is copied from them.