Calculator Inputs
The page stays in a single-column flow, while the input area uses 3 columns on large screens, 2 on medium screens, and 1 on mobile.
Example Data Table
These rows are illustrative sample runs to show how the calculator output can be organized and compared.
| Case | c | Zoom | Grid | Max Iterations | Bounded Ratio | Mean Escape Iteration | Seed Status |
|---|---|---|---|---|---|---|---|
| Reference A | -0.8000 + 0.1560i | 1.00 | 140 × 140 | 120 | 31.48% | 18.72 | Escaped |
| Reference B | -0.4000 + 0.6000i | 1.80 | 160 × 160 | 180 | 42.11% | 24.65 | Bounded |
| Reference C | 0.2850 + 0.0100i | 2.20 | 180 × 120 | 220 | 27.96% | 31.48 | Escaped |
| Reference D | -0.7269 + 0.1889i | 3.50 | 200 × 200 | 300 | 36.54% | 47.03 | Bounded |
Formula Used
The calculator uses the classic Julia iteration:
zn+1 = zn2 + c,
where c = a + bi is fixed and every grid point starts as
z0 = x + yi.
A point is treated as escaped when
|zn| > R,
where R is the escape radius.
The escape-time count estimates how quickly each point leaves the bounded region.
The calculator also estimates:
- Bounded ratio = bounded points ÷ total sampled points.
- Estimated bounded area = viewing window area × bounded ratio.
- Boundary density proxy = points with mid-range escape values ÷ total points.
- Seed orbit diagnostics by repeatedly iterating the chosen starting point.
When smoothing is enabled, the plotted value uses a continuous coloring adjustment to reduce harsh color banding near boundaries.
How to Use This Calculator
- Enter the real and imaginary parts of the fixed complex constant c.
- Set the complex-plane center and choose a zoom factor.
- Adjust plane aspect ratio to widen or tighten the visible region.
- Choose grid width and height for sampling density.
- Set the maximum iterations and escape radius.
- Enter a seed point to inspect one orbit in the same system.
- Select smoothing and a color scale for clearer plotting.
- Press Generate Julia Set to show summary results above the form.
- Use the CSV button for raw sampled output and the PDF button for a summary report.
FAQs
1) What does this calculator actually compute?
It samples many starting points across the complex plane, iterates each one with a fixed constant, and records whether the point escapes or remains bounded within the selected iteration limit.
2) Why do different constants create different shapes?
The constant controls the nonlinear feedback in the recurrence. Small changes can move the system between disconnected dust-like sets, filament structures, or more connected boundaries with dense self-similar detail.
3) What does bounded ratio mean?
It is the fraction of sampled grid points that did not escape before the iteration limit. It estimates how much of the visible window behaves like the filled Julia region.
4) Why is the estimate sensitive to grid size?
A Julia set boundary can be extremely intricate. Coarse grids miss tiny structures, while finer grids capture more detail and usually improve stability in area and density estimates.
5) What is the role of escape radius?
The escape radius defines when a point is safely considered divergent. For quadratic Julia sets, values above 2 are usually sufficient, though larger radii can still be explored for comparison.
6) What is the seed orbit overlay?
It shows the path of one chosen starting point under repeated iteration. This helps you study local behavior, escape timing, and how nearby points can separate quickly around sensitive regions.
7) What does smoothing change?
Smoothing replaces abrupt integer bands with continuous values derived from the escape magnitude. The structure stays the same, but the color transitions become more informative and visually cleaner.
8) Is this a proof-level fractal analysis tool?
No. It is a numerical exploration and visualization tool. It provides strong intuition, comparative metrics, and exportable data, but rigorous fractal proofs require deeper mathematical analysis.