Plot Settings
Example Data Table
This sample shows typical settings and what they influence.
| Setting | Example | What you get |
|---|---|---|
| Center (x, y) | (-0.5, 0.0) | Classic full-body Mandelbrot overview. |
| Zoom | 15 | Closer detail around the selected center point. |
| Max iterations | 1200 | Sharper boundaries and finer filament structure. |
| CSV step | 6 | Smaller CSV with a coarser sampled grid. |
Formula Used
- Escape test: if |zₙ| > R (bailout radius), the point escapes.
- Iteration count: the first n where it escapes is used for coloring.
- Smooth coloring: uses a continuous value to reduce color banding near boundaries.
How to Use This Plotter
- Start with center (-0.5, 0.0) and zoom 1.
- Increase Max Iterations if edges look blocky.
- Change Zoom and tweak the center to explore details.
- Use Bailout Radius (usually 2) for standard behavior.
- After plotting, download PNG, a sampled CSV, or a PDF report.
Rendering Model and Membership Test
The plot is built by scanning pixels and mapping each one to a complex point c = x + yi. Starting from z₀ = 0, the iteration zₙ₊₁ = zₙ² + c runs up to a chosen limit. If |zₙ| exceeds the bailout radius, that point is treated as escaped and colored by its escape time.
Viewport Controls for Targeted Exploration
Center coordinates define which region of the complex plane is inspected, while zoom scales the viewing window. This tool uses a default horizontal span of about 3.5 units at zoom 1, preserving aspect ratio for the vertical span. Small changes in center at high zoom reveal miniature bulbs, filaments, and repeating structures.
Iteration Depth and Boundary Detail
Max iterations strongly affect boundary sharpness because points near the edge can escape very slowly. Raising iterations improves accuracy of fine tendrils and reduces false “inside” classifications at deep zooms. For quick previews, lower iterations speed up rendering, then increase iterations for final exports and reports.
Coloring, Smoothing, and Visual Readability
Escaped points are colored with a palette driven by the escape count and an optional smooth value. Smooth coloring applies a continuous estimate based on the magnitude of the final z, reducing banding. Multiple schemes help separate dense bands from low-iteration regions, improving interpretation of gradients and structure.
Exports and Data-Driven Analysis
PNG provides a lossless plot for sharing or publishing, while CSV exports sampled grid points for analysis in spreadsheets. Each CSV record includes pixel coordinates, mapped (x,y), iteration count, and an escape flag. The PDF report packages key settings and render statistics alongside the generated image for documentation.
Performance Guidance and Practical Limits
Rendering cost grows with resolution and iterations, so the calculator caps total pixels to keep response time stable. Use smaller previews to locate interesting regions, then increase resolution for final output. Adjust CSV step upward to reduce exported rows while retaining representative samples of the plotted region.
FAQs
1) Why is bailout radius usually set to 2?
For the Mandelbrot set, if |z| ever exceeds 2, the sequence diverges. Larger values work, but they slow escape detection without improving correctness for standard plotting.
2) What does “inside percentage” mean?
It is the fraction of rendered pixels that did not escape within the selected iteration limit. It is a practical estimate, not a proof of membership at infinite iterations.
3) Why do deep zooms look noisy at low iterations?
Near the boundary, many points escape very slowly. Low iteration limits can misclassify points and create blocky edges. Increase iterations to improve boundary fidelity.
4) What is smooth coloring and when should I use it?
Smoothing uses a continuous escape estimate based on the final magnitude of z. It reduces visible bands and produces cleaner gradients, especially at higher iterations.
5) Why is the CSV “sampled” instead of full resolution?
Full-resolution export can be extremely large. Sampling keeps files manageable while still providing representative iteration data for profiling, filtering, and charting in external tools.
6) The plot is blank or errors appear. What should I check?
Ensure the server has the GD extension enabled for image creation. Also keep width×height under the pixel cap, and try a moderate zoom before pushing extreme values.