About

The Burning Ship fractal is generated by iterating z_n+1 = (|Re(z_n)| + i|Im(z_n)|)² + c over the complex plane. The absolute value operation on real and imaginary parts before squaring breaks the symmetry found in the standard Mandelbrot set, producing asymmetric structures that resemble burning ships and other chaotic formations. Insufficient iteration depth produces false convergence: regions that appear "inside" the set may actually escape at higher counts, distorting the boundary. This tool computes escape-time values in a Web Worker to avoid UI thread blocking, applies smooth iteration normalization via log₂(log₂(|z|²)), and maps results through cyclic HSL palettes. Coordinates and zoom state persist across sessions.

Computation cost scales linearly with pixel count and iteration cap. At deep zooms beyond 10¹⁴, IEEE 754 double-precision introduces rounding artifacts. This renderer uses standard 64-bit floats, so visual accuracy degrades past approximately 13 decades of magnification. For most exploratory purposes this limit is not reached.

Formulas

The Burning Ship fractal belongs to the family of escape-time fractals. Each pixel maps to a point c = c_re + i c_im on the complex plane. The recurrence relation is:

z_n+1 = (|Re(z_n)| + i|Im(z_n)|)² + c

Expanding the squared term with x_n = |Re(z_n)| and y_n = |Im(z_n)|:

x_n+1 = x_n² − y_n² + c_re

y_n+1 = 2x_ny_n + c_im

Iteration continues until x_n² + y_n² > 4 (escape radius squared) or the iteration count n reaches the maximum. For smooth coloring, the normalized escape count is:

n_smooth = n + 1 − log(log(x_n² + y_n²))log(2)

where n = discrete escape iteration, x_n, y_n = final coordinates at escape, c_re, c_im = real and imaginary components of the pixel's complex coordinate, and the escape bailout radius is 2 (squared check: 4).

Reference Data

Landmark	Center Re	Center Im	Zoom	Min Iterations	Description
Full View	−0.4	−0.6	3.5	100	Complete fractal overview showing main ship body
Main Antenna	−1.756	−0.028	0.05	300	Filament extending left from the hull
Bow Detail	−1.694	−0.0003	0.005	500	Fine structure at the ship's prow
Stern Spiral	0.82	−1.28	0.15	200	Spiral arm near lower right
Mini Ship	−1.762	−0.028	0.008	500	Self-similar copy embedded in antenna region
Upper Mast	−0.4	−1.3	0.3	200	Vertical structure above main body
Hull Crack	−1.0	−0.15	0.2	250	Branching fracture line along the hull
Keel Spike	−0.5	−0.2	0.1	300	Sharp protrusion beneath main body
Wake Region	−1.5	−0.05	0.15	300	Turbulent boundary west of hull
Crow's Nest	−0.3	−1.5	0.08	400	Isolated detail above the mast
Porthole	−0.1	−0.65	0.05	350	Circular feature on the hull surface
Deep Zoom Filament	−1.7611	−0.0283	0.0001	1000	Requires high iteration count for resolution

Frequently Asked Questions

The critical difference is the absolute value operation applied to the real and imaginary parts of z before squaring. In the Mandelbrot set, z_{n+1} = z_n² + c preserves the sign of both components, producing symmetric structures. The Burning Ship applies |Re(z_n)| and |Im(z_n)|, which folds the complex plane along both axes at every iteration. This folding creates the characteristic asymmetric, angular structures and the ship-like silhouette. The operation is non-analytic (not complex-differentiable), which is why the boundary lacks the smooth rotational symmetry of the Mandelbrot set.

Iteration count must scale roughly with the logarithm of the zoom factor. At the default overview (zoom span ≈ 3.5), 100-200 iterations suffice. At 10⁴× magnification, 500-800 iterations are typical. At 10⁸×, you may need 2000-5000. Insufficient iterations cause boundary detail to collapse into uniform color bands. This tool caps at 5000 iterations; beyond that, computation time per frame on a single Web Worker exceeds acceptable thresholds for interactive use.

JavaScript uses IEEE 754 double-precision floating point, which provides approximately 15-16 significant decimal digits. When the viewport span shrinks below roughly 10⁻¹³, adjacent pixels map to the same floating-point value, producing rectangular artifacts. This is a fundamental precision limit, not a rendering bug. Arbitrary-precision libraries or perturbation theory (not implemented here) can extend the useful zoom range by several additional orders of magnitude.

Without smoothing, each pixel is colored by its integer escape iteration, creating visible banding. Smooth coloring computes a fractional iteration count: n_smooth = n + 1 − log₂(log₂(|z|²)). This maps escape values to a continuous range, allowing gradient interpolation across the palette. The log₂(log₂()) term normalizes the exponential growth of |z| near the escape boundary. The result is visually continuous color transitions that reveal fine boundary structure otherwise hidden by discrete bands.

The traditional orientation plots Re(c) on the horizontal axis and Im(c) on the vertical axis with Im increasing downward. In this convention the ship appears upright with the hull at bottom and masts pointing up. Some renderers flip the imaginary axis, inverting the image. This tool uses the mathematical convention (negative Im downward), which shows the ship in its commonly recognized orientation. No rotation transform is needed.

A 1920×1080 canvas at 500 iterations requires up to 1,036,800,000 floating-point operations (1920 × 1080 × 500 × ~2 multiplications + additions per iteration). Running this on the main thread blocks all UI events - scrolling, clicking, and even browser tab responsiveness - for several seconds. The Web Worker runs on a separate OS thread, keeping the UI responsive. Progress is reported back via postMessage so the progress bar updates in real time.