Burning ShipEasy

Burning Ship

A quadratic escape-time fractal: fold both coordinates with absolute values before squaring, and the color bands form an asymmetric ship-like outline.

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$Burning Ship fractal: dark main hull, flame-colored escape bands$

The Burning Ship: a wedge-shaped hull, flame-like ridges above the real axis, and layered escape-time bands.

Overview

The Burning Ship stands close to the Mandelbrot set, but one move changes everything. Before each square, it replaces both the real and imaginary parts of z with their absolute values. The orbit folds toward the axes, and the quadratic map loses its rotational symmetry. The map is no longer analytic—it does not satisfy the Cauchy–Riemann equations—and hard corners appear along its edge.

Use the usual display convention, with the real axis horizontal and the imaginary axis inverted, and the fold suggests a burning ship. A wedge-shaped hull sits below flame-like ridges. Along the real line to the left, smaller ships repeat the same jagged profile.

The usual escape-radius test still works: once |z| exceeds 2, divergence is certain. Yet this non-analytic orbit no longer behaves like a complex-analytic function. Its escape-time bands stack into hard-edged layers and nested rectangular boundaries instead of the Mandelbrot set’s smooth radial gradients.

The Mathematics

Fold, then square

z_{n+1}=(|\operatorname{Re}z_n|+i|\operatorname{Im}z_n|)^2+c

Both real and imaginary components are reflected to nonnegative values, combined into a complex number, then squared. In expanded form, the real part is xₙ₊₁ = xₙ² − yₙ² + Re(c) and the imaginary part is yₙ₊₁ = 2|xₙ||yₙ| + Im(c). At every iteration, the absolute values fold the orbit back toward the axes: if xₙ or yₙ is negative, it is mirrored to positive before the squaring step. That fold produces the sharp, rectangular boundaries that set the Burning Ship apart from the Mandelbrot set’s smooth contours.

Try c = −1. For the Mandelbrot set, the orbit z₀=0, z₁=−1, z₂=0, z₃=−1, … stays bounded (period 2). For the Burning Ship, z₁ = (|0| + i|0|)² − 1 = −1, then z₂ = (|−1| + i|0|)² − 1 = 0, and the orbit settles into the same 2-cycle. Now take a point such as c = i: the fold redirects its orbit where the Mandelbrot iteration would not, producing a different escape decision.

History

In 1992, Michael Michelitsch and Otto E. Rössler first described and rendered the Burning Ship in a Computers & Graphics paper. Both were then researchers at the University of Tübingen (Eberhard Karls Universität Tübingen). Their paper gave the iteration and showed that the absolute-value fold produces what they called a “quasi-Julia set”: it resembles a classical Julia set, but arises from a non-analytic map.

The name “Burning Ship” comes from a close view near the real axis: a dark hull below, flame-like ridges above. Paul Bourke independently implemented the fractal in October 1993, crediting Michelitsch and Rössler; his page became an early online reference.

Because it is a non-analytic alteration of the Mandelbrot recurrence, the formula falls outside the classical complex-dynamics theory Fatou and Julia developed for analytic functions. It has since become one of the most widely explored non-analytic escape-time fractals. Buffalo, Airship, Celtic, and Perpendicular variants rearrange which coordinates are folded and when.

Visual Characteristics

Dark hulls, flame-like masts, and chimney shapes rise through folded layers. Set beside the Mandelbrot set, the asymmetry is immediate.

Remix and Examples

Start from the canonical Document

Open the same approved formula state used by this guide, then change the view, coloring, transforms, or animation in the interactive Explorer.

Remix in Explorer Open Formula Editor

Cinder Rift

View artwork

Frequently Asked Questions

What changed from the Mandelbrot set?

Taking absolute values component by component breaks the Mandelbrot set’s rotational symmetry and makes the map non-analytic. Rounded boundaries become angled edges and nested rectangles; escape-time color settles into bands rather than smooth radial gradients.

Why is the ship sometimes upside down?

The canonical formula puts the ship upside down because absolute values turn negative imaginary coordinates positive before squaring. Most renderers, including FractalPark, reflect the image vertically for display. A screen whose Y-axis points down can add a second flip, so the final orientation depends on where the renderer applies that reflection.