How does the Burning Ship differ from the Mandelbrot set?

The Mandelbrot set uses z_{n+1} = z_n² + c. The Burning Ship replaces z_n with (|Re(z_n)| + i|Im(z_n)|), which flips all four quadrants into the first quadrant before squaring. This destroys conjugate symmetry (the Mandelbrot set is symmetric about the real axis; the Burning Ship is not) and produces qualitatively different boundary behaviour despite the same escape criterion |z| > 2.

What notable features should I zoom into?

Use the bookmark buttons to explore: the 'Main Mast' around c ≈ (−1.755, −0.029) shows the primary vertical spike from which the ship's name derives; 'Hull Detail' near (−0.348, −0.658) reveals intricate spiral decorations; 'Mini Ship' near (−1.940, 0.001) contains an embedded miniature copy of the entire fractal; 'Rigging' at extreme zoom near (−2.000, 0.00015) shows infinite self-similar filaments. The bookmarks in the simulator load all these views automatically.

Burning Ship Fractal Explorer

Iterate z₁ = (|Re z| + i|Im z|)² + c and zoom into the iconic mast, hull, and embedded miniships — or switch to Julia mode for uniquely jagged Burning Julia sets.

Mode

Julia c

Re -1.7557

Im -0.0280

Bookmarks

Max Iterations

Iter 128

Colour Palette

Shortcuts
Drag → pan
Scroll / pinch → zoom
R → reset view
S → save PNG
J → toggle Julia mode

🔥 The Ship Shape

The whole fractal, viewed upside-down (Im axis flipped), looks like a galleon ablaze. The vertical “mast” at c ≈ (−1.755, −0.029) is the most recognisable feature — a long spike rising from the main body, decorated with miniature copies of the whole ship.

↔️ Broken Symmetry

Unlike the Mandelbrot set (symmetric about the real axis), the Burning Ship has no conjugate symmetry. The absolute value operation folds all four quadrants into the positive one before squaring, destroying the reflection that makes Mandelbrot symmetric. The result is a strikingly asymmetric structure.

🔭 Self-Similarity

Embedded throughout the fractal are miniature copies of the full ship at smaller scales — a hallmark of all complex-iteration fractals. The “Mini Ship” bookmark at c ≈ (−1.940, 0.001) shows one such copy in the main antenna region. Zoom in further for copies within copies.

🌀 Julia Mode

Switch to Julia mode to iterate the Burning Ship map with a fixed c parameter (adjust with sliders) and colouring each starting pixel z₀ by escape time. Burning Julia sets are jagged, highly filamentous fractals with rich spiralling boundary structure — qualitatively different from standard Julia sets.

The Burning Ship Formula

The Burning Ship fractal was published by Michael Michelitsch and Otto E. Rössler in 1992 in Computers & Graphics. It replaces the plain complex squaring in the Mandelbrot iteration with squaring of the absolute-value complex number:

Standard Mandelbrot: z_{n+1} = z_n² + c Burning Ship: z_{n+1} = (|Re(z_n)| + i·|Im(z_n)|)² + c Let z_n = x + iy. Then: |Re| = |x|, |Im| = |y| (|x| + i|y|)² = x² − y² + 2i|x||y| So in real components: x_{n+1} = x² − y² + Re(c) y_{n+1} = 2|x||y| + Im(c) Note: the x_{n+1} formula is identical to Mandelbrot. Only the imaginary part differs: 2|x||y| vs 2xy. Escape condition: x² + y² > BAIL² (BAIL = 256) Smooth colouring: t = n + 1 − log₂(log₂(|z|))

Anatomy of the Ship

The Burning Ship fractal (parameter plane) has several identifiable regions. Moving from the main body outward:

Feature	Approximate c	Description
Full Ship	(−0.40, −0.55), zoom×160	Overview of the entire fractal resembling a galleon
Main Mast	(−1.755, −0.029), zoom×2000	The primary vertical spike; fractal spirals at base
Prow (bow)	(−1.762, −0.034), zoom×8000	The foremost point; fine filamentary structure
Mini Ship	(−1.940, 0.0014), zoom×3000	A miniature copy of the entire ship embedded in antenna
Hull Detail	(−0.3475, −0.658), zoom×4000	Intricate spiral chains along the lower hull
Rigging	(−2.000, 0.00015), zoom×30000	Infinite self-similar filaments in the outermost tendrils

Mathematical Background: Non-Analytic Iteration

A key distinction from the Mandelbrot/Julia family is that the Burning Ship iteration g_c(z) = (|Re z| + i|Im z|)² + c is not analytic (not complex-differentiable) because the absolute value operation is not holomorphic. The theory of analytic complex dynamics (Montel’s theorem, normal families, the Fatou–Julia dichotomy) does not directly apply.

Nevertheless, the same topological escape criterion works: the complement of the filled parameter set is still connected (verified computationally), and the boundary exhibits self-similar structure that appears qualitatively similar to that of analytic fractals. The lack of analyticity means the Julia/Fatou theory cannot classify the connected components precisely, and many results proven for z²+c remain open for the Burning Ship.

The fractal dimension of the Burning Ship boundary has been estimated numerically at approximately D ≈ 1.68, slightly lower than the Mandelbrot boundary dimension of 2. However, no rigorous proof of this value is known, in contrast to the Shishikura (1998) result D=2 for the Mandelbrot set.

Smooth Colouring Formula

Integer escape time n (when |z_n| > BAIL) produces visible concentric bands. Smooth real escape time t removes these: t = n + 1 − log₂(log₂(|z_n|) / log₂(BAIL)) (equivalent form: t = n + 1 − log₂(log₂(|z_n|)) for BAIL = 2) For BAIL = 256: t = n + 1 − log₂(log₂(|z_n|)) + log₂(8) t is then mapped into the palette via t/maxIter ∈ [0,1]. This produces continuous, band-free colour gradients at any zoom level.

FAQ

What is the Burning Ship fractal?

A fractal generated by iterating z_n+1 = (|Re(z_n)| + i|Im(z_n)|)² + c, where taking absolute values before squaring “folds” the plane and creates asymmetric, ship-like shapes. First published by Michelitsch & Rössler (1992). Viewed upside-down, the main body resembles a burning galleon.

How does it differ from the Mandelbrot set?

The real part iteration (x_n+1 = x²−y²+c_r) is identical, but the imaginary part uses 2|x||y| instead of 2xy. This small change destroys conjugate symmetry and produces an entirely different fractal boundary despite the same escape radius. The Burning Ship is not analytic, so the full machinery of holomorphic dynamics does not apply.

What are Burning Julia sets?

For each fixed c one may iterate z₀=pixel under g_c(z)=(|Re z|+i|Im z|)²+c to get a “Burning Julia set”. These are jagged, asymmetric fractals with rich filamentary structure. Toggle to Julia mode in the simulator and drag the sliders to explore them.

Why does the fractal look like a ship?

When the fractal is rendered with the imaginary axis pointing upward (standard mathematical orientation), the main body appears inverted. But when viewed with Im pointing downward (“screen convention”), the main body of the parameter set at the bottom and the long vertical antenna at the top resemble a sailing ship — hull at the bottom, mast rising up. The “fire” comes from the intense orange/amber gradient often used in colour schemes showing the rapid escape regions near the hull.