Burning Ship fractal¶

The escape-time set generated by iterating a quadratic complex map after replacing both real and imaginary components with their absolute values at every step.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8294
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Fractal Geometry, Complex Dynamics → Mathematics

Core Idea¶

The Burning Ship fractal is an escape-time parameter set in the complex plane. For each parameter c, begin with z0=0, replace the real and imaginary components of the current iterate by their absolute values, square the resulting complex number, add c, and repeat.

A parameter belongs to the set when its orbit remains bounded; rendering algorithms color escaping parameters by iteration count or a smoothed variant. The componentwise fold before every squaring step is load-bearing. It creates the characteristic asymmetry and makes the map non-analytic, unlike the Mandelbrot recurrence.

Display conventions are separate from identity. Images are often reflected vertically and sometimes horizontally to place the ship-like form in a familiar orientation. Iteration limit, viewport, palette, smoothing, and reflection change the picture but not the underlying recurrence and bounded-orbit definition.

How would you explain it like I'm…

The Flaming Ship Picture

The Burning Ship fractal is a picture made by playing a number game at every spot on a map. At each spot you start at zero, flip any negative parts of your number to positive, square it, add the spot's own number, and do it again and again. If the number stays small forever, the spot is part of the shape; if it zooms off, it's colored by how fast it zoomed. When turned the right way up, the shape looks like a ship on fire.

The Fold-and-Square Fractal

The Burning Ship fractal is made using complex numbers, which have a 'real' part and an 'imaginary' part and can be drawn as points on a flat map. For each point c, you start with z = 0 and repeat a rule: make both parts of z positive, square it, then add c. If the numbers stay bounded forever, the point belongs to the set; if they grow without limit, the point escapes and is usually colored by how many steps it took. It's close to the rule for the famous Mandelbrot set, but the 'make both parts positive' step changes the shape and makes it lopsided. Pictures are often flipped so the ship looks upright, but flipping or recoloring doesn't change the set itself.

Absolute-Value-Folded Escape-Time Set

The Burning Ship fractal is an escape-time set in the complex plane, similar in spirit to the Mandelbrot set. For each parameter c, you start with z = 0 and repeatedly apply a rule: replace the real and imaginary parts of z by their absolute values, square the resulting complex number and add c. A parameter c belongs to the set if this sequence stays bounded; if it escapes, the point is colored by how many steps it took to escape, or by a smoothed version of that count. The absolute-value fold before each squaring is essential: it causes the set's asymmetric, ship-like shape and makes the map non-analytic, unlike the ordinary Mandelbrot iteration z squared plus c. Images are often flipped vertically, and sometimes horizontally, so the ship appears upright. Changing the iteration limit, zoom window, colors, smoothing or reflection changes the picture, but not the set, which is defined only by the rule and the bounded-orbit test.

The Burning Ship fractal is an escape-time parameter set in the complex plane defined by the recurrence z₀ = 0, z_{n+1} = (|Re z_n| + i|Im z_n|)² + c. A parameter c is in the set when its orbit remains bounded; rendering colors escaping parameters by iteration count or a smoothed escape measure. The componentwise absolute-value fold applied before each squaring is load-bearing: it breaks the symmetry of the Mandelbrot recurrence z² + c, producing the set's characteristic asymmetry, and it makes the map non-analytic (it is no longer a holomorphic function of z), so tools that rely on analyticity in Mandelbrot theory do not carry over automatically. Identity must be kept separate from display: images are often reflected vertically, and sometimes horizontally, to place the 'ship' in a familiar orientation, and the iteration limit, viewport, palette, smoothing, and reflection all change the rendered picture. The underlying object is fixed by the folded recurrence and the bounded-orbit criterion alone.

Structural Signature¶

Sig role-phrases:

complex parameter c. Selects one orbit and one point in the parameter plane. Constitutive input. If altered: Changing c changes the tested orbit and set membership.
zero initial state. Fixes z0=0 for the parameter-set definition. Constitutive initialization. If altered: Varying the initial state produces a Julia-style object rather than the same parameter set.
componentwise absolute value. Folds every iterate into a chosen quadrant before the quadratic step. Identity-bearing transformation. If altered: Removing it yields the Mandelbrot recurrence instead.
quadratic iteration. Squares the folded complex value and adds c repeatedly. Constitutive dynamics. If altered: A different power or recurrence defines another fractal family.
escape criterion and rendering. Classifies bounded versus escaping orbits and assigns finite-iteration color without confusing display reflection with mathematics. Constitutive test plus representational layer. If altered: A low iteration cap or reflected image can alter appearance without changing the defining recurrence.

What It Is Not¶

Not the Mandelbrot set. The absolute-value fold occurs before every quadratic iteration.
Not a quadratic Julia set. The parameter varies across the plane while the initial state remains zero.
Not defined by its picture. Palette and reflection are presentation choices.
Not analytic complex dynamics. Componentwise absolute value breaks the Cauchy–Riemann conditions.

Scope of Application¶

The object applies in escape-time fractal mathematics, complex-plane computation, visualization, and comparative study of non-analytic iterated maps.

Fractal generation. Computes bounded and escaping parameter orbits.
Dynamical systems. Studies folded non-analytic iteration.
Numerical visualization. Explores scale through viewport and iteration controls.
Algorithm design. Implements stable escape tests and coloring.
Comparative dynamics. Contrasts the fold with Mandelbrot and Julia families.

Clarity¶

The abstraction separates the mathematical set from its famous rendering. It fixes the recurrence, parameter role, initial state, and escape criterion, so rotated images, alternative palettes, or low-resolution imitations cannot substitute for the defining dynamics.

Manages Complexity¶

Millions of pixel orbits produce a visually elaborate boundary. The role structure reduces the generator to parameter, initialization, fold, quadratic update, and escape test, while relegating viewport and coloring to a reversible display layer.

Abstract Reasoning¶

Map each pixel to a declared complex parameter c and set z to zero.
Apply absolute value separately to real and imaginary components before every square.
Iterate the quadratic update while tracking magnitude and a maximum iteration count.
Classify boundedness conservatively and color only as a rendering choice.
Verify formula identity with test points before interpreting visual symmetry or detail.

Knowledge Transfer¶

The recurrence transfers literally among software implementations and higher-resolution renderings. Its ship-like visual language does not transfer as a general fractal metaphor; neighboring formulas with different powers, folds, or initial-state roles define distinct dynamical objects.

Examples¶

Canonical¶

A renderer maps c across a complex viewport, starts z at zero, uses xtemp=x²−y²+Re© and y=|2xy|+Im©, and marks points that do not escape before the iteration cap as inside.

Mapped back: complex parameter c → pixel coordinate; zero initial state → z0=0; componentwise absolute value → absolute folded components; quadratic iteration → square plus c; escape criterion and rendering → radius test and iteration color.

Applied / In Practice¶

Two images look vertically reversed because one renderer reflects the final bitmap. Comparing orbit values before display shows identical membership, demonstrating that orientation is not a recurrence change.

Mapped back: complex parameter c → same sampled parameters; zero initial state → same initialization; componentwise absolute value → same fold; quadratic iteration → same recurrence; escape criterion and rendering → same classification, different reflection.

Structural Tensions¶

T1: mathematical identity vs. visual convention. Reflection and coloring shape recognition while leaving set membership untouched. Diagnostic: Did the recurrence change or only the display transform?

T2: finite computation vs. infinite boundedness. Rendering stops after a finite cap although membership concerns the whole orbit. Diagnostic: How does the iteration cap affect uncertainty near the boundary?

T3: formula similarity vs. dynamical difference. One absolute-value operation separates the object from an analytic quadratic family and reorganizes the boundary. Diagnostic: At what exact point in the update is the fold applied?

Structural–Framed Character¶

Burning Ship fractal is structural within dynamical systems. Its recurrence and membership are formal; palette and orientation are conventions. It is non-evaluative and independent of institution. Vocabulary transfers among implementations only when the equation is preserved. Its character: a folded non-analytic quadratic parameter set whose visual identity emerges from bounded-orbit computation.

Structural Core vs. Domain Accent¶

Skeletal core. Repeatedly transform a state under a parameter and classify parameters by long-run boundedness.

Domain-bound accent. Complex components, absolute-value folding, squaring, escape radius, iteration count, and parameter-plane rendering define this set.

Why not prime. Iteration and escape classification travel; the Burning Ship is one exact formula-defined fractal.

This entry is a kind of Dynamical Set.

Iteration. Repeated application generates the orbit whose behavior is tested.
Symmetry breaking. The non-analytic fold changes the symmetry and geometry of the quadratic family.
No canonical parent edge is asserted in the current DAG.

Relationships to Other Abstractions¶

Current abstraction Burning Ship fractal Domain-specific

Parents (1) — more general patterns this builds on

Burning Ship fractal is a kind of Dynamical Set Domain-specific

Burning Ship fractal satisfies the defining boundary of Dynamical Set: A dynamical set is a subset of a dynamical system's state space defined or characterized by the behavior of points, orbits, iterates, images, preimages, recurrence, escape, stability, or invariance under a specified transformation or group or semigroup action.

Hierarchy path (1) — routes to 1 parentless root

Burning Ship fractal → Dynamical Set → Set and Membership

Neighborhood in Abstraction Space¶

Burning Ship fractal sits in a moderately populated region (55^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Dynamical Systems & Differential Structures (37 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Mandelbrot set. Tell: Are real and imaginary components folded by absolute value before squaring?
Burning Ship Julia set. Tell: Does c vary while z0 is fixed, or is c fixed while initial states vary?
Tricorn. Tell: Is the map conjugated or componentwise folded?
Reflected rendering. Tell: Has the formula changed or only the displayed orientation?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Burning_Ship_fractal (revision 1365247420).
Preserved source candidate: https://www.hpdz.net/StillImages/BurningShip.htm
Preserved source candidate: https://theory.org/fracdyn/burningship/
Preserved source candidate: http://paulbourke.net/fractals/burnship/
Preserved source candidate: https://web.archive.org/web/20051001095221/http://www.graphicandfractalworld.com/Me.html
Preserved source candidate: http://jet.ro/video_burningship
Preserved source candidate: https://web.archive.org/web/20110821220641/http://michelitsch-fractals.webs.com/Michelitsch_Fractals.htm
Preserved source candidate: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=4057
Preserved source candidate: https://7affer.github.io/fractalTS/

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.