FractalPark
Burning ShipMedium

Airship

A Burning Ship relative with an uneven fold: the real part folds directly, while the imaginary part couples signed x to |y|.

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Airship fractal with wing-like lobes and asymmetrically folded detail
The Airship: wing-like lobes leave narrow central seams, broad across and spiky upward.

Overview

Airship changes both where the Burning Ship folds and what the other coordinate does. Only the real part is directly folded with an absolute value. The imaginary part multiplies signed x by |y|. That uneven pairing makes an orbit geometry that no simple rotation or reflection of another folded fractal can supply.

From narrow seams, the set spreads wing-like lobes, suggestive of an airship or flying wing. Its broadest parts sit near the center, then taper into slender projections. The uneven coordinate treatment shows in the picture: horizontal forms stay broad and continuous; vertical ones break into spikes.

An inversion or another spatial transform can open nested layers of wings. That is a view choice, not a sign that the formula is incomplete — it already makes its characteristic shapes.

The Mathematics

Airship fold

z(n+1) = (absolute x(n) + i times x(n) times absolute y(n))^2 + c

Fold the real component with an absolute value, then square the complex number (|xₙ| + i·xₙ·|yₙ|) before adding c. In expanded form, the real part after squaring is xₙ₊₁ = (|xₙ|)² − (xₙ·|yₙ|)² + Re(c), while the imaginary part is yₙ₊₁ = 2·|xₙ|·xₙ·|yₙ| + Im(c). Notice that xₙ appears in the cross term both inside and outside the absolute value. Its sign therefore travels through the iteration.

Try c = i. For the Burning Ship, the orbit diverges quickly. For the Airship, z₁ = (|0| + i·0·|0|)² + i = i, z₂ = (|0| + i·0·|1|)² + i = i, and the orbit remains bounded at i indefinitely — a fixed point that the Burning Ship does not share. The asymmetric fold can therefore stabilize or destabilize orbits that behave identically in the Mandelbrot set.

Visual Characteristics

Wing-like lobes rise from narrow central seams. Across the image, forms are broad and continuous; vertically, they break into spikes. Inversion can expose nested layers of those same wing-like forms.

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.

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Frequently Asked Questions

How is the Airship fold different?

The Airship folds only the real component with an absolute value; the imaginary component couples signed x with the absolute value of y. In the Burning Ship, both coordinates are folded independently before squaring. In the Airship, the fold is asymmetric — x is mirrored, but y retains its sign within the product x·|y| before the final squaring step.

Does the formula require an inversion transform?

No. Inversion can open layered wing-like forms from the Airship’s asymmetric lobes. But the Airship formula is complete on its own and produces its characteristic shapes without any view transform.

References

Source walkthrough

  1. frm-v1:824eda8dc180b39ab6c7e32f187530e1c35edeaec7db3f9d00c7a7acef78df63:init/0

    The init source node begins with `if ismand`; it establishes the starting path before the loop.

  2. frm-v1:824eda8dc180b39ab6c7e32f187530e1c35edeaec7db3f9d00c7a7acef78df63:loop/0

    The loop source node executes `safeY = (1, 0)` on each pass.

  3. frm-v1:824eda8dc180b39ab6c7e32f187530e1c35edeaec7db3f9d00c7a7acef78df63:loop/1

    This loop branch is the source operation `if abs(imag(z)) >= 1e-10`; its condition selects whether its nested assignment runs.

  4. frm-v1:824eda8dc180b39ab6c7e32f187530e1c35edeaec7db3f9d00c7a7acef78df63:bailout

    The bailout source node tests `|z| <= 256`; the iteration continues while that condition holds.

Syntax features

source-sequence

The cited source nodes for airship—`if ismand`, `safeY = (1, 0)`, its guarded safeY branch, and `|z| <= 256`—form its source-ordered path from setup through update to termination.

Parameter experiment

Steps

  1. Use the pinned profile (mode parameter-plane; center (-0.5, 0); zoom 0.4; rotation 0; iterations 96). Run it once and record the visible result, then repeat with the same profile.

What to observe

Compare the airship runs and record whether `safeY = (1, 0)` has matching visible results and iteration behavior.

Exercise

Use the cited annotations for airship—`if ismand`, `safeY = (1, 0)`, its guarded safeY branch, and `|z| <= 256`—to describe the source order.

Completion check

Your response lists the 4 cited annotation IDs and orders airship’s `if ismand`, `safeY = (1, 0)`, its guarded safeY branch, and `|z| <= 256` from initialization through loop to termination.

Sources and rights

Grounded in the pinned runtime source revision 824eda8dc180b39ab6c7e32f187530e1c35edeaec7db3f9d00c7a7acef78df63 and its semantic anchors.

rightsStatus: project-owned; implementationBasis: project-owned; publicationDecision: publish; decisionReason: publish-project-owned-native-recipe. This does not make any broader claim about a third-party original.

Formula Record

airship

This canonical FractalPark implementation is published and runnable.

Deterministic preview of airship

Canonical source

Language
frm-like/1
Standard library
1
Profile mode
parameter-plane
Profile center
-0.5, 0
Profile zoom
0.4
Profile iterations
96
Profile evidence
mechanical

Parameters

No declared parameters

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Canonical source

This verified canonical revision is read-only. Remix creates a separate editable fork.

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Source and implementation

Formula ID
38c3b212-c33b-5dd3-aae5-a514fc7e37e3
Canonical name
airship
Original name
airship
Historical source
FractalPark
Current implementation
Project-owned

This is a FractalPark project-owned implementation. The linked file is the pinned canonical Definition.

Rights or attribution concern?

Send the Formula ID and supporting evidence. The maintainer can hold or withdraw the implementation while the claim is reviewed. contact@fractalpark.com