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|.

Open in Explorer
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.

Remix in ExplorerOpen Formula Editor

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