Zubieta

Zubieta is an “absolute-value” cousin of the classic Mandelbrot iteration. It performs the usual update zₙ² + c, then takes absolute value separately on the real and imaginary parts. This is not complex magnitude, which would squash the two parts into one real number. Think of folding a sheet along both axes: the complex structure remains, but each coordinate returns to the nonnegative quadrant.

Repeat that fold and an orbit keeps returning to, or passing through, the first quadrant. Mirror symmetry piles up. The standard Mandelbrot set has one mirror axis, the real axis; Zubieta reflects across both axes, laying out four mirrored quadrants like a kaleidoscope.

Burning Ship and Buffalo use similar component-wise folds at other moments in the iteration. Zubieta folds after both squaring and adding c. That timing gives it its own compact lobes, tiled wedges, and mirrored filaments.

Zubieta belongs to a broader family of "abs" fractal variants — maps that apply absolute-value folds at different stages of the quadratic iteration. The best-known member of this family is the Burning Ship fractal, discovered by Michael Michelitsch and Otto E. Rössler in 1992, which folds the components before squaring. The Buffalo fractal, another relative, applies the fold after squaring but before adding c.

The Zubieta variant applies the component-wise absolute value after both squaring and the addition of c. Its precise origin is difficult to pin to a single published academic source. Variants of this kind circulated among fractal-rendering communities and software packages (Ultra Fractal, Fractint, XaoS) in the 1990s and 2000s, often named after whoever first implemented or popularized them in a given program.

FractalPark's implementation follows the standard component-wise fold and renders the parameter plane in Mandelbrot mode.

Repeated component-wise folding imposes double-mirror symmetry. The image reflects across both real and imaginary axes, making mirrored wedges, compact mandala-like lobes, and tiled filaments.

Compared with Buffalo, which folds before squaring, Zubieta's post-update fold packs its lobes more tightly. In FractalPark's canonical rendering ("Kaleido Amber Mandala"), amber colour fills boundaries patterned with repeated angular motifs.