FractalPark
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Zubieta

Un mapa cuadrático que pliega ambas partes de z² + c en el cuadrante no negativo en cada paso, produciendo doble simetría especular y formas compactas tipo mandala.

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El fractal Zubieta: cuñas espejadas y lóbulos tipo mandala en ámbar cálido («Kaleido Amber Mandala»)
El fractal Zubieta renderizado por FractalPark («Kaleido Amber Mandala»): la doble simetría especular acomoda lóbulos compactos y filamentos en mosaico en un mandala caleidoscópico.

Resumen

Zubieta es un primo de «valor absoluto» de la iteración clásica de Mandelbrot. Hace la actualización habitual zₙ² + c y luego toma el valor absoluto por separado de la parte real y de la imaginaria. Esto no es la magnitud compleja, que aplastaría las dos partes en un solo número real. Piensa en doblar una hoja por ambos ejes: la estructura compleja se mantiene, pero cada coordenada regresa al cuadrante no negativo.

Repite ese pliegue y una órbita no deja de regresar al, o pasar por, el primer cuadrante. La simetría especular se acumula. El conjunto de Mandelbrot estándar tiene un eje de espejo, el eje real; Zubieta refleja a través de ambos ejes, desplegando cuatro cuadrantes espejados como un caleidoscopio.

El Burning Ship y el Buffalo usan pliegues similares por componentes en otros momentos de la iteración. Zubieta pliega después de elevar al cuadrado y de sumar c. Ese momento le da sus propios lóbulos compactos, cuñas en mosaico y filamentos espejados.

Las matemáticas

Component-folded iteration

z(n+1) is the component-wise absolute value of z(n)^2 + c

Here the bars mean component-wise absolute value, not complex magnitude. For u = x + iy, |u| means |x| + i|y|: take the absolute value of each part. Picture folding the point into the first quadrant.

Try c = −0.5 + 0i. In the standard Mandelbrot iteration, z₀ = 0 produces z₁ = −0.5, z₂ = −0.25, z₃ ≈ −0.4375, and so on, converging to a fixed point. Zubieta folds those same values: after one step, |−0.5| = 0.5, so z₁ is 0.5 rather than −0.5. One small fold redirects the orbit—and changes how c is classified.

The escape threshold comes from the quadratic Mandelbrot test: once |zₙ| exceeds 2, divergence is guaranteed. The component-wise fold can keep some orbits bounded longer than the unfolded rule would.

Historia

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.

Características visuales

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.

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Preguntas frecuentes

Is the absolute value a complex magnitude?

No. FractalPark takes absolute value separately on the real and imaginary parts. It takes |x| and |y| independently, reflecting each coordinate into the nonnegative range without collapsing the complex number to one real value.

Why is the image strongly mirrored?

Folding both output components into the first quadrant on every iteration imposes double reflection symmetry. The orbit is repeatedly sent back toward the nonnegative region, and the parameter map mirrors across both axes. Burning Ship gains its flame-like symmetry by a related fold, though Zubieta packs the result into tighter mandala-like patterns.

Referencias