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Mandelbox

Um mapa de dobra e escala inspirado no Mandelbox de 2010 de Tom Lowe: reflexões de caixa, dobra radial e escala transformam uma iteração em cômodos e paredes.

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O mapa inspirado no Mandelbox do FractalPark: paredes retas, cantos em ângulo reto e recuos em camadas como o interior de uma fortaleza cristalina.

Visão geral

A maioria das fórmulas fractais começa elevando ao quadrado, como em z² + c. O Mandelbox escolhe outro caminho: uma dobra de caixa envia as coordenadas fora de um cubo fixo de volta em direção às suas faces; uma dobra radial, ou esférica, então inverte ou escala um ponto conforme sua distância da origem; uma escala uniforme estica ou comprime o resultado antes de somar c. Repita a sequência, e são as dobras — não a quadratura — que constroem a estrutura.

Reflexões e inversões não exigem um número específico de dimensões, então a construção pode ser feita em qualquer uma delas. O FractalPark desenha uma variante bidimensional que mantém o mesmo núcleo de dobra e escala no plano complexo.

Este mapa de parâmetros não se suaviza nas curvas familiares dos polinômios. Ele recorta câmaras de bordas duras, fortificações aninhadas e passagens que parecem saídas de uma planta. Sua borda divide as órbitas limitadas — as que permanecem ao alcance — das que escapam. Aproxime-se e a borda irregular e autossimilar traz de volta o motivo da caixa em escala menor.

A Matemática

Fold-and-scale iteration

z_{n+1}=s\,\operatorname{ballFold}(\operatorname{boxFold}(z_n))+c

Each round puts the orbit through four moves: box fold, radial fold, scale, then a shift by c.

The box fold handles one coordinate at a time. Take a component a of z. Above 1, it reflects as a → 2 − a; below −1, as a → −2 − a. Components in [−1, 1] stay put. The outer plane folds back into the central square.

Next comes the ball fold, checking the squared magnitude r² = |z|². If r² < 0.25 (|z| < 0.5), scale the point by 4 and push it away from the origin. If 0.25 ≤ r² < 1, invert it through the unit sphere: z → z/r², so a radius of m becomes 1/m. Points with r² ≥ 1 stay put. The near points move outward; the middle band turns across the unit sphere. These moves make the unsettled interface behind the Mandelbox's boxy layers.

Try z₀ = (1.5, 0) with scale s = 2. The box fold reflects x = 1.5 to x = 0.5. Since r² = 0.25 falls in the middle band, the ball fold inverts it: z = z/r² = (2, 0). Scaling by 2 gives (4, 0), and adding c completes the step. Start at the origin instead and you are in the innermost band: the ball fold multiplies it by 4—still the origin—so the first iterate lands exactly on c.

História

The Mandelbox was first presented by Tom Lowe (known online as Tglad) in early 2010 on the FractalForums community. Lowe was experimenting with iterative folding operations in three dimensions — box reflections and spherical inversions — to see whether they could produce bounded, self‑similar structures analogous to the Mandelbrot set. The results were immediately recognised as a new class of fractal, and the name "Mandelbox" was adopted both as an homage to the Mandelbrot set and because of the boxlike shape of the set when visualised.

The discovery followed closely on the 2009 Mandelbulb by Daniel White and Paul Nylander, which had sparked renewed hobbyist interest in finding non‑trivial 3D fractals. Within months, contributors on FractalForums — including Knighty, Jesse, and others — extended the Mandelbox idea to different symmetry planes and polyhedral folds.

Despite its popularity in fractal art software (Mandelbulber, Mandelbulb3D, Fragmentarium, Ultra Fractal), the Mandelbox has received relatively little formal mathematical analysis. One of the few peer‑reviewed treatments is Gregg Helt's 2018 Bridges Conference paper, which generalises the spherical inversion step to arbitrary shape inversions.

Características Visuais

Repeated folds produce hard-edged chambers, nested fortifications, and corridors with depth. The default parameter s = 2 gives the structure straight walls, right-angle corners, and layered recesses like the inside of a crystalline fortress.

At finer scales, the box theme returns: small copies of the overall shape appear in recesses and along boundaries, all with the same rectilinear geometry. The canonical image uses cool blues and cyans, which suit the mechanical geometry of the folds.

Parâmetros

Escala: Scale sets how strongly the folded orbit expands or reverses before c is added. A scale of 2 (the default) roughly doubles the folded value; negative values reverse direction and produce mirror-image variations.

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Perguntas Frequentes

What are box and ball folds?

A box fold reflects coordinates outside a fixed interval toward the origin—like folding a sheet’s edges inward. A ball fold (or sphere fold) makes a conditional inversion: points too close to the origin are pushed outward, points at an intermediate distance invert through a sphere, and distant points remain unchanged. Together they redirect the orbit at every step and make the Mandelbox’s layered geometry.

Is this the three‑dimensional Mandelbox?

No. FractalPark renders a two-dimensional map inspired by the same fold-and-scale construction. The original Mandelbox is most often explored in 3D, where the box fold reflects across a cube’s faces; here the same logic works in the complex plane with a square boundary.