FractalPark
MagnetDifícil

Magnet Type 2

Um mapa racional elevado ao quadrado com z² dentro do quociente, dando ao terreno do Type 1 uma forma mais íngreme e mais dobrada.

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Visão de Julia do Magnet Type 2 com simetria de cruz e quatro setores que se repetem de anéis, cadeias de contas e lobos
O Magnet Type 2 do FractalPark em c = −0.56 + 0.11i: simetria de cruz quádrupla, anéis em camadas e cadeias de contas, com cristas partindo dos polos gêmeos.

Visão geral

O Magnet Type 2 itera ((z² + c) / (z² − c))². Ele mantém o quociente quadrado do Type 1, mas z já está ao quadrado tanto no numerador quanto no denominador. Elevar essa razão ao quadrado dá um mapa racional de grau quatro, o dobro do grau do Type 1.

O grau extra aparece em dois lugares. O denominador z² − c se anula em dois valores de raiz quadrada de c, não um, produzindo dois polos. A quadratura repetida também afia a dobra angular em uma simetria em forma de cruz. Com mais polos e mais giros, o Type 2 fica mais denso e mais angular.

O mecanismo da família do ímã permanece. Uma órbita que passa perto de um polo é lançada a um valor grande e depois se dobra de volta em direção a 1 no passo seguinte. Ela não vai ao infinito: na esfera de Riemann, ela é mapeada para 1. Bacias se reúnem pelo plano, enquanto as pré-imagens dos polos montam a borda fractal entre elas.

A Matemática

Type 2 rational iteration

z(n+1) = ((z(n)^2 + c) / (z(n)^2 - c))^2

Each step computes (z² + c) / (z² − c), then squares it. The numerator is zero when z² = −c; the denominator is zero when z² = c. After squaring, every zero and pole is double, leaving the map with two double zeros and two double poles on the Riemann sphere. Its rational degree is four.

As |z| grows, (z² + c) / (z² − c) approaches 1, so the next value lands near 1 no matter how large z becomes. Only a close pass by one of the two poles produces a large value. FractalPark treats |z| above 16 as escaped, so a coloured “fast escape” region records a close encounter with a pole, not a one-way trip to infinity.

Suppose c = 1. The poles are at z² = 1, i.e. z = 1 and z = −1. Choose z₀ = 0.98, very close to the pole at 1. Then (z₀² + c) / (z₀² − c) ≈ (0.9604 + 1) / (0.9604 − 1) = 1.9604 / (−0.0396) ≈ −49.5, and squaring gives z₁ ≈ 2450. The large value then folds back: on the next step, with |z₁| large, the ratio tends to 1 and z₂ lands near 1. The shader colours the pixel according to how quickly the orbit returns near 1 after its pole encounter.

História

In 1986, Heinz-Otto Peitgen and Peter H. Richter brought magnet fractals into the literature in *The Beauty of Fractals: Images of Complex Dynamical Systems*. Their chapter “Magnetism and Complex Boundaries” (pp. 129–138) presented rational maps derived from renormalization transformations in a model of magnetic materials. The boundary between magnetic and non-magnetic phases is fractal; iterating the transformation draws it, giving the images their clustered, field-like structure.

Fractint later included the maps as “magnet1” and “magnet2,” in both Mandelbrot-style and Julia-style views, and the name spread through fractal software. POV-Ray’s documentation still describes its magnet1 and magnet2 patterns as “derived from some magnetic renormalization transformations,” crediting the Fractint help files.

The classic Type 2 formula from that lineage is ((z² + c − 1) / (2z + c − 2))². FractalPark implements a different simplified squared quotient, ((z² + c) / (z² − c))², under the same name; every render and every mathematical statement on this page refers to the formula as FractalPark actually computes it.

Características Visuais

The guide image is a Julia-mode view at c = −0.56 + 0.11i. Cross-like symmetry holds the frame together: two intersecting folding axes divide the plane into four repeating sectors, each packed with layered rings, bead-like chains, and lobes. Two opposing bright nodes mark the poles, with angular ridges radiating outward.

At the same parameter, Type 2 has finer angular detail and a more articulated basin boundary than Type 1. The warm core and cool exterior emphasise its radial structure, but the geometry comes from the map: the extra degree reorients the basins along the cross axes set by the two poles.

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Perguntas Frequentes

How does Type 2 differ from Type 1?

Type 2 squares z before forming the quotient, so its numerator and denominator carry z² rather than z. This raises the rational degree from two to four, takes the number of poles from one to two, and strengthens the cross-like angular symmetry. At the same parameter, its images are denser and more folded than Type 1’s.

Where do its poles occur?

Poles occur where the denominator z² − c vanishes: z = √c and z = −√c. Type 1 has one pole at z = c; Type 2 has two, except when c = 0, when the pole at zero is double. Their positions move with c and strongly influence the fractal boundary’s layout.

Referências