MagnetMedia

Magnet Type 1

Un cociente de Möbius al cuadrado con un polo en z = c, que reúne el plano en cuencas agrupadas y cadenas tipo cuenta.

Una vista Julia de Magnet Type 1: cuencas redondeadas y suaves se agrupan en racimos, mientras rosetas tipo estrella llenan sus fronteras fractales.

Resumen

Magnet Type 1 parte de ((z + c) / (z − c))². El cociente entre paréntesis es una transformación de Möbius, la función racional no trivial más simple; al elevarlo al cuadrado se obtiene un mapa racional de grado dos. A diferencia de z² + c, este mapa no manda los puntos lejanos cada vez más lejos. Ellos regresan hacia 1.

La dinámica se centra en dos ubicaciones opuestas. El cociente es cero en z = −c y tiene un polo en z = c. Una órbita que roza el polo sale volando muy lejos y luego regresa cerca de 1 en su siguiente paso. Las cuencas estables se extienden por el plano; entre ellas hay una frontera fractal hecha de capa tras capa de preimágenes del polo.

FractalPark ofrece un plano de parámetros, donde c se mueve, y vistas Julia, donde c se mantiene fija. La imagen guía es un primer plano Julia. Sus cuencas redondeadas se agrupan en montones como limaduras de hierro alrededor de un imán —de ahí el nombre de la familia.

Las matemáticas

A Möbius quotient, squared

z_{n+1}=\left(\frac{z_n+c}{z_n-c}\right)^2

Each step forms (z + c) / (z − c), then squares it. The ratio is zero at z = −c and blows up at z = c. Squaring makes −c a double zero and c a double pole, while leaving the map with rational degree two.

Infinity behaves differently here. As |z| grows, (z + c) / (z − c) approaches 1, so the next value lands near 1 however large z becomes. Huge values arise only near z = c. FractalPark treats |z| above 16 as escaped, so a “fast escape” marks a close pass by the pole, not a one-way trip to infinity.

On the Riemann sphere, the pole maps to infinity and infinity maps to 1, so the orbit remains well-defined at the singularity. A numerical renderer cannot divide by zero: when the denominator is exactly zero, the shader substitutes 0, and the next step continues from f(0) = 1. That is a computational safeguard, not another mathematical definition.

Historia

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. There, the boundary between magnetic and non-magnetic phases is fractal. Iterating the transformation draws that boundary, 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; the name was then adopted by other fractal software. POV-Ray’s documentation still calls its magnet1 and magnet2 patterns “derived from some magnetic renormalization transformations,” crediting the Fractint help files.

The classic Type 1 formula from that lineage is ((z² + c − 1) / (2z + c − 2))². FractalPark implements a simplified squared quotient, ((z + c) / (z − 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 visuales

In this Julia-set close-up, large smooth copper-orange spheres gather on the right in clusters of very different sizes. The deep blue at left and below holds smaller spheres and blue-white rosettes, each star-shaped around a faint orange core.

The rounded blobs are basins, where orbits settle. Rosettes and dense grain trace the fractal border between them. This arrangement belongs to this parameter and framing—other c values rearrange it. The warm/cool palette is a colouring choice, not a property of the formula.

Remix y ejemplos

Empieza desde el Documento canónico

Abre el mismo estado de fórmula aprobado que usa esta guía y luego cambia la vista, la coloración, las transformaciones o la animación en el Explorar interactivo.

Remix en Explorar Abrir el Editor de fórmulas

Magnet Copper Spores

Ver obra

Preguntas frecuentes

Why is it called Magnet?

The name comes from a family made widely known by Fractint. Its maps trace back to renormalization calculations for a model of magnetic phase transitions, presented in Heinz-Otto Peitgen and Peter H. Richter’s 1986 book *The Beauty of Fractals*. The squared-quotient formulas form clustered, domain-like basins; their boundaries mark the transition between magnetic and non-magnetic phases.

FractalPark’s Type 1 keeps the name and the squared-quotient shape but uses the simpler ratio ((z + c) / (z − c))². The formula shown on this page is exactly the one the engine computes.

What happens near z = c?

At that point the denominator z − c vanishes. Thus z = c is a pole of the quotient, and a double pole after squaring. Orbits that skim past it leap to very large values, then fold back near 1 a step later.

This behaviour builds the boundary: a growing web of points whose orbits eventually hit the pole. For the rare pixel that lands exactly there, the shader uses a safe value rather than divide by zero, matching the orbit’s continuation on the Riemann sphere.