마그넷보통

Magnet Type 1

z = c에 극 하나를 가진 제곱 뫼비우스 몫이에요. 평면을 뭉친 분지와 구슬 같은 사슬로 모아요.

Magnet Type 1의 줄리아 뷰: 매끄러운 둥근 분지가 무리를 지어 모이고, 별 모양 장미 문양이 프랙탈 경계를 빽빽이 채워요.

개요

Magnet Type 1은 ((z + c) / (z − c))²에서 시작해요. 괄호 안의 비는 뫼비우스 변환, 즉 가장 단순한 비자명 유리 함수예요. 이를 제곱하면 2차 유리 사상이 돼요. z² + c와 달리 이 사상은 먼 점을 점점 더 멀리 보내지 않아요. 대신 1 근처로 되돌아와요.

동역학은 서로 반대편의 두 지점을 중심으로 돌아가요. 몫은 z = −c에서 0이 되고 z = c에서 극을 가져요. 극을 스치듯 지나는 궤도는 멀리 날아갔다가 다음 단계에서 1 근처로 돌아와요. 안정된 분지가 평면에 퍼지고, 그 사이에는 극의 원상(preimage)이 겹겹이 쌓여 만든 프랙탈 경계가 있어요.

FractalPark에는 c가 움직이는 매개변수 평면과 c가 고정된 줄리아 뷰가 있어요. 가이드 이미지는 줄리아 확대 사진이에요. 둥근 분지들이 자석 주위의 쇳가루처럼 뭉쳐 있는데, 그래서 가족 이름이 붙었어요.

수학

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.

역사

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.

시각적 특징

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.

리믹스와 예제

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Magnet Copper Spores

작품 보기

자주 묻는 질문

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.

개요

수학