마그넷어려움

Magnet Type 2

몫 안에 z²이 들어 있는 제곱 유리 사상이에요. Type 1의 지형을 더 가파르고 더 많이 접힌 형태로 만들어요.

c = −0.56 + 0.11i에서의 FractalPark Magnet Type 2: 4중 십자 대칭, 층층의 고리와 구슬 사슬, 쌍극에서 뻗어 나오는 능선.

개요

Magnet Type 2는 ((z² + c) / (z² − c))²을 반복해요. Type 1의 제곱 몫 구조는 유지하지만, 분자와 분모 모두에서 z가 이미 제곱돼 있어요. 그 비를 제곱하면 Type 1의 두 배인 4차 유리 사상이 돼요.

차수가 늘어난 효과는 두 곳에서 드러나요. 분모 z² − c는 c의 제곱근 값 두 곳에서 0이 되는데, 하나가 아니라 둘이라서 극도 두 개예요. 반복된 제곱은 또 각진 접힘을 십자 모양 대칭으로 날카롭게 다듬어요. 극도 많아지고 굽이도 많아지면서 Type 2는 더 빽빽하고 더 각져요.

자석 계열의 메커니즘은 그대로 남아요. 극 근처를 지나는 궤도는 큰 값으로 내던져졌다가 다음 단계에서 1 쪽으로 되접혀요. 무한대로 가는 게 아니라 리만 구면 위에서 1로 사상돼요. 분지가 평면에 모이고, 극의 원상들이 그 사이의 프랙탈 경계를 조립해요.

수학

Type 2 rational iteration

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

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. 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.

시각적 특징

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.

리믹스와 예제

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Rust Cross

작품 보기

Moonstone Reef

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자주 묻는 질문

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.

개요

수학