이색보통

Circle Inversion

매 단계마다 가까움과 멂을 뒤바꾸는 역수 이차 사상이에요. 작은 값은 바깥으로 날아가고, 큰 값은 원점 쪽으로 돌아와요.

Circle Inversion 매개변수 평면: 겹겹의 고리, 축 방향 가시, 원점의 극.

개요

Circle Inversion은 익숙한 이차 성장을 역수 이차 단계로 바꿔요. 1/z²을 취한 다음 c만큼 옮기는 거죠. 작은 맞바꿈처럼 보이지만, 그 행동은 전혀 작지 않아요.

보통의 다항식 반복에서는 궤도가 일정 크기 이상 자라면 계속 원점에서 멀어져요. 여기서는 그 논리가 뒤집혀요. 0에 가까운 값은 어마어마하게 커지고, 어마어마하게 큰 값은 0 근처로 돌아와요. 그러면 c가 그 맞바꿔진 값이 어디에 닿을지를 결정해요.

그림은 방사형이에요. 겹겹의 고리와 날카로운 축 방향 흔적이 보여요. 원점은 2차 극, 즉 사상이 정의되지 않는 점이고, 그 끌림은 매개변수 평면 전체에 미쳐요.

수학

Reciprocal quadratic iteration

z_{n+1}=\frac{1}{z_n^2}+c

At each step, the map takes the reciprocal of z²—geometric inversion through the unit circle with a doubled angle—then shifts the result by the complex parameter c. That inversion gives the formula its name; it is classical circle inversion applied to z² rather than to z.

Try c = 0 and start at z₀ = 2. Then z₁ = 1/4 = 0.25, z₂ = 1/(0.0625) = 16, z₃ = 1/256 ≈ 0.0039. The orbit ricochets between large and small values rather than settling at a fixed point. For c = 0.5, start at z₀ = 1: z₁ = 1/1 + 0.5 = 1.5, z₂ = 1/2.25 + 0.5 ≈ 0.944, z₃ = 1/0.892 + 0.5 ≈ 1.62. This orbit oscillates without diverging, suggesting c = 0.5 belongs to the bounded set. Escape needs a different test from the quadratic case: a huge value may collapse on the next step, so the bailout must allow for this alternation.

역사

The map z → 1/z² + c sits near the broader rational family known as McMullen maps, z → zⁿ + λ/zⁿ. Curt McMullen introduced that family in the late 1980s, showing rational maps whose parameter spaces contain several kinds of hyperbolic components, including ones with Sierpiński-carpet Julia sets.

With n = 2, λ = 1, and a translation parameter c, this gives the Circle Inversion map. Its name comes from 1/z², which inverts the complex plane with respect to the unit circle and doubles the angle. Robert L. Devaney and others later studied the McMullen family, tracing how escape loci and Julia sets change with parameters. FractalPark renders exactly this map — the n = 2, λ = 1 case with c as the translation parameter.

시각적 특징

Concentric bands and radial spines run outward from the pole at the origin. The bands mark regions whose orbits stay bounded or escape at different rates, making nested rings like a target or the ripples from a dropped pebble.

Sharp axial structures form distinct spokes. They come from the inversion’s angular part: 1/z² doubles z’s angle, creating preferred directions where the dynamics line up. The reciprocal quadratic form gives this parameter plane a more rigid radial symmetry than most.

리믹스와 예제

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Citrine Spine

작품 보기

자주 묻는 질문

Why is the origin special?

The origin is a pole of order 2: 1/z² is undefined at z = 0 because division by zero creates a singularity. An orbit landing exactly there cannot continue; one passing nearby gets a very large next value. That is why the rendered image grows rings and spines around the origin.

How does FractalPark handle values near zero?

The renderer clips the denominator at a small epsilon value to prevent division by zero. The pole still leaves its rings and spines visible, while the calculation stays numerically stable. Its effect remains part of the rendered structure.

개요

수학