Multi-Phoenix

Multi-Phoenix는 마지막 단계를 기억해요. 이 2차 점화식은 현재의 z와 바로 직전 궤도 값을 함께 사용하고, 직전 값에는 복소 기억 계수 p로 가중치를 줘요. 그 기억이 특징의 방향과 지속 시간을 바꿔요.

표준 줄리아 단계 z → z² + c에는 기억이 없어요. Phoenix는 이전 z를 p·zₙ₋₁로 끌어와요. p를 바꾸면 과거가 다른 가중치를 갖게 되고, p가 복소수라서 그 가중치는 커지거나 작아질 뿐 아니라 회전할 수도 있어요. 깃털 모양의 살, 불꽃 같은 부채, 나침반 장미, 숯불 같은 무리들이 따라 나와요.

실 모양의 구조가 신화 속 새의 날개와 꼬리를 떠올리게 해서 “Phoenix”라는 이름이 붙었어요. 교토대학의 우시키 시게히로(Shigehiro Ushiki)가 발견해 1988년에 발표했어요.

Shigehiro Ushiki, a mathematician in Kyoto University’s Graduate School of Human and Environmental Studies, discovered the Phoenix fractal. In 1988, he published the Phoenix iteration in IEEE Transactions on Circuits and Systems. He presented it as a complex-one-dimensional section of a Julia-like set derived from a complexified Hénon map. The paper is simply titled "Phoenix." The classic parameter pair (c = 0.5667, p = −0.5) produces the named Phoenix curve, later well known in fractal-rendering communities.

Ushiki’s broader research covers complex dynamical systems, Julia sets with polyhedral symmetries, higher-dimensional complex Hénon maps, and the dynamics of surface automorphisms. His Kyoto University homepage lists publications from the 1980s to the present.

The Phoenix family later appeared in Ultra Fractal and other fractal software in both Julia-mode and Mandelbrot-mode variants. FractalPark’s Multi-Phoenix uses the same core recurrence and adds a multi-preview parameter interface.

That one-step memory is visible in feathered spokes radiating from centres, flame-like fans curling along filaments, and compass forms with a sense of direction. A standard quadratic Julia does not make these in the same way; they rely on the directional persistence of p·zₙ₋₁.

With different parameters, the same rule can close into ember-like clusters, open into radiating pinwheels, or dissolve into dust-like regions. FractalPark’s “Ember Compass” foregrounds the compass spokes and warm filament clusters.