FractalPark
PhoenixMedia

Multi-Phoenix

Una recurrencia cuadrática con memoria: un coeficiente complejo lleva el valor de la órbita anterior al siguiente paso, produciendo formas emplumadas tipo llama.

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Fractal Multi-Phoenix con radios emplumados y racimos de filamentos tipo brasa («Ember Compass»)
El Multi-Phoenix de FractalPark («Ember Compass»): radios tipo brújula irradian a través de racimos brillantes de filamentos.

Resumen

Multi-Phoenix recuerda su último paso. Esta recurrencia de segundo orden usa tanto el z actual como el valor inmediatamente anterior de la órbita, ponderando este último por el coeficiente complejo de memoria p. Esa memoria cambia la dirección de un rasgo y cuánto tiempo persiste.

El paso Julia estándar, z → z² + c, no tiene memoria. Phoenix trae el z anterior como p·zₙ₋₁. Cambia p y el pasado pesa distinto; como p es complejo, ese peso puede rotar además de crecer o encogerse. Siguen radios emplumados, abanicos tipo llama, rosetas de brújula y racimos tipo brasa.

Los filamentos sugirieron las alas y la cola del ave mitológica, de ahí «Phoenix». Shigehiro Ushiki, de la Universidad de Kioto, lo descubrió y lo publicó en 1988.

Las matemáticas

Phoenix recurrence

z(n+1) = z(n)^2 + c + p times z(n-1)

Take the standard quadratic step zₙ² + c, then add p times the value from two steps ago. A second-order recurrence needs two starting values before its first iteration: z₀ and the “previous” value, usually 0. FractalPark exposes p as an adjustable complex control.

Try the classic Phoenix parameter pair c = 0.5667 and p = −0.5. Starting with z₀ = 0 and z₋₁ = 0, the first few values stay bounded and trace angular, wing-like arcs. Make p positive with a similar magnitude and the feathered structure can break apart or rotate: the memory term now reinforces the earlier direction instead of opposing it.

Historia

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.

Características visuales

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.

Parámetros

Memoria
The complex memory coefficient p sets both the strength and the rotational direction of the previous orbit value’s influence on the next step.

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Preguntas frecuentes

Why does this formula need the previous value?

The recurrence is second-order: its next state depends on the current and immediately preceding orbit values. It is a discrete analogue of a delay differential equation, where the past feeds back into the present. The extra parameter p sets how much of that past continues forward.

Does the initial previous value matter?

Yes. A second-order recurrence requires two initial conditions — z₀ and the value of z before iteration begins (z₋₁) — for a deterministic orbit. FractalPark initializes z₋₁ to zero by default, the standard convention for the Phoenix family.

Referencias