Multi-Phoenix

Multi-Phoenix remembers its last step. This second-order recurrence uses both the current z and the immediately previous orbit value, weighting the latter by the complex memory coefficient p. That memory changes a feature’s direction and how long it persists.

The standard Julia step, z → z² + c, has no memory. Phoenix brings the earlier z forward as p·zₙ₋₁. Change p and the past carries a different weight; because p is complex, that weight can rotate as well as grow or shrink. Feathered spokes, flame-like fans, compass rosettes, and ember-like clusters follow.

The filaments suggested the wings and tail of the mythical bird, hence “Phoenix.” Shigehiro Ushiki of Kyoto University discovered it and published it in 1988.

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.