The primary formula for the Phoenix fractal, a fascinating modification of the well-known Mandelbrot and Julia sets, is given by:
zn+1 = zna + c znb + p zn-1
This recursive formula defines the iterative process that generates the complex and often stunning visual patterns characteristic of the Phoenix fractal.
Understanding the Phoenix Fractal Formula
The Phoenix fractal formula introduces a unique dependency on the previous iteration (zn-1), which distinguishes it from simpler fractal formulas like those for the Mandelbrot or Julia sets. Here’s a breakdown of its components:
- zn+1: The next complex number in the sequence.
- zn: The current complex number in the sequence.
- zn-1: The complex number from the previous iteration.
- c: A complex constant parameter that significantly influences the shape and structure of the fractal.
- p: Another complex constant parameter, often referred to as the "Phoenix parameter," which provides the historical dependency on zn-1.
- a, b: Exponents, typically integers, that dictate the power to which zn is raised. For many common Phoenix fractals, 'a' and 'b' are set to specific values like 2, resulting in terms like zn2.
Initial Parameters and Variations
While the core formula remains consistent, the specific values of the parameters c and p, along with the initial values for z0 and z1 (or zn-1), determine the unique appearance of each Phoenix fractal.
For example, an initial Phoenix fractal formula utilized specific real-valued parameters:
- c = 0.5667
- p = -0.5
In many implementations, z0 is initialized to a specific complex number (often 0) and z1 (or zn-1 for the first iteration) is also initialized. The c parameter can sometimes be the complex plane point being tested, similar to how 'c' works in the Mandelbrot set.
Key Characteristics
- Recursive Nature: The iterative calculation means that each point's fate depends on its previous values.
- Parameter Sensitivity: Slight changes to c or p can lead to dramatically different visual outcomes, showcasing the rich complexity inherent in fractal geometry.
- Connection to Julia/Mandelbrot Sets: The Phoenix fractal is considered a variant that adds an extra dimension of complexity by incorporating a historical term, differentiating it from fractals that only depend on the current iteration and a constant.
- Visual Patterns: Phoenix fractals are known for their distinctive "eye" or "head" shapes, often with intricate tendrils extending outwards.
For further exploration of various fractal formulas and their visual representations, you can delve into resources on computational geometry and fractals.
