Visualize the dynamics of zn+1 = zn2 + pzn + c and explore the fascinating patterns of fractals and metallic means
p = 0.0 + 0.0i, c = 0.0 + 0.0i
zc = -p/2 = 0.0 + 0.0i
Δ = p² + 4c = 0.0 + 0.0i
B = (p + √Δ)/2 = 0.0 + 0.0i
Fixed points: z² + (p-1)z + c = 0
0.0 + 0.0i (Multiplier: 0.0)
0.0 + 0.0i (Multiplier: 0.0)
The map zn+1 = zn2 + pzn + c defines the iteration process.
- z is a complex number representing the point being iterated
- p is a complex parameter controlling the linear term
- c is a complex parameter controlling the constant term
For each complex c (while keeping p fixed), we compute whether the critical point zc = -p/2 remains bounded under iteration. The set of all such c values forms the generalized Mandelbrot set.
For fixed complex parameters p and c, a Julia set consists of all complex numbers z0 for which the orbit remains bounded under iteration. The Julia set is the boundary between points that escape to infinity and those that remain bounded.
The metallic means are solutions to B2 - pB - c = 0, with positive root:
B = (p + √(p² + 4c))/2
These are distinct from the fixed points of the iterated map which solve z² + (p-1)z + c = 0.