The iterated map
zₙ₊₁ = zₙ² + c
Julia: c fixed · z₀ moves through the dynamical plane
Primary plane
Coloring method
Smooth escape uses ν = n + 1 − log(log|zn|)/log p.
Color palette
Orbit laboratory
z₀
z₈——Select z₀ in the Julia plane.
Point information
Move across the fractal
Coordinate · escape iteration · magnitude · periodicity
Mandelbrot anatomy
Select a bulb to connect its period with the critical orbit and Julia set.
Julia galleryexact c-values
Deep Zoom Journeys
Whole parameter planecenter = −0.55 + 0iChoose a destination to fly into the Mandelbrot boundary.
Iterated zN field
For bounded points: H = 2.4s|zN|/R. Scaling changes only the displayed height.
Starting pointz₀ = hover the surface
Iterated statezN = —
Field value|zN| = —
Display heightH = —
Live mathematical state
zₙ₊₁ = zₙ² + c
N = 100 · R = 2 · smooth escape time