One recurrence, z → zpower + c, iterated at every point of the complex plane until it escapes or it doesn't. Where it doesn't escape is the Mandelbrot or Julia set; where — and how fast — it does is colored with Palette Synth's own cosine-gradient formula. Pan, zoom, and change the exponent live — or just drag directly on the picture.
Pick a point c in the complex plane, start with z = 0, and repeat z = zpower + c. For most starting points z shoots off to infinity — some quickly, some after wandering for a while — but for others it stays bounded forever. Color every pixel of the plane by how many iterations it took to escape (and leave the never-escaping points a fixed dark color) and you get the Mandelbrot set: exponent 2 is the classic shape, exponent 3+ gives the "multibrot" variants with more-fold symmetry. Flip z and c — start z at the pixel and fix c at one chosen point (the Julia knobs) — and the same recurrence traces the Julia set for that point instead, a different picture for every c.
The coloring is Palette Synth's cosine-gradient formula, reused rather than reinvented: escape speed becomes t, bands and offset rescale it before it's fed through a + b·cos(2π(c·t+d)), and the palette picker borrows that instrument's own six presets directly — computed inside a real compiled GLSL fragment shader running on your GPU, the site's second, after the plasma demo on Media Servers II, whose compile/link setup this reuses directly. Every pixel of the full picture is re-iterated from scratch on every single frame; a GPU running thousands of pixels in parallel makes that cheap enough that dragging a knob, panning, and Animate's color drift all just mean "draw one more frame," with no separate low-resolution or cached-field shortcut needed. If your browser can't run WebGL, the instrument says so plainly instead of showing a blank canvas.
The picture itself is also a control, not just a display: drag on it and, in Mandelbrot mode, it pans the view; in Julia mode, it sets c to whatever point is under the pointer — a 2D pad generalized down from a single drag point. Either way it's driving the same View and Fractal knobs, not a separate code path, so the knobs and the drag never disagree.
Fifth and last instrument in the visual-synth family that started with Palette Synth — after this one, all five from the original roadmap (color gradient, additive waveform, harmonograph, audio-reactive chemistry, and this fractal mapper) are built.