About

In flat · 2d view — where this opens — the plotter draws the orbit set of

for a single choice of the unit-circle parameters $\alpha$ and $\beta$.

Here we treat each such picture as a slice. Pick a trajectory through the $(\alpha, \beta)$ parameter square — sweep $\beta$ with $\alpha$ held fixed, or the other way around — and stack the orbit sets along it, at height $z = t$ for trajectory time $t$. The union of slices is a 3D point cloud: a bifurcation diagram of the whole family.

The cut controls slice this solid open with clipping planes. A thin slab perpendicular to the $t$ axis recovers a single 2D fractal; slabs along $x$ or $y$ are sections transverse to time, revealing the filament structures traced by the orbits as the parameters move — a view that does not exist in any single 2D picture.

Everything is generated on the fly (a few million points in well under 100 ms) and rendered as an additively-blended point cloud with WebGL.

More about the math (PDF in Spanish)