GenieForge Tools — Beautiful Structures, No. 01
Drag the zeros inside the disk. Sweep a point around the boundary and watch its three preimages trace a triangle that never crosses one hidden ellipse.
A Blaschke product takes the unit disk and wraps it around itself exactly n times, once for each zero — every point on the boundary circle gets hit exactly n times. Domain coloring paints each point z by the direction (phase) of f(z): same color means same direction. The dark points are the zeros you’re dragging — each is a small vortex where every hue on the wheel meets at once. The faint rings and spokes are just a reference grid over the disk, for scale.
The small filled dot on the rim is a single boundary point w. The darker dots connected by a line are its preimages — every z with f(z) = w. With three zeros there are exactly three of them: that’s the inscribed triangle from the original post.
Turn on Sweep and Trail: as w goes all the way around, the triangle rotates and re-inks its edges on every pass. The patch it never once touches is the “lacuna” — drawn here by evidence, not by formula. That gap is Poncelet’s closure theorem, hiding inside a formula about complex numbers.
The idea and the original visualization are by Simone Conradi, Ph.D. — computer science teacher and theoretical physicist — who posted it with a lovely bilingual explanation of the Poncelet connection, including his own note that the fixed ellipse’s foci sit at two of the zeros of f.
This is an independent, interactive re-build for GenieForge Tools — not a copy of his code, a homage to the idea, built to be dragged, turned, and broken.
GenieForge Tools · Beautiful Structures