Behind a curved like a saddle screen, r puppeteers work. Each flick of the wrist births a pattern on the canvas. Mathematicians call such patterns Schottky groups — they arise in topology when studying global shapes. Schottky space is a giant archive of all possible performances. Its structure is a network of all conceivable shadows and the transitions between them.

When the screen’s size becomes exactly twice the number of puppeteers (d = 2r), unusual switches appear in the theater. Each puppeteer can flip their figure, creating a loop. A double snap — and everything returns. There are exactly r such independent switches. At first glance, this makes the system nonlinearly tangled. But once you allow the shadows to touch, all loops vanish. Any motion can be smoothly contracted to a point by dragging it through the seam where patterns merge. The boundary works like glue.

It turns out that in this theater, all performances form one continuous play. Any pattern can be smoothly transformed into another, briefly blending the shadows. This result links differential geometry and gravitational physics, where similar curved spaces describe the Universe. The first to think along these lines was Henri Poincaré. In the end, the separate shadows turn out to be parts of a single dance, where every movement extinguishes.