Penrose Unilluminable Room
A room whose walls are perfect mirrors, yet no single point light source can illuminate every corner. Drag the light source around the room and watch rays reflect off the curved walls. The mushroom-shaped protrusions create shadow regions that persist no matter where you place the light — a striking result from the geometry of specular reflection.
About this lab
In 1958, Roger Penrose posed a remarkable question in mathematical billiards: does there exist a room with mirrored walls such that a point light source placed at any position leaves some region of the room unilluminated? He answered affirmatively by constructing a room with mushroom-shaped protrusions along its walls. The key insight is that the curved and concave sections of these protrusions create "trapped" regions where light from certain positions can never reach, regardless of how many times rays reflect off the walls.
The Penrose unilluminable room belongs to a family of problems in computational geometry and dynamical billiards. In a convex room, every point can illuminate every other point (possibly after reflections), but non-convex rooms break this guarantee. The mushroom geometry is particularly effective because the concave undersides of the mushroom caps create regions that are shielded from direct illumination, and the curved surfaces redirect reflected rays away from these shadow zones. This is not merely a matter of finite ray count — it is a proven mathematical property of the room's geometry.
This simulation traces rays from a point light source, reflecting them specularly off the room's boundary segments. The illumination map reveals which areas of the room are lit and which remain in shadow. As you drag the light source, notice how shadow regions shift but never entirely vanish — there are always dark corners the light cannot reach, no matter how many reflections are permitted. This elegant result connects to deep questions about ergodicity in billiard systems: while generic convex billiards are ergodic (every trajectory eventually visits every region), non-convex boundaries can create stable shadow zones that persist indefinitely.