Illumination Problem

In the early 1950s, Ernst Straus asked

1. Is every Polygonal region illuminable from every point in the region?

2. Is every Polygonal region illuminable from at least one point in the region?

Here, illuminable means that there is a path from every point to every other by repeated reflections. Tokarsky (1995) showed that unilluminable rooms exist in the plane and 3-D, but question (2) remains open. The smallest known counterexample to (1) in the Plane has 26 sides.

See also Art Gallery Theorem

References

Klee, V. ``Is Every Polygonal Region Illuminable from Some Point?'' Math. Mag. 52, 180, 1969.

Tokarsky, G. W. ``Polygonal Rooms Not Illuminable from Every Point.'' Amer. Math. Monthly 102, 867-879, 1995.