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42 votes
2 answers
3k views

Can one "hear" the shape of a polygon via external reflections?

This question is a rough analog of Kac's "Can One Hear the Shape of a Drum?" A closer analog is the recent "Bounce Theorem" that says, roughly, the shape of a polygon is determined by its billiard-...
Joseph O'Rourke's user avatar
10 votes
0 answers
177 views

Minimum reflection paths in a mirror polygon

Let $P$ be a simple, orthogonal polygon of $n$ edges, i.e., one whose edges meet at right angles, and is non-self-intersecting; also known as a rectilinear polygon. Treat every edge of $P$ as a ...
Joseph O'Rourke's user avatar
4 votes
1 answer
330 views

Billiard circuits in pentagons

A billiard circuit in a convex $n$-gon is a closed billiard path of $n$ segments reflecting from consecutive edges of the polygon. Every regular $n$-gon has such a billiard circuit: Recently a ...
Joseph O'Rourke's user avatar
4 votes
0 answers
232 views

Illuminating a just-barely irrational polygon

As has been discussed earlier on MO,1,2 recently an impressive advance was proved concerning internally illuminating a mirrored polygon. Here is the result: Let $P$ be a rational polygon. Then for ...
Joseph O'Rourke's user avatar