The Illumination Problem: Exploring Rational Billiards
Delve into the intriguing realm of the Illumination Problem and Rational Billiards, examining questions of illuminability from every point in a region and the fascinating world of translation surfaces. Discover different room types, from convex to non-convex, along with insights from renowned mathematicians like Guy, Klee, Tokarski, and Castro. Unravel the complexities of folding, unfolding, and rational polygons, including isosceles right-angled triangles. Venture into the unique properties of translation surfaces and their geometric intricacies.
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Presentation Transcript
The Illumination Problem and Rational Billiards An Introduction to Translation Surfaces
The Illumination Problem Question 1 Is a region illuminable from every point in the region? Question 2 Is a region illuminable from at least one point in the region?
Question 1 Is a region illuminable from every point in the region? Ans (Guy and Klee) No. There are smooth regions not illuminable from any point.
Polygonal Rooms There is no pool shot from the yellow point to the black point. Tokarski (1995) Castro (1997)
Folding Animation See Animation
Rational Polygon All angles are rational multiples of ?
II II III Back Front V V VI IV V V Front Back III II II
(?/5, 3?/10, ?/2) Triangle A non-convex example (McMullen-Mukamel-Wright)
Fix a polygon T. Suppose the interior angles of T are of the form ?? Let N be twice the lcm of ??. Take N copies of T to make a surface ?. Then the genus is given by ???. ? ? ? = 1 +? k 2 ?? 4 ?=1
Consequence: Everything is illuminated! (with exception of finitely many points)
Thank you! References: 1. Everything is illuminated, Samuel Lelievre, Thierry Monteil, Barak Weiss 2. Three-Cornered Things, Zachary Abel's Math Blog 3. Rational billiards and flat structures, Howard Masur and Serge Tabachnikov 4. Isolation theorems for SL(2,R)-invariant submanifolds in moduli space, Alex Eskin, Maryam Mirzakhani, and Amir Mohammadi
