Project IV 2020-2021

Billiards and Kaleidoscopes

Anna Felikson


A mathematical billiard is in idealisation of a well-known game, where balls are moving along straight lines on a table, reflecting from the boundaries of the table so that the angle of incidence is equal to the angle of reflection. Some simple questions one could ask about billiards are: if a ball starts from a given point to a given direction, will it ever come back to this point? are there periodic trajectories? how does this depend on the shape of the table? can one work with billiards in geometries other than Euclidean one? These questions may bring us to many different areas of mathematics, such as dynamical systems, reflection groups, geometry of surfaces.

A kaleidoscope is a kids toy which features reflections - it allows to obtain beautiful patterns but also a geometric idea used often to work with billiards or to discuss tessellations. Kaleidoscopes are related to Coxeter groups and to regular polytopes.

Prerequisites and Corequisites: Algebra II is essential, taking Geometry III/IV would greatly help.


More reading on billiards can be found here and on reflection groups here.

email: Anna Felikson