Imagine an observer standing in a rectangular room whose walls, floor, and ceiling are mirrors.
It would appear that they are standing in an infinite space, with infinitely many copies of
themselves standing at regularly spaced intervals throughout. Geometrically this space is the
familiar 3dimensional Euclidean space. Algebraically there are symmetries that shift the point
of view between the infinitely many mirror images of the observer.
Now imagine that the room is not rectangular. There are still finitely many mirrors surrounding
the observer, but intersecting one another in various angles. It will still appear to the observer
that they are standing in an infinite space, with infinitely many identical people standing
throughout the space in a symmetric pattern, but depending on the angles of the mirrors the
space can appear to bend and curve in different ways.
The symmetries of spaces that occur this way, also in dimensions other than three, are called
reflection groups or Coxeter groups. They occur naturally in various different branches of
Mathematics and Physics. This project is about understanding the algebraic properties of
Coxeter groups and relating them to the geometric properties of the infinite space that the
observer in the mirrored room sees.