Bistellar Flips
Graduate Student Mentors: Jonathan Alcaraz & Ethan Kowalenko
Telly Monster loves triangles. He tries to make every shape he sees into something made out of triangles. He does this with stop signs, doors, honeycombs, The Pentagon, what-have-you.
Telly has recently come across the idea of bistellar flips, which is a way of moving between different triangulations of the same shape. For instance, he's holding two triangulations of a hexagon. Can you think of how one might get to from one to the other?
The process of recreating a polygon by gluing together triangles is called creating a triangulation of the polygon. It is actually true that there is a way to flip between any two triangulations of the hexagon.
However, Telly's imagination is so good that he even tries to do the same with convex shapes in any dimension, such as platonic solids, coffins filled with concrete, and the tesseract (4-dimensional cube), using tetrahedra and higher dimension analogues of triangles called n-simplices. Even more generally, he can consider triangulations of the convex hull of any finite set of point in Euclidean space. It turns out that it is not always guaranteed that one can find a sequence of flips between any two triangulations of the same shape.
In this project, we'll consider the following problem: for which finite sets of points can one flip any triangulation to any other? Can we do this for the tesseract? This is still an open problem.
We only require linear algebra as the background for this project, but we will be thinking about convex geometry quite a bit as well. The book Triangulations: Structures for Algorithms and Applications by Jesus De Loera, Joerg Rambau, and Francisco Santos will be our main resource.