Presentations:
Tiling the Plane with Squares
Presenters: Sasha and Amy
Adviser: Jim Henle
Abstract: In [1] it is shown that it is possible to tile the plane using exactly one square of each integral side-length. We will present new results stemming from this discovery dealing with the nature, the possibility, and the impossibility of square tilings.
[1] Henle, F. V. and Henle, J. M., “Squaring the Plane,” The Am. Math. Monthly, 115(1): 3-12, 2008.
Unfolding Convex Polyhedra
Presenters: Jette and Emma
Adviser: Joe O'Rourke
Abstract: It is a long-unsolved problem to decide whether or not the surface of every convex polyhedron may be sliced along its edges and unfolded flat to one connected piece without overlap. (Such a planar shape is sometimes called a *net* for the polyhedron.) Cutting any spanning tree of the 1-skeleton of the polyhedron permits the surface to be unfolding flat, but no one has found a way to guarantee there will not be overlap. Nor is there a counterexample to the hypothesis that all convex polyhedra have such an unfolding.
We prove that a subclass of the prismatoids do indeed have a non-overlapping unfolding. A prismatoid is the convex hull of two convex polygons lying in parallel planes. We hope to extend our proof to all prismatoids.
Coloring Graphs
Presenters: Nora and Marissa
Adviser: Ruth Haas
Abstract: Two colorings of a graph, G, are isomorphic if by permuting the colors in one of them, we can obtain the other. The set of nonisomorphic colorings of G is the set of isomorphism classes of proper colorings. Define the graph of nonisomorphic colorings of G, I(G), to have vertex set equal the set of nonisomorphic colorings of G, with an edge between two colorings if they are isomorphic on V(G-x) for some x in V(G). Similarly, define the graph of canonical colorings of G, Can(G) on the same set of vertices, but with an edge between two colorings if they are identical on V(G-x). In this talk we explore
properties of I(G) and Can(G).
Configuration Spaces in Phyllotaxis
Presenter: Cordelia
Adviser: Chris Gole
Abstract: Phyllotaxis is the study of plant organ arrangements such as the scales on a pine cone or the florets on a sunflower. A simple model for these arrangements consists of stacking disks on the surface of a cylinder. The resulting configuration is determined by the initial chain of disks around the cylinder. In this talk, we will examine some of the configuration spaces of those chains.