A Thing I Made Up: Kelly Diagrams

In my Vector Calculus class recently, we were tasked with writing a paper detailing all rhombic tilings of the sphere with at most two distinct "types" of vertices. A "rhombic tiling of the sphere" effectively means a polyhedron made out of rhombi. I found it difficult to visualize these polyhedra because they exist in three dimensions. Drawing them didn't help much since drawing is a 2D medium. However, I came up with a new way of diagramming polyhedra: the Kelly diagram.

The Kelly diagram for a cube.

To create a Kelly diagram for any polyhedron, draw all of its faces out flat. Then, draw loops around the corners of faces that meet at a vertex. This means that, in a Kelly diagram, each vertex is represented by a loop that encloses the corners of the polygons that meet at the vertex. For example, the Kelly diagram above represents a cube. Note that there are 12 loops, and each one encloses the corners of three squares (for the outermost loop, assume that the area around the drawing is the inside). This is because there are 12 vertices on a cube, and each one has 3 squares meeting at it.

Kelly diagrams were a very useful visualization tool in my paper. Using this new form of diagram, I was able to prove the non-existence of the rhombic octadecahedron. If you want to read my full paper, you can download it here. The file is an open document text (.odt) so you may have to download Openoffice to view it.