Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
Tile inflations
As outlined in Chapter 3, each tile may be re-tiled after a τ2 inflation in more than one way depending on how the short 5-fold and 3-fold edges are subdivided. Here, these different inflation rules are outlined in detail. For each way that a given tile can be inflated, a list of the vertices formed after inflation is given. Then the re-tiling of the inflated tile is listed by tile type with vertices. Each tile's ordering of vertices is significant when oriented surface patches need to be associated with a tile, and the ordering of the vertices is the same as in Table 3.1 . For the O 2 tile, only the 4 vertices actually in the tiling are listed; a fifth vertex is necessary to realize this tile as a tetrahedron, and this vertex may be deduced by context since two or more O 2 s are always glued together in a tile's inflation.
The possible number of ways to inflate a tile depends only on the small faces of the tile. These are the faces that have long and short 5-fold edges and a short 3-fold edge. It is useful to introduce a shorthand for the number of ways to inflate one of these faces. The vertex that has the two short edges emanating from it defines an unambiguous reference point for this face, and the inflation of the face may be specified by saying whether the two vertices placed on the short edges after inflation are closer to the reference point or its opposite vertex along the edge. ``S'' for ``short'' will be used to denote the case when the vertex added after inflation is closest to the reference point. ``L'' for ``long'' will denote the other case. By convention, the inflation of a face will be specified by first specifying the inflation of the 5-fold short stick and then the inflation of the 3-fold short stick. Thus, ``SL'' means that the face is inflated so that the vertex added on the 5-fold edge is closest to the reference point and the one on the 3-fold edge is closest to the other endpoint.
Subsections Chris Kimmer 2011-06-01