(Slide 16)

The subfaces of a given cell c is a set S defined by starting with the set containing only c, and adding any faces of elements of S until there are no more cells to be added. The image above shows the set of subfaces for a 3-cube. By the procedure described above, we would construct this set by starting with the cube, and adding all of its faces, the rectangles. We would then see if there are any cells in S whose faces are not in S. The answer being yes, we would then add the line segments that are the faces of the rectangles, and taking one more step add the 0-cells that are the faces of the line segments. Since the 0-cells have no faces, we have constructed the subfaces. The value of the notion of the subfaces of a k-cell will be come apparent when we consider the problem of defining self contained definitions of behavior in a single cell -- in that case, the various subfaces will correspond to all the ways this cell can interact with other cells