Using the Model

This chapter explores some consequences of the results in Chapter 2, especially Theorem 2.3. It suggests that Theorem 2.3 gives a way to extract information from the geometry of the low capacity lines. Section 3.2 proves that, relative to the parameter p/q, the 0th block always contains a polygon whose projection onto the X-axis has diameter at least (p + q)/2. Section 3.3 elaborates on the theme in Section 3.2 to show how to extract increasingly fine scale information about the plaid polygons. Section 3.4 explains how to augment the idea in Section 3.3 to make it more useful. Section 3.5 shows how the ideas from Section 3.3 sometimes explain why the plaid model looks similar at different rational parameters.