III: Images without Subdivisions
Part III of PwP brings together extensions of the basic model in File 1. The four files comprising Part III create images in a similar fashion in that all images are from line segments connecting one vertex to another. Put differently, no subdivisions are used.
The Spirals file allows the size of the polygon to reduce by a constant amount each time a line segment is created. Let r be a whole number larger than 0. If each time a segment is created the radius of the parent polygon is reduced by 1/r then the center will be attained after r segments. This file comes naturally after File 1 because it employs only n and J, in addition to r.
The Cardioids file is much more like File 9, the most general file in the second part of the book, in that individual lines are laid out one at a time rather than needing to be in a circuit. A simple rule produces a cardioid, connect vertex j to 2j (MOD n) for j = 1, … n-1. The file also examines what happens when 2 is replaced by a different scaling factor.
The Stacked Stars files allow multiple jump sets of stars stacked on top of one another, with each vertex having a set of lines that look kind of like the cables in a suspension bridge. Two versions are provided, one that quickly allows you to scan across images by controlling the lines on either side of center via an equation (c is how far off center the closest in line lies, and d is the decline in vertex jumps thereafter). The other allows manual input of each vertex jump. These files are a natural extension of File 5 which examined clock arithmetic by allowing 2 and 3 jump sets on a dodecagon.