**Part III** of ** PwP** brings together extensions of the basic model in File 1. The three 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

*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**

*r***and**

*n***, in addition to**

*J***.**

*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

**to 2**

*j***(MOD**

*j***) for**

*n**= 1, …*

**j***-1. The file also examines what happens when 2 is replaced by a different scaling factor.*

**n**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 (

**is how far off center the closest in line lies, and**

*c**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.*

**d**