Introduction

This website has a number of sheets organized into three major parts, all of which are based in one way or another on regular polygons. Regular polygons have vertices that are equally spaced around a circle. The sides of a polygon are created by connecting adjacent vertices with line segments around the circle. Since vertices are equally spaced, all sides are the same size. If there are n vertices (and sides) around the circle, we call the resulting polygon an n-gon. We follow the convention of always starting at the top of the circle and count clockwise around the circle because we live in a world with clocks!

One can imagine drawing the sides of the n-gon one at a time without regard to the order in which we draw the sides. Eventually, all of the sides will be drawn, and the polygon will connect all n sides. But one can also imagine drawing the polygon by connecting vertices around the circle in a step-wise continuous fashion. We conceptualize this as jumping from one vertex to the next, drawing line segments as we go, without lifting our pencil until the end point of last segment connects to the starting point of the first segment. This occurs after n single-vertex connected jumps. We call this being “continuously drawn.” We create continuously drawn stars in the same fashion except now we jump more than one vertex before drawing the next line. The simplest star, the n = 5 pentagram, ⛤, involves connecting every other vertex (J = 2 so the order in which the pentagram is drawn is Top-2-4-1-3-Top) of a 5-gon (instead of adjacent vertices (J = 1 in which the 5-gon is drawn as Top-1-2-3-4-Top) which produces a pentagon). In both of these examples, we can think of Top as both vertex 0 and n = 5.

Chapters in PwP are associated with Excel files and chapters are often referred to by the term File rather than Chapter. These files are supplemented by Excel-free web versions of these files. Eventually, all chapters will include small (1-2 page) documents called Explainers that are not required reading but are offered to help users understand why images look like they do.

Prior to PART I we have a chapter, File 1, dealing with polygon and star basics. We also include a mathematical appendix here because a number of the files employ pieces of mathematics that are not commonly taught in a K-12 setting, most notably modular arithmetic. This need not be mastered to enjoy the materials discussed here but it seems more natural to have this early rather than late in the book, even though it should be used as needed (or as interest arises).