## Introduction

**Polygons and Stars**

Regular polygons are obtained by using line segments to connect closest vertices that are equally spaced around a circle. A common term for a regular polygon with* n* vertices (and sides) is an

*-gon. One can label these*

**n****vertices 0, 1, …,**

*n**-1 or 1, 2, …,*

**n***. In either event, a polygon is created once the circuit is closed by connecting the last vertex to the first (via the*

**n**

**n**^{th}line segment). It is worth noting that there are two ways to enumerate these vertices: clockwise and counter-clockwise. We restrict ourselves to clockwise enumeration and we always start at the top (at 0 or 12 o’clock depending on how you want to talk about it) because that has the most intuitive appeal given that we live in a world with clocks!

Stars are obtained when multiple vertex jumps are employed (rather than a single (closest) vertex jump pattern) until the initial vertex is once again obtained as part of the vertex jump pattern. Two things are immediately clear about stars: a) there are multiple ways to get the same image, and b) not all multiple jump patterns produce a star.

The smallest * n* that produces a star is

*= 5 and*

**n***= 2. This produces a pentagram, ⛤, by going around the circle two times. But this same VISUAL image occurs when*

**J****= 3, the only difference is that now, one needs to go clockwise around the circle three times to create the image. More generally, for any**

*J***-gon, the same image occurs for jumps of**

*n**and*

**J***–*

**n***, the only difference is the number of times one counts clockwise around the circle to complete the image.*

**J**All * n* larger than 4, except for

*= 6 produce at least one continuously drawn*

**n***-pointed star. In general, as n increases so does the number of distinct star images that are possible. But it also depends on whether*

**n****is composite or prime. If**

*n**is prime, then there are (*

**n***-1)/2-1 distinct stars, one for each jumps 2, 3, … , (*

**n***-1)/2, so that for*

**n****= 11, there are 4 distinct star images. But**

*n**= 10 and*

**n***= 12 produces only 1 possible 10 and 12 point stars each (*

**n***= 3 for*

**J***= 10 and*

**n***= 5 for*

**J****= 12).**

*n*The * n* = 6 case is worth considering because, of course, we know that there is a regular six point star, ✡, created by superimposing one equilateral triangle over another that has been turned 180°. But such a star cannot be drawn in one continuous motion using multiple jumps. The six point star requires two circuits one connecting even vertices, the other odd.