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 n-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 nth 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 n = 5 and J = 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 n-gon, the same image occurs for jumps of J and nJ, the only difference is the number of times one counts clockwise around the circle to complete the image.

All n larger than 4, except for n = 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 (J = 3 for n = 10 and J = 5 for n = 12).

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.