*star and compound polygons*

on 2 to 9 vertices

on 2 to 9 vertices

You might draw ten pointed polygons while exploring the multiplication table, for example. In the picture below, skip counting by 6 while drawing a line between the last digits of consecutive numbers gives us a pentagon: counting 0, 6, 12, 18, 24, 30 we draw lines connecting 0, 6, 2, 8, 4, and 0.

*skip counting by 6 draws {5/2}*

When drawing star and compound polygons by hand, you start with

*n*points spaced evenly around a circle, and then from each point connect to another, always skipping over the same number of points. If you skip over 0 points, you get the regular

*n*-gon. If you skip over

*k*points, and

*k+*1 is relatively prime with

*n*, you will get a star polygon, if

*n*and

*k*+1 share factors, you get a compound of several regular or star polygons.

*On 9 points, skipping over 0, 1, 2, and 3*

vertices

vertices

It is interesting how an easy to describe algorithm like this, skipping around points on a circle, translates into a program.

The polygons on this page are drawn using some JavaScript (code here), which includes some use of trig functions to place the initial vertices (like points around a unit circle) and modular arithmetic to help traverse the list of points in a circular way. It's a nice example of how math makes its way into how we implement even simple algorithms.

When we go fully over to a mathematical way of expressing how to draw these by using

**desmos**, we can see how mathematics can, in this case, express the algorithm in a surprisingly compact way. You can check out the graph here.

*desmos sketch of {7/2},*

graph here

graph here

**Related links and posts***star polygon page*

*star polygons in desmos*

*polygons in the multiplication table*