String Art using Subdivisions between Vertices

Part I keys off of the notion of creating S equally spaced subdivisions between vertices. The image is created by connecting subdivision points that are P subdivisions from one another via connected line segments until the starting point is once again attained.

Given n vertices and S subdivisions between vertices there are a total of n*S possible endpoints for line segments rather than n possible endpoints used in the Polygons and Stars file. There are P subdivisions between points used to create the line segments in each image and the image is created one segment at a time until the initial vertex (top of the polygon) is once again reached and the circuit is completed.

Completing the circuit need not take n*S jumps. For example, let n = 3 and S = 2 and consider P = 1, 2, and 3. When P = 1, it takes six segments to complete the image with two segments on each side of the equilateral triangle. If P = 2, the same equilateral triangle is mapped out in three segments. If P = 3, the image is the vertical line from the top to the midpoint of the base and then back to the top.

If we consider P = 4 and 5 we obtain exactly the same images as P = 2 and 1, but if you follow the order in which lines are drawn, they appear to be drawn counterclockwise. The same is true whenever n/2 < P < n. In general, the image with P = nj is a mirror image of the image with P = j. Another initial point to note is that if n*S is even, then P = n*S/2 always produces in image of a vertical line.