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 = n–k is a mirror image of the image with P = k. Another initial point to note is that if n*S is even, then P = n*S/2 always produces an image of a vertical line.
The files in PART I provide the ability to create a very wide array of images, many of which may pique your interest.
Once you find an image you like, a natural question arises: Are there other values of n, S, P and J that produce SIMILAR (but not the same) image?
For example, can you create a similar image but with a different number of vertices to the image you originally found interesting. As you search for similar images, you will undoubtedly see patterns emerge in the numbers and images and this process strengthens mathematical intuition and, eventually, mathematical understanding. Click here to see an example of this process using a star-in-a-star pattern discussed as explainer 2.8.