String Art on Polygons and Stars

3 versions of the Excel File: 2.String Art          2.String Art (with algebra)          2.String Art (for teachers)

To see some of the range of images you can create using 2.String Artclick here.

How it works: All images are created using 4 parameters: n, the number of vertices in polygon; S, the number of equally spaced subdivisions between vertices; P; the number of subdivisions between points; and J, the number of clockwise counted polygonal vertex jumps. Two parameters, n and J, create the vertex frame. Each piece of the vertex frame is subdivided into S equal pieces. These subdivision endpoints are the only possible endpoints of segments in the image. The image is created, starting at the polygon’s top vertex, by connecting every Pth subdivision endpoint with a line from the previous endpoint until the circuit is completed by having an endpoint at the top starting point.

Old-school exercises: One way to learn about string art is to do some comparative examples using the Pencil and Ruler sheets to the right. Two versions are provided, one that includes the vertex frame and one that does not. Print out the sheets and make sure to use a pencil and ruler. Check your answers using the Excel file.

About Explainers. Explainers are short documents (most one page, some two pages) that explain specific aspects of the file. Think of these as supplementary material that will help you better understand the file.

Notes: MA (Mathematical Approach) explainers dive more deeply into the material and may not be accessible to primary school audiences. In PwP, the identifier E# identifies this material as being in Chapter # of ESA.

It may be worthwhile to at least look at 2.1 while initially learning about File 2. You need not look at other explainers until you want to learn more about why an image looks the way it does. Some explainers require greater mathematical understanding but none have to be read to enjoy the file.

The two worksheets in this file introduce the idea behind Aestheometry, or string art, on a polygon. The file adds two additional variables, S and P, to the two introduced in 1.Polygons and Stars, n and J. The first sheet, Subdivisions, focuses attention on S, the number of subdivisions between vertices (with P in the background) and the second sheet, Points, allows independent manipulation of S and P, the number of subdivisions between points. A wide range of images are possible as a result. The Points sheet includes click boxes that allow the user to show or hide attributes of the image as well as vertex labels. The vertex labels are useful for those interested in discussing notions of symmetry as well as a number of other issues.

The Explainers above provide a reasonably complete view of what you need to understand this file. They can be thought of as short sections, suitable for use as handouts, that provide annotated primers on the distinction between subdivisions, S, and points, P as well as a discussion of a number of other topics. Many of the issues discussed in these documents apply to the other files in PART I, sometimes with minor modifications as will be noted on those files.

E13. MA. 60-Second Images: One particularly interesting set of images are created in 60 cycles with each cycle ending one vertex past the start of the cycle. This 720 line 60-second image is a 12-point spinning star.  The Ticking Clock is 6-page paper, longer than other explainers but it could be viewed as a quick way into the discussion above.

Paper: The Ticking Clock     Excel file opening to show first cycle of 12-point spinning star: String-Art-MMI

The first half of this paper quickly covers the basics of electronic string art (discussed in the explainers above) and the last half focuses attention on what is required to create 60-second images. In short, P and J must be modular multiplicative inverses MOD 60. You can read more about MMI in the Mathematical Odds and Ends page. If interested, you can compare the above 6-page paper to the final 8-page paper presented at Bridges 2023.

MA. This provides an alternate view of 60-Second images using 2.8. 60 Polygons and Stars