2. String Art
String Art on Polygons and Stars
To see some of the range of images you can create using 2.String Art, click 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.
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.
- Using subdivisions to create images (done without using multiplication): 2.1a. Explaining Subdivisions
- Additional explanation using multiplication: 2.1b. Subdivisions (including multiplicative explanation)
- Lines used, vertex common factor, VCF, and subdivision common factor, SCF: 2.2a. Lines Used with VCF and SCF
- Vertex frame and why lines might appear less than calculated: 2.2b. Number of Lines – Take 2
- The number of lines in a Cycle: 2.2c. Number of lines in a Cycle
- Optional Excel file for locating placement of lines in the first cycle: 2.2c. Tracking Lines in the First Cycle
- How cycles fill in the image: 2.2d. Number-of-times-around images
- Showing cycles on images: 2.2e. Using cycles to understand an image
- How many distinct images can you make? Click here to find out: 2.3, Symmetry about P = S*n/2 and Distinct Images
- On simplifying the values that created an image: 2.4a. Simplifying Parameter Values
- Symmetry: Take 1, 2.5a. Lines of Symmetry
- How subdivision points on the vertex frame create Levels of concentric circles: 2.6a. Subdivisions Create Concentric Circles
- Shape-shifting Polygons: 2.7a. Shape-shifting Polygons
- Take 2, 2.7b. Changing J to Compare Cycles for Shape-shifting Polygons
- Take 3, First version 2.7c1, Switching S and n to Compare Cycles for Shape-shifting Polygons (simple version)
- Take 4, 2.7d. Three Shape-shifting Triangles
- An overarching question: How can you find similar images? Here is an example: 2.8 Finding Similar Images: A Star In a Star
- If you want a preview, click here for some examples of Image Archetypes.
- Curved-tip stars: 2.Curved-tip Stars
- Near-halfway images on polygons: 2.Porcupine Polygons
- Near-halfway images on stars: 2.Porcupine Stars
- Sunburst polygons: Take 1, 2.Sunbursts
- Take 2, 2.Additional Sunburst Polygons
- Quivering polygons: Take 1, 2.Quivering Polygons
- Odd needle stars: 2.Odd Needle Stars
- Spinning stars: 2.Spinning Needle Stars
- Stacked circles: Take 1, 2.Stacked Circles
- Chrysanthemums: 2.Chrysanthemums
- On being an Image Detective (with Challenge Questions): 2.10a. Reverse Engineering an Image
- Image Detective: Take 2, 2.10b. Additional Image Detective Strategies
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.
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.
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.
Pencil and Ruler Exercises
-About Vertex Frame VF
Instructor Excel File