2. String Art

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 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 P^th 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.

E3. String Art basics. Creating S Subdivisions per line: 2.1a. A primer on S
- Drawing the image using P subdivisions between lines: 2.1b. A primer on P
- An alternative explainer combining S and P (without using multiplication): 2.1c1. Explaining Subdivisions
  - Additional explanation using multiplication: 2.1c2. Subdivisions (including multiplicative explanation)
E4. 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
  - 2.2c. Drawing the Vertex Frame
  - MA. 2.2d. Analyzing Patterns in Continuously Drawn Stars
    - MA. 2.2d2. Sequential Polygons and Stars Excel file
  - MA. 2.2e. An Analytical Approach to Line Placement
- How many distinct images can you make? 2.2f, Symmetry about P = S*n/2 and Distinct Images
E5. About Cycles: 2.3a. Number of lines in a Cycle
- - MA. Excel file for locating placement of lines in the first cycle: 2.3a2. Tracking Lines in the First Cycle
- How cycles fill in the image: 2.3b. Number-of-times-around images
- Showing cycles on images: 2.3c. Using cycles to understand an image
- How Image Density relates to VCF and SCF: 2.3d. Image Density
  - 2.3d2. Image Density, Take 2
- 2.3e. Single Cycle Images
E6. On simplifying the values that created an image: 2.4a. Simplifying Parameter Values
- 2.4b. There are only two ways to draw an image but there are four ways to describe it
- 2.4c1. A Primer on Vertical Symmetry
  - 2.4c2. Vertical Symmetry – Take 2
- 2.4d1. Rotational Symmetry versus Lines of Symmetry
  - 2.4d2. Additional points about Lines of Symmetry
E7. How subdivision points on the vertex frame create Levels of concentric circles: 2.5a. Subdivisions Create Concentric Circles
- More on concentric circles of subdivision points: 2.5b. One Level Change Images
- Level patterns across a cycle: 2.5c1. Level patterns across a cycle
  - MA. Excel file for Level patterns: 2.5c2. Level Jump pattern for S up to 21
- 2.5d. Symmetry across a Cycle
E8. Shape-shifting Polygons: 2.6a. Shape-shifting Polygons
- Take 2, 2.6b. Changing J to Compare Cycles for Shape-shifting Polygons
- Take 3, First version 2.6c1. Switching S and n to Compare Cycles for Shape-shifting Polygons (simple version)
  - Second version 2.6c2. Switching S and n to Compare Cycles for Shape-shifting Polygons (more complex version)
- Take 4, 2.6d. Three Shape-Shifting Triangles
- MA. 2.6e1. Single-Step Images
  - MA. Applying the difference between squares formula: 2.6e2. Creating Multiple Single-Step Polygons and Polygrams
- 2.6f. Kicking the tires of 3SST
E9. An overarching question: How can you find similar images? Here is an example: 2.7a. Finding Similar Images: A Star In a Star
- On being an Image Detective (with Challenge Questions): 2.7b1. Reverse Engineering an Image
  - Image Detective: Take 2, 2.7b2. Additional Image Detective Strategies
- Considering how the curve got there: 2.7c. Curves from Lines and Points
- Using Drawn Lines to explore the image: 2.7d. Smallest-Step Images
- 2.7e. Comparing Single-Step with Smallest-Step using 3 Shape-Shifting Triangles
- 2.7f. Pushing the Bounds of an Image Type
- 2.7g. Using Shape-Shifting Stars to Explore Curves from Lines
E10. Functionally modified String Art files can help you find similarity in images: 2.8a. Functionally Enabling n, S, P, and J
- Waves of images and the Donut Hole: 2.8b1. Analyzing Waves of Images
  - MA. 2.8b2i. Automating P to Avoid the Donut Hole
    - MA. Subdivision numbered and automated P Excel file: 2.8b2ii. String-Art-Numbered Dots and P no Donut
  - Other uses for the automated P file: 2.8b3. Exploring Modified Brunes Stars
- Placing the first line part of the way around the circle: 2.8ci. Partial Way Around Images
  - Excel file: 2.8cii. String-Art-numbered subdivisions P Fractional Around Circle
- Automating S and P: 2,8di. Pulsing Polygons
  - Excel file: 2.8dii. String-Art-functional S (Pulsing images)
- MA. 2.8ei. Polygons and Stars in a Cycle
  - MA. Functionally modified Excel file: 2.8eii. String-Art-S_and_P_Linear_Functions
E11. Named images: Click here for 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
  - Take 2, 2.Quivering Polygons: The role of J
  - Take 3, 2.Quivering Polygons: On the relative size of S and P
  - MA. 2.On Quivering Polygons Peak Rotation
- Odd needle stars: 2.Odd Needle Stars
- Spinning stars: 2.Spinning Needle Stars
  - Take 2, 2.Spinning Needle Stars: Even and Odd Dancing Partners
- Ultra-needles: Ultra-Needles and SLD mode
- Stacked circles: Take 1, 2.Stacked Circles
  - Take 2, 2.Stacked Circles – Take 2
  - Take 3: 2.Stacked Circles – Take 3
- Chrysanthemums: 2.Chrysanthemums
- 2. Clothespins as Spinning Needle Stars Turned Inside-Out
- 2. Can you find Similar Images
E12. n = P, a special class of images: 2.10 Introduction to n = P . Examples of n = P Images: 2.10 n=P Image Archetypes
- Drawing the elusive 6,2 star: 2.10 Creating Divisible Stars
- 2.10 Small Images
- 2.10 Counting Strands and Loops
- 2.10 Two Footballs
  - MA. 2.10 Modular Analysis of Two Footballs Cycles
- A Quivering ∆ that is 3 Shape-Shifting ∆s and more: 2.10 Variations on Quivering Triangles
- 2.10 Finger Traps
- About seemingly parallel curves: 2.10 Suspension curves
- 2.10 Generalized Stars
- 2.10 n = P Porcupines
  - MA. Excel file: 2.10-Tracking Line Locations (up to 560 lines)

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

EXTRA MATERIALS

1&2. GLOSSARY

FOR USERS

Pencil and Ruler Exercises

-About Vertex Frame VF

-About n with VF no VF

-About J with VF no VF

-About S with VF no VF

-About P with VF no VF

Short Videos

About S, part 1