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.
Oldschool 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.
MA (Mathematical Approach) explainers dive more deeply into the material and may not be accessible to primary school audiences.
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.
 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)
 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
 How many distinct images can you make? 2.2f, Symmetry about P = S*n/2 and Distinct Images
 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. Numberoftimesaround 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
 2.3e. Single Cycle Images

 On simplifying the values that created an image: 2.4a. Simplifying Parameter Values
 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
 Shapeshifting Polygons: 2.6a. Shapeshifting Polygons
 Take 2, 2.6b. Changing J to Compare Cycles for Shapeshifting Polygons
 Take 3, First version 2.6c1. Switching S and n to Compare Cycles for Shapeshifting Polygons (simple version)
 Second version 2.6c2. Switching S and n to Compare Cycles for Shapeshifting Polygons (more complex version)
 Take 4, 2.6d. Three ShapeShifting Triangles
 MA. 2.6e1. SingleStep Images
 MA. Applying the difference between squares formula: 2.6e2. Creating Multiple SingleStep Polygons and Polygrams
 2.6f. Kicking the tires of 3SST
 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. SmallestStep Images
 2.7e. Comparing SingleStep with SmallestStep using 3 ShapeShifting Triangles
 2.7f. Pushing the Bounds of an Image Type
 2.7g. Using ShapeShifting Stars to Explore Curves from Lines
 On being an Image Detective (with Challenge Questions): 2.7b1. Reverse Engineering an Image
 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. StringArtNumbered Dots and P no Donut
 Other uses for the automated P file: 2.8b3. Exploring Modified Brunes Stars
 MA. 2.8b2i. Automating P to Avoid the Donut Hole
 Placing the first line part of the way around the circle: 2.8ci. Partial Way Around Images
 Excel file: 2.8cii. StringArtnumbered subdivisions P Fractional Around Circle
 Automating S and P: 2,8di. Pulsing Polygons
 Excel file: 2.8dii. StringArtfunctional S (Pulsing images)
 MA. 2.8ei. Polygons and Stars in a Cycle
 MA. Functionally modified Excel file: 2.8eii. StringArtS_and_P_Linear_Functions
 Waves of images and the Donut Hole: 2.8b1. Analyzing Waves of Images
 Named images: Click here for examples of Image Archetypes
 Curvedtip stars: 2.Curvedtip Stars
 Nearhalfway images on polygons: 2.Porcupine Polygons
 Nearhalfway 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
 Ultraneedles: UltraNeedles 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 InsideOut
 2. Can you find Similar Images
 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
 A Quivering ∆ that is 3 ShapeShifting ∆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: 2.10Tracking 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.
MA. 60Second 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 60second image is a 12point spinning star. The Ticking Clock is 6page 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 12point spinning star: StringArtMMI
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 60second 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 6page paper to the final 8page paper presented at Bridges 2023.
MA. This provides an alternate view of 60Second images using 2.8. 60 Polygons and Stars
Generative Arts Erfle, Generative Art 2023, provides a 10page overview of electronic string art.
FOR USERS
Pencil and Ruler Exercises
About Vertex Frame VF
Short Videos
Challenge Questions
2.Modified Brunes Star Internal Squares
2.Areas of Internal Squares in MBS
FOR INSTRUCTORS
Instructor Excel File
Additional Material
2. Common Core Standards Met by Grade
James Marks explains symmetry