I was introduced to string art (aestheometry) about 50 years ago, probably by my high school math teacher, Art Fruhling. I was fascinated by the idea that one could create curved images using straight lines (click to see annotated examples) and I created many pieces during my teen years. Most were abstract mirror-image pieces contrasting black string on white art-board with white string on black art-board. One of the most intricate pieces I created was a three-dimensional model using threaded steel rods, fishing line and string that adorned the picture window of my undergraduate dorm room at U.C. Davis until I traded it for the painting that hangs in my office today. I did my graduate work in economics at Harvard and I am currently a professor in the International Business and Management program at Dickinson College.
I did not think about aestheometry very much again until 2008 when my son’s fifth grade teacher, Denise Eschenmann, asked if I would like to share with class some of the things you can do with Excel. (At the time, I was writing an intermediate microeconomics text in which all figures in the text had interactive counterparts in Excel and I had gotten pretty good at doing graphical things in Excel.) Two of the files I shared with this class, and with my daughters’ classes once they too were in her fifth grade class, are versions of files that are part of PwP. The aestheometry file encouraged kids to explore (x, y) graphing by mimicking string art on a closed set of vertices and the spirals file created polygons, stars, and spirals using three parameters.
Once my daughter Kate decided to study math at Dickinson, I decided to work with her on papers using the aestheometry file as a starting point. The article “Exploring Symmetry Using Aestheometry in Classrooms and Beyond,” Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture, pp. 547-554 is based on this file. This article also provides a link to an earlier unpublished piece we did with Denise Eschenmann and David Jackson using the idea behind the aestheometry file. Neither piece is readily accessible to younger viewers because graphing in the coordinate plane is initially discussed in about the fifth grade according to CCSS, 5.G.
My other daughter, Vera, was interested in early elementary education. As a result, I worked with her, together with two other colleagues on a “pre-aestheometry” piece targeted at grades K-2. The article “Connecting Geometric Patterns to Numeric Patterns using the Polygons and Stars Excel File,” Spreadsheets in Education, 2021, pp. 1-11 forms the basis for the introductory part of PwP. As I worked on this piece, I started to think about how I could create aestheometry materials that were accessible to younger viewers. My epiphany came in recognizing that if I restrict my focus to regular polygons (which are obtained by connecting successive points that are equally spaced around a circle), I could make the files accessible to young children since all they need to do is click and watch what happens. The files comprising PART I of PwP, Files 2-5, all key off of this idea.
PART II of PwP is an outgrowth of some images created using the multiple jump models. These images were mathematically interesting enough that I was able to entice my mentor, U.C. Davis professor emeritus of mathematics, Don Chakerian, into working together on some counting papers using these images. File 6, for example, shows that one can create innovative interactive materials accessible to late elementary students showing connections between geometric and numeric patterns.
PART III of PwP emerged from conversations that I have had with Liam Myles, a computer programmer from the UK I met online because we both used the phrase “playing with polygons” in describing materials we had created. His original materials can be seen here. Happily, Liam was interested in working with me to try to create an “Excel-free” version of my materials. In less than a month, Liam has created an online app that is able to create many of the images in PART I of PwP (and indeed, it can now do some of the things in PART III as well). Click here to go to that app. As importantly from my perspective, he has spurred me to create a couple of new files that round out the materials in this book. One file deals with jump sets (stacked stars) and the other deals with cardioids. These files, together with the spirals file, share a common attribute: they create images without using subdivisions between vertices.
As I have begun working on this as a book, rather than a series of academic papers, I have come to appreciate how synergistic are the two methods of delivery. The two delivery mechanisms complement one another as each has features not available in the other. As a result, you will find a number of short documents in File 2, the first file in PART I of PwP, that utilize both delivery mechanisms.