Welcome to Playing with Polygons, PwP, a website I started in early 2021 that I am using to organize materials I am writing that are at the intersection of art, mathematics, and education. The first book to come out of this work is Electronic String Art: Rhythmic Mathematics, CRC Press, an imprint of Taylor and Francis Group, February 2024.

This material started with Excel files using the vertices of regular polygons (obtained by connecting successive points that are equally spaced around a circle) to frame each image. The files are controlled by Up/Down ?/? Scroll Arrows and Click-Boxes so users need not be familiar with Excel to play with the files. These Excel files are supplemented by web materials created by Liam Myles (for example, watch the Spinning Needle Star image in the upper left corner above (and discussed in File 2 explainers) get created as a connected series of 147 line segments using the various Drawing Modes). By playing with the arrows and watching what happens, even very young users begin to see patterns … that is, they begin to make connections between numbers and images. If you want to very quickly dive into the string art material, I suggest you click here. Alternatively, you can watch one or more of these four short videos.

Such interactive play strengthens mathematical intuition and encourages learning even in the absence of formal discussion of a topic. To take the simplest example, if each regular polygon with n sides has a vertex at the top of the circle then the first few look like: △  ◇  ⬠  ⬡ and we see that the bottom of the polygon will be flat if n is odd and pointed if n is even. Kindergarteners can see this pattern … two years before even and odd is formally introduced into the curriculum according to Common Core State Standards, CCSS, 2.OA.3, or when that alternating pattern is more forcefully examined in grade four according to 4.OA.5, Generate and analyze patterns. Click here to read more about such interactive playing.

There are multiple audiences for PwP. One could consider PwP a book of recreational mathematics, not mathematical research or a mathematics textbook. It is certainly useful for those interested in mathematics education. But it would also be helpful in bridging the gap between mathematics and art. Although mathematics and art teachers in K-12 could certainly incorporate these materials into their classroom, they were initially developed for independent exploration outside the classroom. The idea was to create materials that were easy enough to use that students would want to explore each model and create their own patterns by manipulating the parameters of each model.

This link shows the general structure of PwP including main worksheets within each file. Current versions of the files as well as links to papers that have been created to supplement these files are located on separate pages. This link is to the book proposal itself; the first two pages lay out the overall structure and rationale, the remaining pages show images from each file to provide a glimpse of the range of images available using each file. Finally, if you would like to watch a video instead, this 17 minute video provides introductory glimpses of 4 the files that comprise this book (Files 1, 2, 6, and 10).

It is worth drawing attention to the material in File 2 (the first file in I. IMAGES USING SUBDIVISIONS) because it represents my current thinking regarding how the material will be presented in the final product. Instead of having long papers that provide an overview of many features at once, this chapter is composed of a series of short one to two page documents which I call Explainers based on Stein’s Triex. Think of these as short sections of the chapter. These documents are, by necessity, conceptually linked to one another, but the goal is to provide information in small bite-sized pieces targeting specific topics. Many provide annotated images that explain why an image looks the way it does. Those wishing to teach with PwP could use these documents as hand-outs that could be distributed to students electronically, or in hard copy. Alternatively, they could be projected onto smart-boards and used in much the same way that PowerPoint slides are used to support lectures introducing material.

The images stand on their own, without need for mathematical explanation. But part of the fun is in understanding those explanations and applying it in new situations. Some of those explanations are more mathematically challenging than others. Those prefaced with the letters MA (for Mathematical Approach) are more challenging but dig more deeply into the underlying structure of the images under consideration. These are typically appropriate for secondary school audiences. Additionally, short 3–5-minute videos will eventually be displayed on the website that will explain the features, patterns, and variables included in each file. On top of all this content, various lesson plans targeting different Common Core State Standards will be posted for each file. This link provides a table showing suggested topic coverage by day for different grades. The goal is to provide students and teachers with materials that will allow them to integrate PwP into a classroom setting. Once students are exposed to these materials, they can simply play and learn as they explore creating images using PwP.