String art and 2D curvature

Jan 2024
craft math-crafts-class

Summary:

Craft: sew thread through evenly spaced holes around the perimeter of a circle according to a simple pattern. Connect each hole to the hole a constant rotation around the circle. Observe the emergent circular patterns.

Math: relate the radius of a polygon to its side length and angles. Understand curvature of a curve as angle per unit distance.

String art and 2D curvature

summary

We’ve seen a couple ways to force crafts to pop out of the plane, a process we called “curvature”. Today we will discover how to quantitatively measure curvature. We start in 2d: for a curved line scribbled on a piece of paper, can we describe how much it curves, and in which direction it curves? We will investigate equilateral polygons and stars, seeing the relationship between curvature and circles. We make polygons by stretching stings across a circle, creating beautiful, intricate patterns called string art.

Materials and setup

materials and setup

Every student should have:

  • 2 string art blanks (see below)
  • 1 mini spool of thread (see below)
  • a protractor

    Making the string art blanks

    Start by creating some number of equally spaced points around a large circle, on some piece of paper. I made 36 equally spaced points around a circle of diameter 6 inches. There are a couple ways to do this:

  • The easier way is probably to find a template online and print it out. I didn’t use this method, but i suspect this site will work
  • I only had a protractor. I put a nail inside the little hole at the center, through the protractor, the paper, and a piece of backing cardboard below. Then, I could trace a circle by swiveling the protractor around the nail. Using the degree markers on the outside, I subdivided the circle into 36 equal parts (10 degree increments) Next, place your marked template on a stack of 6 sheets of cardstock, placed utop a layer of cardboard. Hammer a nail at every marked point through all the paper into the cardboard. Make sure the body of the nail goes through each sheet, not just the tip.

Separate the template from the rest of the sheets, but keep all 6 nailed sheets lined up. Cut out a larger circle around the circle of holes, about a centimeter larger radius. Then, cut slits from the outer border into each hole.

Giving students string

For a 6 inch diameter blank with 36 holes, each piece of string art needs at most 24 feet of string – on average, they’ll take about half that. It turns out, it’s tricky to get students individual units of string without it tangling. You want to use fairly thin string to get good results. A couple options:

  1. Probably the best option: Buy a pack of embroidery floss skeins. These are around 8 yards each, which is perfect for 1 6-inch string art. They’re also embarrassingly cheap. On amazon, you can find 100 for 10 buck.
  2. I used thread. To give it to the students, I respooled the thread on segments of cardboard tube (i used toilet paper tubes). This works muuuch better then trying to wind the up individually, which invariably tangled.


Activity

Circular string art

Part 1: thinking about curvature

Mathematical definitions are designed to capture our intuitions. A quantitate measurement of curvature will eventually serve as our definition. Before we can quantify, we need to understand our intuition better – what curves are curvy and which ones aren’t?

Pedagogical note: If you have blackboards, give each student chalk and strongly encourage them to write on the blackboards.

Question 1: Draw some curvy pencil strokes, some straight pencil strokes, and some straight pencil strokes. Compare with others at your table. What curves are the curviest? What attributes are shared by the curviest strokes?