Sewing r = cos(2θ/3) into my Backpack with Electroluminescent Wire

I recently finished sewing the polar curve r = cos(2θ/3) into my backpack with electroluminescent (EL) wire. It was neat seeing how math, sewing, and electronics could come together in an inexpensive project, so I thought I’d write a short blog post about its development.

The finished product.

The finished project: a glowing approximation to a mathematical curve.


Back in November I was sewing EL wire into an elderly MEC backpack. It was an idea inspired by the Adafruit Tron Bag, but the bag’s zipper disintegrated before I finished the project. Not wanting to give up, I decided to find a new backpack and start again with a more mathematical design.

Over the past few months I was a teaching assistant for a multivariable calculus course at Georgia Tech. My students and I were finding the areas bounded by polar curves using double integrals, one of which was a specific Rose Curve.

The Rose Curve

The equation r = cos(2θ/3) is a specific Rose Curve with four overlapping petals.

r = cos(2theta/3)

The curve r = cos(2θ/3) in Cartesian coordinates. And what the curve was supposed to look like after I finished sewing it onto my bag …

Knowing that this was the design I wanted to use, my challenge was figuring out how to shape EL wire into that pattern, especially as I am not tremendously experienced with hand sewing. Luckily, the fabric of my backpack has a repeating pattern of “dots” that resemble points on a Cartesian grid. If you look closely at the photograph above you can make them out.

I wanted to place the curve on the front of the backpack in an area that was about 21 dots wide. I decided to trace the curve onto that space with chalk, using the dots as a guide. I could then follow the chalk lines as I sewed the EL wire. But how would I make sure that my chalk lines were precise?

Mapping the Curve onto the Fabric

Getting out the old ruler, I estimated the horizontal distance between each dot in the fabric by measuring the horizontal length between multiple dots. I made 10 measurements, and then took the average.

Number of Dots Horizontal Length (mm)
3 9.5
4 12.5
5 15.75
6 19
7 22
8 25
9 29
10 32
16 50
20 62.5

On average, the white dots in the fabric are 3.15 mm apart in the horizontal direction. I then made a number of measurements in the vertical direction.

Number of Dots Vertical Length (mm)
6 13
7 15
8 17
10 21.5
13 28
14 30
20 43
24 52.5
29 62
36 77
Estimating locations of fabric dots with Desmos.

Estimating locations of fabric dots with Desmos.

On average, the dots are 2.15 mm apart in the vertical direction. Using these averages and scaling the curve to r = 10cos(2θ/3), I plotted the locations of fabric dots in the first quadrant in Cartesian coordinates using Desmos. The blue dots in the figure to the right are approximate locations of white dots on the fabric.

Essentially, if the curve is scaled so that each dot is exactly 1 unit apart in the horizontal direction, then each dot is 2.15/3.15 units apart in the vertical direction. And the petal curve I was using is symmetric about the horizontal and vertical axes, so I only need to find a few dots in one of the four quadrants to sketch the chalk lines, below.

chalk lines

Chalk lines drawn, ready for sewing!

The Finished Project

The finished project

The curve isn’t perfect. Oh well – nothing in life really is.

Some of the threads are already loose, and because I didn’t follow the chalk lines exactly I ended up with something that resembles the perfectly symmetrical graph above. Oh well. I learned a bit about electronics, how to hand sew end knots, and hopefully it’ll help drivers see me at night when I’m riding my bike around Atlanta.


Sewing with EL wire is actually fairly easy and safe. I used invisible thread for the sewing, and all of the electronics  I purchased online through Adafruit for about $30: they have an EL Wire Starter Pack that has everything I needed. And EL wire doesn’t get hot when it’s on (unlike LEDs), and I didn’t have to solder anything.

What’s Next?

Solar panels! Building a solar powered AA battery charger for my EL Wire design looks fairly straightforward. One design on claims to cost $4. We’ll see.

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