Math, Desmos, and Artglass Windows

In a earlier post, I explained the steps involved in making an artglass window using lead came. In this one. I’ll show you how to make a window using the copper foil technique. This technique is good for smaller pieces, and designs that have a lot of detail.

I’ve been using Desmos to brainstorm window designs. It’s so easy to plot polar graphs with it, and they usually have a lot of symmetry. For this particular design, I played around with a tangent plot. In Desmos, I entered r = a*tan(b*theta) + c, and created sliders for a, b, and c. Then I adjusted them until I found a promising design; in this case a = 1.8, b = 1.6, and c = 5.3:

desmos tangent window

Next, I printed out the design and traced it onto a large sheet of paper. This will be my working pattern, called a cartoon:

Tangent Window CartoonNow comes the most time-consuming step: cut all the pieces of glass to fit into the cartoon. I decided to go with green in the center, then alternate clear, blue, and yellow pieces as you work out from the center. Here are the pieces of glass as I cut them in stages:

Tangent Window Cuts 1 Tangent Window Cuts 2 Tangent Window Cuts 3

Once all the pieces are cut (make sure there are no pieces overlapping their boundaries), I wrapped them in copper foil tape. It’s exactly like it sounds: copper with a sticky backing.

Tangent Window Taped

Now the window is ready for soldering. I brush all the copper with flux (a chemical that enables the lead/tin solder to adhere to the copper tape), and then use a soldering iron to melt the solder onto the tape. I do this on both sides. On the front side, I add more solder to “raise a bead” and make it look finished. Here’s the result:

Tangent Window Final

The ease with which Desmos plots complicated polar equations makes it an ideal tool to design symmetric artglass windows. I think this is the beginning of a beautiful relationship!

Update: I’ve made three more windows using Desmos.

This one uses the polar plot of r = 1.9tan(0.3θ) – 5.1:

tan window

r = 0.3sec(1.6θ) – 3.65:

sec window

r = sqrt(10sin(3.3θ)) – 6:

sqrt(sin) window


And two more:

r = -2.8³√(csc(0.6θ) + 0.8):


r = tan(0.5θ) + sin(0.8θ):








5 thoughts on “Math, Desmos, and Artglass Windows

  1. I’m teaching a workshop on math in fashion designing. I found this page really interesting as another way to show math in design. Thanks!

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