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:

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

Now 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:

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.

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:

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:

r = 0.3sec(1.6θ) – 3.65:

r = sqrt(10sin(3.3θ)) – 6:

And two more:

r = -2.8³√(csc(0.6θ) + 0.8):

r = tan(0.5θ) + sin(0.8θ):

Exploring Some Polar Graphs with Sketchpad

I really like the new version 5 of Geometer’s Sketchpad. One of the most useful new features is the ability to quickly create parameters that can change over a specified range continuously or in specified jumps. Then you can use those parameters in functions that can be plotted. In the screencast below, I used parameters and animation to illustrate properties of some special polar graphs, specifically “rose curves”, cardioids, limaçons, and lemniscates: