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):








Embedding Interactive Graphs in Haiku

I’m a big fan of the online graphing calculator at My students and I use it all the time instead of graphing calculators because it is so much faster, and it is easier to enter functions. And now I just figured out that teachers who use Haiku can embed interactive graphs into their Haiku pages!

First, create a function with sliders. For example, in the function pane, enter y = m*x + b. Desmos will automatically ask you if you want to create sliders for m and b, so click “All”.


Next, copy the embed code Desmos provides by clicking on the “Share” button at the upper right (you have log into Desmos to access the Share feature).


Now go to the Haiku page where you want to embed the graph. Click “Add Content Block” and choose “Embed the Web”. Paste the code into the yellow box:


Haiku will say it doesn’t recognize the code, but go ahead and click “OK”. Give it a title and place your content block where you wish, then hit “Save”. You should now see your Desmos graph in its own content block. When your students click on it, it will load a fully interactive grapher!



Creativity + Desmos = A Rewarding Math Project

A couple of weeks ago, I assigned a project to my Honors Precalculus students that made use of the fantastic online calculator, Desmos :

Honors Precalculus Desmos Project

In this project, you get to combine your mathematical knowledge with your artistic creativity.

Use the Desmos online graphing calculator ( to plot a set of functions that create a picture. You must use at least 25 functions. You may use any type of function we’ve learned so far this year: polynomial, rational, piece-wise, trigonometric. You can add shading by using inequalities.

Save your finished project (include your name in the title), and submit it to me by using the “share graph” button on the top right:

 This project is worth 40 points.

Your masterpiece is due at the beginning of class Monday, November 25. We’ll view everyone’s submissions, and vote on the “Best in Show”. The winner will get a special prize!

The results far exceeded my expectations. The students threw themselves into the task with amazing enthusiasm. They learned all about restricting domains of functions, using inequalities for shading, and transformations. One student even researched how to rotate conic sections, and shared her new knowledge with her classmates.

If you are concerned about spending a lot of time learning a new program, fear not: Desmos is one of the easiest and most intuitive graphers I’ve ever worked with. They provide a brief but excellent user guide that can be downloaded here, as well as lots of video tutorials.

The gallery below contains all of my students’ final submissions, but I have to spotlight a couple students’ masterpieces. In the Beauty and the Beast one, the student used 406 equations to create it, and it is simply spectacular!


And here is a magnificent rendering of the Taj Mahal by another gifted mathematical artist:


Here are the rest of their creations. Clicking on a thumbnail brings up the full-size image. Enjoy!

Update: Desmos featured one of my student’s work on Twitter!


Math Is Everywhere (cont.)

Being a math teacher is both a blessing and a curse (to paraphrase Adrian Monk). I’ve just returned from an overnight trip to Decatur, AL with my cross country team (where our varsity crushed all competition with an incredible team score of 20, and our JV earned third with an excellent 69 points). As I got off the elevator last night, I noticed the carpet in the hallway had a very interesting pattern. I’d be willing to bet the carpet company has an undercover mathematician working in its art department! Parts of it looked like Riemann Sums:

Carpet1 Carpet2

and the overall pattern looked like a sum of trigonometric functions:

Carpet3 Carpet4

I’ve been playing around on desmos, trying to match it (and not being very successful). Does anyone have any suggestions?