# A Stained-Glass Fractal

I have been invited to participate in Harpeth Hall’s faculty art show this fall, so I’m planning on spending some time this summer creating three or four pieces that use mathematics in their design.

I just completed the first one, which uses the fractal property of self-similarity: each outer circle is split into two smaller circles. Of course, if it were a true fractal, the process would repeat ad infinitum, but due to the limitations of working with glass I had to stop after four iterations.

# Stained Glass Mathematics, 2019 Edition

As I promised in the previous post, these are the final projects of my Mathematician As Artist students. They each researched a mathematical topic and created a design to illustrate it with stained glass. The results are pretty good, I think!

The bisector of an angle is equidistant from the sides of the angle.

The medians of a triangle intersect at a point called the centroid.

The perpendicular bisectors of a triangle intersect at a point called the circumcenter. It is the center of the circle that circumscribes the triangle.

You can construct an equilateral triangle by using two congruent circles that share a common radius.

The Four-Color Theorem states that in a map, no more than four colors are required so that no two adjacent regions have the same color.

The altitudes of a triangle meet at a point called the orthocenter.

The exterior angles of any polygon add up to 360 degrees.

If you stack right triangles so that the hypotenuse of the previous one is a leg of the next one, you create a Pythagorean Spiral.

The Two-Color case of Ramsey’s Theorem

# Light, Math, and Color – 2017 Projects

I just finished teaching another Light, Math, and Color minicourse. Twenty students researched a math topic and illustrated it with a stained glass window. Their projects this time around are really spectacular, especially considering these are first tries. (Note: If you hover over a picture, the math topic it illustrates will show up.)

# 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 aÂ 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 forÂ 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:Â I’ve made three more windows using Desmos.

This one uses the polar plot ofÂ r = 1.9tan(0.3Î¸) – 5.1:

r = 0.3sec(1.6Î¸) – 3.65:

r = sqrt(10sin(3.3Î¸)) – 6:

And two more:

r = -2.8Â³âˆš(csc(0.6Î¸) + 0.8):

r = tan(0.5Î¸) + sin(0.8Î¸):