Faith's Summer Research

Pop-Up Cards as a Vehicle to Teach Mathematics

Week One

We were introduced to the projects that Professor O'Rourke is currently working on. Ana, Nell, and I were assigned to pop-up books. Our goal is to make a manual for teachers to use pop-up books as a tool to teach mathematics. The level will span from elementary (basic geometry) to high school (trigonometry) to college (differential geometry). To better understand pop-up books we looked at some examples, including fascinating books by Robert Sabuda. We then began our making our own creations following models illustrated in The Pop-Up Book by Paul Jackson and The Elements of Pop-Up by David A. Carter and James Diaz.

The first models I made included the square and the rectangle by making a single cut and parallel folds. These can be centered by making one additional cut. I then added additional squares to make a more complex generations model. Finally I switched from cuts to pasting "tents." Through trial and error we learned that there are certain constraints this model must follow. These include:

• A+B=C+D;
• A+D is less than or equal to B+C;
• A+B is less than or equal to "a";
• alpha is greater than or equal to beta,

• where
1. A, B, C, D are the sides of the quadrilateral that pops up when the card folds;
2. "a" is the short side of the base of the card;
3. alpha is the angle of the card that changes when the card folds and beta is the opposite angle across from alpha in the quadrilateral ABCD.

I began experimenting with how to make 3D letters using parallel cuts and folds. I tried making them by looking at and attempting to duplicate examples. However, this proved to be rather difficult. I then searched the web via Google and found a website http://niagaracalligraphyguild.netfirms.com/news0300/popup.htm where I could download the font consisting of templates for each letter. The letters (or words) can then be printed on card stock. Ana and I each created our names.

Our next task is to create animated images of our pop-ups in Mathematica. To do this we must define the path of all the vertices in space, with respect to the back plane of the card. We spend the next couple of days creating Mathematica 3D graphics for the basic square made from two cuts, the generations of squares, and the tent made by pasting. To put these graphics into this website and the teaching manual we need to convert Mathematica animation into an animated GIF. Please see Nell's instructions on how to accomplish this.

We learned how to use Dreamweaver in order to keep a journal of our daily activities, which we will update every Friday.

Week Two

While we are continuing to work in Dreamweaver, we have also started to learn Adobe Illustrator. The figures that we create in Adobe will be used as illustrations in the teaching manual. These illustrations are much more detailed and professional than those created in Mathematica. The following figures illustrate the constraints we discovered in Week One.

Along with the figures, we are also using Adobe to create templates to be used to create the figures, including a color code for cuting, folding, and pasting.

Click on our Photo Studio to see some of the pictures of our pop-up models. We edited the images in Photoshop.

In addition, we are working on the text for the teaching manual. We want to include questions refering to the mathematics (specifically geometry) that can be discussed in the classroom. We spent time rereading and editing the current draft of the manual. We are almost finished with the parallel folds section, and next week we will be moving on to rotation.

On the side, I explored the website of Eric Demaine (who is currently working with Joe O'Rourke), and thru his site I found out about Scott Kim, Puzzle Master.

Week Three

This week we moved on to rotary motion. To make an object rotate as you open the card you must make a V cut. It is fairly easy to make the cards, but we learned that it is quite challenging to make the animations in Mathematica. It is even more difficult to animate the displaced V-fold. .

Week Four

We are working on the POV-Ray animation of the intersection of a cone and a sphere. This intersection makes a circle, which lies in a plane. The image made in POV-Ray is quite impressive.

I also researched David Huffman, who created complex folded structures using paper. I examined pictures of his work, and attempted to reproduce them. I was unable to find directions on how to make any of his structures. For this, I am disappointed, but I am still amazed and even more intrigued about his work. Here are a few of the links that I found.

http://www.skypape.com/huffman.htm

http://www.cs.dartmouth.edu/~robotics/papers/thesis.pdf

Huffman, David A., Curvature and creases: a primer on paper, IEEE Transactions on Computers, Vol. C-25, No. 10 (Oct. 1976), 1010-1019.

Stephanie and Duc are working on the Knight's Visor pop-up. We created a template to make this pop-up to include in the manual. I enjoy making these templates using Adobe Illustrator. Below is a picture of the knight's visor, as well as a "real" knight's visor.

Week Five

Our next task was to further explore the geometry of the V-fold. In particular, we wanted to show the path in space of the line extending from the central rib of the V-fold. The tip of this line, called p, moves in a plane as the card opens. This plane, known as the medial plane, is the plane through the card’s centerline, midway between the front and back faces of the card. Moreover, p traces out a quarter-circle as it moves along the medial plane centered on the V-fold apex, with radius equal to the length of the rib. We created an animation in POV-ray, one still of which is shown to the left, but we also wanted to make a card model, as shown to the right. To create the medial plane, we used two square parallel folds. We then placed the medial plane along the centerline and into a slit in each of the mountain folds, so the medial plane would continuously bisect the card angle as the card opens. Finally, it was necessary to cut a hole in the medial plane in order to place the V-fold and extended tip along the centerline.

Week Six

We are now working on the animation for rotary motion. We examined and duplicated a card made by one of Joe's former employees, Gail Parsloe. Rotary motion is created by a V-fold in a valley crease. The line extending from the central rib of the V-fold is forced to stay flat by the platform lying over the V-fold. The angle of rotation is equal to the angle of the apex of the V.

We also went to the AWIS Conference, which was held at Smith this year. We heard Dr. Rita Colwell, the former Director of NSF, speak. She funded our current project, so it was interesting to hear her speak. It was inspiring to hear about her and other past female leaders in the sciences. She also stressed the importance of mathematics as a gateway to engineering and the sciences. On the side, Dr. Colwell noted that while the sciences have advance dramtically in the last century, our educational system has not. I wonder if this is a national or international occurence? Could this explain the increasing disparity between the number of well-educated scientists in the U.S. and abroad? I should research more.

Week Seven

This week is our last week of work. Each of us wrote a one page summary of our project for the Women in Science publication. Here is my summary.

We also finished making our last cards using rotary motion. Here there are: