## Origami-Math Science Fair Project Advice

By far, the most frequent email questions I get about origami-math are from parents or students (elementary, middle, and high school) asking me for science project ideas for themselves or their children. I also get emails from college students (and sometimes graduate students!) looking for project ideas. I have to say that I'm always a bit mystified by these questions; there's a lot of information and ideas for such origami projects on my web pages (and on other web pages out there), so there's no need to ask me such questions. Just pick something you find interesting, and go!

But I suspect that the real question such people have is HOW to start such a project, and since I do not have a bulleted list of project ideas and instructions on how to do them, they appeal to me for help. Thus, I've created this "Origami-Math Project" advice page to help people get started. Note that I only give advice here. What I won't do is tell you exactly what you should do. Sorry, but I have too many students of my own at Western New England University to work with, and they take priority. (If you really want me to be your or your child's teacher, apply to WNEU! :) (Or apply to MathILy, where I sometimes teach in the summer.)

Major advice: Every good science fair, in-class, or other math or science project has two main components.

2. Answering it. (Or partially answering it, if it's a particularly hard question.)

In other words, you should frame your project around one good question. This provides focus to your project and usually impresses teachers and science fair judges. (You pose an interesting question, and you answer it. Bingo!) The details are up to you -- you can spend some time writing about how important or interesting the question is (this is called motivating the question), and in your answer you may need to provide either a mathematical proof or informative data to support your conclusion.

That said, here are some project ideas. All of them require looking at other parts of my web page (or other web pages) for further details.

### Explorations with modular origami

There are lots of things to do with this. The "math part" would involve the geometry of polyhedral solids (like the cube, tetrahedron, dodecahedron, etc.) and/or graph theory. Here are some ideas:

### Explorations with origami geometry

The geometry of paper folding is very interesting, and there are many different things one could do in this area. My main resource is here, but there are lots of other web pages that deal with origami geometric constructions. A Google search will turn up lots. (Try searching for "origami geometry" as a start.) But here are some specific ideas:

• Explore how to fold an equilateral triangle from a square or rectangle sheet of paper. Here is one way, but there are many others.
• Show how origami can trisect an angle. This is interesting because angle trisections cannot be generally done using the classic tools of an unmarked straightedge and compass. Instructions can be found here (scroll down to the middle of the page), but proving that this method works would be needed for a good project. (You could also try to expand the method to handle obtuse angles.)
• Show how origami can double a cube, which means constructing the cube root of two. Again, look here, and scroll to the bottom.
• Describe the list of "folding axioms", as seen here. A more up-to-date list and explanation can also be found on HATORI Koshiro's web page.
• Describe how one can "fold" a parabola (and prove it). My web page, again, describes this; see the "Exercise for Axiom O5" section.

### Other topics

There are many, many things you could do for such a project. This page should give you some ideas. But please explore other web pages as well.

Part of doing a science or math project is to gather information and understand it on your own. This is what it's like to be a scientist. So please do not email me and ask me to explain everything or do the work for you. If you have a specific question about something on my web page, then please do email me. But any emails along the lines of, "I need to do a project. Please send me ideas," will probably be ignored.

I hope this helps!

Last updated: 4/22/05