Project Origami

Hello! If you're visiting this web site, then you must be interested in my Project Origami book. This book is a workbook of in-class projects that you can insert into various college-level math classes. (Some might be appropriate for advanced high school classes.) Each project includes a brief description, handouts, solutions, and follow-up ideas. Most of these projects I've used myself in classes at Merrimack College, and all of them were, collectively, beta-tested by over 30 college and university mathematics professors from around the country.

The idea is that origami is an active, hands-on way to engage students directly with mathematical concepts. Origami not only offers an easy way to inject active-learning into your classes, but it's also a fertile field for discovery-based learning. Furthermore, it's amazing how many different mathematical topics can turn up in paper folding. Students will be amazed to see how the math topic they are studying actually has applications in, of all places, origami!

Below is a list of the activities in the book (an informal table of contents) along with a brief description. The book can be ordered at amazon.com or at the A K Peters web page.

The Projects:

Folding Equilateral Triangles in a Square (geometry, calculus)
This asks, "How do you fold a perfect, equilateral triangle in a square piece of paper? And how do you make it as big as possible?

Dividing a Length into nths: Fujimoto Approximation (calculus, number theory, discrete dynamical systems, modeling)
This is a really cool way to approximate dividing the side of a square into n equal pieces. It can either be a simple example of error converging to zero in calculus, or a wild application of number theory, depending on how much you do.

Dividing a Length into nths: Exact Methods (geometry, precalculus)
Using simple precalculus methods, students can see how to divide the side of a square into perfect 3rds via paper folding. Or 5ths, 7ths, etc.

Folding a Parabola (geometry, precalculus, calculus, abstract algebra, modeling )
This can be an easy calc or precalc exercise on folding tangent lines to a given parabola. Or it can be an example of how performing geometric operations can be equivalent to solving algebraic equations. Field theory, anyone? This one is also born to be done on Geometer's Sketchpad.

Can Origami Trisect an Angle? (geometry, abstract algebra)
The answer is YES!

Solving Cubic Equations (geometry, abstract algebra)
Yes, origami can solve general cubic equations, and I have an activity here that demonstrates this. This is another one that's great to use with Geometer's Sketchpad.

Folding a Strip into Knots (geometry, number theory, abstract algebra)
It can be wild fun to tie a strip of paper into a flat, regular pentagon-shaped knot. But can we also make hexagon knots, heptagon knots, etc? Amazingly, this can be translated into a problem involving the Euler phi function. And if we allow ourselves to use multiple strips of paper, we get a great appplication of cosets! (Of all things!)

Haga's "Origamics" (geometry, math for liberal arts, intro to proof )
These are very creative little exercises that explore the intrinsic geometry of paper as we fold it. They require no previous knowledge of ANYTHING and offer loads of chances for discovery-based conjecturing and proofing.

Folding a Butterfly Bomb (geometry, math for liberal arts)
This is a tricky model to put together, but when you're done you can toss it in the air, wack it, and it'll explode in a shower of paper! Building this requires learning about the structure of the cubeoctahedron.

Business Card Modulars (geometry, math for liberal arts)
This VERY simple module is made from standard business cards. Two of them make a regular tetrahedron. Four make an octahedron. Ten make an icosahedron. Six can make a snub disphenoid. (Ha!)

Folding Five Intersecting Tetrahedra
(geometry, math for liberal arts, vector calculus)
The units of this model are easy to make, but putting it all together is a real puzzle. Building this stunning model offers a chance to study numerous symmetries of the dodecahedron. And exploring the specifics of this model is a challenging vector calculus project.

Making Origami Buckyballs (geometry, graph theory)
This includes instructions for my PHiZZ unit which is very good at making geodesic dome structures. Lots of graph theory to explore here (assumes knowledge of Euler's formula).

Making Origami Tori (geometry, graph theory, topology)
The PHiZZ unit can make "Bucky tori" too! Planning these kinds of structures requires messing around with tori fundamental domains and Euler's formula for the torus.

Modular Menger's Sponge (fractal geometry, discrete math, combinatorics)
In a fractals course, this is a way to easily build iterations of this fractal. But the combinatorial questions that arise (basically, how many business cards will we need to make a Level n sponge?) can involve recurrence relations and generating functions.

Folding and Coloring a Crane (discrete math, graph theory)
The mathematics of flat origami models, like the classic crane (or flapping bird) actually leads to some fun questions on graph colorings.

Exploring Flat Vertex Folds (geometry, discrete math, combinatorics, math for liberal arts, intro to proof, modeling)
This open-ended, discovery-based project can give students at all levels a taste of what mathematical research is like. The students play with paper, make conjectures, and try to prove them.

Impossible Crease Patterns (geometry, discrete math, combinatorics, math for liberal arts, intro to proof, modeling)
A follow-up to the previous activity, students are asked to fold up crease patterns that are given to them. Ooops! They are impossible to fold flat! The question is why...

Folding a Square Twist (geometry, discrete math, combinatorics, math for liberal arts, intro to proof, modeling, abstract algebra)
This little fold is one of the most counter-intuitive ways in which paper can fold flat. The question asked is, "How many different ways can we fold this up?" Approaches to this can utilize either brute force, combinatorics, or Burnside's Theorem.

Matrix Model of Flat Vertex Folds (geometry, linear algebra, modeling)
Let's face it–when we fold a piece of paper in half, we're actually reflecting half of the paper. So folding flat is really doing a reflection. Thus we should be able to model flat paper folding using reflection matricies. What can we do with these matrices?

Matrix Model of 3D Vertex Folds (geometry, linear algebra, modeling)
For that matter, even when we fold a piece of paper in a non-flat manner (leaving a dihedral angle in the paper), we can still use a matrix model–but now we need to use 3D rotation matrices! This gets much more confusing and really tests students' knowledge of matrix operations and 3D rotation matrices.

Rigid Folds 1: Gaussian Curvature (geometry, differential geometry)
How can we model rigid paper folding? That is, if we used sheet metal instead of paper and hinges instead of creases, could we still fold? The matrix model gives one way to look at this, but it only tells us about the finished folded state, not how we got there. Gaussian curvature can help us fill in the blanks.

Rigid Folds 2: Spherical Trigonometry (geometry)
By wading through some messy trig and the spherical law of cosines, we end up with a powerful theorem about the dihedral angles in 4-valent rigid folds as they open and close. This can tell us when some multiple-vertex crease patterns won't be rigidly foldable.

Questions? Comments? Email me: Thomas.Hull@merrimack.edu

Thanks for visiting!

Last updated: 7/1/06