Origami and Mathematics

Such geometry, this mathematics of origami, has been studied extensively by origamists, mathematicians, scientists and artists. The intersection between the subjects of origami and mathematics is rich with interesting results and applications. It's also a field that's becoming more and more popular, especially the use of origami in math and science education. The Italian-Japanese mathematician Humiaki Huzita has formulated a list of axioms to define origami geometrically. Physicist Jun Maekawa has discovered some fundamental theorems about origami and used them to design origami models of surprising elegance. Mathematician Toshikazu Kawasaki has a number of origami theorems to his name and has even generalized some of them to describe paper folding in higher dimensions. (Origami in the fourth dimension!) Robert Lang of California has developed an ingenious way to algorithmatize the origami design process, using a computer to help him invent models of amazing complexity. Educator Shuzo Fujimoto and artist Chris Palmer have discovered amazing parallels between origami and tessellations. And an enormous number of teachers have been developing ways to use origami to teach concepts in math, chemistry, physics and architecture. There are many ways to make polyhedra from paper without scissors or tape. Sonobe polyhedra are a great first method to learn. Sonobe polyhedra are easy to make and put together, require a relatively small number of modules to achieve impressive results, and can be used to construct an amazing number of different polyhedra.

Lesson

Goals 1: Geometry - The learner will recognize and use basic geometric properties of two- and three-dimensional figures. Goal 2: Geometry - The learner will understand and use properties and relationships of plane figures.

Objectives :

1) Identify, define, describe, and accurately represent triangles, quadrilaterals, and other polygons. 2) Make and test conjectures about polygons involving: Sum of the measures of interior angles, Lengths of sides and diagonals, Parallelism and perpendicularity of sides and diagonals. 3) Classify plane figures according to types of symmetry (line, rotational). 4) Use appropriate vocabulary to compare, describe, and classify two- and three-dimensional figures. Materials/resources

Physical Resources

Paper precut to the appropriate size for the origami sculpture . A copy of the directions for the chosen origami sculpture (one for each student, pair, or group) Paper, pencil, calculator for solving follow-up mathematics problems .

Activities

Orally review fractional parts 1/2, 1/3, 1/4, and geometric terms: symmetry, faces, edges, triangle, rectangle, square, rhombus. Distribute Origami paper and instructions . Students may work as individuals, pairs or in small groups. Task for students:The students will follow the following steps:

Using the site: http://nuwen.net/poly.html

A freshly cut square of paper. Dividing the square in half.

Here is the process halfway through completion.

The actual purpose of this fold is just to give you a reference to make the next two folds. Unfold the paper and lay it flat. Take the bottom edge of the paper and fold it to the center crease; then spin the paper 180 degrees and do the same. Here is what I mean: The folds that you'll be making.

Here is the process halfway through completion (both folds are shown simultaneously; you should make them one at a time, of course).

Now, unfold the paper and lay it flat. (You will be folding the paper here again; you just need to do some things in the meantime. I'll refer to these folds, rather uncreatively, as "the second and third folds" later on.) Take the bottom-right corner of the paper and fold it into a triangle so that what was the left side of the paper now lies on top of the second fold you made. Leave that folded, spin the paper 180 degrees and make the same fold. Here is what I mean: Folding two triangles.

This is the traditional fold you make when producing a (lousy) needlenose paper airplane. Now, take the bottom-right corner of the paper and make another needlenose-type fold. That means bringing the fold that you just made to lie exactly on top of the second fold you made. Then rotate the paper 180 degrees and make the same fold. The following image's bottom-right corner shows the end result of this process; the upper-left corner shows it halfway through completion. Another "needlenose" type fold.

That fold was hard to describe but easy to perform; it's used in the production of lousy needlenose paper airplanes everywhere. Now is the time to remake the second and third folds you made:

Now, take the bottom-left corner of the paper and fold it so that what was the left edge of the paper now lies on top of the top edge of the paper, producing a triangle, like this:

Rotate the paper 180 degrees and repeat. A parallelogram! Now, you must tuck in that large triangle fold into the paper. I have no way to easily describe this in words. Here is what I mean: The left fold is tucked in, while the right fold is not.

Then rotate the paper 180 degrees and tuck in the other fold, resulting in:

flip the paper over and rotate it so that it looks like this:

The backside of the paper.

Fold the bottom point of the paper straight up to meet another vertex of the parallelogram, like this:

Then rotate the paper 180 degrees and repeat, producing this:

Now you need to give the paper a bend in the middle. (This is actually a mountain fold, but I could have you flip the paper over again and make a valley fold; so what?) You will end up with this: Now, as you can see, you have a finished piece:

Congratulations.

Now, here's a bit of useless trivia: you can actually make mirror-images of these pieces. The crucial decision comes when you make the triangle folds after the second and third folds. If you choose to make them on the bottom-left and upper- right corners of the paper, and modify subseqent folds accordingly, you end up with a left-handed piece instead of a right-handed piece. (The pieces are chiral, in other words.) Being right-handed and very used to making right-handed pieces such as I've shown here, making a left-handed piece takes a lot of time for me. (I have no idea if left-handed people find left-handed pieces easier to make, or if right-handed people who've never made pieces before find left- or right-handed pieces easier to make.) However, and this is the important part, left-handed pieces and right-handed pieces cannot be used in the same model! They just won't fit together. So if you make one left-handed piece, all of the pieces in your model will be left-handed. (I also have trouble assembling left-handed pieces into models, because everything is reversed.) Stick with right-handed pieces.

Making models:

Now that you know how to make pieces, you need to choose which model you want to construct so that you'll know how many pieces to make. There are four models that I know how to construct (though I could derive how to make many other types of models).

1. The cube. A boring cube. The easiest to construct, it takes 6 pieces. 2. The octahedron. (A stellated octahedron, actually.) Takes 12 pieces. Not difficult. 3. The icosahedron. (A stellated icosahedron.) Takes 30 pieces. Also not difficult. 4. The stellated truncated icosahedron. Takes 270 pieces... I think. Difficult (though not overly so), but incredibly time-consuming.

I suggest that you start off with the cube and work onwards. Now that you have enough pieces constructed to make the model of your choice, you need to learn the basics of model construction. A piece has two sharp corners and two pockets, which allow them to interlock. Here are two pieces placed to illustrate this:

And here they are locked together, corner in pocket:

Here is a third piece, placed over the first two:

And here the third piece is locked in: There is a free corner and free pocket that can be locked together. Doing so necessitates forming the three pieces into a three-dimension configuration that I call a peak:

It is vitally important to understand what I mean when I say "peak", because peaks are the founding blocks of your models. (Although you should assemble your models piece-by-piece and not make a bunch of 3-piece peaks and then assemble the peaks. The former works; the latter won't. Example: the cube contains three peaks, but only requires six pieces. Solution: pieces can form more than one peak. Trust me: go piece-by-piece.) Now you should be able to assemble a cube. Here is a cube, pictured with a peak at the center of the image:

A cube.

If you were unable to assemble the cube, then read on, because I will make it even clearer. (I need to demonstrate the next fact with an octahedron; cubes are too small). Now, here is a what I call (somewhat confusingly), a "point". This is my own terminology; call it what you will. Around every "point" in a model there are three or more peaks. The lens flare in the following image shows where a point is located relative to a peak: This is what I mean by "point".

Here is a picture of an octahedron, with a point more or less in the center of the image. See how four peaks are arranged around the point?

An octahedron.

And here is an icosahedron. Icosahedra have five peaks around every point:

An icosahedron.

4. Have students describe the folding process using geometric terms, e.g., faces, symmetry, edges, square, triangle, etc. as they apply to your chosen Origami.

Sample math questions pertaining to origami geometry:

The students will measure the side of the icosahedron, cube and octahedron and they will calculate area A and volume V.

Geometry

Paper Folding Constructions Theorem: All of the constructions of Plane Euclidean Geometry that can be performed by straightedge and compass can also be performed by folding and creasing paper. --- James R. Stuart, Modern Geometries, Second Edition

Add a hands-on component to the study of geometry... Paper folding provides a challenging and interesting avenue for discovery which allow students to physically manipulate (play with) geometric figures. All that is needed is a flat sheet of paper. Much of the introductory work in Geometry associated with developing the basic concepts of a point, a line, and a plane, are easier to illustrate using paper folding activities.

Try This:

Show that the sum of the interior angles of a triangle add up to 180 degrees (a straight line).

First, cut a triangle from a piece of paper and mark the three vertices as shown in the figure at the right.

Second, fold the top vertex of the triangle, in this case, vertex C, so that it touches the opposite side of the triangle, side AB. Make sure that the crease in the paper is parallel to side AB.

Finally, fold the other two vertices, A and B, so that they fall on the same point where vertex C touches side AB. Check your results...

Use a "paper ruler" that you get when you fold a piece of paper. See if the results of the three folds in the excercise above does form a straight line.

Mathematics teacher : Constanta Goja