# Paper Polyhedra

I highly recommend making paper models, both for the elegant beauty of the models themselves and for gaining insight into three-dimensional space as you construct and then study them. They are great in a classroom. Related types of paper constructions, other constructions and the Zometool are also recommended.

At right is one of the 59 stellations of the icosahedron. It is made by cutting out paper triangles with tabs on their edges, and assembling with glue. I made it almost 20 years ago, following plans given in Magnus Wenninger's book Polyhedron Models, listed in the references. Click on the picture for a 3D version.

Modular origami is a construction technique in which many similar or identical pieces are individually folded and then assembled together into a model. For the example shown here, each edge piece is folded from a separate square of paper, and then the edge pieces slide into each other and interlock (a method I learned from Jeannine Mosely). The polyhedron is a skewed version of the truncated icosahedron, a well-known Archimedean "semi-regular" solid which provides the pattern used for the soccer ball. Often glueless, modular origami techniques are described in several of the books listed in the references.

There are also books you can buy in which the cardboard pages are printed with full-size pieces to cut out, labeled so that you just glue tab A to A, and tab B to B, etc. Some of these are excellent, especially for compound models. This compound of five cubes I made from one of the cut-and-paste books in the "Tarquin" series, listed under Jenkins in the references.

However, for quickly constructing many polyhedra models, I like the "no-tab, taping" method (illustrated further below). This pentagonal hexecontahedron (an interesting example of an Archimedean dual) is constructed in this manner. Here is a template you can print out to make your own.

### Four techniques

There are four basic cardboard construction techniques. The methods with glued tabs result in the strongest models, but it is difficult to avoid visible glue fingerprints. Each of these has its advantages and adherents:
• Glueless origami methods. This has a certain Zen purity.
• The no-tab, taping method. It is fastest and easiest to cut paper polygons and join them using tape on the inside. This is detailed below.
• The one-tab method. One of the two faces that meet at an edge of the polyhedron is cut out with a protruding tab. It is folded back and glued to the inside of the face it joins.
• The two-tab method. All edges of all faces are cut out with tabs. Two are glued to each other to make a rib on the inside of each edge. The advantage of this is in its symmetry and the fact that a clip can hold the two tabs together as the glue dries and you work on something else.

### The no-tab, taping method

Here are step-by-step instructions for quickly making paper models. As an illustration, I am including pictures of this 53rd stellation of the icosahedron which I made recently. However, this model is not recommended for the inexperienced; start with the Platonic solids.

1.) Obtain good (acid free) colored stock from an art supply store if you want a long-lasting model. Note that inexpensive "construction paper" and "oak tag" will fade badly in less than a few years (This was once brightly colored like this.) I am using five colors in this model.

2.) Draw one copy of each face carefully on a piece of paper, or at least locate the vertices. This will serve as a template. (Here I made two copies because of its unusual shape; usually one is sufficient. I sized the full size version to make four faces from a folded sheet of paper.)

3.) Stack up pieces of stock and place the paper template on top. A couple of binder clips will hold everything firmly together. (This model requires 12 pieces (3 folded sheets) of each of five colors.)

4.) Use a pin or needle to poke vertically through the stack at each of the vertices marked on the template. Even better is a very fine drill bit. (You can replace steps 3 and 4 by simply xeroxing your template onto card stock or printing from your computer onto card stock, but that costs more and won't work with stock too thick to go through the machine.)

5.) Using the pinholes as guide points, cut the stock one sheet at a time. A slicing paper-cutter is quite handy. The white paper under the cutter lets you see through the holes easily, to line the paper under the knife for the cut.

A rotary razor cutter (available at quilting supply stores) has the advantage that it cuts accurately through several layers at once, but it is hard to end the cut exactly where you want on an inside corner.

6.) Tape the faces together on the inside. (For this model, one must keep in mind the five-color pattern for the icosahedron, but beginners might start with solid-color models.)

Taping the inside while you are on the outside takes some creativity for the last few edges. Make an explorer from a bent paper clip to get inside and secure the tape through the corners.

That's it. Then make a couple more. Decorate your home. Show your friends. Impress your neighbors. Start a club. Make lots more. Pile them high on every horizontal surface you own. Put them in large boxes and carry them with you every time you move. Make even more. They're like tribbles...

After mastering paper, you will be ready to try making wooden models.