# Pentagon-Hexagon Zig-Zag (PHiZZ) Unit

 Over the past several years I've been playing with a certain member of the Zig-Zag family of modular origami units. Zig-Zag units are modular origami folds whose locking mechanism is based on an accordion pleat. That is, you accordion pleat a square, typically into 4ths or 3rds, making a rectangle. The short ends of the rectangle become the flaps, and the layers at the sides created by the accordion folds become the pockets. Of course, other folds in the rectangle are needed to make the flaps hook and stay, but that's the basic concept. Lots of modular origami units exist that fall into this category, units by creators such as Robert Neale, Lewis Simon, Jeannine Mosely, and Jun Maekawa, just to name a few. The unit I've been using extensively is what I call the pentagon-hexagon zig-zag unit (or PHiZZ unit). I call it this because you can use it to make any polyhedron that is cubic - each corner of the polyhedron has three edges meeting it, and has only pentagon and hexagon faces (the faces need not be regular). (Of course, you can break these rules in various ways, say, by making square faces. But the units tend to buckle when forced to do this. Or you could make the vertices have degree 4 (I know of two ways you can do this), and that's interesting too. But for this exposition I'll just stick to the above rules.) The unit itself is very simple, and the first thing I made with it was a dodecahedron. But then I noticed that the locking mechanism in this unit is particularly strong, allowing one to make much larger structures. Thus began my quest to see exactly what other polyhedra you could make with this unit and, later on, how to properly 3-color them. But first let's see how to fold and lock the PHiZZ units together.

## How to make the PHiZZ unit

 (1) Take a square piece of paper, white side up, and accordion pleat it into fourths. (2) Fold the top left corner down. (3) Fold the right end of the strip down to meet the folded edges.
 (4) Now fold the strip up, making the bottom flush. (5) Fold the strip down to the right. (6) Mountain-fold the upper right corner behind, and you're done with one unit!

## Putting the units together

 (1) Slide the end of one unit into the side of another, as shown. The flap should go in between the layers of paper, and crease lines should line up. (2) It takes three units to make a "corner" of the polyhedron. Slide the third unit into the second unit, and slide the first into the third. (3) This is what you should see. Notice how exactly half of each unit was used to make this "pyramid." Continue adding units to this to make your polyhedron.