### Modular Origami with Sonobe Units

The other month, I volunteered at Seattle Universal Math Museum's corner of Free First Thursdays at the Museum of Flight. Part of this was helping kids make origami paper airplanes, which reminded me of some origami challenges I used to give students. I'll present one of my favorite arcs here.

## Setup

For this activity, you'll need some paper squares. Origami paper can be gotten pretty cheaply. Here are some options:

In a pinch, you can cut printer paper or construction paper into squares, but from a time and precision standpoint, it's easier to just get some origami paper.

Every object we'll make here will be built from one basic object, called a Sonobe unit or Sonobe module. The way this unit is made is as follows:

- Fold your square in half and open it so your square has a vertical crease down its center.
- Fold each half into halves and open it so your square is divided into four equal vertical strips divided by three vertical creases.
- Fold the upper right corner down so its edge aligns with the rightmost vertical fold line.
- Similarly, fold the lower left corner up so its edge aligns with the leftmost vertical fold line.
- Fold the two outer quarters back in.
- Fold the upper left corner down so its edge rests on the horizontal midline.
- Similarly, fold the lower right corner up so its edge rests on the horizontal midline.
- Tuck the upper triangular flap under the vertical flap behind it to its right.
- Similarly, tuck the lower triangular flap under the vertical flap behind it to its left.

- Flip the unit over to reveal its flat side. Fold the two triangular arms in to make a square.
- Flip the unit back over the flappy side. Fold the center square diagonally in half so that when the arms from the previous step are folded back, the unit looks like a right triangle.

Each Sonobe unit has two hands and two pockets. The way they fit together, is the hand of one unit goes into the pocket of another:

The standard way we'll fit them together is three coming together in a swirl to form a pyramid. (There are ways to connect four or more in a swirl, which I'll allude to in the extensions.)

In preparation for running this, I would recommend making a few models in advance, both for practice and to have examples that kids can mimic. If you want display models that last, I've had decent luck making them myself out of cardstock but had to cut the squares myself.

### Example: Bipyramid

This model requires 3 Sonobe units. Despite its small number of units, it can be a little tricky as a first model. It holds its shape more cleanly if you make the fold in Step 11 backward for all units.

### Example: Cube

This model requires 6 Sonobe units. It holds its shape more cleanly if you don't make the fold in Step 11 for any of the units.

### Example: (Elevated) Octahedron

This model requires 12 Sonobe units.

### Example 4: (Elevated) Icosahedron

This model requires 30 Sonobe units.

## Initial Questions

As students look at the models, some natural questions to ponder are:

- What do the objects we've built have in common?
- Is there an easy way to tell how many units these objects take to build?
- What other sorts of objects can be built using these units?

## Explorations

There are a few ways to run this activity.

- One option is to have each student try to copy one of our examples. The octahedron is a nice mid-range object that feels like an accomplishment but won't take an afternoon while the bipyramid and cube are nice smaller goals. With some care, smaller objects can be disassembled and recycled into parts for larger objects, which can create a checkpoint system.
- A more small-scale exploratory activity is to note that three of these examples required 3, 6, and 12 units. Each student could make 9 units and try to design an object using those 9. The object they would find looks sort of like two cubes fused together:
A far more open-ended exploration is to simply ask them what other sorts of objects they could make. It's usually good to start them out making one of the example models just so they get some practice, but otherwise Sonobe units are building blocks and there are many things that can be constructed with them.

If students have some familiarity with the platonic solids, they might suspect that besides the cube, octahedron, and icosahedron, they might be able to make a tetrahedron and a dodecahedron. The answer turns out to be a little stranger, since our cube model is actually best not thought of as a cube! Note that our octahedron and icosahedron are

**elevated**, meaning they have pyramids growing out of each face while the cube is apparently not elevated. However, if you think of a tetrahedron, if you elevate each of its four faces with pyramids of just the right pitch, the faces of those pyramids will be flush with one another making it look like a cube! So our cube is actually an elevated tetrahedron. Similarly, with the bipyramid, one way to think of it is as a triangular dihedron (think a triangular-shaped coin with two faces and no edge) with a pyramid sprouting out of each face.The common thread among our models is each is based on a polyhedron whose faces are all equilateral triangles. These polyhedra are called

**deltahedra**. If students figure out this pattern after playing around for a bit, I usually point them to the family of eight strictly convex deltahedra and see if they can make an elevated version of one of the five we haven't seen before. Some of these look rather surprising when they're done. For example, the elevated triangular bipyramid is actually the model above that looks like two cubes fused together. This makes sense because the elevated tetrahedron looks like a cube and the triangular bipyramid is two tetrahedra pasted together!If you want to have some models of deltahedra for students to look at while they try to create them with Sonobe units, I've made some printable worksheets. The first three pages are the strictly convex deltahedra while the last three pages are some small coplanar and concave models. Cut each object out around its outline. Bold internal lines are fold lines while faint internal segments are to mark coplanar triangles.

- Plain models
- Models with tabs for gluing
- Models with interlocking tabs (make sure to cut lines shared by small triangular tabs and along the bold half-edges)

These are a little flimsy, but get the point across. If you have ready access or funds, I recommend a hinged polygon playset like one of the following:

- Geometiles, Polydrons, or other hinged tiles
- PicassoTiles or other magnetic tiles
- ITSPHUN construction sets

As students build with these units, have them keep track of how many units each object takes to build. As long as they stick to these deltahedra, they might notice the number of units is always a multiple of 3. Why is that the case? (Hint: How many units are involved in a pyramid and how many pyramids involve a given unit?) This line of inquiry can lead them to see why a deltahedron with \( f \) faces requires \[ \frac{3}{2} f \] units to build. It is also a gateway to noticing that deltahedra must always have an even number of faces! (Why?)

Since larger objects require a lot of units, a good cooperative angle is to have students work together to build as large of an object as they can. This distributes the job of making units and requires them to coordinate or delegate for the actual building.

## Wrap-Up Questions

- Given an elevated deltahedron you want to build, how can you predict the number of units you'll need?
- What was the largest object you were able to create?
- Are there any things you didn't build but are pretty confident you could, given enough time?

## Extensions

As hinted, there are ways to swirl more than three units together, and in the course of building some of these objects we've already done it! Here is what a swirling of five units looks like:

This might look familiar, since this 5-pointed star appears in our elevated icosahedral model. Just as we viewed a pyramid as a triangle, if we view this star a pentagon, we can instead reimagine the icosahedron made from 20 elevated triangles as a dodecahedron made from 12 starry pentagons. (This is a nice manifestation of the duality of these two shapes in origami!) Similarly, we can see what a four-unit swirl looks like if we reinterpret our octahedron as its dual, the cube.

If we swirl six units together, we can get something strange that might be thought of as a hexagon. Building up a store of regular polygons (triangles, squares, pentagons, and hexagons so far), what sorts of objects do these enable us to build that we might not have considered?

*This post was sponsored by the Julia Robinson Mathematics Festival*

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