Modular Origami: How to Make a Cube, Octahedron & Icosahedron from Sonobe Units

Modular origami is a technique that can be used to build some pretty interesting and impressive models of mathematical objects. In modular origami, you combine multiple units folded from single pieces of paper into more complicated forms. The Sonobe unit is a simple example unit from modular origami that is both easy to fold and compatible for constructing a large variety of models. Below are a few models that are easy to make using this unit.

Cube

Octahedron

Icosahedron

In the rest of this post we are going to learn how to make the Sonobe unit and each of these models.

Materials and Tools

• Paper (square origami paper is preferred as it folds much better)
• Scissors (if paper isn't square)

How to Make Square Paper

All of the instructions below assume square paper is being used.  If you need to make square paper, here is a fast way to do it.

Take an ordinary piece of paper and fold it diagonally. Cut off the remaining strip.

Unfold your square piece of paper.

The square created above is 8.5 inches on a side.  I like using smaller paper so I usually fold in half vertically, and then horizontally. This makes crease marks that I cut along to make 4 square sheets that are 4.25 inches on a side. This seems to be a good size for modular origami.

How to Make a Sonobe Unit

Take a square piece of paper:

If you are using colored origami paper, flip it so it is colored side down:

Fold in half horizontally:

Unfold. You should have a horizontal crease in the middle:

Now you are going to fold the paper in fourths. Start by folding the top down to meet the crease in the middle:

Fold the bottom up to meet the crease in the middle:

Unfold the top:

Fold the bottom corner up to meet the upper crease:

Unfold. Rotate 180 degrees. Fold the bottom flap up to meet the middle crease:

Fold the bottom corner up to meet the upper crease:

Fold the top down:

Refold the top corner down along the previously formed crease. Tuck it into the bottom flap:

Flip the paper over:

Fold the corners over:

Fold the corners in to form a square. First one corner:

Now the other corner:

Flip the unit over. It is complete. There should be two triangular pockets that you can use to insert other units into.

When making the octahedron, or icosahedron you need to make an additional diagonal fold across the square part of the unit.

This post/video shows the process of making a Sonobe unit in more detail.

How to Make a Cube from 6 Sonobe Units

Insert one Sonobe unit into the pockets of another unit:

Insert another Sonobe unit:

Now fold the squares together to form right angles and connect the units. You should have 3 faces of the cube and one corner.

Continue forming the cube. It should look like this before inserting the last unit:

Finished cube. They can be fun to number as large dice.

Colored version:

This post/video shows how to make this origami cube in more detail.

How to Make an Octahedron Out of 12 Sonobe Units

You must use the units that have an additional fold down the middle of the square. Place three units together to form a triangular pyramid.

Build another pyramid connected to the first:

Add a third pyramid:

For an octahedron, there should always be cycles of four pyramids around a point. Use one Sonobe unit to connect the three pyramids, forming a fourth. The whole object will now bend into the third dimension.

To finish the octahedron, you have to keep attaching units, always forming triangular pyramids in cycles of 4.

Here's a colored version. Note that if you look at it this way you can see triangles.

But if you look at it this way, then you see squares. This is because 4 triangles meet at each vertex in a octahedron.

This post/video shows how to make the octahedron in more detail.

How to Make an Icosahedron from 30 Sonobe Units

You must use the units that have an additional fold down the middle of the square. Place three units together to form a triangular pyramid.

Keep forming pyramids connected to each other until you have a cycle of five of them connected around a point.

Keep connecting pyramids together in cycles of 5. It will begin to curve into a ball:

Connecting the last couple up can be a little challenging sometimes. Here's the completed model:

The colored version. Notice the triangular figures.

From this angle, notice how you can see pentagons and pentagrams. This is because on an icosahedron, 5 triangles meet at each vertex.

This post/video shows how to make the icosahedron in more detail.

Show Off Your Work

If you complete any of these projects, any of the variations, or any of the other previous Math Craft projects, please share with us by posting to the corkboard. Maybe you just have something cool that you made or saw on the web—you can share that, too!

If you like these types of projects, let me know in the comments. If you have any other ideas you would like to pursue, let me know in the forum.

Next post, we're going to create slightly more complicated objects from Sonobe Units.