How to Make Nested Cube and Octahedron Boxes

These boxes are inspired by a comment from Imaatfal Avidya on a corkboard post on Platonic polyhedra from sonobe units. Imaatfal was commenting about how the cube and octahedron are related to each other.

"There is something cool and special about the platonic solids—there is something so simple, yet so complex about them. For instance, if you look at some of the drawings of Leonardo da Vinci, you will see that he recognized that the cube and octahedron are kinda opposites. If you cut the corners off a cube, you get and octahedron—and vice versa; just a bit weird!"

The cube and the octahedron are dual polyhedra of each other. If you replace the faces of a cube by vertices at the center then you end up with a octahedron. If you replace the faces of an octahedron by vertices at the center then you end up with a cube. I think one of the easiest ways to really see this is to build a paper model of an octahedron nested in a cube and a cube nested in an octahedron.  

All of these boxes fit "perfectly" into the next larger box as you can see in this video.

Note from the video: Taping a hinge onto the boxes works really well for the cubes, but not as great for the octahedron since it really doesn't allow the octahedron to open correctly because the motion is impeded by the cube inside.

Materials and Tools

• 3 sheets of paper
• Templates
• Glue

The Templates

For the simpler versions of either a octahedron in a cube or a cube in an octahedron use these templates from Gijs Korthals Altes' awesome site.

I took and modified these templates so that there would be a complete nesting sequence.

For some reason, my .pdf templates weren't being made correctly so here are some image templates. You will need to print out the first image twice and the second image once. First click on the full size image. Then either copy or print it from there.

How to Make Nested Cube and Octahedron Boxes

Cut out the net for the half cube with half octahedron impression. Fold all the solid lines away from you while making the cube. Fold all the dotted lines toward you while making the half octahedron impression.  

Glue all the tabs to construct the cube. You will be left with the octahedron impression.

Glue the tab that connects the octahedral parts, and then glue the tabs connecting the octahedral impression into the cube. Now you will need to repeat all of these steps to construct the other half of the box.

Now cut out and fold the nets for the half octahedron with a cube impression. Make sure you cut out the square hole.  

The half cube impression is a separate net. Note that all of these lines fold toward you except the top tabs.

Glue up the side tabs to form the box.

Glue the top tabs of the box onto the half octahedron top, making sure it is well lined up with the square hole.

Now glue up all the tabs to make the half octahedron. You will need to repeat all of these steps with the other half of the octahedron.

Now continue to make all of the other boxes that nest inside. You can color these before printing out by using a paint program or afterwards using your method of choice.  I used gold and silver metallic leafing paint on mine.

Show Off Your Work

If you make the nesting boxes or any of the other previous Math Craft projects, please share with us by posting to the corkboard. Perhaps you have some original project or something you've seen on the web that you'd like to share.

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.