Nets for Polyhedron: Cube, Tetra-, Octa-,and Icosahedron

In geometry, the net of a polyhedron is an arrangement of edge-joining polygons in two dimensions that can be folded along edges to become the faces of the polyhedron.

This has both practical and theoretical applications. Every product you have ever purchased is packaged in some form of polyhedron or cylindrical solid with the exception of plastic packaging that is formed around the product. For example, most boxes are rectangular prisms or hexahedra (six faced). Some boxes for guitars are trapezoidal prisms (still six faces). Helmets are often packaged in triangular prisms.

One of the most challenging puzzles in geometry is looking at multiple nets for the same polyhedron. For instance, there are only two distinct nets for a triangular pyramid or tetrahedron (four faces). Cubes have exactly 11 distinct nets. Octahedra also (usually, looks like two rectangular pyramids stuck together at the bases) have exactly 11 distinct nets.

During class, the children explored finding all 11 nets for the cube. I gave them a worksheet with 20 possible nets only 11 of which actually work.

Possible homework activities:

1.    If they did not find all of the nets during class, they can use the nets I gave them to construct these polyhedra (the cube, the tetrahedron, the octahedron, and the icosahedron). Pdfs for these nets also are attached.

2.    For the advanced student, if they want to keep exploring these different nets, the children can copy the net pages of squares and triangles and tape them together to test for actual net solutions.

3.    The last pdf attached here is one where they can sketch the tetrahedron, octahedron, and icosahedron nets. Please do not have them attempt to find all of the nets for the octahedron.

4.     In fact, the last page has all of the solutions; they can copy down these solutions from the answer key. This is a challenge in and of itself.

The children and I were amazed that there are exactly 43,380 distinct nets for the icosahedron (20 triangular faces). I was going to attach a file with all of these solutions. But thought it would better to just give you a link to the site where I found this colosal file. This is a Japanese website http://www.al.ics.saitama-u.ac.jp/horiyama/research/unfolding/r05.pdf

Do not print this document, as it is 434 pages. I assume that you know this, but you also can print selected pages (click on Pages instead of All and hit a range of pages like 1-5).

5.   It might be fun if they try to copy a few of these solutions onto the triangular paper I gave them.