Nets for Geometric Models
Inspired by the book Adventures Among the Toroids by B.M. Stewart, I searched for a polyhedron net with "holes." What I found was a 3 dimensional flexagon, which is hinged together at common sides. As opposed to typical flexagons, this one is not a static figure, rather it is a kaleidocycle that can move and transform into different shapes. The type of kaleidocycle I made is a type of hexaflexagon, which displays six triangular faces at the same time and can be transformed to display a different set of six triangles. I am posting pictures of some of the steps I took in creating my kaleidocycle.
Here is the link for the instructions which I used to make my 3D hexaflexagon.:
This kaleidocycle inverts to display a different pattern on the other side. This was a really interesting and fun activity that I hope to bring to my students someday. I could incorporate lessons about vertices, edges, faces, polygons, perimeter, area, and volume.
As I was exploring on the Internet, I came across this great math lesson for exploring area, perimeter, and volume, using polyhedra and flexagons. This lesson aligns with 5th grade Common Core State Standards.
Understanding Polygons and Polyhedrons Using Flexagons
Here is a real world example of a polyhedron with a hole. It is a little distorted, but it is a lamp. This is different than a flexagon since its shape is static in nature. It does not change but it was created with a hole in the middle, staying true to the properties of the polygon.