Sunday, May 25, 2014

Doing Math

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.:
3D Hexaflexagon

My 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.


  1. Interesting - never seen these called hexaflexagons before. Those are usually flat - see Vi's explanation.

    The 3D one is not really a polyhedron, I think, because of the common edge between adjacent tetrahedrons.

    Fun with the designs, it would be interesting to see what students make of them. Good job describing some of what you noticed. Flesh out the ideas you have (volume, etc.) to have a super strong exemplar.
    5C's 4.5

  2. I really like the model that you created. This idea would be great to incorporate into a classroom where students would be given the opportunity to create their very own shape and then present it to the classroom. The last photo that you presented was a nice way to wrap up your post. I like how you related your model to an actual real life model that students could use to understand that math is all around us in our daily lives.