A guest post by Maria Rainier
Understanding Platonic Solids with Modular Origami
Solid geometry is perhaps one of the best mathematical applications of origami, but of course, there are many other ways to use it in improving students' understanding of math's processes, concepts, and underpinnings. For anyone who has difficulty with the abstract components of math, origami can help provide both visual aids and the opportunity to arrive at mathematical conclusions through trial and error. It's an especially effective way to help visual and kinesthetic learners to understand basic geometric concepts.
You can teach two of the platonic solids with a simple demonstration or a more elaborate project, depending on how much time you'd like to spend. With a demonstration, you'll be doing most of the origami module construction, allowing students to experiment with it. If you assign a project, you can have different groups working to construct their own modular components and the more difficult module itself. Either way, it will help to become familiar with the model before you use it to teach solid geometry, but constructing the components isn't difficult and you'll be able to envision the model easily. Take a look at the following instructions and images to determine how you would adapt this idea to your teaching style.
5 Intersecting Tetrahedra = 1 Dodecahedron
Constructing the five tetrahedra is a relatively easy task, but weaving them together to form a dodecahedron is both challenging and fascinating. Your students will almost certainly need your help if you decide to have them complete this part, but accomplishing something so difficult is great for self confidence and a stronger grasp of solid geometry.
Basic Unit
You'll need ten squares of paper to complete this model – two for each tetrahedron. Divide each square into equal thirds, then cut them into strips so that you have 30 small 1X3 rectangular pieces. To create one modular unit, fold one of the pieces in half along the longer side, unfold, and bring the edges into the center crease. To form a 60° pointed end, fold the top right edge into the center and give the resulting new edge a light pinch (this is just to form a guidance crease). Now, fold the top left corner to meet the crease you've just made on the right side, taking care to form a corner at the top of your midline crease. Fold the top right corner down over it to get a triangular point. Now, unfold both of the flaps you've just made and reverse fold the left flap so that it's inside of your unit, creating a small pocket. Fold the top edge of the right flap down to meet the 60° crease and unfold. Turn the unit 180° and repeat at the other end to finish your first unit, then give it a good crease along the midline. Make five more, and you'll be ready to make your first tetrahedron – see this helpful Merrimack College page for diagrams.
Tetrahedron
To construct a tetrahedron, simply insert the right-hand projection of one unit into the left-hand pocket of another. Now, add a third unit to join the first two, forming one of the tetrahedral frame's four points and three of its six edges. Use the remaining three units to complete the tetrahedron.
Dodecahedron
Now, the tricky part is weaving your five tetrahedra together to form a dodecahedron. The rule of thumb is that the peak of each tetrahedron should come through the base of another – it's also helpful to keep in mind that the 20 points of the combined tetrahedra form the pentagonal points of the dodecahedron. The diagrams described above are especially helpful in assembling the final platonic solid, but the peak-base rule can also be used to successfully weave the dodecahedron.
Wrap-Up Questions
- Can you make any other platonic solids using the modular units that form the tetrahedra?
- Why is the 60° angle important? Could you complete this model with units formed by any other angles?
- Could thinner units be made with the 60° angles intact?
- Why does the method used to form the 60° angle in the construction of the basic units work? (Hint: Check out Huzita's fifth axiom.)
About Author
Maria Rainier is a freelance writer and blog junkie. She is currently a resident blogger at First in Education, where recently she's been researching online mechanical engineering degrees and blogging about student life. In her spare time, she enjoys square-foot gardening, swimming, and avoiding her laptop.
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