Paper Folding and Polyhedron Ashley Shimabuku 8 December 2010 1 Introduction There is a connection between the art of folding paper and mathematics. Paper folding is a beautiful and intricate art form that has existed for thousands of years. Origami is a great example of something people generally consider to be non-mathematical to have hidden math aspects. Origami folding has recently found practical applications in industry. Car companies have been trying to find the best way to fold the airbag up into the dashboard that will provide the fastest and most efficient unfolding upon impact. Geneticist have been looking at folding to help them think about DNA strands and protein chains [2]. There is an assortment of paper folding activities concerning geometry. However, some of my students at the Thurgood Marshall Academic High School Math Circle have not taken a geometry class. This activity is accessible to grade levels 8 12. This activity is designed so that all the students can participate and the instructors do not have to worry about the different levels of mathematical ability. The activity is based around a shape called a Sonobe unit from [1]. Figure 1: Sonobe unit A Sonobe unit is a very basic shape and is easy to make quickly. Using several of these Sonobe units we can construct polyhedra. Students of all mathematical abilities will be 1
Ashley Shimabuku Paper Folding and Polyhedron Math 728 able to recognize and understand these basic three-dimensional objects. And from this basic polyhedron we can talk with the students about counting and coloring. I think that the students will have fun figuring out the connection between the number of Sonobe units and faces of polyhedron and how to piece together the Sonobe units to color the polyhedron in a certain way. 2 Learning Objectives We will start the activity with a shape called a Sonobe unit. A Sonobe unit is a simple shape requiring only a dozen folds to construct. Although it is simple we can use several of them to build a three-dimensional polyhedron. Starting with only six Sonobe units we will construct a cube. After constructing a cube we can ask mathematical questions about its construction and coloring. 3 Materials Required 1. Origami paper 2. Diagram on how to make a Sonobe unit 3. Example of a constructed cube 4 Mathematical Background Definitions: A polyhedron is a three dimensional solid with straight faces and edges. A vertex is a corner of the polyhedron. An edge of a polyhedron is a line that connects two vertices. A face of a polyhedron is the two dimensional polygon created by the edges. A cube is a three dimensional solid with 6 square faces, 8 vertices and 12 edges. The origami cube is picture in Figure 2a. An octahedron is a three dimensional solid with 8 faces, where three squares meet at a vertex. The cube has 8 vertices and 12 edges. An icosahedron is a three dimensional solid with 20 faces, 12 vertices and 30 edges. 2
Ashley Shimabuku Paper Folding and Polyhedron Math 728 Stellate means to make or form into a star. A stellated octahedron is an octahedron with a triangular pyramid on each face. The origami stellated octahedron is picture in Figure 2b. A stellated icosahedron is an icosahedron with a triangular pyramid on each face. The origami stellated iscosahderon is picture in Figure 2c. A polyhedron is n-colorable if there is a way to construct the polyhedra from n different colored Sonobe units where no Sonobe of the same color are inserted into each other. 5 Examples (a) Cube (b) Stellated Octohedron (c) Stellated Iscosahedron Figure 2: Examples of constructed polyhedron 6 Lesson Plan This lesson plan may need to be done over two class periods. The lesson will begin with a classroom introduction to the Sonobe unit. The folding and construction part of the activity can be done in groups. 1. We will begin with an introduction to the Sonobe unit. Each table should have a copy of the instruction sheet. The instructions are clear but some of the students may have a hard time associating the pictures to the physically folding their own paper. We will take the class step by step through the construction. They will need to make six Sonobe units to make a cube. 2. Have the students work in groups to finish their Sonobe units and build a cube. While the students are making all of their Sonobe units the ones having difficultly will get a chance to watch their peers. The handout has questions about the cube. They can also make and talk about the stellated octahedron and stellated icosahedron. 3
Ashley Shimabuku Paper Folding and Polyhedron Math 728 3. After the students have built a their own cube and have talked through some of the question on the handout we ll bring the class back together to introduce coloring. A cube is 3-colorable. 4. The students can get back in their groups to talk about what it means to be colorable. There are questions about this topic on the handout. 5. If students finish early, they can talk about coloring a stellated octahedron and stellated icosahedron in their groups. 6. The last part of the lesson will be a class discussion about the following topics. (a) What kinds of polyhedron were you able to make? (b) What were your favorite polyhedron? (c) Was a cube 2-colorable? What polyhedron were 3-colorable? (d) Did you expect to find math while doing origami? 7 Teacher s Reflection The activity went very well in Math Circle. All of the students were engaged and asking questions about how to construct the cube. Almost everyone made their own cube and a few of them were able to make the stellated octahedron with minimal guidance. The students enjoyed constructing something they could bring home and show their family. Even the teachers enjoyed making the cubes and one of the TMAHS teachers was interested in using this activity in her own class. My partner teacher and I did the first part of the lesson plan with his Algebra Project class and his freshmen algebra class. We wanted to see how difficult it would be for them to build a cube. Some of the students only needed a short introduction and were able to build their own cubes quite quickly. Others were much slower at understanding how to fold the Sonobe unit. We found that the third and ninth step in the instruction sheet on building the Sonobe unit was hardest for the students. These steps needed special attention. My partner teacher and I decided the best thing for Math Circle would be to have the students pair up and work on one cube between them. This plan would hopefully save some time and enable the students to start answering questions on the worksheet. However, this plan did not work very well in Math Circle. All the students, except for the junior high students, wanted to make their own cubes. The students were so focused on building their own cubes that they completely ignored the worksheet that went along with it. We realized that this project is best done over two days. One day to build the cube and get used to the Sonobe units and another day to ask questions about the cube. 4
Ashley Shimabuku Paper Folding and Polyhedron Math 728 8 References 1. http://riverbendmath.org/modules/origami/sonobe_polyhedra/info 2. http://nuwen.net/poly.html 3. Between the Folds : PBS documentary, http://www.greenfusefilms.com/ 5
Questions and Challenges: 1. How many Sonobe units does it take to build a cube? 2. What is the least number of Sonobe units you need to make a polyhedron? Can you make a polyhedron from 1 Sonobe unit? 3. What other polyhedron can you make? 4. A stellated octahedron is a three dimensional solid with 8 triangular faces, 8 vertices and 12 edges. How many Sonobe units would you need to build a stellated octahedron? A stellated icosahedron? 5. Can you construct a cube that is 3-colorable? 2-colorable? 6. Can you construct a polyhedron with a 2-coloring? How about something other than a cube with a 3-coloring? 6
Further Questions: 1. A polyhedron is n-colorable if there is a way to construct the polyhedron from n different colored Sonobe units where no Sonobe of the same color are inserted into each other. Can you construct a polyhedron that is 4-colorable? 2. Can you construct a stellated octahedron with a 3-Coloring? How about a stellated icosahedron? 3. Not all polyhedrons are 3-colorable. Can you construct a polyhedron without a 3- Coloring? How about 4-Coloring? 7
Sonobe Origami Units Crease paper down the middle and unfold Fold edges to center line and unfold Fold top right and bottom left corners do not let the triangles cross crease lines Fold triangles down to make sharper triangles do not cross the crease lines Fold along vertical crease lines Fold top left corner down to right edge Fold bottom right corner up to left edge Undo the last two folds Tuck upper left corner under flap on opposite side and repeat for lower right corner Turn the paper over, crease along dashed lines as shown to complete the module Corners of one module fit into pockets of another A cube requires six modules Sonobe modules also make many other polyhedra Figure 3: http://riverbendmath.org/modules/origami/sonobe_polyhedra/activity_ Directions/ 8