Business Card Origami
The purpose of this lesson is to illustrate symmetry with its manifestation in modular origami.
The adjective modular means that the origami objects are constructed by putting together several pieces of folded paper.
squares can be constructed from business cards by laying one card crosswise across another to use as a template, and then folding up the sides of the lower card. (see illustration at http://www.math.uni.edu/mdm/square.tif). The resulting object can be viewed as a square table or trough depending on whether the edges are pointing down or up. What is of interest to us is that there is only one way to make such a square: if you reversed the direction of the folds, you could rotate the resultant object in 3-space so it looked like the original object. (Indeed, if you only reversed one of the folds, you would have a "Z", but no matter which fold you reversed you could rotate it in space to get the same "Z" shape.)
Six of these squares can be formed into a cube. The way in which you combine them will depend on whether you have bent the sides so that they press in (as is generally assumed) or press out (as I prefer, but there is good reason that pressing in is generally assumed). If you have succeeded in forming a reasonably stable cube, you will notice that it is symmetric in the sense that every piece is in the same position relative to the other pieces.
What we have illustrated in building a cube is a module which is inherently symmetric with respect to itself.
A triangle can be formed from a business card by folding the lower left corner to the upper right corner, creasing, and then folding the free flaps across the double thick triangle (see illustration at http://www.math.uni.edu/mdm/triangle.tif). The resultant triangle is approximately equilateral because the standard dimensions for a business card are 2 inches by 3.5 inches, and arctan(4/7)=29.74 degrees. What is of interest to us is that if you reverse all the folds, you will not be able to rotate the resultant object in 3-space to coincide with the original object. (Reversing the folds is equivalent to initially folding the lower right corner to the upper left corner.) [To understand this, you should make two modules, keeping one for reference.]
Each module contains two triangles which can be used in constructing tetrahedra, octahedra, and icosahedra (among other polyhedra). A tetrahedron will require two modules, an octahedron four, and an icosohedron ten, but the symmetry is important. How many modules of each type are necessary to construct the regular polyhedra? When more than one set of modules will work, how is the choice of modules manifested in the appearance of the polyhedron?
Reflection: What can you form putting 3, 4, or 5 modules together to form a vertex? Why cannot you use 2 or 6? is the symmetry of the modules important?
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