# Paper Folding

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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When people see my polyhedra constructed of paper, they often ask if I'm interested in origami. With all respect to aficionados of origami, the answer is no. Some origami constructions are incredibly complex and astonishing; nevertheless, it strikes me as pretty obvious that if you start with a big enough sheet of paper and make enough folds, you can derive any imaginable shape. There are some interesting modular shapes that can be folded to construct polyhedra, but the modules turn out to be far more complex than simply drawing the faces and cutting them out with scissors.

## Stars and Snowflakes

A number of years ago my wife was decorating a classroom for winter and asked me if there was a simple way to cut 6-pointed snowflakes. As it happens, there is. That got me wondering about simple ways to fold other angles, and a bit of experimentation turned up plenty of simple constructions. I wrote the results up and published them as "Folding N-Pointed Stars and Snowflakes" (Mathematics Teacher, v. 87, no. 8, November, 1994.) Those constructions and a few new twists are shown below.

Everybody knows how to fold paper to make patterns with 4-, 8-, or 16-fold symmetry. Pretty simple constructions exist for any value of n up to at least 11. We really need concern ourselves only with odd numbers, since we can always bisect any angle to get a pattern with 2n symmetry.

The problem boils down to constructing an angle 180/n by folding. If n is fairly large, rather than finding 180/n, it's better to find some large multiple (4 or 8 times the angle) and repeatedly bisect it. Bisection of thick wedges of paper may not be very accurate, so it may pay, once down to twice or four times the desired angle, to unfold the paper and bisect angles separately.