## Project 20: Polyhedra Folded from Regular Polygons

A (convex) regular n-gon is a plane polygon with n sides and for which all the sides and all the angles are equal.

Problems:

1. Consider one (convex) regular n-gon R drawn in the plane. Consider all the triangulations possible for R. (By a triangulation we mean a division of the n-gon into triangles using lines joining vertices of the n-gon that lie within the polygon. Thus, an equilateral triangle has no triangulations other than itself. Note, however, that folding an equilateral triangle along one of its medians will create two congruent triangles.) What polyhedra can be formed by folding the triangulations of R? (Of particular interest are the situations where convex polyhedra can be obtained.)

2. Given a supply of (convex) regular n-gons R. Consider identical triangulations of R. When can these be assembled to form polyhedra? Convex polyhedra? When can a mixture of different triangulations of R be assembled to form polyhedra? Convex polyhedra?

3. Given a supply of (convex) regular n-gons R for different choices of n. When can these be folded along diagonals (i.e. lines joining vertices) in an identical way, and assembled to form polyhedra? Convex polyhedra? What about allowing different foldings to be used together.

Extensions:

1. All folding to occur along lines other than those joining two vertices. One can consider the case where each fold goes through one vertex, and the more general case where folds need not go through any vertex. One can allow folding with several layers of paper. For example, given 4 2x2 squares, one can create a 1x1x2 brick as follow: Fold two 2x2 squares along a vertical mirror. Fold two 2x2 squares into a 1x1 square. (This can be done by folding along a vertical mirror and then taking the resulting rectangle and folding it in half).

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451-0001
email: malkevitch@york.cuny.edu
(Comments and results related to the project above are welcome.)

Acknowledgements
Some of this work was prepared with partial support from the National Science Foundation (Grant Number: DUE 9555401) to the Long Island Consortium for Interconnected Learning (administered by SUNY at Stony Brook, Alan Tucker, Project Director).