### Mathematics Research Projects

## 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).

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**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).