### Mathematics Research Projects

## Project 13: Drawing Graphs

It is possible to represent a convex 3-dimensional polytope by
a graph in the plane. Such graphs can be plane graphs (i.e. have
crossings only at vertices) or show crossings. It may be desirable
for a drawing to have crossings if by doing so it displays some
aspect of the original polytope, for example, more of its symmetries.

**Problem**

1. Examine the relation between the isometries of 3-dimensional
polytopes and the symmetries that are displayed by different graphs
that might be used to represent them, with or without crossings,
in the plane. As specific examples one might consider drawings
of the tetrahedron, cube, and octahedron in the the plane.

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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).