# Zigzag walk

A **zigzag walk**, also called a **k -zigzag**, is a polygon (usually skew) formed by following the edges of a polyhedron in a particular way.

A 1 -zigzag, that is a path alternating between the leftmost and rightmost possible edges, is also called a Petrie polygon or sometimes just a zigzag, and is integral in the construction of the Petrie dual.

## Definition[edit | edit source]

### Geometric[edit | edit source]

A k -zigzag is a polygon formed by following the edges of a polyhedron in a particular way. Starting from a particular vertex and edge the polygon follows that edge to another vertex. It then follows along the k th rightmost edge to the next vertex, and then follows along the k th leftmost edge, repeating this pattern of left and right turns until the path loops back on itself.^{[1]}

### Distinguished generators[edit | edit source]

For a regular polyhedron with distinguished generators , the k -zigzag is the polygon given by the distinguished generators:

## See also[edit | edit source]

## External links[edit | edit source]

- Wikipedia contributors. "Petrie polygon".
- Weisstein, Eric W. "Petrie Polygon" at MathWorld.

## References[edit | edit source]

## Bibliography[edit | edit source]

- Coxeter, Harold Scott MacDonald; Moser, William Oscar Jules (1972).
*Generators and Relations for Discrete groups*(4 ed.). Springer-Verlag. doi:10.1007/978-3-662-21946-1. - McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF).
*Discrete Computational Geometry*(47): 449–478. doi:10.1007/PL00009304.