# 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

### Geometric

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

For a regular polyhedron with distinguished generators ${\displaystyle \langle \rho _{0},\rho _{1},\rho _{2}\rangle }$, the k -zigzag is the polygon given by the distinguished generators:

${\displaystyle \left\langle \rho _{0},(\rho _{2}\rho _{1})^{k}\right\rangle }$