Zigzag walk

A , also called a $1$-zigzag, is a polygon (usually skew) formed by following the edges of a polyhedron in a particular way.

A $k$-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.

Geometric
A $1$-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.

Distinguished generators
For a regular polyhedron with distinguished generators $$\langle\rho_0,\rho_1,\rho_2\rangle$$, the $k$-zigzag is the polygon given by the distinguished generators:

$$ \left\langle\rho_0, (\rho_2\rho_1)^k\right\rangle $$