Zigzag walk

A octahedron with a 1 -zigzag (Petrie polygon) highlighted in orange and red

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.

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:

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

See also

• Petrie dual
• Hole

External links

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

Bibliography

• 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.