Zigzag walk

From Polytope Wiki
Jump to navigation Jump to search
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[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]

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.