# Skewed muoctahedron

Skewed muoctahedron
Rank3
SpaceEuclidean
Notation
Schläfli symbol$\{\infty ,4\}_{\cdot ,\ast 3}$ , $\left\{{\frac {3}{1,0}},4:{\frac {3}{1,0}}\right\}$ Elements
FacesN  triangular helices
EdgesN×2M
VerticesN×M
Vertex figureSquare
Petrie polygonsN  triangular helices
HolesApeirogons
Related polytopes
RegimentSkewed muoctahedron
Petrie dualSkewed muoctahedron
Abstract & topological properties
Schläfli type{∞,4}
Properties
ChiralYes
ConvexNo

The skewed muoctahedron is a regular skew polyhedron within 3-dimensional space. It is an infinite polyhedron consisting of triangular helices, with 4 at a vertex. Like the hemidodecahedron, the skewed muoctahedron is self-Petrie; it is its own Petrie dual. Its vertices and edges are a subset of those found in the cubic honeycomb.

The skewed muoctahedron can be obtained by skewing ($\sigma$ ) the muoctahedron, an operation abstractly equivalent to $\pi \delta \eta \pi \delta$ where $\pi$ is the Petrie dual, $\delta$ is the dual, and $\eta$ is halving. Despite the fact that some intermediate steps in the process of skewing cannot exist in 3D space, the final result ends up having a realization.

The skewed muoctahedron is a chiral polyhedron; its helices are either all clockwise or all counterclockwise.