Skew muoctahedron

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