Skewed muoctahedron

Skewed muoctahedron
Rank	3
Space	Euclidean
Notation
Schläfli symbol	$\{\infty ,4\}_{\cdot ,\ast 3}$, $\left\{{\frac {3}{1,0}},4:{\frac {3}{1,0}}\right\}$
Elements
Faces	N  triangular helices
Edges	N×2M
Vertices	N×M
Vertex figure	Square
Petrie polygons	N  triangular helices
Holes	Apeirogons
Related polytopes
Regiment	Skewed muoctahedron
Petrie dual	Skewed muoctahedron
Abstract & topological properties
Schläfli type	{∞,4}
Properties
Chiral	Yes
Convex	No

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.

External links

• jan Misali (2020). "there are 48 regular polyhedra".

Bibliography

• McMullen, Peter; Schulte, Egon (1997). "Regular Polytopes in Ordinary Space" (PDF). Discrete Computational Geometry (47): 449–478. doi:10.1007/PL00009304.