Halved mucube

Halved mucube
Rank	3
Space	Euclidean
Notation
Schläfli symbol	$\{6, 6\}_4$, $\left\{ \frac{6}{1,3}, 6 : \frac{4}{1,2} \right\}$
Elements
Faces	∞ skew hexagons
Edges	∞
Vertices	∞
Vertex figure	Hexagon
Petrie polygons	Skew squares
Holes	Triangular helices
Related polytopes
Dual	Halved mucube
Petrie dual	Petrial halved mucube
Convex hull	Tetrahedral-octahedral honeycomb
Abstract & topological properties
Schläfli type	{6,6}
Orientable	Yes
Genus	∞
Properties
Convex	No

The halved mucube is a regular skew apeirohedron within 3-dimensional space. It is an infinite polyhedron consisting of skew hexagons with 6 meeting at a vertex, and it can be obtained by alternating (also known as halving) the mucube. It has the Schläfli symbol {6,6}₄ and it is self-dual.

The halved mucube's Petrie polygon is a skew square, and the petrial halved mucube is its Petrial. One can also apply a second-order facetting onto the halved mucube to obtain the tetrahelical triangular tiling.

Vertex coordinates

The vertex coordinates of the halved mucube of edge length 1 are the same as the vertex coordinates of the tetrahedral-octahedral honeycomb, being:

• $(i \frac{\sqrt{2}}{2}, j \frac{\sqrt{2}}{2}, k \frac{\sqrt{2}}{2})$

where i, j and k are integers where i+j+k is even.

Gallery

• 

  A section of the halved mucube superimposed on a section of the mucube. A single face of the halved mucube is shown in red.

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.