Halved mucube

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.