# Halved mucube

Halved mucube
Rank3
SpaceEuclidean
Notation
Schläfli symbol$\{6, 6\}_4$ , $\left\{ \frac{6}{1,3}, 6 : \frac{4}{1,2} \right\}$ Elements
Facesskew hexagons
Edges
Vertices
Vertex figureHexagon
Petrie polygonsSkew squares
HolesTriangular helices
Related polytopes
DualHalved mucube
Petrie dualPetrial halved mucube
Convex hullTetrahedral-octahedral honeycomb
Abstract & topological properties
Schläfli type{6,6}
OrientableYes
Genus
Properties
ConvexNo

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.