# Halved mucube

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

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 halving or alternating the mucube. It has the Schläfli symbol {6,6}4 and it is self-dual.

## 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:

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

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

## Related polytopes

The halved mucube's Petrie polygon is a skew square, and its Petrial is the Petrial halved mucube.

One can apply a second-order facetting onto the halved mucube to obtain the tetrahelical triangular tiling.

Rectifying the halved mucube gives the muoctahedron.