Halved mucube

From Polytope Wiki
Jump to navigation Jump to search
Halved mucube
Rank3
SpaceEuclidean
Notation
Schläfli symbol,
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[edit | edit source]

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

  • ,

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

Gallery[edit | edit source]

Related polytopes[edit | edit source]

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.

External links[edit | edit source]

Bibliography[edit | edit source]

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