Compound of cube and octahedron

From Wikipedia, the free encyclopedia
Compound of cube and octahedron
Compound of cube and octahedron.png
Type Compound
Coxeter diagram CDel nodes 10ru.pngCDel split2-43.pngCDel node.pngCDel nodes 01rd.pngCDel split2-43.pngCDel node.png
Stellation core cuboctahedron
Convex hull Rhombic dodecahedron
Index W43
Polyhedra 1 octahedron
1 cube
Faces 8 triangles
6 squares
Edges 24
Vertices 14
Symmetry group octahedral (Oh)

The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound.

Construction[edit]

The 14 Cartesian coordinates of the vertices of the compound are.

6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2)
8: ( ±1, ±1, ±1)

As a compound[edit]

It can be seen as the compound of an octahedron and a cube. It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot polyhedron and its dual.

It has octahedral symmetry (Oh) and shares the same vertices as a rhombic dodecahedron.

This can be seen as the three-dimensional equivalent of the compound of two squares ({8/2} "octagram"); this series continues on to infinity, with the four-dimensional equivalent being the compound of tesseract and 16-cell.

A cube and its dual octahedron
The intersection of both solids is the cuboctahedron, and their convex hull is the rhombic dodecahedron.
Seen from 2-fold, 3-fold and 4-fold symmetry axes
The hexagon in the middle is the Petrie polygon of both solids.
If the edge crossings were vertices, the mapping on a sphere would be the same as that of a deltoidal icositetrahedron.

As a stellation[edit]

It is also the first stellation of the cuboctahedron and given as Wenninger model index 43.

It can be seen as a cuboctahedron with square and triangular pyramids added to each face.

The stellation facets for construction are:

First stellation of cuboctahedron trifacets.pngFirst stellation of cuboctahedron square facets.png

See also[edit]

References[edit]

  • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 978-0-521-09859-5.