hexoctahedra

A hexoctahedron is a polyhedron composed of eight triangular faces, twelve edges, and six vertices. It is one of the five convex deltahedra, meaning all its faces are equilateral triangles, and it is also one of the Catalan solids, the duals of the uniform polyhedra known as cuboctahedra. The hexoctahedron is notable for its symmetry and geometric properties, making it a subject of interest in mathematics and crystallography.

The hexoctahedron can be constructed by truncating the vertices of a regular octahedron at one-third of the edge length, resulting in a polyhedron with eight equilateral triangular faces. Alternatively, it can be seen as the dual of the cuboctahedron, where the vertices and faces are swapped. Its Schläfli symbol is {3, 4}, indicating that each face is a triangle (3) and four triangles meet at each vertex (4).

In terms of symmetry, the hexoctahedron belongs to the octahedral symmetry group, which includes the cube and octahedron as its primary members. This group is one of the highest-order symmetry groups in three-dimensional space, reflecting its rich geometric structure. The hexoctahedron is also a zonohedron, meaning it can be formed by the Minkowski sum of line segments.

While less commonly encountered than other Platonic or Archimedean solids, the hexoctahedron appears in mathematical studies of polyhedral symmetry and as a theoretical model in crystallography. Its properties make it useful in exploring the relationships between dual polyhedra and the geometric transformations that connect them.