The truncated octahedron is a three-dimensional uniform solid produced by truncating an octahedron. It is a space-filling polyhedron with 8 regular hexagonal faces, 6 regular square faces, 24 vertices and 36 edges. It is an Archimedean solid and a zonohedron, with each of its faces having point symmetry or 180° rotational symmetry. The truncated octahedron can be constructed by the bitruncation of a cube, or as the omnitruncation of the tetrahedron.
It has a radius of inradius r and a dual midradius rho. The canonical coordinates for the vertices of a truncated octahedron centered at the origin are (±2, ±1.0), (0, ±2, ±1.0), (±1, ±2, 0), (0, ±1, ±2), (±2, 0, ±1). These coordinates form many rectangles parallel to the axes of the coordinate system. The truncated octahedron can be thought of as an octahedron from which six half octahedrons have been removed.
This relationship between the cube and the truncated octahedron helps us understand its space-filling property. It is formed by removing the 6 pyramids from the vertices of a regular octahedron. The truncated octahedron is an interesting and beautiful polyhedron with many interesting properties. It has been studied extensively by mathematicians and is used in many applications such as architecture and engineering.
It is also used in art and design to create interesting shapes and patterns.