Truncated octahedron

Basic characteristics

• Faces: 14 (8 hexagons, 6 squares).
• Edges: 36.
• Vertices: 24.
• Vertex configuration: 4.6.6 (one square and two hexagons meet at each vertex).
• Symmetry: octahedral symmetry related to the parent octahedron.

This solid is one of the family of Archimedean solids; for a general introduction to that class see Archimedean solids. The hexagonal faces are regular when the truncation depth is chosen appropriately; for shallower or deeper truncations the hexagons can be irregular.

Construction and geometry

Formally, the truncated octahedron can be described as the uniform truncation t{3,4} of the octahedron: cut off each vertex of the octahedron at one-quarter of the original edge length so that the original triangular faces become hexagons and the former vertices become squares. The truncation produces a single vertex orbit, which is why the shape is called uniform. It may also be generated as a zonohedron or constructed from coordinate sets in three-dimensional space for modeling and computation.

Space-filling and applications

An important feature of the truncated octahedron is that it tiles three-dimensional space without gaps or overlaps. The arrangement of identical truncated octahedra fills space to form the truncated octahedral honeycomb, a simple and efficient tessellation. This property makes the shape relevant in crystallography and materials science: the truncated octahedron is the Voronoi cell of the body-centered cubic packing and appears in models of foams and alloys. Historically, a partition of space using truncated octahedra was proposed in discussions of Kelvin’s problem; it remains a classical example of a space-filling polyhedron.

Related forms and distinctions

The truncated octahedron is one of thirteen Archimedean solids and is distinct from other truncations such as the truncated cube or truncated tetrahedron by its combination of hexagonal and square faces and its specific vertex configuration. It is also related to other uniform constructions and to dual Catalan solids; studying these relationships reveals how truncation operations transform face and vertex structure while preserving overall symmetry. For more on the octahedron that underlies this form, see octahedron. For more on hexagonal and square faces in polyhedra, see hexagon and square.

The truncated octahedron combines mathematical elegance with practical relevance: it provides a clear example of uniform truncation, a natural space-filler, and a shape that appears in both theoretical geometry and applied science.