romboéder

A rhombohedron is a polyhedron with eight faces, where each face is a rhombus. It is one of the five Platonic solids, characterized by having congruent regular polygonal faces and identical vertices. The rhombohedron is distinct from the cube, though it shares the same number of faces, edges, and vertices. While a cube has square faces, a rhombohedron’s faces are rhombi, meaning all sides are of equal length, but adjacent faces meet at non-right angles.

The rhombohedron can be visualized as a cube that has been uniformly stretched or compressed along one or more axes. Its dual polyhedron is the cuboctahedron, which has alternating square and triangular faces. The rhombohedron is also a type of zonohedron, as it can be formed by the Minkowski sum of line segments.

In crystallography, the rhombohedral system describes one of the seven crystal systems, where the unit cell is a rhombohedron rather than a cube or other parallelepiped. This system is notable for its threefold rotational symmetry, often observed in minerals like calcite and hematite. The rhombohedron’s geometric properties make it relevant in fields ranging from mathematics to materials science.

Mathematically, a rhombohedron can be defined by three vectors of equal length originating from a single vertex, with the angle between any two vectors being equal but not necessarily 90 degrees. Its volume can be calculated using the scalar triple product of these vectors. Despite its simplicity in definition, the rhombohedron serves as a fundamental shape in both theoretical and applied geometry.