Hexahedral

A hexahedron is a polyhedron with six faces. The term is formed from the Greek hex = six and hedra = faces. The cube is the regular or archetypal hexahedron, featuring six square faces, equal edges, and right-angled dihedral angles. However, the category includes many other six-faced solids that may be convex or concave, and whose six faces can be polygons of various shapes and sizes.

In general, a hexahedron is defined by having exactly six faces, but it need not be highly regular. Convex hexahedra form a wide class of shapes, and concave six-faced polyhedra are also possible. The combinatorial structure of a simple six-faced polyhedron is constrained by Euler’s formula, which for convex polyhedra states V − E + F = 2. With F = 6, this gives E = V + 4. The sum of the face degrees equals 2E, so the average number of sides per face varies with the particular shape. If all six faces are quadrilaterals, then E = 12 and V = 8, as in the cube.

Applications of the term extend beyond pure geometry. In computer graphics and finite element analysis, “hexahedral” refers to cube-like elements or cells used in 3D meshing, known as hex elements. Hexahedral meshes are valued for their favorable numerical properties and efficiency in representing certain geometries, though generating high-quality hex meshes can be challenging.

Overall, the hexahedron is a fundamental six-faced polyhedral form, with the cube standing as its most familiar example.