Ahexahedron (pl.:hexahedra orhexahedrons) orsexahedron (pl.:sexahedra orsexahedrons) is anypolyhedron with sixfaces. Acube, for example, is aregular hexahedron with all its facessquare, and three squares around eachvertex.
There are seventopologically distinctconvex hexahedra,[1] one of which exists in two mirror image forms. Additional non-convex hexahedra exist, with their number depending on how polyhedra are defined. Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.
A hexahedron that is combinatorially equivalent to a cube may be called acuboid, although this term is often used more specifically to mean arectangular cuboid, a hexahedron with six rectangular sides. Different types of cuboids include the ones depicted and linked below.
| Cuboids | ||||||
|---|---|---|---|---|---|---|
 |  |  |  |  |  |  |
| Cube (square) | Rectangular cuboid (three pairs of rectangles) | Trigonal trapezohedron (congruentrhombi) | Trigonal trapezohedron (congruentquadrilaterals) | Quadrilateralfrustum (apex-truncated square pyramid) | Parallelepiped (three pairs of parallelograms) | Rhombohedron (three pairs of rhombi) |
There are seven topologically distinct convex hexahedra,[1] the cuboid and six others, which are depicted below. One of these ischiral, in the sense that it cannot be deformed into its mirror image.
| Image |  |   |  |  |  |  |
|---|---|---|---|---|---|---|
| Name | Triangular bipyramid | Pentagonal pyramid | Doubly truncated tetrahedron[2] | |||
| Features |
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| Properties | Simplicial | Dome |
Three further topologically distinct hexahedra can only be realised asconcaveacoptic polyhedra. These are defined as the surfaces formed by non-crossingsimple polygon faces, with each edge shared by exactly two faces and each vertex surrounded by a cycle of three or more faces.[3]
| Concave | ||
|---|---|---|
 |  |  |
| 4.4.3.3.3.3 Faces 10 E, 6 V | 5.5.3.3.3.3 Faces 11 E, 7 V | 6.6.3.3.3.3 Faces 12 E, 8 V |
These cannot be convex because they do not meet the conditions ofSteinitz's theorem, which states that convex polyhedra have vertices and edges that form3-vertex-connected graphs.[4]For other types of polyhedra that allow faces that are not simple polygons, such as thespherical polyhedra of Hong and Nagamochi, more possibilities exist.[5]