A hexahedron (: hexahedra or hexahedrons) or sexahedron (: sexahedra or sexahedrons) is any
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
with six
faces. A
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
, for example, is a
regular hexahedron with all its faces
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, and three squares around each
vertex.
There are seven
topologically distinct ''convex'' hexahedra,
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.
Convex
Cuboid
A hexahedron that is combinatorially equivalent to a cube may be called a
cuboid, although this term is often used more specifically to mean a
rectangular cuboid
A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped.
Many writers just call these ...
, a hexahedron with six rectangular sides. Different types of cuboids include the ones depicted and linked below.
Others
There are seven topologically distinct convex hexahedra,
[ the cuboid and six others, which are depicted below. One of these is chiral, in the sense that it cannot be deformed into its mirror image.
]
Concave
Three further topologically distinct hexahedra can only be realised as ''concave'' acoptic polyhedra. These are defined as the surfaces formed by non-crossing simple polygon
In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...
faces, with each edge shared by exactly two faces and each vertex surrounded by a cycle of three or more faces.
These cannot be convex because they do not meet the conditions of Steinitz's theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedron, convex polyhedra: they are exactly the vertex connect ...
, which states that convex polyhedra have vertices and edges that form 3-vertex-connected graphs.
For other types of polyhedra that allow faces that are not simple polygons, such as the ''spherical polyhedra'' of Hong and Nagamochi, more possibilities exist.
References
{{Polyhedra
6 (number)
Polyhedra