Definitions

# hexahedron

[hek-suh-hee-druhn]
hexahedron: see cube; polyhedron.
A hexahedron (plural: hexahedra) is a polyhedron with six faces. A regular hexahedron, with all its faces square, is a cube.

There many kinds of hexahedron, some topologically similar to the cube, and some not. Three are briefly examined below:

Parallelogram faced:

Parallelepiped
(Three pairs of
parallelograms)

Rhombohedron
(Three pairs of
rhombi)

Trigonal trapezohedron
(congruent rhombi)

Cuboid
(Three pairs of
rectangles)

Cube
(square)
Others:

Pentagonal pyramid
(pentagon and triangles)

Triangular dipyramid
(triangles)

(apex-truncated
square pyramid)

## Topologically distinct hexahedra

There are seven topologically distinct convex hexahedra, one of which exists in two mirror image forms. (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.)

An example of each type is depicted below, along with the number of sides on each of the faces and the numbers of vertices and edges.

Cube and topological equivalents.

• Faces: 4,4,4,4,4,4
• 8 vertices
• 12 edges

Pentagonal pyramid.

• Faces: 5,3,3,3,3,3
• 6 vertices
• 10 edges

• Faces: 5,4,4,3,3,3
• 7 vertices
• 11 edges

• Faces: 5,5,4,4,3,3
• 8 vertices
• 12 edges

Triangular dipyramid.

• Faces: 3,3,3,3,3,3
• 5 vertices
• 9 edges

• Faces: 4,4,4,4,3,3
• 7 vertices
• 11 edges

Tetragonal antiwedge. Chiral – exists in "left-handed" and "right-handed" mirror image forms.

• Faces: 4,4,3,3,3,3
• 6 vertices
• 10 edges

There are three further topologically distinct hexahedra that can only be realised as concave figures:

• Faces: 4,4,3,3,3,3
• 6 vertices
• 10 edges

• Faces: 6,6,3,3,3,3
• 8 vertices
• 12 edges

• Faces: 5,5,3,3,3,3
• 7 vertices
• 11 edges