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)	Quadrilateral frustum (apex-truncated square pyramid)

Topologically distinct hexahedra

There are seven topologically distinct convex hexahedra,^[1] 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

References

1. ↑ Counting polyhedra

See also

• Prismatoid

External links

• Polyhedra with 4-7 Faces by Steven Dutch

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