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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:
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.
There are three further topologically distinct hexahedra that can only be realised as ''concave'' figures:
1. Counting polyhedra
★ Prismatoid
★ Polyhedra with 4-7 Faces by Steven Dutch
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) | ||
| Contents |
| Topologically distinct hexahedra |
| References |
| See also |
| External links |
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.
| ★ Faces: 4,4,4,4,4,4 ★ 8 vertices ★ 12 edges | ★ 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 |
| ★ Faces: 3,3,3,3,3,3 ★ 5 vertices ★ 9 edges | ★ Faces: 4,4,4,4,3,3 ★ 7 vertices ★ 11 edges | ★ 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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psst.. try this: add to faves
