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

Parallelogram faced:
Parallelepiped.svg
Parallelepiped
(Three pairs of
parallelograms)
Rhombohedron.svg
Rhombohedron
(Three pairs of
rhombi)
Trigonal trapezohedron.png
Trigonal trapezohedron
(congruent rhombi)
Cuboid.png
Cuboid
(Three pairs of
rectangles)
Hexahedron.png
Cube
(square)
Others:
Pentagonal pyramid.png
Pentagonal pyramid
(pentagon and triangles)
Triangular bipyramid.png
Triangular dipyramid
(triangles)
Usech kvadrat piramid.png
Quadrilateral frustum
(apex-truncated
square pyramid)

Contents

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.

Hexahedron1.GIF

Cube and topological equivalents.

  • Faces: 4,4,4,4,4,4
  • 8 vertices
  • 12 edges
Hexahedron2.GIF

Pentagonal pyramid.

  • Faces: 5,3,3,3,3,3
  • 6 vertices
  • 10 edges
Hexahedron3.GIF
  • Faces: 5,4,4,3,3,3
  • 7 vertices
  • 11 edges
Hexahedron4.GIF
  • Faces: 5,5,4,4,3,3
  • 8 vertices
  • 12 edges
Hexahedron5.GIF

Triangular dipyramid.

  • Faces: 3,3,3,3,3,3
  • 5 vertices
  • 9 edges
Hexahedron6.GIF
  • Faces: 4,4,4,4,3,3
  • 7 vertices
  • 11 edges
Hexahedron7.GIFHexahedron7a.GIF

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:

Hexahedron8.GIF
  • Faces: 4,4,3,3,3,3
  • 6 vertices
  • 10 edges
Hexahedron9.GIF
  • Faces: 6,6,3,3,3,3
  • 8 vertices
  • 12 edges
Hexahedron10.GIF
  • Faces: 5,5,3,3,3,3
  • 7 vertices
  • 11 edges

References

  1. ^ Counting polyhedra

See also

External links


Simple English

Redirecting to Cube








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