Heptahedron

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A heptahedron (plural: heptahedra) is a polyhedron having seven sides, or faces.

A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven equilateral triangle faces form a rudimentary projective plane. No heptahedra are regular.

Topologically distinct heptahedra[edit]

Convex[edit]

There are 34 topologically distinct convex heptahedra, excluding mirror images.[1] (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. The images are ordered by descending number of six-sided faces (if any), followed by descending number of five-sided faces (if any), and so on.

Heptahedron01.GIF
  • Faces: 6,6,4,4,4,3,3
  • 10 vertices
  • 15 edges
Heptahedron02.GIF
  • Faces: 6,5,5,5,3,3,3
  • 10 vertices
  • 15 edges
Heptahedron03.GIF
  • Faces: 6,5,5,4,4,3,3
  • 10 vertices
  • 15 edges
Heptahedron04.GIF
  • Faces: 6,5,4,4,3,3,3
  • 9 vertices
  • 14 edges
Heptahedron05.GIF
  • Faces: 6,5,4,4,3,3,3
  • 9 vertices
  • 14 edges
Heptahedron06.GIF
  • Faces: 6,4,4,4,4,3,3
  • 9 vertices
  • 14 edges
Heptahedron07.GIF
  • Faces: 6,4,4,3,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron08.GIF
  • Faces: 6,4,4,3,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron09.GIF
Hexagonal pyramid
  • Faces: 6,3,3,3,3,3,3
  • 7 vertices
  • 12 edges
Heptahedron10.GIF
  • Faces: 5,5,5,4,4,4,3
  • 10 vertices
  • 15 edges
Heptahedron11.GIF
  • Faces: 5,5,5,4,3,3,3
  • 9 vertices
  • 14 edges
Heptahedron12.GIF
  • Faces: 5,5,5,4,3,3,3
  • 9 vertices
  • 14 edges
Heptahedron13.GIF
Pentagonal prism
  • Faces: 5,5,4,4,4,4,4
  • 10 vertices
  • 15 edges
Heptahedron14.GIF
  • Faces: 5,5,4,4,4,3,3
  • 9 vertices
  • 14 edges
Heptahedron15.GIF
  • Faces: 5,5,4,4,4,3,3
  • 9 vertices
  • 14 edges
Heptahedron16.GIF
  • Faces: 5,5,4,3,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron17.GIF
  • Faces: 5,5,4,3,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron18.GIF
  • Faces: 5,4,4,4,4,4,3
  • 9 vertices
  • 14 edges
Heptahedron19.GIF
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron20.GIF
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron21.GIF
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron22.GIF
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron23.GIF
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
Heptahedron24.GIF
  • Faces: 5,4,3,3,3,3,3
  • 7 vertices
  • 12 edges
Heptahedron25.GIF
  • Faces: 5,4,3,3,3,3,3
  • 7 vertices
  • 12 edges
Heptahedron26.GIF
  • Faces: 4,4,4,4,4,3,3
  • 8 vertices
  • 13 edges
Heptahedron27.GIF
  • Faces: 4,4,4,4,4,3,3
  • 8 vertices
  • 13 edges
Heptahedron28.GIF
Elongated triangular pyramid
  • Faces: 4,4,4,3,3,3,3
  • 7 vertices
  • 12 edges
Heptahedron29.GIF
  • Faces: 4,4,4,3,3,3,3
  • 7 vertices
  • 12 edges
Heptahedron30.GIF
  • Faces: 4,4,4,3,3,3,3
  • 7 vertices
  • 12 edges
Heptahedron31.GIF
  • Faces: 4,4,4,3,3,3,3
  • 7 vertices
  • 12 edges
Heptahedron32.GIF
  • Faces: 4,4,4,3,3,3,3
  • 7 vertices
  • 12 edges
Heptahedron33.GIF
  • Faces: 4,3,3,3,3,3,3
  • 6 vertices
  • 11 edges
Heptahedron34.GIF
  • Faces: 4,3,3,3,3,3,3
  • 6 vertices
  • 11 edges

Concave[edit]

Heptahedron concave 03.GIFHeptahedron concave 04.GIFHeptahedron concave 05.GIFHeptahedron concave 06.GIFHeptahedron concave 07.GIFHeptahedron concave 08.GIF

Six topologically distinct concave heptahedra (excluding mirror images) can be formed by combining two tetrahedra in various configurations. The third, fourth and fifth of these have a face with collinear adjacent edges, and the sixth has a face that is not simply connected.

Heptahedron concave 01.GIFHeptahedron concave 02.GIF

13 topologically distinct heptahedra (excluding mirror images) can be formed by cutting notches out of the edges of a triangular prism or square pyramid. Two examples are shown.

Heptahedron concave 09.GIFHeptahedron concave 10.GIF

A variety of non-simply-connected heptahedra are possible. Two examples are shown.

References[edit]

  1. ^ Counting polyhedra

External links[edit]