Heptahedron


A heptahedron is a polyhedron having seven sides, or faces.
A heptahedron can take a large number of different basic forms, or topologies. The most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, which can be seen as a tessellation of the real projective plane. No heptahedra are regular.

Topologically distinct heptahedron

Convex

There are 34 topologically distinct convex heptahedra, excluding mirror images.
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, followed by descending number of five-sided faces, and so on.

  • Faces: 6,5,5,5,3,3,3
  • 10 vertices
  • 15 edges
  • Faces: 6,5,5,4,4,3,3
  • 10 vertices
  • 15 edges
  • Faces: 6,5,4,4,3,3,3
  • 9 vertices
  • 14 edges
  • Faces: 6,5,4,4,3,3,3
  • 9 vertices
  • 14 edges
  • Faces: 6,4,4,4,4,3,3
  • 9 vertices
  • 14 edges
  • Faces: 6,4,4,3,3,3,3
  • 8 vertices
  • 13 edges
  • Faces: 6,4,4,3,3,3,3
  • 8 vertices
  • 13 edges

    • Faces: 6,3,3,3,3,3,3
    • 7 vertices
    • 12 edges
  • Faces: 5,5,5,4,4,4,3
  • 10 vertices
  • 15 edges
  • Faces: 5,5,5,4,3,3,3
  • 9 vertices
  • 14 edges
  • Faces: 5,5,5,4,3,3,3
  • 9 vertices
  • 14 edges

    • Faces: 5,5,4,4,4,4,4
    • 10 vertices
    • 15 edges
  • Faces: 5,5,4,4,4,3,3
  • 9 vertices
  • 14 edges
  • Faces: 5,5,4,4,4,3,3
  • 9 vertices
  • 14 edges
  • Faces: 5,5,4,3,3,3,3
  • 8 vertices
  • 13 edges
  • Faces: 5,5,4,3,3,3,3
  • 8 vertices
  • 13 edges
  • Faces: 5,4,4,4,4,4,3
  • 9 vertices
  • 14 edges
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
  • Faces: 5,4,4,4,3,3,3
  • 8 vertices
  • 13 edges
  • Faces: 5,4,3,3,3,3,3
  • 7 vertices
  • 12 edges
  • Faces: 5,4,3,3,3,3,3
  • 7 vertices
  • 12 edges
  • Faces: 4,4,4,4,4,3,3
  • 8 vertices
  • 13 edges
  • Faces: 4,4,4,4,4,3,3
  • 8 vertices
  • 13 edges

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

    • Faces: 4,4,4,3,3,3,3
    • 7 vertices
    • 12 edges
  • Faces: 4,4,4,3,3,3,3
  • 7 vertices
  • 12 edges
  • Faces: 4,3,3,3,3,3,3
  • 6 vertices
  • 11 edges
  • Faces: 4,3,3,3,3,3,3
  • 6 vertices
  • 11 edges
    		• -

    Concave

    One particularly interesting example is the Szilassi polyhedron, a Toroidal polyhedron with 7 non-convex six sided faces.