Heptahedron


A heptahedron 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 faces form a rudimentary 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: 6,6,4,4,4,3,3
  • 10 vertices
  • 15 edges
  • 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
    • Hexagonal pyramid

    • 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
    • Pentagonal prism

    • 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
    • Elongated triangular pyramid

    • 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