A heptahedron (plural: heptahedra) is a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

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 In geometry, a hexagonal pyramid is a pyramid with a hexagonal base upon which are erected six isosceles triangular faces that meet at a point (the apex). Like any pyramid, it is self- dual. A right hexagonal pyramid with a regular hexagon base ...

and the pentagonal prism. Also notable is the

tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...

, 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. (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.

Concave

One particularly interesting example is the

Szilassi polyhedron In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. Coloring and symmetry The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surf ...

, a Toroidal polyhedron with 7 non-convex six sided faces.

References

External links

Polyhedra with 4–7 Faces
by Steven Dutch * {{Polyhedra Polyhedra