A heptahedron is a polyhedron

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 having seven sides, or faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

.

A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid

Pyramid

A pyramid is a structure whose outer surfaces are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces...

 and the pentagonal prism

Pentagonal prism

In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :...

. Also notable is the tetrahemihexahedron

Tetrahemihexahedron

In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....

, whose seven equilateral triangle faces form a rudimentary projective plane

Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

. No heptahedra are regular

Regular polyhedron

A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...

.

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.

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 Hexagonal pyramid In geometry, a hexagonal pyramid is a pyramid with a hexagonal base upon which are erected six triangular faces that meet at a point... 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 Pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :... 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 Elongated triangular pyramid In geometry, the elongated triangular pyramid is one of the Johnson solids . Norman Johnson discovered elongated triangular pyramids. As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base... 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

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

External links

• Polyhedra with 4-7 Faces by Steven Dutch