The geometric term “octahedron” refers primarily to a regular octahedron, with eight triangular faces. The term “octahedron” is commonly associated with a regular octahedron, but the original definition is ignored: any polyhedron made up of eight planes is called an octahedron.

Among all convex octahedra, there are 257 convex octahedra with significantly different topologies, including their mirror images. Of these, there are 2 convex octahedra with 6 vertices, 11 with 7 vertices, 42 with 8 vertices, 74 with 9 vertices, 76 with 10 vertices, 38 with 11 vertices and 14 with 12 vertices.
Common octahedra edit Podcast
Common octahedra include the ortho-octahedron, the Dioctahedral smectite supplier hexagonal column, the heptagonal cone, the truncated tetrahedron, the orthotriangular tent tower, the heterogeneous double triangular column, the side-cone triangular column, and the triangular antiprism. 
Hexagonal column
A hexagonal column, also known as a hexagonal prism, is a type of column with a hexagonal base. All hexagonal columns have 8 faces, 18 sides and 12 vertices. A positive hexagonal prism represents a hexagonal column where each face is a positive polygon and each of its vertices is the common vertex of 2 squares and 1 positive hexagon, thus having the property of equal angles at each corner and can be classified as a semi-positive octahedron.
Heptagonal cone
A heptagonal cone is a cone with a heptagonal base, which has 7 faces, 14 sides and 7 vertices, and whose dual polyhedron is itself. A positive heptagon is a heptagonal cone with a positive heptagonal base.
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The corners and sides of these polygons that meet are identical everywhere. It is because of the existence of such regularity that the planes in which the individual polygons lie are equally likely to occur. The octahedron is one of the five orthopolyhedra known as Platonic cubes, all of which share this regularity.
The octahedron, consists of six vertices with eight square triangles, four of which intersect at a single vertex. Plato considered the octahedron to be somewhere between the tetrahedron (fire) and the icosahedron (water), and therefore believed that the element it represented was air. The octahedron has six secondary axes of rotation, through the midpoint of the opposite side; four tertiary axes of rotation, through the opposite centre; and three quadratic axes of rotation, through the opposite vertex. Any polyhedron that conforms to these axes of rotation is said to have octahedral symmetry.
Monomorphic names edit Podcast
Occurs only on equiaxed crystal systems. It is a positive octahedron formed by two pairs of eight identical crystalline faces parallel to each other in an equilateral triangle. The three lines passing through the centre and connecting the tops of each pair of corners are perpendicular to each other and of equal length. The angle between adjacent crystalline faces is 109°; each pair of faces is perpendicular to one of the three axes of symmetry in the crystal; each face is octahedrally truncated and equal to all three crystallographic axes. The monomorphic symbol is {111}. Crystals of magnetite, spinel, etc. often have this monomorph. In addition, octahedra are a very common form of coordination polyhedra in crystal structures. However, when used to represent a coordination polyhedron, the octahedron may not be an ortho-octahedron but may have aberrations.