In Geometry, What Is an Octahedron?

An octahedron is a pyramid consisting of 6 vertices and 8 regular triangles. It has eight faces, and each face is an equilateral triangle. Just like a cube, the eight faces of an octahedron are all regular polygons. ^[1]

Although a polyhedron with 8 faces is called an octahedron . However, the geometric term "octahedron" mainly refers to

Among all convex octahedra,

Common octahedrons are

These ones

Appears only on equiaxed crystals. It is a regular octahedron consisting of two identical crystal planes in the form of equilateral triangles. It passes through the center and the three lines connecting each diagonal top are perpendicular and equal in length to each other. The included angle between adjacent crystal planes is 109 °; each pair of crystal planes is perpendicular to a third-order symmetry axis of the crystal; each crystal plane and the three crystal axes are octahedral-section-equivalent. The simplex symbol is {111}. Crystals such as magnetite and spinel often have this single shape. In addition, octahedron is a very common form of coordination polyhedron in the crystal structure. However, when used to represent coordinated polyhedrons, the so-called octahedron may not be a regular octahedron and may allow distortion.