Mathematics Animation: movie of an octahedron by Russell Kightley Media

GEOMETRY: Mathematics Animation: movie of an octahedron within its circumscribed sphere

An octahedron is a solid with 8 faces each composed of (in a regular octahedron) equilateral triangles. It has 6 vertices and 12 edges. It is equivalent to two square pyramids joined base to base.

In its regular form it is one of the five Platonic solids:

tetrahedron (Greek tetras = 4)
hexahedron (cube) (Greek hex = 6)
octahedron (Greek okto = 8)
dodecahedron (Greek dodeka = 12)
icosahedron (Greek eikosi = 20)

These are all regular (edges are all the same length) polyhedra (Greek: poly = many + hedra = seat) whose vertices (the sharp points) all touch the surface of an associated sphere called the circumscribed sphere (drawn as translucent red in this graphic). Each polyhedron thus sits precisely inside its circumscribed sphere. Another sphere (the inscribed sphere) sits precisely within each polyhedron and so glancingly touches each facet. The Platonic solids therefore lie sandwiched between their two associated spheres.