GEOMETRY: Mathematics Animation: movie of an icosahedron within its circumscribed sphere
An icosahedron is a solid with 20 faces each composed of (in a regular icosahedron) equilateral triangles. It has 12 vertices and 30 edges and it is an efficient way of constructing a spheroidal container using simple repeating units, which is why it is the favoured form of viral capsids.
In its regular form it is one of the five Platonic solids:
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.