Akshat Rathi in The Conversation:
The work of the Greek polymath Plato has kept millions of people busy for millennia. A few among them have been mathematicians who have obsessed about Platonic solids, a class of geometric forms that are highly regular and are commonly found in nature.
Since Plato’s work, two other classes of equilateral convex polyhedra, as the collective of these shapes are called, have been found: Archimedean solids (including truncated icosahedron) and Kepler’s solids (including rhombic polyhedra). Nearly 400 years after the last class was described, researchers claim that they may have now invented a new, fourth class, which they call Goldberg polyhedra. Also, they believe that their rules show that an infinite number of such classes could exist.
Equilateral convex polyhedra need to have certain characteristics. First, each of the sides of the polyhedra needs to be of the same length. Second, the shape must be completely solid: that is, it must have a well-defined inside and outside that is separated by the shape itself. Third, any point on a line that connects two points in a shape must never fall outside the shape.
Platonic solids, the first class of such shapes, are well known. They consist of five different shapes: tetrahedron, cube, octahedron, dodecahedron and icosahedron. They have four, six, eight, twelve and twenty faces, respectively.