After 400 years, mathematicians find a new class of solid shapes

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 regular convex polyhedra, as the collective of these shapes are called, have been found: Archimedian solids and Kepler solids. 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.

more…

I am very familiar with Platonic, Archimedean and Kepler-Poinsot polyhedra. At one stage in my life I attempted to list all four-dimensional Archimedean hypersolids – if you think that’s easy then try it.

> a fourth class of convex polyhedra

There are huge numbers of convex polyhedra that don’t fit in the three classes mentioned in the article – the most familiar are the prisms, others are called “antiprisms”, another big class is the “zonohedra”, and that’s not even scratching the surface of what is already known.

I’m looking at this article and trying to figure out if these solids are just ones that have been known for hundreds of years (like soccer ball), are extensions of one I stumbled across in my youth and couldn’t place, or are completely new.

Looking further, they’re not extensions of ones that I found in my youth. That had a curved faces, in particular it had six places where four irregular pentagons joined at a point.

The “new” ones appear to be just glorified dodecaheda, all of them have 12 pentagons in the same positions as in a dodecahedron. Around the edges of the pentagons extra hexagons are added. In order to get the “bottom right” to be a polyhedron, with flat faces, all the hexagons have to be irregular, so not like a Platonic, Archimedean or Kepler-Poinsot solid where all the faces are regular polygons.

The “new” ones are also well known from geodesic domes, they are simply the duals of the triangular-faced structures used in geodesic domes.

mollwollfumble said:

The “new” ones are also well known from geodesic domes, they are simply the duals of the triangular-faced structures used in geodesic domes.

For example, it you connects the points of all the triangles in this picture then you get something very similar