In this article, mathematicians are creating a “Periodic Table of Shapes” in three, for, and five dimensions.

The shapes are Fano varieties, but I don’t really understand what that means. Apparently in two dimensions there are nine of them, and in four dimensions they estimate 500 million of them. About how many of them are there in three dimensions? What do they look like in two and in three dimensions? If they’re shapes, it ought to be possible to draw them, especially in 2 dimensions. What do these shapes look like? Are those four shapes thee-dimensional ones, or slices of four- or five-dimensional ones? If they’re three-dimensional ones, what makes them elemental shapes?

Since the number is finite in four dimensions, it presumably is also finite in three, so they can’t just mean “sphere, torus, two-holed surface, three-holed surface, etc.”

Googling, there are several other articles on this, but they all seem to have about the same level of information, and don’t answer the questions I have above.

Here’s another article with a couple more images.