How To Make Sculptures of 4D Objects

Humans evolved in a three-dimensional environment.  As a result we are very good at visualising three-dimensional objects.  But what about four-dimensional objects?  The best we can do is look at their three-dimensional "shadows".

Just as the shadow of a three-dimensional object squishes it into a two-dimensional plane, we can squish a four-dimensional shape into three-dimensional space. If the four-dimensional object is not too complicated, and if we choose a good way to squish it, then we can get a very good sense of what it is like.  We will explore the sphere in four-dimensional space, the four-dimensional polytopes (which are the four-dimensional versions of the three-dimensional polyhedra - the shapes of dice), and various sculptures and puzzles that have come from thinking about these things.