We only have 3 spatial dimensions in our world, but mathematically one can work with any number. There's no inherent restriction to 3 dimensions, it's just the way our particular universe appears to be.
The term polytope is used to refer to the equivalent of polyhedra in any number of dimensions, eg polygons in 2D, polyhedra in 3D, and what we call polychora in 4D.
In 2D, each side of a polygon is a 1D line. In 3D each side of a polyhedron is a 2D polygon. So in 4D each side of a polychoron is a 3D polyhedron.
4D is very hard to imagine for the brains we have which evolved in this 3D world. And since we live in 3D, you can't view a 4D polytope directly, but Stella4D provides several ways to view them:
- The whole polytope can be projected into 3D. This is much like how you look at a 3D object on a 2D screen.
- A 3D cross-section through the polytope can be taken. This is similar to taking a 2D cross-section through a 3D polyhedron. And by changing the depth of the cross-section, the resulting shape morphs and changes.
- A 4D polytope can be unfolded into a 3D net. This is similar to unfolding a 3D polyhedron into a 2D net.
Projection into 3D.
3D cross-sections. The animation comes from changing the depth of the slicing hyperplane. You can control this depth in real-time in Stella4D.
4D polytopes unfolded into 3D nets.
Any of these can be converted to a 3D model in Stella4D which can then be treated as any 3D polyhedron can (eg create and print nets).
You can see some of the paper models I've made based on 4D polytopes here: