Slicing is a good way to understand shapes better because it breaks them down into a series of lower-dimensional objects. Another good way to understand a shape is to try to build it from a lower-dimensional "fold-out," like a cardboard box unfolded and placed flat on the floor. To define the original shape, all one needs is the fold-out and a set of instructions for putting it together again. This is how Euclid, ca. 300 B.C., proved the existence of exactly five regular polyhedra. To see a discussion of the five Platonic solids that uses the same proof method that Euclid did, click here.[Getty's platonic solids page]
Consider this picture of an unfolded cube:
Does it look like a cube to you? What if we were to give you instructions that said "connect the blue edges" to build a cube? The best Flatlanders could do would be if they stretched the squares and aligned the blue edges together in the plane.
But this was not what we were looking for. We wanted them to fold up the fold-out without stretching the squares, something that requires three dimensions, something they would not be able to do. The fold-out and instructions for putting it together could nonetheless help the Flatlanders gain a better appreciation for what a real cube is.
What about other three dimensional polyhedra? What are the shapes of these fold-outs? What are the instructions for putting them together? What could a Flatlander learn from looking at them? For answers and images to some of these questions click here.
For us, who live in Spaceland, we can gain a better understanding of four-dimensional polytopes by examining their three-dimensional fold-outs and the instructions that go along with them.
The minimum material required to build a cube, as Euclid outlined, is three squares around a point (the red-and-blue squares above). When we talk about polytopes, the situation is analogous: the minimum material required to build a hypercube is three cubes arranged around an edge with instructions to connect the blue faces without stretching the cubes.
As spacelanders we cannot complete this task, but a four-dimensional creature could fold the fold-out into a hypercube with no problem if they filled in the spaces. Here is the complete hypercube fold-out:
Notice the similarities of this fold-out to the cube fold-out: we have a central object (square or cube) surrounded on each face with other identical objects, and then a final one stuck on the bottom. It is easy to think of the center object as being the "base" of the folded-up polytope, and the extra object on the end of the fold-out as being the "top" or "cap" of the polytope.