Slicing Polyhedra

The Flatland Perspective

Most people know that we live in a three-dimensional world. But have you ever wondered what it would be like to live in a world that wasn't three-dimensional? In 1885, Edwin A. Abbott wrote Flatland, a novel describing the life of creatures in a flat space, limited to only two dimensions. At one point in the story, A. Square, the protagonist, encounters a sphere of our three-dimensional world. As it passes through Flatland, A. Square sees the sphere as a single circle, growing gradually until A. Sphere's equator intersects the plane, then diminishing once again to a point.

Through the circular slices, A. Square gains insight into the nature of the higher dimensional object. In some respects, polyhedra are more complicated than spheres, but we too can gain insight into the nature of solid objects, both three dimensional a nd higher dimensional, through examining their slices.

If we start with a cube, a polyhedron that is very familiar to us, we notice that we can look at it from three different perspectives: from a face, an edge, or a vertex. Friedrich Froebel, the inventor of kindergarten, noticed the importance of these different perspectives back in the early 1800's when he was building gifts for his children to play with. So what do we get if we "slice" the cubes from these different perspectives? First, what do we mean by "slicing"? In thre e dimensions slicing is analogous to what happened with A. Sphere in Flatland: the intersection of a two dimensional plane with a three dimensional object. Think again about A. Square now. Did it matter which way the Sphere was facing when it entered Fl atland? No, it didn't. But it does matter which way we slice the cube.

When we slice a cube starting with a face and moving parallel through the solid, we simply get a series of squares. What about when we start parallel to an edge? This is a bit more involved, but we realize that the first slice will be a line (the edg e), and the line will grow into rectangles, and then shrink back to a line (the opposite edge). Now, what about slicing from a vertex? This is more complicated, but also very interesting.

To slice a cube from its vertex, click here.

When we move to four-dimensional shapes --polytopes-- our method of slicing changes a bit. Instead of slicing with a two-dimensional plane, we can gain the same kind of information if we slice with a three-dimensional hyper-plane. Thus our slices will be three-dimensional rather than two-dimensional.

So what happens when we slice the hypercube? For instance, what would a visiting hypercube look like if it were to pass through our three-dimensional space? Does the orientation of our slicing hyper-plane matter? Yes, it does. In fact, the slices o f the hypercube very much resemble those of the cube. For example, what do you think a hypercube looks like when it passes through our space head-on (starting with one of its eight three-dimensional cubic faces)? You guessed it, it looks like a series o f cubes, all equal in size, just like slicing the cube face first gave a series of squares. See the parallels?

Slicing the hypercube from a two-dimensional side first gives analogous results to the cubic slices from an edge. Cubic slices from a vertex, as seen in the interactive model, has its analogy when a hypercube is sliced from a one-dimensional edge. Bu t we still have one more "perspective" from which to slice the hypercube that returns a new phenomenon that we did not experience with the cube. To think about the shape of the hypercube by slicing it from a vertex, click here.