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 and higher dimensional, through examining their slices.

If we start with a cube, a polyhedra 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 three 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 Flatland? 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 edge), 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. ****Check out Applet****

When we move to four dimensional polytopes, our method of slicing changes a bit. Instead of slicing with a two dimensional plane, we can gain much more 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 of 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 of 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. But we still have one more "perspective" to slice the hypercube from that returns a new phenomenon that we did not experience with the cube. Try playing with the picture below and learn about the shape of a hypercube. *****applet*****

To see an interesting discusion (and pictures) on rotating polyhedra that incorporates slicing, click ****Link right here to Dave's rotation page*****