__Beyond__ __the__ __Third__ __Dimension__--Chapter 3

This chapter explores the idea of slicing three dimensional objects by parallel planes to reveal certain characteristics about the object. By showing that an object is made of stacked-up figures of one less dimension, the book attempts to explain a new way of visualizing four-dimensional figures.

One confusing point in this concept is that by orienting the object in a different manner, the slices can look very different, and therefore reveal different aspects of the figure. For instance, a cube sliced with a square first will prove to be a stack of congruent squares. But if it is cut first at a vertex, then it will be seen as a stack of two points, several triangles, and one perfect hexagon. These two views are surprisingly different, and someone viewing each slice individually would find it hard to understand that the same cube is being represented.

In the same way, it is difficult for us (or at least me), existing in three dimensional space, to visualize what a sliced hypercube would look like. While the book explains very well why, theoretically, a series of a line, several triangualr prisms, one hexagonal prism, more triangualr prisms, and finally another line could indeed be a hypercube, it is doubtful that we would recognize this if we were in such a situation as A Square during his encounter with the sphere. We have the ability to rationalize the parts, but putting them together in our mind to create a whole is much more challenging.

But more seriously, let me see if I can create a link. If it works, you'll see a