Tuesday, July 26, 2011

Math: Pyramids, Prisms, and Infinity

   I like math, because you can formulate questions that you know can be answered, even if you haven't always enough knowledge to answer them yourself.
   For example:  Take a tetrahedron, a pyramid of three triangles on a triangular base. If all four triangles are the same, then it's a regular tetrahedron, the smallest of the five regular solids. Like the other solids, you can, within limits, change the proportions of its faces in some way. You can increase the height of the triangles that make up the sides of the pyramid. How much? As much as you like.
   Now here’s the question: What happens when the height is infinite? Well, that depends on how you define “infinite” in this context. If by “infinite height” you mean “without limit”, then the tetrahedron becomes a pyramid of infinite height. As the height of the triangular sides increases, the pyramid becomes more and more like a prism of triangular cross-section. That is, the edges become closer and closer to being parallel. We can say that the difference between three edges that converge on a point (the apex of the pyramid) and three edges that are parallel becomes smaller and smaller. This difference approaches zero. “At the limit” it is zero: then the pyramid has become a triangular prism.
   Does it make sense to talk about a limit here, when we are talking about a pyramid of infinite height? Yes, on the same grounds that the differential calculus uses the concept of a limit. But this question, and its answer, are beyond my ability to explicate or justify. The best I can do is to notice that stretching the pyramid towards an infinite height is the same as rotating each edge about a point (the corner) so that they become parallel.  So the pyramid “eventually” morphs into a prism. That “eventually” is “at the limit”, when the difference between converging and parallel edges has become zero. It corresponds to a pyramid of infinite height. This implies that prisms as we conceive of them (of finite height)  are sections of infinitely high pyramids.
   I don’t know whether the above line of thought is mathematically acceptable. Maybe I’m mixing two branches of math illegitimately. But it feels right. So I’ll state my conclusion as a theorem:
   “A finite prism of N sides is a section of an infinitely high pyramid of N sides.”