Conclusion

So we see that the sphere is a fascinating object. There several general characteristics of the infinite family of $\Bbb{S}^{n}$. Stereographic projection, the two balls method, and the triangulation method all generalize to $\Bbb{S}^{n}$.

There are also some unique structures on $\Bbb{S}^{3}$ which do not occur on other spheres, like the symmetrical division into two tori. And finally, there is this construction, mentioned in section 7, which takes advantage of both unique and general characteristics of $\Bbb{S}^{3}$ and encapsulates them in a single object.

Acknowledgements

I cannot thank Richard Montgomery enough for the time he spent working on this paper with me. Without his encouragement, both in and out of class, I would never have even asked the questions that led me to this paper. And without his dedication and patient help I would never have been able to find the answers. He did far more for me than any student could expect of a teacher.

I'd also like to thank, in order of appearance, my family, for putting up with me and my ravings, Virginia Bale, for making math fun, Paul Lockhart, wherever he may be, for his What is Mathematics class, Jeremy Avnet, without whose friendship I would never have made it this far in school, let alone mathematics, Tom Lehrer, for schooling me all about Infinity ($\infty$), Richard Mitchell, for his superior teaching and honest advice, Marcia Levitsky, for her dedication to the students, Alpha Schram, for the mouse-pad abuse, and Hirotaka Tamanoi for his assistance with the proof in section 3.2.1.

This paper was composed using entirely free software, on a Debian GNU/Linux system (http://www.debian.org). It was typeset using Donald Knuth's L^ATEX typesetting system. The graphics were created using Sketch 0.6.4 (http://sketch.sourceforge.org). This paper, in postscript, will be available at http://www.theory.org/~matt/ or by emailing me.