Charting the 3-Sphere--An Exposition for Undergraduates

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Most people are familiar with the use of latitude and longitude in locating and naming points on the surface of the Earth, which, mathematically at least, is considered to be a perfectly round two-dimensional sphere. This article is about a similar coordinate system on the 3-sphere 53, the set of points in four-dimensional Euclidean space that lie exactly one unit from the origin. 53 iS a rich and beautiful space, and an exploration of it, even at the level of this introductory article, involves a wealth of interesting mathematics. Indeed, here are just some of the mathematical players who will appear in the brief exposition we're about to present: numbers real, complex and quaternion; matrices real and complex, orthogonal and unitary; linear algebra vector spaces, subspaces, inner products, traces, eigenvalues, eigenvectors, diagonalization and the matrix exponential; group theory-conjugation and conjugacy classes; geometry intrinsic distance, tangent spaces and the exponential map. Quite a lot of mathematics in a short expository paper about a single three-dimensional space-and we could have gone much further. In fact, the main difficulty in writing this paper was knowing when to say 'Enough!'. Which brings me to the point of this paper. I wrote this paper as a paper to be studied by undergraduate Mathematics Majors in a Senior Seminar. Of course, I hope that this article will be read by other people in other contexts as well, but my focus while writing this paper was fixed on the Senior Seminar, and I think that this focus is reflected in the paper's style. In particular, this paper is meant to be read and discussed by a group of students, so that all might benefit from the sharing of knowledge, and so that confusions might be quickly resolved and obstacles overcome. (I also assumed that there would be a faculty member present to help the students navigate through some of the paper's more treacherous passages.) I did not attempt to make this paper either self-contained or linear. This is because few topics in mathematics are truly self-contained, and because mathematics as a whole is decidedly non-linear. Throughout the paper there are terms and facts borrowed from many branches of mathematics, and there is also the occasional tidbit, usually parenthesized, meant to entice the readers to explore new territories in the mathematical kingdom. There are also many verifications left to the readers; most of these verifications are calculations. Having said all this, I feel that I should add that I tried to organize the paper so as to assist the readers in learning the material, and that I attempted to explain, at least informally, most of the technical terms in the article. I also tried to give proofs, or sketches of proofs, of most of the claims made within. At the end of the paper is a short list of references, which I include mainly as an act of basic decency. My hope is that the readers already have their own favorite sources of mathematical knowledge, and that they will consult these favorite sources as needed. I was inspired to write this