This is monograph number 20 in a series of undergraduate monograph. The purpose is to introduce the Euclidean spaces $E^n$, and briefly their infinite dimensional version $l^2$ using topics in physics as motivation and background. The topic of expanding universes (or more precisely the inflating balloon model of an expanding universe) and some 4-dimensional models of space-time, provide motivation for a discussion of several 3-dimensional manifold in $E^4$ including 3-spheres, 3-tori, and 3-hyperboloids. The topic of wormholes, and parallel universes provide background for $l^2$. We include a number of mathematical exercises of varying difficulty which require only analytic geometry and differential calculus (albeit in higher dimension). We include brief discussion of the relevant physics and numerous references for further reading.