Visualising spheres in higher dimensions

Spheres are one of the simplest geometric objects. In simple geometric terms, spheres can be described as the set of points which are a fixed distance away from a specified point, the origin. This definition extends very naturally to higher dimensional spaces, where the usual notion of Euclidean distance has a very simple generalisation. Higher dimensional spheres are very easy to define, but trying to visualise them is, perhaps, a little more difficult. In this talk, I will discuss different ways of thinking about higher dimensional spheres, and how we can construct higher dimensional spheres from lower dimensional spheres, paying special attention to the three-sphere (a sphere in four dimensional space). I will conclude with an explanation of the Hopf fibration - a magnificent tool for visualising the three-sphere. This talk will (hopefully) have something in it for everyone!