An n-dimensional sphere, often denoted as S^n, is a generalization of the familiar concept of a circle and a standard sphere to higher dimensions. For example, a 1-dimensional sphere is a circle, while a 2-dimensional sphere is the surface of a ball in three-dimensional space. The n-dimensional sphere consists of all points that are equidistant from a central point in an (n+1)-dimensional space. The mathematical definition of an n-dimensional sphere can be expressed using coordinates. For instance, the equation for a 2-dimensional sphere in three-dimensional space is x² + y² + z² = r², where r is the radius. This concept is important in various fields, including geometry, topology, and physics, as it helps in understanding complex shapes and spaces.