the boundary of an ordinary ball (3-ball). a 3-sphere is a 3-dimensional sphere in 4-dimensional Euclidean space. Euclidean coordinates in -space The set of points in -space, , that define an -sphere, , is represented by the equation: where is a center point, and is the radius. The above -sphere exists in -dimensional Euclidean space and is an example of an -manifold. The volume form of an -sphere of radius is given by where is the Hodge star operator; see for a discussion and proof of this formula in the case . As a result, -ball The space enclosed by an -sphere is called an -ball. An -ball is closed if it includes the -sphere, and it is open if it does not include the -sphere. Specifically: A 1-ball, a line segment, is the interior of a 0-sphere. A 2-ball, a disk, is the interior of a circle (1-sphere). A 3-ball, an ordinary ball, is the interior of a sphere (2-sphere). A 4-ball is the interior of a 3-sphere, etc. Topological description Topologically, an -sphere can be constructed as a one-point compactification of -dimensional Euclidean space. Briefly, the -sphere can be described as , which is -dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an -sphere, it becomes homeomorphic to . This forms the basis for stereographic projection. Volume and surface area and are the -dimensional volume of the -ball and the surface area of the -sphere embedded in dimension , respectively, of radius . The constants and (for , the unit ball and sphere) are related by the recurrences: The surfaces and volumes can also be given in closed form: where is the gamma function. Derivations of these equations are given in this section. In general, the volume of the -ball in -dimensional Euclidean space, and the surface area of the -sphere in -dimensional Euclidean space, of radius , are proportional to the th power of the radius, (with different constants of proportionality that vary with ). We write for the volume of the -ball and for the surface area of the -sphere, both of radius , where and are the values for the unit-radius case. The volume of the unit -ball is maximal in dimension five, where it begins to decrease, and tends to zero as tends to infinity. Furthermore, the sum of the volumes