A mathematical surface can be defined like a curve, using three mutually perpendicular axes in space, x, y, z. The equation of the sphere is x2 + y2 + z2 = r 2. All points outside it have x2 + y2 + z2 greater than r 2; those inside have x2 + y2 + z2 less than r 2; the equation balances on the boundary. An equation such as x2 + y2 + z2 = r 2- 2x - 8z + 17 also defines a sphere but its center is not at the intersection of the three axes.
A sphere, roughly speaking, is a ball-shaped object. In everyday usage a sphere is often considered to be solid; mathematicians call this the interior of the sphere. In mathematics, a sphere is a quadric consisting only of a surface and is therefore hollow. More precisely, a sphere is the set of all points in three-dimensional Euclidean space that lie at distance r, the radius, from a fixed point. In coordinate geometry a sphere with center (x0, y0, z0) and radius r is the set of all points (x, y, z) such that
(x - x0)2 + (y - y0)2 + (z - z0)2 = r 2
The surface area of a sphere is 4πr 2 and its volume is 4πr 3/3. The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. In nature, bubbles and water drops tend to form spheres because surface tension always tries to minimize surface area.
The circumscribed cylinder (see circumscription) for a given sphere has a volume which is 3/2 times the volume of the sphere. If a spherical egg were cut up by an egg-slicer with evenly spaced wires, the bands between the cuts (on the surface of the sphere) would have exactly the same area. Spheres can be generalized to other dimensions. For any natural number n, an n-sphere is the set of points in (n + 1)-dimensional Euclidean space that lie at distance r from a fixed point of that space. A 2-sphere is therefore an ordinary sphere, while a 1-sphere is a circle and a 0-sphere is a pair of points. An n-sphere for which n = 3 or more is often called a hypersphere.
If a cone is sliced through by a plane, the two spheres that just fit inside the cone, one on each side of the plane and both tangent to it and touching the cone, are known as Dandelin spheres. They are named after the Belgian mathematician and military engineer Germinal Pierre Dandelin (1794–1847) who gave an elegant proof that the two spheres touch the conic section at its foci. In 1826, Dandelin showed that the same result applies to the plane sections of a hyperboloid of revolution.
|Dandelin spheres. Image by Hop David, used with permission.