sphere
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 x^{2} + y^{2} + z^{2} = r^{ 2}. All points outside it have x^{2} + y^{2} + z^{2} greater than r ^{2}; those inside have x^{2} + y^{2} + z^{2} less than r^{ 2}; the equation balances on the boundary. An equation such as x^{2} + y^{2} + z^{2} = 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 ballshaped 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 threedimensional Euclidean space that lie at distance r, the radius, from a fixed point. In coordinate geometry a sphere with center (x_{0}, y_{0}, z_{0}) and radius r is the set of all points (x, y, z) such that
(x  x_{0})^{2} + (y  y_{0})^{2} + (z  z_{0})^{2} = r^{ 2}
A sphere can also be defined as the surface of revolution formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.
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 intersection of a sphere and any plane is circular; should the plane pass through the center, the intersection is a great circle.
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 eggslicer 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 nsphere is the set of points in (n + 1)dimensional Euclidean space that lie at distance r from a fixed point of that space. A 2sphere is therefore an ordinary sphere, while a 1sphere is a circle and a 0sphere is a pair of points. An nsphere for which n = 3 or more is often called a hypersphere.
Dandelin spheres
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.
