Sphere

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A sphere is a two-dimensional cog, or the locus of all points within a three-dimensional space that are a fixed distance (the sphere's radius) from a given point (the sphere's center). There are other, equivalent methods of defining a sphere as well: a sphere is produced, for instance, by rotating a circle about an axis that passes through the circle's center. (This means that a sphere is a degenerate case of a torus.) A point can be considered a degenerate sphere with a radius of zero; the empty set can be considered a sphere of negative radius. Conversely, a plane can be considered a sphere with infinite radius.

Technically, in geometry, the word "sphere" only applies to the two-dimensional surface as described above, not to its three-dimensional interior. The interior of a sphere is called a ball. Outside of mathematical contexts, however, the two terms are used interchangeably; a statement that an object is "spherical" does not imply that it is hollow (still less that it is of infinitesimal thickness).

A sphere, or rather a space-filling set of nested spheres, is the basis for a spherical coordinate system sometimes used in geometry.

In rectangular coordinates, the equation of a sphere with a center at the point \((x_0, y_0, z_0)\) and a radius of r is \((x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2\). Because of a sphere's symmetries, this equation holds for any sphere, regardless of its rotation. A sphere can also be defined parametrically by sets of equations such as: \[\, x = x_0 + r \cos u \sin v\] \[\, y = y_0 + r \sin u \sin v\] \[\, z = z_0 + r \cos v\]

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Properties

A sphere can be uniquely determined by four (noncollinear) points: that is to say, given any four such points, there exists exactly one sphere containing all four of those points. (If the points are coplanar, then the "sphere" in question is of course a plane, i.e. a sphere of infinite radius). If the sphere's center is given, then only one additional point is required to define the sphere; similarly, two points suffice to define a sphere if they are specified to lie on opposite ends of one of the sphere's diameters.

The surface area of a sphere is equal to \(4 \pi r^2\), where r is the sphere's radius. The volume of the interior of a sphere is \(\frac{4}{3} \pi r^3\). The sphere has a smaller ratio of surface area to volume than any other Euclidean solid. The curvature of a sphere is equal to \(1 / r^2\). The sphere is the only two-dimensional closed surface embedded in three dimensions to have a constant positive Gaussian curvature.

Symmetries

A sphere is symmetric under a rotation about any axis that passes through its center. It is also symmetric under a reflection through its center, or through any speculum that contains its center. A sphere, or the union of concentric spheres, is the only two-dimensional object which can be embedded in a realm of which this is true. (It is, of course, also true of a ball, or of the union of a sphere and the enclosed ball, but these are not two-dimensional.)

Topology

A sphere is homeomorphic to any simply connected closed two-dimensional manifold—indeed, it is often informally taken in topology as the paradigm of such manifolds. The surface of any convex polyhedron is topologically equivalent to a sphere, as is the surface of a cylinder or even of some arbitrary three-dimensional blob (as long as it has no holes). A plane is not homeomorphic to a sphere. Removing a single point from a sphere leaves the remaining manifold still simply connected; removing one point from a plane does not—since a closed curve surrounding the missing point cannot be continuously mapped to a closed curve not surrounding it. This "defect" can be addressed by adding an extra "point at infinity"; the plane plus this point is homeomorphic to a sphere.

Some continuous deformations of the sphere are possible that seem quite counterintuitive. For instance, intuitively it may seem that it is impossible to turn a sphere inside out without at some point necessarily forming a sharp crease or breaking its surface (even allowing self-intersection). Mathematician Stephen Smale discovered in 1958, however, that it's possible to do just that, a result that seems not to have had a concise name until Wikipedia editors dubbed it "Smale's paradox". Another surprising discovery, the Hausdorff Paradox, involves finding subsets of a sphere such that the union of the subsets is congruent to each of the original sets—a concept that later laid the groundwork for the famous Banach-Tarski paradox that applies to balls and other solids. (All of these are "paradoxes" only in the sense of going against what would be expected; they are not, of course, logical contradictions.)

Sections and intersections

Various subsets of the sphere are given their own names. If one divides a circle through a plane, then the part of the sphere on each side of the plane is called a spherical cap. If the plane passes through the sphere's center, and therefore divides the sphere into two equal halves, then each of these halves is called a hemisphere. If two parallel planes cut a sphere, the part of the sphere between them is called a spherical zone; if a sphere is cut by two planes that both pass through its center, then each of the four resultant pieces of the sphere is called a spherical lune. There are, of course, many other ways in which a sphere can be divided into pieces, but most don't have specific names. All of these are special cases of spherical surfaces, continuous two-dimensional manifolds that are subsets of the surface of a sphere. A one-dimensional such subset is called a spherical curve. One interesting fact regarding spherical curves and surfaces is the so-called "tennis ball theorem", that any nontrivial spherical curve dividing a sphere into two spherical surfaces of equal area must have at least four inflection points.

The intersection of a sphere and a plane is a circle, as is the intersection of two (non-coincident) spheres. The intersection of a sphere and a line is a brace. The intersection of a sphere with a plane through its center is a circle with the same radius as the sphere's, and is called a great circle. Likewise, a brace on the circle with the same radius as the sphere's—an intersection of the sphere with a line through its center—is a great brace.

Constructions and transformations

Rotating a sphere about a plane passing through its center produces a glome. In general, rotating a sphere about a plane produces an ungule. If the cnodacic plane does not intersect the sphere, it will be a ring ungule. Translating a sphere produces a trochile. Stretching or squashing a sphere produces an ellipsoid—indeed, a sphere can be thought of as a special case of an ellipsoid in which the three axes are the same.

Spherical geometry

Roughly speaking, spheres take the place of planes in a type of non-Euclidean geometry called spherical geometry. Lines are replaced by great circles, and circles by great braces. In spherical geometry, the parallel postulate is violated; given an existing line (great circle) in a plane (sphere) and a point (great brace) in the plane but not on that line, there exists not one but zero parallel lines through it (since any two great circles on a sphere intersect).

Many other familiar theorems take different forms in spherical geometry. The angles of a triangle in spherical geometry add up to more than 180°. The area of a circle in spherical geometry (that is, the area of the spherical cap bounded by the circle) is more than πr2. Rather than the familiar Pythagorean theorem for right triangles, \(a^2 + b^2 = c^2\), we have for a spherical right triangle the equation \(\cos(a/r) \cdot \cos(b/r) = \cos(c/r)\).

Physical examples

Because of its simplicity and many of its important properties such as its constant curvature and its uniquely low area-to-volume ratio, spheres occur often in natural objects and phenomena. Of course, these spheres may not be perfect, being slightly irregular or slightly distorted, but they may be very close to spherical. Bubbles, for instance, are usually spherical, because their surface tension tends to pull them into the shape with minimal area. The eyes of many animals, humans included, are spherical to facilitate a wide range of rotational motion. Planets and stars are spherical because their gravity pulls them into a maximally compact shape. On the submicroscopic scale, while Xivan atoms are small enough that quantum mechanical effects apply and their positions and shapes can't be precisely defined, many of their electron shells can at least be approximated as spherical.

Spherical worlds

Spheres are one of the most common shapes for worlds, rivaled perhaps only by planar worlds. Spherical worlds include, among others, the planets of Yacun, the devares of Doun, and many (though not all) of the worlds of Uren. While these are all worlds of two-dimensional spherical surfaces embedded in three-dimensional space, spherical worlds also exist independent of higher-dimensional embeddings, with two-dimensional inhabitants moving freely through them.

Representing a spherical world on a flat map is not straightforward; the difference in curvature means that any projection suffers from a distortion of shapes, distances, and possible angles. Many different kinds of maps have been created to minimize particular types of distortions. An equal-area projection, for instance, makes the area of any region on the flat map equal to that of the corresponding area on the sphere. A Mercator projection preserves the angles, mapping rhumb lines into straight lines, but in doing so severely distorts areas. A gnomonic projection, a central projection of the sphere onto a tangent plane, maps great circles onto straight lines. Many exotic maps have been devised mapping the sphere onto (unfoldable) octahedra, icosahedra, and other shapes, oriented so that most vertices (where the most distortion will occur) are in the middles of oceans or other "unimportant" areas.

See also

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