| Sphere | |
|---|---|
 A perspective projection of a sphere | |
| Type | Smooth surface Algebraic surface |
| Euler char. | 2 |
| Symmetry group | O(3) |
| Surface area | 4πr2 |
| Volume | 4/3πr3 |
In geometry, a sphere (from Greek σφαῖρα, sphaîra)[1] is an idealization of a physical ball, a perfectly round surface, the solid analog of a planar circle.
As a two-dimensional surface embedded in three-dimensional space, the sphere consists of all points that are at the same distance (the radius) from a given centre point, a property typically taken as its definition[2] It is the basic setting for spherical geometry, in which straight lines are replaced by great circles, intersections of the sphere with planes passing through its centre.
The sphere's shape does not change under any three-dimensional rotation. It is a surface with uniform curvature.
Spheres and nearly-spherical shapes also appear in nature and industry. The earliest known mentions of spheres appear in the work of ancient Greek mathematicians and astronomers, who used the celestial sphere as a model for explaining the motions of stars and planets and modeled the Earth as a sphere. Bubbles such as soap bubbles take a spherical shape in equilibrium. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings.
A sphere can be formed by revolving a semicircle around its diameter, the definition of sphere used in Euclid's Elements (c. 300 BC). A solid sphere can be made this way by turning a semicircular profile with a lathe, and the lathe metaphor was used in Ancient Greece to describe creation of the heavens and the earth.
- ^ σφαῖρα, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus.
- ^ Albert 2016, p. 54.