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* For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere. All geodesics of the sphere are closed curves. Geodesics are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of ...
It is called the 2-sphere, S 2, for reasons given below. The same idea applies for any dimension n; the equation x 2 0 + x 2 1 + ⋯ + x 2 n = 1 produces the n-sphere as a geometric object in (n + 1)-dimensional space. For example, the 1-sphere S 1 is a circle. [2] Disk with collapsed rim: written in topology as D 2 /S 1
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
The key observation that leads to Lie sphere geometry is that theorems of Euclidean geometry in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangential contact have a more natural formulation in a more general context in which circles, lines and points (resp. spheres, planes and points) are treated on an equal footing.
The only shape that casts a round shadow no matter which direction it is pointed is a sphere, and the ancient Greeks deduced that this must mean Earth is spherical. [ 8 ] The effect could be produced by a disk that always faces the Moon head-on during the eclipse, but this is inconsistent with the fact that the Moon is only rarely directly ...
For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another.
In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed n {\displaystyle n} -dimensional ball is often denoted as B n {\displaystyle B^{n}} or D n {\displaystyle D^{n}} while the open n {\displaystyle n} -dimensional ball is int B n {\displaystyle ...
The n-sphere S n is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n -sphere S n can be constructed by gluing together two copies of R n {\displaystyle \mathbb {R} ^{n}} .