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The above -sphere exists in (+) -dimensional Euclidean space and is an example of an -manifold. The volume form ω {\displaystyle \omega } of an n {\displaystyle n} -sphere of radius r {\displaystyle r} is given by
The sphere is an example of a surface. The unit sphere of implicit equation. x 2 + y 2 + z 2 – 1 = 0. may be covered by an atlas of six charts: the plane z = 0 divides the sphere into two half spheres (z > 0 and z < 0), which may both be mapped on the disc x 2 + y 2 < 1 by the projection on the xy plane of coordinates. This provides two ...
This is an index to notable programming languages, in current or historical use. Dialects of BASIC, esoteric programming languages, and markup languages are not included. A programming language does not need to be imperative or Turing-complete, but must be executable and so does not include markup languages such as HTML or XML, but does include domain-specific languages such as SQL and its ...
There are 4 symmetry classes of reflection on the sphere, and three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.) Point groups:
The sphere theorem of Papakyriakopoulos gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres. One example is the following: Let M {\displaystyle M} be an orientable 3-manifold such that π 2 ( M ) {\displaystyle \pi _{2}(M)} is not the trivial group.
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their ...
For example, the isometry group of the n-sphere is the orthogonal group O(n + 1). Given any finite subgroup G thereof in which only the identity matrix possesses 1 as an eigenvalue , the natural group action of the orthogonal group on the n -sphere restricts to a group action of G , with the quotient manifold S n / G inheriting a geodesically ...
It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a 3-manifold.