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Orientation is a function of the mind involving awareness of three dimensions: time, place and person. [1] Problems with orientation lead to disorientation, and can be due to various conditions.
The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of R 4. The Euclidean metric on R 4 induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3-sphere has constant positive sectional curvature equal to 1 / r 2 where r is the radius.
A torus is an orientable surface The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. The Roman surface is non-orientable.. In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". [1]
Three spheres, triple spheres, and related terms may refer to any of the following: Architecture ... a 3-sphere, an n-sphere whose surface is three-dimensional;
A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called unoriented. In mathematics , orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left ...
Another method for solving such sphere problems was to write down the coordinates together with the sphere's radius. [19] This was employed by Lie (1871) in the context of Lie sphere geometry which represents a general framework of sphere-transformations (being a special case of contact transformations) conserving lines of curvature and transforming spheres into spheres.
The connected sum of two oriented homology 3-spheres is a homology 3-sphere. A homology 3-sphere that cannot be written as a connected sum of two homology 3-spheres is called irreducible or prime , and every homology 3-sphere can be written as a connected sum of prime homology 3-spheres in an essentially unique way.
That is, the 3-torus is R 3 modulo the action of the integer lattice Z 3 (with the action being taken as vector addition). Equivalently, the 3-torus is obtained from the 3-dimensional cube by gluing the opposite faces together. A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is also an example of a compact abelian ...