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Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point.
Consider the linear subspace of the n-dimensional Euclidean space R n that is spanned by a collection of linearly independent vectors , …,. To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : (), = ….
S 3: a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centered at the origin is denoted S n and is often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.
This is an accepted version of this page This is the latest accepted revision, reviewed on 27 February 2025. American multinational home improvement supplies retailing company The Home Depot, Inc. A Home Depot in Onalaska, Wisconsin Company type Public Traded as NYSE: HD DJIA component S&P 100 component S&P 500 component Industry Retail (home improvement) Founded February 6, 1978 ; 47 years ...
We assume that is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension , we can apply the divergence theorem, i.e. =, and substitute for the volume integral of the divergence with the values of () evaluated at the cell surface (edges / and + /) of the finite volume as follows:
Let be any 3-manifold and a ()-invariant subgroup of (). If f : S 2 → M {\displaystyle f\colon S^{2}\to M} is a general position map such that [ f ] ∉ N {\displaystyle [f]\notin N} and U {\displaystyle U} is any neighborhood of the singular set Σ ( f ) {\displaystyle \Sigma (f)} , then there is a map g : S 2 → M {\displaystyle g\colon S ...
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An illustration of the iterative construction of a Menger sponge up to M 3, the third iteration. The construction of a Menger sponge can be described as follows: Begin with a cube. Divide every face of the cube into nine squares in a similar manner to a Rubik's Cube. This sub-divides the cube into 27 smaller cubes.