Δημοσιεύτηκε στις 27 Ιαν 2014
A
basic introduction to the idea of m-dimensional space, m-dimensional
manifolds, and the strong Whitney embedding theorem. I explain the idea
of high dimensional Euclidean space in a simple way, by describing how
to make an m-dimensional hypercube, starting with one dimensional space.
I also explain the basic idea of an m-dimensional manifold by
illustrating low dimensional cases.
Technically, when I say
'm-dimensional manifold' I mean a smooth real m-dimensional manifold
that is Hausdorff and second-countable. The strong Whitney embedding
theorem states that any such m-dimensional manifold can be smoothly
embedded in real 2m dimensional space. The Klien bottle is a 2
dimensional manifold which cannot be smoothly embedded in 3 dimensional
space (i.e., any 3D model of a Klien bottle involves self-intersection).
The Whitney Embedding Theorem guarantees that the Klien bottle can be
embedded in 4D space without self-intersection.
basic introduction to the idea of m-dimensional space, m-dimensional
manifolds, and the strong Whitney embedding theorem. I explain the idea
of high dimensional Euclidean space in a simple way, by describing how
to make an m-dimensional hypercube, starting with one dimensional space.
I also explain the basic idea of an m-dimensional manifold by
illustrating low dimensional cases.
Technically, when I say
'm-dimensional manifold' I mean a smooth real m-dimensional manifold
that is Hausdorff and second-countable. The strong Whitney embedding
theorem states that any such m-dimensional manifold can be smoothly
embedded in real 2m dimensional space. The Klien bottle is a 2
dimensional manifold which cannot be smoothly embedded in 3 dimensional
space (i.e., any 3D model of a Klien bottle involves self-intersection).
The Whitney Embedding Theorem guarantees that the Klien bottle can be
embedded in 4D space without self-intersection.
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