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Many ideas and algorithms come from imagining points on 2d and 3d spaces. Be it in function analysis, machine learning, pattern matching and many more.

How do mathematicians think about higher dimensions? Can intuitions about the meaning of dot-product, angles and lengths transfer from 2d geometry to a 100d?

If so, would it be enough to fully understand the higher dimesions, namely, could the same problem in 100d have properties\behaviours that are not seen in 2d\3d?

Leo
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  • Usually in algebraic geometry there's a "local-global" principle, where we study how a curve behaves locally and try to extend the results globally. – Eugene Jun 13 '12 at 21:41
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    To think about $100$ dimensions, think of $n$-dimensions and set $n=100$. –  Jun 13 '12 at 21:42
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    Related: Intuitive crutches for higher dimensional thinking on MathOverflow. –  Jun 13 '12 at 21:44
  • let's say I set n=100 just to be concrete – Leo Jun 13 '12 at 21:44
  • A joke: we just think about $\mathbb{R}^n$, with $n$ set to $100$, for example. Seriously, though, dot-products, norms, and the like are vastly generalized in linear algebra, even to infinite-dimensional spaces. The geometry of Euclidean 3-space helps visualize some concepts, but they are usually made rigorous in an algebraic way, a bit disconnected from the geometric intuition. This intuition sometimes aids in understanding, and sometimes works against us, because certainly not all low-dimensional phenomena hold for higher-dimensional spaces. – talmid Jun 13 '12 at 21:45
  • Is there really no escape but to disconnect from geometry and move to algrbra when dealing with high dimensions? Can you give an example of the mentioned phenomena? – Leo Jun 13 '12 at 21:52
  • No, a connection still remains. In the link posted by Rahul you'll find how one can try to visualize higher dimensions using geometric intuition from Euclidean 2-space and 3-space. Regarding examples of phenomena that fail in higher dimensions, see: http://math.stackexchange.com/questions/48301/examples-of-results-failing-in-higher-dimensions. The thing is: $\mathbb{R}^3$ and $\mathbb{R}^{100}$ are similar, but not the same. Surely intuition from one object can shed light on how to understand the other, but sometimes intuition can be wrong and ideas have to be verified algebraically. – talmid Jun 13 '12 at 22:01
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    Just to give you a very concrete example: two lines in the plane are either parallel or intersect in exactly one point. However, in three-dimensional space, one can have two lines that aren't parallel and have empty intersection: http://en.wikipedia.org/wiki/Skew_lines. – talmid Jun 13 '12 at 22:08
  • At least for the dot product of two vectors, you can think in normal geometric ways. Given the two vectors, you consider the 2-dimensional subspace generated by them, and after that, you can visualize. But that happens only because this operation is local in the sense that it keeps itself inside the space (it is basically a projection). Clearly, thinking about a vector product would't work as it is not defined as an application on pairs. – guaraqe Jun 13 '12 at 22:12
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    Also, a volume ($n$-dimentional measure) of a $n$-dimentional ball as a function of $n$ is increasing upto 5d and then decrasing! – dtldarek Jun 13 '12 at 22:20

1 Answers1

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It vastly depends on the objects you define. Indeed, when talking about vector spaces, we algebraically think about $\mathbb{R}^n$, build up intuition, and then set $n=100$.

However, when you start adding exotic objects, like knots, it becomes less "easy". For example, some knots in $\mathbb{R}^3$ are trivial loops in $\mathbb{R}^4$ (ie. the trefoil knot falls apart in 4D).

Then again, functions and their orthogonality are computed in $\mathbb{R}^{89270}$ just as they are in $\mathbb{R}$ - nothing strange going on there. It's only when you consider infinite-dimensional spaces that this becomes slightly unintuitive again.

So, in short, it completely depends on the objects you talk about. Most finite-dimensional vector spaces over some field $K$ are equal in almost all aspects. Adding more structure can make it much more difficult, and oftentimes all mathematicians have is algebra.

akkkk
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    Are there even any 3D knots that stay knotted in 4D? – hmakholm left over Monica Jun 13 '12 at 22:40
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    @Henning: no, and this is straightforward to see. In general, knotting is a codimension 2 phenomenon... – Pete L. Clark Jun 13 '12 at 23:25
  • @PeteL.Clark From Wikipedia: "In general, piecewise-linear n-spheres form knots only in (n + 2)-space (Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted (4k − 1)-spheres in 6k-space, e.g. there is a smoothly knotted 3-sphere in the 6-sphere (Haefliger 1962)(Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth k-sphere in an n-sphere with 2n − 3k − 3 > 0 is unknotted." – Akiva Weinberger Nov 22 '18 at 18:15