# How Many Movable Ways(Direction) In Pure Nth-Dimensional Space? [closed]

In my opinion, In pure 2th-dimensional space, There is 2 movable ways.

And in pure 3th-dimensional space, There is 3 movable ways.

Am I think in right way?

Any answers will be appreciated, thank you.

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What exactly do you mean by "movable ways"? And what does it mean for a space to be pure? – David Zaslavsky Jul 18 '11 at 5:42
Suppose there is one particle in 2-dimensional space, it can move only by 2ways. – 4545454545SI Jul 18 '11 at 5:42
But what exactly do you mean by a "way"? Are you talking about independent directions? – David Zaslavsky Jul 18 '11 at 5:43
The particle's movable directions are only $\pm x,\pm y$. – 4545454545SI Jul 18 '11 at 5:46
– David Zaslavsky Jul 18 '11 at 7:29

## closed as not a real question by Robert Cartaino♦Jul 18 '11 at 16:55

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.

I guess that by pure $N$-dimensional space you just mean $N$-dimensional space. Indeed, in an $N$-dimensional space, there are exactly $N$ linearly-independent, i.e. mutually perpendicular, directions in which one can move.

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If so, how many directions in real space? – 4545454545SI Jul 18 '11 at 6:07
If 3, it will be a contradiction. – 4545454545SI Jul 18 '11 at 6:10
If by real you mean 3-dimensional, that is, the space around us, then 3. I seriously doubt that that can be a contradiction. – AndyS Jul 18 '11 at 6:10
We can move by Infinity ways in real space. not 3. – 4545454545SI Jul 18 '11 at 6:11
You have to think a little harder before you assign meaningless terminology to already well-defined concepts. The word "pure" has absolutely no content here. – AndyS Jul 18 '11 at 6:19

You seem to be highly uncertain about what an $n$-dimensional space is supposed to be. When we talk about such a thing, we usually mean an $n$-dimensional vector space*. A vector space is a precisely defined mathmatical structure. What you call "ways" are vectors in this space. The special thing about vectors is that you can

• Add two of them together and get again a vector – this means that you can go one way and then another way, and then describe the total way you've walked again as one single way.
• "Scale" them. If you can go, say, $6$ feet in some direction, you can as well go $2\cdot 6\:\mathrm{ft}$ or any other multiple in the same direction.

Now, this means that any vector space contains infinitely many "ways": you can take some vector $\mathbf{v}$ and some vector $\mathbf{w}$, and some number $a$. Then $\mathbf{v}+a\cdot\mathbf{w}$ will always be a vector, and for different numbers $a$ (of which there are of course infinitely many) the result will always be different.

So what does "$3$-dimensional" mean now? It obviously does not mean that there are only 3 ways to go. But, and that's it: it is sufficient to "know" only 3 vectors to describe any way at all. That is, any vector $\mathbf{v}\in\mathbb{R}^3$ can be written as $x\cdot \mathbf{e}_x+y\cdot\mathbf{e}_y+z\cdot\mathbf{e}_z$, where $x,y,z$ are simply numbers and $\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z$ are your known unit-directions.

* it could also be a manifold, which is essentially a (more complicated) generalization of a vector space.
† in fact, you can mathematically consider degenerate spaces with finite elements, but these are of no physical interest.
‡ provided that $\mathbf{w}$ is not the zero-vector, that is, the way where you just stay at the same place.

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If by movable ways one is to understand degrees of freedom then there are Permutations(1,n) + Permutations(2,n) + ...+Permutations(n-1,n) degrees of freedom in n dimensional space, but the question will be more intresting in the case various fractal or topological dimension. For example how many degrees of freedom are there in a space of complex or matrix valued dimensions? but that would be more for the math forum.

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You are considering only the Taxicab geometry
Move the original question to the one: How distances are calculated?
The surface of a sphere (think of the Earth, use latitude and longitude and disregard altitude) is a 2 dimensional one and you can move in any direction between 0º and 360º.

A nice document about the Notion of Distance that I will leave here just for reference:
a few topics: Metric Distance ,Binary Vector Distances,Tangent Distance, Distance Measures, Definition of a Metric, Weighted Euclidean Distance, nearest-neighbor classification, cluster analysis, multi-dimensional scaling, Proximity ,similarity, dissimilarity, Euclidean Distance, Similarities between cups, Sample Covariance between variables X and Y, Correlation Coefficient, Correlation Matrix, Generalizing Euclidean Distance, Minkowski Metric, Vector Space Representation of Document, Cosine Distance between Document Vectors, Distance Measures for Binary Data, Hamming Distance,Tanimoto Metric, k-nn digit recognition accuracy, Tri-Edge Inequality (TEI), Tangent Distance, Learning from a limited set of samples, Euclidean Distance versus Tangent Distance versus Manifold Distance in a picture, (Levenshtein distance is not there).

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