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

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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).