I every so often hear claims like:  M-Theory predicts that there are 10 spatial dimensions!  Now I'm not really sure what these claims mean.  There are three spatial dimensions that I normally observe so 7 of these dimensions must be different in some way.  (Several seconds of researching reveals that the difference is that these dimensions are compact and have very small measure which explains why they are not apparent in every day life) However there are a ton of dimensions that are not quite like our three spatial dimensions.  Time is the one that very clearly springs to mind, but we could also talk about momentum.  In fact dimension is pretty frequently used to mean unit (e.g. dimensionless, dimensional analysis). So there must be some thing relevant about these 7 spatial dimensions that is not relevant with these other dimensions. My question then is: What is the relevant factor that unites the spatial dimensions not present in other dimensions? For example if I discovered (or theorized) a new degree of freedom, how would I determine if it corresponded a spatial dimension?  What properties would it need to count as spatial?  helps explain the difference between spatial and time dimensions with respects to the Minkowski tensor, but I would like to see a more authoritative answer that would allow me to fully categorize a potential new dimension. I am looking for a definition that is rather precise rather than intuitive.  I have plenty of intuition as to what space is since I live in it and all the sources I can find a couple describe the intuition (e.g. the ). I am curious about a more mathematized definition. Some notes in response to comments:  I am already very well aware of what a dimension in general and this question is asking about how we separate dimensions into spatial and non-spatial.  I already understand compactification.  I am not at all confused by the concept and I am not asking for an explanation of compact dimensions.
123143
How do you exactly define what is and isn't a ? I heard somewhere that it is "anything you can move through" but if that is right, why wasn't time and space considered a dimension before Einstein?
118043
Let's say that we want to measure the gravity force in 1D, 2D, 3D  and higher spatial dimensions.   Will we get the same force strength in the first 3 dimensions and then it will go up? How about if we do this with Electromagnetic force?   I've include the Electromagnetic force just to see if I can find an analogy to the gravity force behavior.
68471
I have been reading about multi-dimensional numbers, and found out that it's been proven that the Octonions are the composition algebra of the largest dimension. I was wondering why, despite having infinitely many different dimensions of numbers, the only composition algebras are of 1, 2, 4, and 8 dimensions. What's so special about 8?
28482
What is the chain of reasoning (beginning, of course, from observations about the universe) that leads one to predict that Minkowski space provides an accurate description of space-time in the framework of special relativity?  That is, why should flat space-time have the Minkowski metric  $$\eta = \left(\eta_{\alpha\beta}\right) = \left( \begin{matrix} 1 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; -1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; -1 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; -1 \end{matrix}\right)\,?$$
96667
2 points are from equal distance to each other in dimensions 1,2,3,...  3 points can be equidistant from each other in 2,3,... dimensions   4 points can be equidistant from each other only in dimensions 3,4,...  What is the property of number dimensions that relates the number of points that can be equidistant to all other points?
123146
The last decades there has been a lot of research going on in the the area of three dimensional gravity. The motivation, I understand, is threefold:   Whereas gravity is not perturbatively renormalizable in four spacetime dimensions, in three dimensions it is. To make it even more interesting it has black hole solutions and it is exactly solvable. This opens the way to to study quantum black holes. This make three dimensional gravity a very interesting system on itself. Through the AdS/CFT correspondence there is a connection between conformal field theories (CFT) in two dimensions and gravity in three dimensions. CFT's are important in condensed matter physics and one can use 3D gravity to learn more about them. Gravity in three dimensions is simpler to deal with then gravity in four dimensions. Therefor it can be used as a toy model for gravity in four dimensions.    I am wondering what are the most important insights that 3d gravity brought in these respects? In particular I am interested in point three: did 3d gravity provide any new view on 4d gravity so far?
93884
In dimensional analysis, it does not make sense to, for instance, add together two numbers with different units together. Nor does it make sense to exponentiate two numbers with different units (or for that matter, with units at all) together; these expressions make no sense:  $$(5 \:\mathrm{m})^{7 \:\mathrm{s}}$$  $$(14 \:\mathrm{A})^{3 \:\mathrm{A}}$$  Now my question is plainly this: why do they not make sense? Why does only multiplying together numbers with units make sense, and not, for instance, exponentiating them together? I understand that raising a number with a unit to the power of another number with a unit is quite unintuitive - however, that's not really a good reason, is it?
54183
When I make a claim in a proof that a mathematical entity exists, is this no more than saying that the theory I'm working within is consistent, and that all the steps upto that point in the proof are allowed moves in the theory?   By mathematical entity I mean a number, say 3; or an algebraic system such as the group of rotations of the square or the ring of natural numbers; or the category of varieties (a variety is an algebraic system with a fixed number of operations). It could also be a circle, or the category of all manifolds. It could be set theory axiomatized in ZFC.  Not only do these entities relate to each other (for example, after 2 is 3; the group of rotations of a square is related to that of a cube) but also they have structure (3=2+1 in the ring of natural numbers; or the circle, a 1-dimensional manifold is made up of 0-dimensional points)    Existence is a loaded word, we experience in our everyday lives with the existence of chairs and tables or the colour red. It seems very odd that the same word is used for entities that are unavailable directly to the senses, but at the same time it is 'oddly' appropriate.   Taking the platonic perspective - that these entities exist in their own world - and that we access this world through an effort of will. Then supposing that our own physical universe does have a complete mathematical theory. Then the entities in this theory, as well as the theory itself appear to exist in this other separate world, and our own physical world (with us in it as we are physical beings) appears to be incarnated in someway in this other world. This seems odd. Perhaps this suggests that the universe cannot have an underlying mathematical theory?  On the other hand, taking an entirely formal point of view seems (to me) deeply unsatisying. Its an interesting/entertaining point of view, granted.   For one thing, a formalisation appears to be only a perspective or an approximation. For example I have a circle in mind, I could choose to formalise this as a figure of plane geometry as Euclid originally did or a 1-dimensional smooth manifold. It is approximate, as formalisations have changed over time, and are likely to keep on doing so.
117072
These are two definitions in page 48 of the book an introduction to manifolds by Loring Tu.  Definition 5.1. A topological space $M$ is locally Euclidean of dimension $n$ if every point $p$ in $M$ has a neighborhood $U$ such that there is a homeomorphism $\phi$ from $U$ onto an open subset of $\mathbb R^n$.  Definition 5.2. A topological manifold is a Hausdorff, second countable, locally Euclidean space. It is said to be of dimension $n$ if it is locally Euclidean of dimension $n$.  In the last lines of page 48, we reed,  Of course, if a topological manifold has several connected components, it is possible for each component to have a different dimension.  But this is a bit strange for me. If a topological manifold has several connected components and each component has different dimension, then how this manifold can be locally Euclidean space, say for example of dimension $n$? That is, by the above definition of topological manifolad, can a non-connected toplogical space be a topological manifold?
34309
Given that measurement along the X, Y and Z axes correspond to the terms "width", "height", and "depth", is there an accepted term for spatial measurement along the W axis when dealing in four dimensions?  For that matter, are there such terms for additional dimensional axes?
15314
Let $C_{0}=\{(x_{n}):x_{n}\in\mathbb{R},x_{n}\rightarrow 0\}$ and $M=\{(x_{n})\in C_{0}:x_{1}+x_{2}+\cdot\cdot\cdot+x_{10}=0\}.$ I have to find dimension of the space $\frac{C_{0}}{M}.$ According to me its dimension is infinite. But i am not correct. Please tell me what is the exact dimension of the quotient space and how to find it? Thanks in advance.
46252
As human beings, we observe the world in which we live in three dimensions. However, it is certainly theoretically possible that more dimensions exist.  Is there any direct or indirect evidence supporting a fourth dimension?
26343
It seems to me that although dimensional analysis is useful for demonstrating when a physics equation is wrong (when the dimensions are inconsistent), there is little justification for how it is often used to show that a given equation is correct. I'll motivate this claim with an example I went over in a class I'm TAing. Generally speaking, my question is: How (and under what conditions/assumptions) can we know that a functional relationship predicted by dimensional analysis, specifically by Rayleigh's method (), is the unique functional relationship that can hold between a given set of quantities?  Consider a column of stationary fluid in Earth's gravitational field. We'd like to find a formula for $\Delta P$, the increase in pressure that is measured when moving from the surface of the fluid to a point at depth $h$. Assume that we know that $\Delta P$ depends on only three variables: The depth $h$, the mass density $\rho$ of the fluid, and the acceleration of gravity $g$. These quantities have dimensions of:  $$\Delta P : [M][L]^{-1}[T]^{-2}$$ $$\rho : [M] [L]^{-3}$$ $$g : [L][T]^{-2}$$ $$h : [L]$$  Here, $[M]$ is mass, $[L]$ is length, and $[T]$ is time. Now, the standard argument in dimensional analysis goes as follows (this is basically Rayleigh's method of dimensional analysis). We suppose the equation for $\Delta P$ has the form  $$\Delta P = k \rho^a g^b h^c,$$  where $k$, $a$, $b$, and $c$ are all dimensionless constants. Now, for this equation to be meaningful, the dimensions on the left and right sides must be the same. To ensure this, we can replace each quantity in $\Delta P = k \rho^a g^b h^c$ with its corresponding dimensions, yielding:  $$[M][L]^{-1}[T]^{-2} = ([M] [L]^{-3})^a ([L][T]^{-2})^b ([L])^c = [M]^a [L]^{-3a+1+c} [T]^{-2b}$$  For the dimensions to be the same on both sides, we have to have $a=b=c=1$, and thus $\Delta P = k \rho g h$ for some dimensionless $k$.  At this point, we've certainly shown that $\Delta P = k \rho g h$ is a possible, dimensionally consistent formula for the pressure change. But I don't see how this formula is at all unique. For me, the biggest unjustified assumption enters when we assume that $k$ must be dimensionless. Even if we know that the only variable dimensionful quantities that $\Delta P$ depends on are $\rho$, $g$, and $h$, I don't understand how it follows that $\Delta P$ can't depend on one or more dimensionful constant. But the values we get for $a$, $b$, and $c$ will depend on what dimension we suppose that $k$ has. So if we can't pin down the dimension (or lack thereof) of $k$, then we can't fix $a$, $b$, and $c$, and so we've made no progress beyond the initial assumption $\Delta P = k \rho^a g^b h^c$.  When considering Newton's law of gravitation from the perspective of dimensional analysis, this ambiguity manifests itself in the opposite way. In order to derive the proportionality $F = G m_1 m_2 / r^2$ in the same way as we did $\Delta P = k \rho g h$ (by assuming that $F = G m_1^a m_2^b r^c$), we must assume that our constant $G$ has dimensions $[L]^3 [M]^{-1} [T]^{-2}$. Why is a dimensionful constant reasonable in this case, but not when deriving $\Delta P = k \rho g h$? I think this is the sticky point that this one asker was getting at in their question:   In that question, the answerer provides a method to determine the dimensions of $G$ to make Newton's law dimensionally consistent, but this is only after having shown that $F \propto m_1 m_2 / r^2$, without dimensional arguments. But this is a far more modest result than what is often claimed to be a consequence of dimensional considerations - namely results like $\Delta P = k \rho g h$, or $T \propto \sqrt{l/g}$ for the period of a pendulum as a function of $g$ and its length $l$.  So is there an insight missing in my analysis, which would clear up these ambiguities and ensure that dimensional analysis produces a unique functional form for the relationship between a set of quantities? Or is dimensional analysis just often used in ways that aren't always justified?
87501
In the paper "" we read that,      We show that under certain broad conditions (in terms of data and query distributions, or workload), as dimensionality increases, the distance to the nearest neighbor approaches the distance to the farthest neighbor. In other words, the contrast in distances to different data points becomes nonexistent. The conditions we have identied in which this happens are much broader than the independent and identically distributed (IID) dimensions assumption that other work assumes.   My question is, how I should generate a dataset that produces this effect?   I have created three points each with 1000 dimensions with random numbers ranging from 0-255 for each dimension but points create different distances and do not reproduce what is mentioned above. It seems changing dimensions (e.g. 10 or 100 or 1000 dimensions) and ranges (e.g. [0,1]) do not change anything. I still get different distances which should not be any problem for e.g. clustering algorithms!   Edit: I have tried more samples, based on my experiments distances between points do not converge to any number, on the contrary the max and min distances between points get more apparent. This is also in contrary to what is written in the first post of  and also many other places which claim the same thing like . I would still appreciate it if someone can show me with a piece of code or real dataset that such effect exist in practical scenarios.
121416
I have read and heard in a number of places that extra dimension might be as big as $x$ mm. What I'm wondering is the following: How is length assigned to these extra dimensions?   I mean you can probably not get your ruler out and compare with the extent of an extra-dimension directly, can you? So if not how can you compare one dimension with the other? Does one have some sort of canonical metric? Could one also assign a length (in meters) to time in this way?
45471
Is there any scientific experiment that can lead us to conclude we live in 3 spatial dimensions without the premise of the conception of limited dimensions?  Thank you all who helped in the improvement of this question (which was not clear at first).   EDIT:  I know that this can be a little philosophical, but it is also a scientific question.  Let's consider the scenario where the mankind was not ever able to see.  Let's also consider that this limitation could be surpassed thus not limiting us to reach a scientific and technological knowledge "similar" to what we have today.  Would this civilization of blind people reach the conclusion that they are living in a 3D spatial world?  Is the sense of touch enough to reach that conclusion? Is there any scientific experiment that can lead us to that conclusion without the premise of the conception of limited dimensions?  Would it be easier, harder, or just different to reach a conclusion predicted by the M-Theory? (please do not focus only on this last question)
46362
The concept of dimension is used in many different contexts.   Generally a dimension is a function that has as domain some family of sets ad has value on a set that, in the most common situations, is $\mathbb{N}$ or $\mathbb{R}$. As an example of the first case we can think to the dimension of  (but this can also be infinite) or to the  of commutative rings. As an example of the second case we can think to the  for metric spaces ( and its variants). But we can also define dimensions that have as range a family of ordered sets and as range an interval of $\mathbb{R}$, as in the case of , and it seems that we can also define a dimension functions on super vector space that can have negative values, and a dimension with complex values for self-similar sets ().  All these dimensions are different in their definitions and properties and, if I well understand there is not an axiomatic definition of dimension that can be used to identify  a function as a dimension function (see:).  So my question is why mathematicians call all those different function with the same name? I understand that the name come from our common intuition of dimension but I don't understand how such intuition apply to such sophisticated notions called dimension.  More precisely:     I'm curious to know what is the inspiration that guide a mathematician to recognize that a particular function can be called a dimension.    I see that this is not a question that can have a unique and well defined answer, but I suppose (or I hope) that there is some common mathematical meaning about this so used word.