What is Dimension: Definition and 897 Discussions

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.
In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions (4D) of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. 10 dimensions are used to describe superstring theory (6D hyperspace + 4D), 11 dimensions can describe supergravity and M-theory (7D hyperspace + 4D), and the state-space of quantum mechanics is an infinite-dimensional function space.
The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.

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  1. arivero

    A M theory dimension change and large N_c, are they related?

    Reading old text about the limit of large N_c, there are some remarks telling that for finite N we have a colour tube and that it becomes a string in the limit. It sounds as the extra dimension of M-Theory, and I wonder if the math of both calculations are related somehow. Is the transition to...
  2. C

    B How can a dimension be "curled up" and have a finite extent?

    As I understand it, dimension is a way of describing direction, with the first three spatial dimensions being straight lines which extend infinitely in one direction, perpendicular to each other. In string theories, several additional dimensions are required, sometimes up to nine or 10, I...
  3. H

    B Skew symmetric 1 dimension

    Hi, if I have a equation like (just a random eq.) p_dot = S(omega)*p. where p = [x, y, z] is the original states, omega = [p, q, r] and S - skew symmetric. How does the equation appear if i only want a system to have the state z? do I get z_dot = -q*x + p*y. Or is the symmetric not valid so I...
  4. C

    I Two Dimensional Coordinate Plane with Distance as Third Dimension

    Imagine we draw a two dimensional finite plane with coordinate axes; for simplicity, let's make it a square. Now, suppose we add a third dimension that represents the possible distances between any two points on the square. Now we have a three dimensional space. What shape will that space have...
  5. Monoxdifly

    MHB [ASK] Minimum Dimension of a Map

    The farthest distance of two places in an area is 200 km. If someone wants to make a map of that area on a 1 m × 1 m paper, the possible scale to make it is ... a. 1 : 210 b. 1 : 2.100 c. 1 : 21.000 d. 1 : 210.000 Can you help? The 200 and 210 makes me think that the distance on map won't be an...
  6. R

    Exploring Trigonometric Functions & Physics: Velocity, Distance, & Dimension

    Hello, It has been a long time since I first looked at this, so thought I might ask for some help in clarifying this problem: Is an equation of the form --> Velocity = (Distance) * (Trigonometric function) a valid one in physics? If so, what is the relationship of trigonometric functions...
  7. S

    I Finding the dimension of the span ##u_1-u_2, ...., u_n-u_1##?

    If ##\textbf{u}_1,...,\textbf{u}_n## form a basis in a linear space, how does one determine the dimension of the span ##\textbf{u}_1-\textbf{u}_2, \textbf{u}_2-\textbf{u}_3,...,\textbf{u}_n-\textbf{u}_1##? Since ##\textbf{u}_1,...,\textbf{u}_n## form a basis, they're linearly independent. If one...
  8. J

    Average kinetic energy in 1 dimension according to the M-B Distribution

    Summary: Integrating the 1 dimensional MB Distribution in terms of translational kinetic energy up to infinity, does not yield ##\frac{1}{2}k_BT## as it should be. The format for the 3 dimensional Maxwell-Boltzmann Distribution is ##A\cdot e^{-\frac{E}{k_BT}} \cdot g(E)## in which ##A## can be...
  9. T

    B Why is E=MC2 Squared and not Cubed?

    Problem Statement: Trying to understand the principles for the equation e=mc2. Relevant Equations: E=mc2 So just started at a physics A level and so far loving it. I understand this question has indeed been asked before, however for different reasons. I understand the purpose of e=mc2 and I...
  10. Kairos

    I Adding Extra Dimension for Calculations in Curved Spaces

    as calculations are technically difficult in curved spaces, I wonder if we would obtain the same results by adding one additional (virtual) dimension in order to embed the space in a higher order Euclidean volume, just to facilitate the treatments? (for example embed a 3D hypersphere in a 4D...
  11. P

    B Are we past due to declare EM/Light/Waves its own dimension?

    If we set a dimension for the unobservable, we may stumble on a unifying theory for the large and small. 3D + Time + Waves When I say Waves, I'm talking about the waves a particle becomes when it is unobserved and going through the double slit. If waves only exist as math, observation pulls...
  12. T

    I Exploring the Dimension of Space: The Beauty of Cosmology and Unsolved Mysteries

    Motivated by some apparently intractible unsolved problems, I see cosmology as a beautiful mathematical description of a strongly flawed paradigm. This forum wisely does not allow laying out alternate paradigms, so I try to ask questions to guide my immature understanding , in the spirit of...
  13. jk22

    A Fractional spacetime, dimension equation

    Suppose we use fractional derivatives (https://en.m.wikipedia.org/wiki/Fractional_calculus) in GR, hence we have a local group symmetry ##SO(3-\epsilon,1+\epsilon)## does any reference exist about an equation for ##\epsilon## ?, since it could depend on coordinates too.
  14. U

    Vector space has dimension less than d

    Homework Statement Problem given to me for an assignment in a math course. Haven't learned about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below. [/B] Homework Equations Unsure of relevant equations The Attempt at...
  15. Mutlu CELIKKOL

    Calculating the dimension of intersection of two matrices

    <Moderator's note: Moved from a technical forum and thus no template.> I am at the beginners level of linear algebra and having problem of the intersection of matrices. Your kind help is much appreciated for the following question Let\quad M1=\begin{Bmatrix} x & -x \\ y & z \end{Bmatrix},\quad...
  16. nomadreid

    I Hausdorff dimension of Riemann zeta function assuming RH

    In several places, for example https://xxx.lanl.gov/pdf/chao-dyn/9406003v1, it is claimed that the Riemann zeta function is a fractal under the assumption of a positive result for the Riemann Hypothesis, because (1) the Voronin Universality Theorem, and (2) if the RH is true, then the zeta...
  17. karush

    MHB Null Space of A: Find Rank & Dim.

    Let $$\left[\begin{array}{rrrrrrr} 1 & 0 & -1 & 0 & 1 & 0 & 3\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 4 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \end{array}\right]$$ Find a basis for the null space of A, the dimension of the null space of A, and...
  18. jk22

    B Dimension of the metric of a projection of a sphere

    Let ##(x_1,x_2,x_3)=\vec{r}(\theta,\phi)## the parametrization of a usual sphere. If we consider a projection in two dimension ##(a,b)=\vec{f}(x_1,x_2,x_3)## Then I don't understand how to use the metric, since it is ##g_{ij}=\langle \frac{\partial\vec{f}}{\partial...
  19. nomadreid

    Is fractal nature dependent on the number of steps?

    In looking at the definition of a Hausdorff dimension of a space S = inf{d>0: inf{Σirid: there is a cover of S by balls with non-zero radii} =0} where i ranges over a countable set, it would appear that it would be acceptable to take the index set to be finite, but I am not sure how you would...
  20. Spacepantz86

    B Understanding the First Dimension: Exploring the Unknown

    I do not understand the principle, to me it would make more sense if there was no line, there is nothing. The best way I can think of it would be, the first dimension is the observer, or nothing. I really don't know, its just this way makes more sense in my head, though I am lacking any...
  21. T

    B Time as a Dimension: Exploring Questions

    Is time its own dimension or is it a constant that remains through all dimensions? Also, are there multiple dimensions of time, and how do we know the answers to these questions?
  22. F

    I Dimension of a set with vector function

    I have a vector ##\textbf{v} \in \mathbb{R}^{3N}## and a function ##\textbf{Ψ} : \mathbb{R}^{3N} \longrightarrow \mathbb{R}^p## such that ##\textbf{Ψ}(\textbf{v})=0##. Why the set ##T=\{ \textbf{x} \in \mathbb{R}^{3N} \ | \ \textbf{Ψ}(\textbf{x})=0 \}## has dimension ##n=3N-p##?
  23. S

    I Fractal Dimension & Degree of Freedom

    I have been learning about the idea of "dimension" of a shape in terms of how its "mass" scales down when we cut it into self-similar parts. For example, However, the term "dimension" is closely linked with the idea of degree of freedom. So my question is, is there any sense in which the...
  24. R

    I Spectrum classification and dimension reduction

    I am trying to distinguish two different plant species based on their chemical signature using statistical signal classification. This is what the average curves of two species look like with their standard deviation bounds: These spectra are made up of ~1400 points. So it was suggested to...
  25. nomadreid

    I General definition of spatial dimension?

    The question is not to distinguish space from time, but in general, what distinguishes a spatial dimension from other types of dimensions? For example, Hilbert space has an infinite number of dimensions, but they are not spatial; string theories add extra spatial dimensions. Is there a...
  26. DracoMalfoy

    Motion in One Dimension: The Tortoise and The Hare Race

    Homework Statement A turtle and a rabbit engage in a footrace over a distance of 4.00km. The rabbit runs 0.500km and then stops for a 90min nap. Upon awakening, he remembers the race and runs twice as fast. Finishing the course in a total time of 1.75h, the rabbit wins the race. A) Calculate...
  27. T

    MHB What is the minimal dimension of a complex realising a group representation?

    This question is inspired by one question, which was about representations that can be realized homologically by an action on a graph (i.e., a 1-dimensional complex). Many interesting integral representations of groups arise via homology from a group acting on a simplicial complex that is...
  28. Y

    Fortran How to use Real Array Index in Matrix dimension?

    I am developing a FORTRAN code (.f90) which "ll calculate some matrix in some time interval (dt1=0.001) and these matrices have to be integrated in some time steps (dt=0.1). Though I am experience in FORTRAN 77, new to FORTRAN 90. I am unable to make dimension of matrix real (I think that is the...
  29. L

    Solve Q14: Linear Dimension Question

    Hello everyone .I cannot understand what Q14 is asking about. May I know how to solve Q14 shown in the pic？
  30. bornofflame

    [LinAlg] Find the dimension of the subspace

    Homework Statement Find the dimension of the subspace of all vectors in ##\mathbb{R}^3## whose first and third entries are equal. Homework EquationsThe Attempt at a Solution So I arrived at two solutions and I'm not entirely sure which is the valid one. #1 Let ##H \text{ be a subspace of }...
  31. thermia

    B Is the time dimension more fundamental?

    If we do the thougt experiment to delete one of the three spacedimentions we end up with a flat-land universe, which is fully possible to imagine (while probably not existing) But if we delete the time dimesion it becomes inpossible to imagin. Whithout time there is no existence. Could that mean...
  32. S

    A Can a 3d holographic universe exist in 3d?

    In a holographic universe model, could our 3D universe be encoded in 3D and still be a holographic universe, instead of 3D information encoded in 2D space? Or is the standard model (non-holographic) of the universe already 3d information encoded into 3d space...
  33. Specter

    Impulse and momentum in one dimension - Two carts are moving

    Homework Statement [/B] In an elastic head-on collision, a 0.60 kg cart moving at 5.0 m/s [W] collides with a 0.80 kg cart moving at 2.0 m/s [E]. The collision is cushioned by a spring (k=1200 N/m). a) Find the velocity of each cart after the collision b) Find the maximum compression of the...
  34. Arman777

    B Spherical Geometry (Two dimension ) Defining a metric

    I am trying to understand how to define a metric for a positively curved two-dimensional space.I am reading a book and in there it says, On the surface of a sphere, we can set up a polar coordinate system by picking a pair of antipodal points to be the “north pole” and “south pole” and by...
  35. Pushoam

    Dimension of length using h,G,c

    Homework Statement Homework EquationsThe Attempt at a Solution Dimension of length using h,G,c [h] = [F r] ##[G] =[ \frac { Fr^2}{m^2} ] \\ [\frac { hG}c] = [L] ## So, the answer is option (b). Is this correct?
  36. S

    A Does a wavefunction in a cartesian system create a one-dimensional subspace?

    Hi, I have heard (or imagined) that a wavefunction, where Psi is on the y-axis and the positions x is naturally on the x-axis, is really a one-dimensional system in Physics (not in mathematics), because the signal or the oscillation of the wavefunction is not really a dimension, and only the...
  37. Spoti112

    Kinematics, One Dimension, Acceleration, 2 objects problem

    Homework Statement Ball A is dropped from the top of a building of height h at the same instant that ball B is thrown vertically upward from the ground. At what height will the balls collide if the collision occurs when the balls are moving in the same direction and the speed of A is 4 times...
  38. M

    Find largest number of linearly dependent vectors among these 6 vectors

    Homework Statement Given the six vectors below: 1. Find the largest number of linearly independent vectors among these. Be sure to carefully describe how you would go about doing so before you start the computation. 2 .Let the 6 vectors form the columns of a matrix A. Find the dimension of...
  39. B

    I The Relation Between Wavefunctions in Dirac Equation

    Can the wave function in four dimensions be expressed as e^i(kx+ky+kz-wt)?
  40. ErenJaeger2

    B What is the True Dimension of Human Perception?

    I have read things saying humans see 2d and also things saying its more like "two and a half d" but that humans don't actually see in three dimension. But when you explain the spatial dimension, you start with the first which can be explained by a line. So anything living in 1d would only be...
  41. M

    Dimension of the electric charge in CGS and in SI?

    The Bohr magneton is (see e.g. Wikipedia) in SI units: $$\mu_B=\frac{e\hbar}{2m_e}$$ and in CGS units: $$\mu_B=\frac{e\hbar}{2m_ec}$$ So the dimension of the electric charge in SI, ##[q_{SI}]##, is related to the dimension of the electric charge in CGS, ##[q_{CGS}]##, by...
  42. S

    Random Walk - 1 dimension

    Homework Statement Suppose a particle moves along the x-axis beginning at 0. It moves one integer step to the left or right with equal probability. What is the pdf of its position after four steps? 2. Homework Equations Binomial distribution ##P(k) = {{n}\choose{k}} p^k (1-p)^{n-k}## The...
  43. D

    Characteristic linear dimension (Reynolds' number)

    How can I choose the characteristic linear dimension? For example in pipe it is its diameter, but on a surface is the length, on a flat plane it can be measured as 4A/P. I was having problems determining the characteristic linear dimension for a diffusion problem in a "rectangular" pipe. I don't...
  44. marce

    I Can Muons or Tau Particles Be Explained by Kaluza-Klein Towers?

    - Does anyone know , why for instance a muon or a tau , can't be considered as Kaluza-Klein tower(s) ? M.
  45. A

    A Anomalous field dimension in the on-shell scheme

    In learning QFT, I've found that the renormalization group is often introduced with the MS scheme. I noticed that if one uses the on-shell scheme instead and calculates the anomalous field dimension using \gamma_{\phi} = \mu \frac{\partial \ln Z_{\phi}}{\partial \mu} one finds that...
  46. E

    B Moving in the 4th Dimension -- Does Time have a velocity?

    Hello all, i read several threads concerning the a.m. topic, but are still not sure if i got it right. Is "passing time" or moving in the 4th dimension in relativistic means a real changing of position in grade 4 spacetime or is it just a mathematical effect ? Why asking: If i understood...
  47. Rodrigo Schmidt

    Conclusion about the dimension of C°(R)?

    [mentor note: thread moved from Linear Algebra to here hence no homework template] So, i was doing a Linear Algebra exercise on my book, and thought about this. We have a linear map A:E→E, where E=C°(ℝ), the vector space of all continuous functions. Let's suppose that Aƒ= x∫0 ƒ(t)dt. By the...
  48. Pushoam

    I Time as a Dimension: Exploring Galilean and Lorentz Transformation

    In Galilean transformation,it is said that time is a physical quantity while space is dimension. What do we mean by this i.e. time is a physical quantity while space is dimension ? Isn't space a physical quantity in Galilean transformation? Here, too, we plot graph of position(on y-axis) as a...
  49. Pushoam

    How do we decide dimension of motion?

    How do we decide dimension of motion? Consider a particle moving along ##\hat x ## direction. This motion is known as one dimensional motion as only one coordinate i.e. x is changing with respect to time. Consider a particle having circular motion in X-Y plane. In Cartesian Coordinate system...
  50. R

    B Dimension of subset containing two circles

    So I am reading a calculus book, and went online to find explanations for why a circle is 1D. Theres the explanations that say something about zooming in very close and seeing that it's indistinguishable from a Real line. Or you can specify any point on it with only one variable Or if there was...
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