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. chwala

    I Determining the dimension of a given PDE

    Now in my understanding from text ...just to clarify with you guys; the pde is of dimension 2 as ##t## and ##x## are the indepedent variables or it may also be considered to be of dimension 1, that is if there is a clear distinction between time and space variables. Your insight on this is...
  2. T

    I Dimension of a vector space and its subspaces

    Can a vector subspace have the same dimension as the space it is part of? If so, can such a subspace have a Cartesian equation? if so, can you give an example. Thanks in advance;
  3. Z

    A How to add higher dimensional operator at higher energy in SM?

    Hi, I Learned that we can add higher dimensional operator but they are non-renormalizable - but effect of higher dimensional operator is vanishes at low energy - my question is than how can we add higher dimensional operator at higher energy - like dimension 5 operator ( weinberg operator)...
  4. Z

    A Dimension of terms in Lagrangian

    hi, when we say in SM , we can add terms having dimension 4 or less than that- in this to what dimension we are refering ? kindly help how do you measure the dimension of terms in Lagrangian. thanks
  5. P

    C/C++ Overload functions by dimension of vector

    vector<OP> negate (vector<OP> a) { a.insert(a.begin(), neg); return a; } vector<vector<OP>> negate (vector<vector<OP>> a) { for (int i=0; i<a.size(); i++) a[i] = negate(a[i]); // reference to 'negate' is ambiguous? return a; } OP is an enum here. Why can't C++...
  6. V

    A Metric of a Moving 3D Hypersurface along the 4th Dimension

    Consider a hypothetical five dimensional flat spacetime ##\mathbb{R}^5## with coordinates ##x, y, z, w, t##. Now imagine that the hypersurface ##\Sigma =\mathbb{R}^3## of ##x, y, z## moves with constant rate ##r## along the coordinate ##w##, i.e. ##dw/dt=r##. Assuming that ##t \in (-\infty, +...
  7. Safinaz

    A The scale of an extra dimension

    Any help to understand how the authors of this paper Fine Tuning Problem of the Cosmological Constant in a Generalized Randall-Sundrum Model calculted this size of the extra dimension Equ. (3.8) from the scale factor defined by Equ. (3.3) ? Specifically, this paragraph after Equ. (3.8) -...
  8. S

    I Operators in finite dimension Hilbert space

    I have a question about operators in finite dimension Hilbert space. I will describe the context before asking the question. Assume we have two quantum states | \Psi_{1} \rangle and | \Psi_{2} \rangle . Both of the quantum states are elements of the Hilbert space, thus | \Psi_{1} \rangle , |...
  9. H

    Vector space of functions defined by a condition

    ##f : [0,2] \to R##. ##f## is continuous and is defined as follows: $$ f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$ $$ f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$ ##V = \text{space of all such f}## What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V##...
  10. O

    A Methods for estimating embedding dimension

    Say that I want to reconstruct the Lorenz system only using one of the original three variables using Takens theorem. What methods can I use to estimate the embedding dimension else from the False nearest Neighbor method (FNN)? The reason I wonder is because I want to compare the accuracy of FNN...
  11. J

    What is the physical dimension of transduction coefficient in photonics?

    I am studying Tunable couplers with Programmable Integrated photonics by Jose Capmany. In this textbook, what is the physical meaning of signal s1 and s2? Is it an electric field intensity or the phase of electromagnetic wave? And what is the physical dimension of transduction coefficient here?
  12. ergospherical

    I Determine Scaling Dimension of Field Theory

    It is given that a theory is invariant under the length scaling:\begin{align*} x &\rightarrow \lambda x \\ \phi(x) &\rightarrow \lambda^{-D} \phi(\lambda^{-1} x) \end{align*}for some ##D## to be determined. The action of a real scalar field is here:\begin{align*} S = \int d^4 x...
  13. pellis

    A How to visualise complex vector spaces of dimension 2 and above

    According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex...
  14. patric44

    Confirming the dimension of induced charge density of a dielectric

    hi guys our professor asked us to confirm the units of volume charge density ρ and also the surface charge density σ of a dielectric material given by $$ \rho = \frac{-1}{4\pi k} \vec{E}\cdot\;grad(k) $$ $$ \sigma= \frac{-(k-1)}{4\pi} \vec{E_{1}}\cdot\;\vec{n} $$ I am somehow confused about the...
  15. P

    A $\phi^4$ in $4 - \epsilon$ dimension renormalization beta function

    Hi all, I am currently studying renormalization group and beta functions. Since I'm not in school there is no one to fix my mis-understandings if any, so I'd really appreciate some feedback. PART I: I wrote this short summary of what I understand of the beta function: Is this reasoning...
  16. B

    Third or fourth dimension?

    If a globe is representative of the third dimension, what does a spinning globe represent? As we move through space and time, are we not interacting within the fourth dimension?
  17. Labyrinth

    A Link between 24 dimension kissing number and Monster group

    I've heard that there is some link between these two values (they're so close!) but I can't seem to find it anywhere. Can someone point me in the right direction? (there's also the J-invariant 196884, well you get the idea)
  18. JD_PM

    Dimension statement about (finite-dimensional) subspaces

    My intuition tells me this is a true statements so let's try to prove it. The dimension is defined as the number of elements of a basis. Hence, we can work in terms of basis to prove the statement. Given that ##U_3## appears on both sides of the inequality, let's get a basis for it. How...
  19. DaniV

    The 1-loop anomalous dimension of massless quark field

    I tried as first step to find Z_q the renormalization parameter, to do so I did the same procedure to find the renormalization parameter of the gauge field of the gluon A^a_\mu when a is representation index a \in {1,2,...,N^2-1} such that A^{a{(R)}}_{\mu}=\frac{1}{\sqrt{Z_A}}A^{a}_{\mu}...
  20. Paige_Turner

    B What's the distance metric for a compactified dimension?

    I asked that question on another forum here. The 2 answers I got before the question was closed by an angry mod said: You wouldn't understand the answer. Don't ask that question. Ask about a Riemann sphere instead. You're too lazy to look up the answer in [a GR textbook that I don't own]...
  21. Paige_Turner

    B What is the nature of dimensionality in 11 dimension M-theory?

    They're dimensions, so they DO have a metric equation, right? Does energy flow cyclically between pairs of dimensions? To me, that's what rotation is.
  22. MahdiI84

    Fortran Dimension of arrays (RESHAPE) in Fortran 90

    program testmatek implicit none integer :: Nc=1000 ,k integer,parameter :: N=2 REAL :: kx ,a0=1.0 ,t0=0.25 DOUBLE PRECISION :: pi=4*ATAN(1.) COMPLEX , PARAMETER :: i=(0,1) complex :: ek1 ,ek2 complex :: MATRIX_ek(N,N) open(1,file='matek.txt') MATRIX_ek(N,N)=(0,0) DO k=-Nc,+Nc...
  23. V

    Dimension of orthogonal subspaces sum

    ##| V_1 \rangle \in \mathbb{V}^{n_1}_1## and there is an orthonormal basis in ##\mathbb{V}^{n_1}_1##: ##|u_1\rangle, |u_2\rangle ... |u_{n_1}\rangle## ##| V_2 \rangle \in \mathbb{V}^{n_2}_2## and there is an orthonormal basis in ##\mathbb{V}^{n_2}_2##: ##|w_1\rangle, |w_2\rangle ...
  24. patric44

    I Dimension of a Linear Transformation Matrix

    hi guys I was trying to find the matrix of the following linear transformation with respect to the standard basis, which is defined as ##\phi\;M_{2}(R) \;to\;M_{2}(R)\;; \phi(A)=\mu_{2*2}*A_{2*2}## , where ##\mu = (1 -1;-2 2)## and i found the matrix that corresponds to this linear...
  25. F

    MATLAB Can I calculate the covariance matrix of a large set of data?

    Hello everyone. I want to calculate the covariance matrix of a stochastic process using Matlab as cov(listOfUVValues) being the dimensions of listOfUVValues 211302*50. I get the following error: Requested 211302x211302 (332.7GB) array exceeds maximum array size preference. Creation of...
  26. DaniV

    RG flow of quadrupole coupling in 6+1 dimension electrostatic problem

    I tried to do a Euler Lagrange equation to our Lagrangian: $$\frac{S_\text{eff}}{T}=\int d^6x\left[(\nabla \phi)^2+(\nabla \sigma)^2+\lambda\sigma (\nabla \phi)^2\right]+\frac{S_{p.p}}{T}$$ and then I would like to solve the equation using perturbation theory when ##Q## or somehow...
  27. potassium_mn04

    Dimension of Eigenspace of A and A^T

    I know that the rank of A and A^T are equal, and that the statement follows from there, but I have no idea how to prove it.
  28. G

    B Newton's Constant in Higher Dim. Spacetimes, Velocity of Light=1

    In the following, I set the velocity of light unity. I refer to theories of gravities in higher-dimensional spacetimes. Newton` s constant converts the curvature scalar with dimension ##lenght^{-2}## into the matter Lagrangian with dimension ##energy/length^3##. So its dimension is...
  29. L

    I Double delta potential -- Degeneracy of bound states in one dimension?

    I have a question from the youtube lecture That part starts after 42 minutes and 47 seconds. Balakrishnan said that if delta barriers are very distant (largely separated) then we have degeneracy. I do not understand how this is possible when in 1d problems there is no degeneracy for bond states.
  30. L

    A Dimension of \lambda constant in \delta potential

    Time independent Schroedinger equation in ##\delta## potential ##V(x)=-\lambda \delta(x)##, where ##\lambda >0## is given by -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi(x)-\lambda \delta(x)\psi(x)=E\psi(x). How to find dimension of ##\lambda##? Dimension of ##V(x)## is [V(x)]=ML^2T^{-2}. Because...
  31. B

    I Dimension of an Angle: Clarifying the SI Standard

    Quickly estimate the value of this expression: sin(1.57). If your answer was about 1, then you assumed that 1.57 was a radian value. If your answer was about 0, you assumed that 1.57 was degrees. If you said that you can’t determine an answer, then you were expecting to see an angle...
  32. hquang001

    Motion in one dimension

    total time: t = 36 mins = 0.6h = t1 + t2 => t2 = t - t1 = 0.6h - 0.1h = 0.5h Vmax = a1 x t1 Vat C = Vmax + a2t2 substitute Vmax in Vat C we have : 0 = a1 x 0.1h + (-600 km/h²) x 0.5h => a1 = 3000km/h² Vmax = a1 x t1 = 3000 x 0.1 = 300km/h I check the result by: x1 = ½ a1 t1² = ½ . (3000)...
  33. S

    B What is dimension used for?

    But what is the actual use of dimension? We can do dimension analysis but it can be simply changed into unit analysis and the result will be the same. So why introduce dimension for physical quantities? Why unit is not enough? Thanks
  34. chucho11028

    Motion in one dimension -- Experiments with a Hot Wheels car rolling down a ramp

    Hello This is not a homework, this is my own experiment to understand how the motion works. Please, follow my question here below: I have a hot wheels race with a slope with 10 degrees where I use a small car which departs from the top to the bottom. I have taken 5 times the time to get an...
  35. F

    Finding the dimension of a subspace

    I am stuck on finding the dimension of the subspace. Here's what I have so far. Proof: Let ##W = \lbrace x \in V : [x, y] = 0\rbrace##. We see ##[0, y] = 0##, so ##W## is non empty. Let ##u, v \in W## and ##\alpha, \beta## be scalars. Then ##[\alpha u + \beta v, y] = \alpha [u, y] + \beta [v...
  36. N

    B Area Increasing as a linear dimension increases -- Looking for intuition on this

    I am working on related rates problems involving figuring out how area of a square increases per second based on how much one side increases per second (or how the area of a circle increases based on increase of the radius, etc.). I was wondering about the practical significance of problems like...
  37. K

    I Prove that dim(V⊗W)=(dim V)(dim W)

    This proof was in my book. Tensor product definition according to my book: $$V⊗W=\{f: V^*\times W^*\rightarrow k | \textrm {f is bilinear}\}$$ wher ##V^*## and ##W^*## are the dual spaces for V and W respectively. I don't understand the step where they say ##(e_i⊗f_j)(φ,ψ) = φ(e_i)ψ(f_j)##...
  38. K

    Comp Sci Kalman filter in one dimension to track a moving object

    Below is the whole code. I can't change the whole code, I can only change the "Kalman class". The Kalman class in the code below is my attempt to solve the problem. But the code doesn't work well. I have written these 5 equations in the Kalman-filter algorithm: State Extrapolation Equation...
  39. L

    I Intersection of a 4D line and a 3D polyhedron in 4D

    Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.
  40. J

    I Three equations of planes, dimension should be 1?

    I always thought that one independent equation cuts down the dimension by 1, so if we had two planes, say x - y - z = 1 and x + y + z = 1, then because these are two independent equations, the dimension of the intersection should be 1 because each plane is cutting down the dimension by 1. Using...
  41. S

    I A 5th Dimension May Explain Quantum Theory

    At least according to Tim Anderson Ph.D who wrote the paper in Physics Review. https://news.knowledia.com/US/en/articles/a-5th-dimension-may-explain-quantum-theory-the-infinite-universe-medium-6f1d6fd371e068a07f357b9babe9ab2eec06d034 What do you make of this? "The paper simply presents...
  42. jk22

    Has the time "dimension" no spatial extension?

    It's just think that if we measure 1 second with a clock we should be able to "see" a 300'000km long piece of something in space or not ? Or does the time extension only has to be understood as a set of numbers indicating timelaps, so that there is no "geometry" of time ?
  43. jk22

    I If a 4th dimension existed would universe expand anisotropically?

    Suppose the universe were described by internal geometry by a ball, i.e. the metric where : $$diag(1,r^2,r^2 sin(\theta)^2)$$ Now if we go to exterior geometry and suppose there existed a 4th timelike dimension the manifold were for example modelized by : $$\left(\begin{array}{c}...
  44. M

    B Expansion of the Universe and the fourth dimension

    The perimeter of a circle increases by radius, the surface area of a ball increase by radius(which is height which is the third dimension if the ball is a planet like the Earth), and the universe is expanding by time, can we say that the fourth dimension is time by this ?
  45. G

    I What's the name of a 2-torus looped in the 4th dimension?

    To form a 2-torus, a narrow tube can be bent into a loop and joined end to end: But instead of forming this loop in our three-dimensional space, the loop can also be formed in a direction perpendicular to three-dimensional space, moving it into the fourth dimension of space. What's the name of...
  46. B

    Dimension of Hilbert spaces for identical particles

    My thoughts are: a) it should just be N^2 b) just N since they're identical c) due to Pauli exclusion would it be N^2 - N since they have to be different states?
  47. E

    Direction of a spring's force in more than one dimension

    adding all the torques around the red circle position (taking clockside direction as positive ): -M*g*L*sin(theta)-k*x*y=I *w (considering that the suspension bar is of negligible mass as the problem indicates )here "x" is the normal "x" of hooke's law (I don't know exactly what it is for a...
  48. O

    MHB Dimension of the cut-out squares that result in largest possible side area

    A topless square box is made by cutting little squares out of the four corners of a square sheet of metal 12 inches on a side, and then folding up the resulting flaps. What is the largest side area which can be made in this way? What information I have so far is that since the side of the...
  49. DaveC426913

    B The Mandelbrot Set has a third dimension: the Bifurcation Diagram

    This is wild. I was always fascinated with the Mandelbrot set, as well as the bifurcation diagram. I had no idea the Mandelbrot diagram was a different visualization of the bifurcation diagram. Question: is this video accurate? I always question the veracity of YouTube science videos.