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. T

    String Vibrations Prove Higher Dimension?

    Is it true that strings can vibrate in certain manners in which they cause a "ripple effect" in spacetime, affecting the topology of the higher dimensions, causing these different timelines to occur in which you have changes so minor as to electrons a plank's unit to the right, or so drastic as...
  2. J

    Exploring 4 Dimensions: Uses of the Fourth Dimension in Analysis

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  3. Islam Hassan

    V: Exploring the Word "Dimension

    The Word "Dimension" I have some difficulty understanding the exact connotation of the word "dimension" as used in string theories re "extra dimentions". Are these extra dimensions really meant to be dimensions of space or are they simply extra parameters that need to be integrated...
  4. D

    Hidden dimension in string theory

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  5. JK423

    Bosonic condensation in 1 spatial dimension

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  6. W

    What characterizes each dimension in M-theory?

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  7. T

    What is the dimension of eigenspaces in a function-based linear transformation?

    Linear transformation f:C^∞(R) -> C^∞(R) f(x(t)) = x'(t) a) I have to set up the eigenvalue-problem and solve it : My solution : ke^λtb) Now I have to find the dimension of the single eigen spaces when λ is -5 and 0. My solution : Eigenspaces : E-5 = ke^-5t E0=k (because ke^0t = k)...
  8. C

    Linearising Eqn for Fractal dimension

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  9. B

    Can We Move in Planck Length and Dimension at the Time of the Big Bang?

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  10. R

    Projectile Motion in One Dimension

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  11. J

    What dimension is the vector {0,0,0} in?

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  12. A

    Motion in one dimension problem

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  13. L

    Dimension of the null space of A transpose

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  14. H

    Using Laplace Equation in one dimension to solve for a charge-free slab

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  15. Y

    MHB Basis & Dimension of 2x2 Matrix Subspaces: W1 & W2

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  16. X

    Dimension Of Physical Quantities

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  17. Q

    Calculating Acceleration and Velocity of a Falling Ball in One Dimension

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  18. Q

    Calculating Distance and Time for Two Cars on a One-Dimensional Road

    An automobile which travels with the constant speed 20 m/s passes in a section in the moment t=0s and 5 seconds later in the same section passes another automobile which travles with the speed of 30 m/s in the same direction.a) Find when the second car meets the first one. My solution : x1=x2...
  19. N

    Finding the Dimension and Basis of the Matrix Vector space

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  20. R

    How to deal with vectors (Motion in two dimension)

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  21. V

    Minimum beam dimension for specific max stress

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  22. D

    WOrk done by a variable force (in two dimension)

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  23. V

    Dimension, fluctuations, and phase transitions

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  24. O

    How to construct gamma matrices with two lower spinor indices for any dimension?

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  25. R

    Instantaneous velocity in one dimension

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  26. D

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  27. N

    Proving that V is a Subspace of P4(x) and Calculating its Dimension

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  28. N

    Help please simple Physics motion in one dimension problem Urgent?

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  29. S

    Is Time the Fourth Dimension?

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  30. V

    Car Crash Motion in 1 dimension Question

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  31. Jalo

    What is the Definition of a Basis in a Vector Space?

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  32. A

    Practical Applications of the 4th Dimension (and beyond)

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  33. A

    Einstein's Equations in Dimension 4

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  34. J

    Nusselt Number dimension question

    So, in Nu=\frac{hD}{k} h is the heat transfer coefficient, and D is the diameter of the pipe in which heat transfer takes place... ..but what are the dimensions of k in terms of mass (M), length (L), time (T) and temperature (\theta)? So far, I've worked out that the units for h...
  35. C

    Basis of a subspace and dimension question

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  36. H

    Lebesgue topological dimension

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  37. S

    Simple math problem in one dimension

    Homework Statement A particle of mass m is subjected to a net force F(t) given by F(t)=F0(1-t/T)i; that is F(t) equals F0 at t=0 and decreases linearly to zero in time T. The particle passes the origin x=0 with velocity v0i. Show that at the instant t=T and F(t) vanishes, the speed v and...
  38. A

    Motion in One Dimension. Need help with these problems.

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  39. S

    Finding base number with Hausdorff Dimension Approximation methods?

    According to the link below, fractal dimension is an exponent of some sort: http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html The Hausdorff Dimension (aka fractal dimension) is denoted as D in the website above. And r is the base number. If we were to look at...
  40. J

    Can not figure out my ? Subscripted assignment dimension mismatch-problem

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  41. E

    Dimension of <xn,tn|xn-1,tn-1>

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  42. Z

    11 dimension, what si going on here ?

    Hi guys I m new to his forums o I hope I post this in the right place. I have a questions about the 11 dimension. As I saw some documentaries about M theory, it seems our universe was created by colision in the 11 dimension. So I was wondering how does it looks like. At the beginning of big...
  43. sankalpmittal

    Objective questions regarding projectile motion - 2 dimension .

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  44. A

    Solving one dimension steady state heat equation with finite differences

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  45. L

    4th and higher dimension perpendicular

    For an arbitrary distance the equation is: \sqrt{\Sigma_{i}^{n}x_{i}^{2}} I would like to know what are the proofs for higher dimensions being perpendicular to our 3-spaital dimensions. If I am wrong in any way, please elaborate. I guess what I'm saying is since: r^{2}=x^{2}+y^{2}...
  46. N

    Matrix dimension of a vector in R4

    Homework Statement Let W be the subspace of R4 defined by W={x:V^TX=0}. Calculate dim(w) where V=(1 2 -3 -1)^T note: V^T means V Transpose, sorry I don't know how to do transpose sign in here. Homework Equations The Attempt at a Solution I tries to do it (1 2 -3 -1)(x1 x2 x3...
  47. E

    Minkowski Metric and the Sign of the Fourth Dimension

    Why is the unit vector for time in Minkowski space i.e. the fourth dimension unit vector always opposite in sign to the three other unit vectors? The standard signature for Minkowski spacetime is either (-,+,+,+) or (+,-,-,-). Is there some particular reason or advantage for making time...
  48. A

    Mass dimension of a scalar field in two dimensions?

    Which is the mass dimension of a scalar filed in 2 dimensions? In 4 dim I know that a scalar field has mass dimension 1, by imposing that the action has dim 0: S=\int d^4 x \partial_{\mu} A \partial^{\mu} A where \left[S\right]=0 \left[d^4 x \right] =-4 \left[ \partial_{\mu} \right]=1...
  49. P

    Higgs Bosons in another dimension?

    Hi Could someone tell me if Higgs bosons exist in another dimension or if there's simply something i don't understand about their existence in our timespace? Meaning - from what I understand - the recent experiments at the LHC smashed together particles with enough energy to create a Higgs...
  50. J

    Dimension of direct sum axler

    Exercise #17 in Linear Algebra done right is to prove that the dimension of the direct sum of subspaces of V is equal to the sum of the dimensions of the individual subspaces. I have been trying to figure this out for a few days now and I'm really stuck. Here's what I have got so far: Choose...
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