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. U

    Typical dimension of an apparatus for electron interference

    Greetings This is my first post in this section of PF. As the title says, I just want to know the typical dimensions of the two-slit experiment, intended for electron interference. That is, what are the slits made of, what is their spacing from the screen,source and from each other. Is...
  2. D

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    Homework Statement A ball is thrown straight up. At the top of its path its instantaneous speed is a) 5m/s b) 20m/s c) 50m/s d) 10m/sHomework Equations The Attempt at a Solution
  3. I

    Dimension Analysis: Find x, y, z To Satisfy Boyles Law

    For what value of x, y, and z, will the equation P^x v^y T^z = constant, become Boyles law?
  4. Seydlitz

    Using Special Case of Elastic Collisions in one dimension

    Is it to possible to use the special case of elastic collisions in one dimension with bodies that posses different mass. Ordinarily I know that if the body has same mass the velocity of the bodies will simply be exchanged but is the fact also hold for body with different masses? v_1 - v_2 =...
  5. T

    What are the Dimensions of a 2.5' Hard Disk Drive?

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  6. M

    2 dimension forces - problem solving for parked car

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  7. G

    Dimension of space composed of powers of a matrix

    Let A be an n*n matrix. Consider the space span \{ I, A, A^2, A^3, ... \} . How would one show that the dimension of the space never exceeds n? I feel like the answer lies somewhere near the Cayley-Hamilton theorem, but I can't quite grasp it.
  8. S

    2 Time Dimension in the Quantum Realm?

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  9. R

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  10. H

    MATLAB [MATLAB] Subscripted assignment dimension mismatch.

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  11. A

    Intuitive understanding of dimension (units)

    I was flipping through a physics text and some of the units seemed pretty 'crazy'. Just wanted to know if they can always be understood intuitively, like you can visualize what's going on e.g. force, mass x acceleration, if I look at it as kg*m/s^2 then it doesn't really make sense to me...
  12. C

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  13. B

    Does the 0 vector not count when determining the dimension?

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  14. J

    Motion in One dimension: Stone

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  15. J

    Motion in One Dimension: Model Rocket

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  16. L

    What is the Time to Reach a Distance of 50 ft at a Velocity of 60 mph?

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  17. T

    Using Dimension Formula of a Matrix

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  18. J

    What was the original speed of the bullet?

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  19. P

    Vector space dimension question

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  20. C

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  21. B

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  22. P

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  23. G

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  24. T

    What is the Dimension of Lie Algebras \mathfrak{g} and \mathfrak{h}?

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  25. A

    I know that line is a one dimension but i cannot understand then why

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  26. J

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  27. M

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  28. W

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  29. M

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  30. U

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  31. P

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  32. N

    Why in two dimension space many QTFtheory become renormalizable?

    Please teach me this: Why in two dimension space many QTF theories become renormalizable.By the way,are there any relation between this and the world-sheet in string theory? Thank you very much in advance
  33. 8

    What is the Dimension of a Vector Space over F?

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  34. S

    Why is the time dimension different from the 3 space dimensions

    General relativity states that our universe is four dimensional curved space so time dimension is not separated from space dimensions .Why then is the time dimension different from the 3 space dimensions ? and why there must be 3 space dimensions and not 4 for example ?
  35. E

    In one dimension there are no degenerate bound states?

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  36. L

    Fortran Fortran 95 question about dimension error

    error 542, J appears in the dimension of a variable,yet is not a dummy argument,a variable available through USE or CONTAINS association,a COMMON variable,a PARAMETER,or a PURE FUNCTION the same error about I i am using i,j in order to point the temperature in length and time T(i,j) it is a...
  37. F

    When a particle in one dimension have discrete spectrum?

    What are the conditions for which it can be concluded that a system has discrete energy levels? For example a system in one dimension with the potential V(x)=b|x| has only a discrete spectrum. How I can prove it? My book says moreover that the energy eigenvalues have to satisfy the...
  38. A

    Phase transition in one dimension

    Hi, I was listening to Susskind's lecture on statistical mechanics (lecture 8). He mentioned in relation to Ising model of magnetized spin systems that there could not be any phase transitions in one dimensions. He mentioned that it has to do with the stability of the system. Can anybody...
  39. J

    Exploring the Fourth Dimension: An Analogy to Flatland and Beyond

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  40. bcrowell

    Defining the dimension of a singularity?

    In general, how do you define the dimension of a singularity? E.g., we think of a Schwarzschild singularity as pointlike, so that its world-line is one-dimensional, and on a conformal diagram we represent it as a spacelike line, which seems to make sense. In point-set topology, we have...
  41. L

    Dimension of the linear span of a vector

    Homework Statement find the dimension of the linear span of the given vectors v1 = ( 2, -3, 1) v2 = ( 5, -8, 3) v3 = (-5, 9, -4) Homework Equations The Attempt at a Solution so all i did was make it a matrix and put it in rref and i got [1 0 0] [0 1 0] [0 0 1] does this...
  42. jfy4

    Dimension of Fields in Physics: What is it?

    Hi, I was not entirely sure where to post this, but I think this will work. With the gravitational field we have that g^{\alpha\beta}g_{\alpha\beta}=4 which is the dimension of the manifold I believe. I have normally heard of g_{\alpha\beta} being interpreted as the gravitational field...
  43. D

    Dimension of a multivariate polynomial space

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  44. M

    Vector Spaces of Infinite Dimension

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  45. sankalpmittal

    A Numerical on motion in one dimension .

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  46. F

    Prove that diffeomorphisms are between manifolds with the same dimension

    My definition of diffeomorphism is a one-to-one mapping f:U->V, such that f and f^{-1} are both continuously differentiable. Now, how to prove that if f is a diffeomorphism between euclidean sets U and V, then U and V must be in spaces with equal dimension (using the implicit function theorem)?
  47. M

    Finding basis and dimension of W = {(a,b,c,0)} where abc are real numbers

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  48. Y

    Dimension alignment in a station in curve

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  49. W

    Perspective of 3-dimensional objects from the 4th dimension

    This question will probably make the most sense to those who have read Edwin Abbott Abbott's novel Flatland. But I'm sure many others know the answer. To explain, I'll have to use some dimensional analogy. Let's say you're a 2-dimensional being. You live in a two dimensional world and thus...
  50. S

    Dimension of a Tensor vector space

    Homework Statement Having a symmetric tensor S^{a_1 ...a_n} forming a vector space V_n with indices taking values from 1 to 3; what is the dimension of such a vector space? Homework Equations The Attempt at a Solution essentially this reduces to picking a tensor of type S^{...
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