Dimension Definition and 906 Threads

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. A

    Hausdorff dimension of the Cantor Set

    Hi everyone! I am thinking about, how can calculate the Hausdorff dimension of the Cantor set? I know, that this dimension is \frac{\log 2}{\log 3} but I cannot prove it. Any ideas?
  2. R

    What is the dimension of the graph of F?

    Hi everyone, The problem that I'm having issues with reads: "Let F: Rk →Rn be a linear map. Recall that the graph G(F) of F is the subset of Rk × Rn = Rk+n given by G(F)={(x,y)∈Rk ×Rn : y=F((x)}" It first asked me to show that G(F) is a vector subspace of Rk+n which I did just by the definition...
  3. I

    Dimension of vector space intersect with one proper subset

    Homework Statement Given is a vector space (V,+,k) over kn with n > 1. Show that with W \subseteq V, U \subset V and dim(U) = n - 1 dim(W \cap U) \geq dim(W) - 1 Homework Equations dim(W+U) +dim(W \cap U) = dim(W) +dim(V) The Attempt at a Solutiondim(V) = n dim(W) \leq dim(V) dim(W+U)...
  4. C

    Dimension of subspace of even and odd polynomials

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  5. S

    Dimension of subspace of trace of matrix

    Let V=Mn(k) be a vector space of matrices with entries in k. For a matrix M denote the trace of M by tr(M). What is the dimension of the subspace of {M\inV: tr(M)=0} I know that I am supposed to use the rank-nullity theorem. However I'm not sure exactly how to use it. I know that the trace is...
  6. H

    What is the basis and dimension of the subspace U of P2?

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  7. V

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  8. F

    Calculating Distance and Displacement in One Dimension

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  9. F

    What is the acceleration of a bullet passing through a board?

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  10. L

    Could someone explain the fourth dimension?

    Could someone explain the fourth dimension to me? I understand that it represents time. Could anyone thoroughly explain the concept? Thank you.
  11. K

    How to find the dimensions of a particle moving in one dimension?

    So a particle of mass M is moving in one dimension given by: x(t) = (alpha)t^2 + (beta)t + (gamma) where alpha, beta, gamma are non zero constants. What are the dimensions of the alpha, beta, gamma? Either the answer is staring right at my face or my physics is rusty :/
  12. B

    Dimension of SL(2,H), SL(2,R), SL(2,C), SL(2,O) - Proving 15 & 45

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  13. N

    Understanding the Dimension of a Variety in Algebraic Geometry

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  14. W

    Motion in 2 dimension (undergrad level)

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  15. S

    How does M-theory explain genesis of the 11th dimension?

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  16. G

    CAPA problem - Kinematics in 1 Dimension

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  17. S

    Kinematics in one dimension - hot air balloon and bullet

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  18. W

    Dirac equation in one dimension

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  19. R

    Is 4th Dimension Time? Exploring the Possibilities

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  20. inflector

    Time as a Dimension or Projection?

    I see lots of references to time being the fourth dimension as well as there being 3 + 1 dimensions to spacetime as we know it, etc. I also see that time has to be treated differently in some of the constructs of physics. So it seems that time seems to be both similar and dissimilar to the other...
  21. I

    Change of basis in R^n and dimension is <n

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  22. M

    Why Do We Use the Speed of Light to Measure Time in Minkowski Space?

    In Minkowski 4-D spacetime the cooridnates of an point are (ct,x,y,z). My question is why we multiply time with speed of light c and not some other speed v (speed of object moving in three spatial dimantions)? If we treat the time as a fourth dimension it would be more reasonable to say that...
  23. E

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  24. sbrothy

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  25. A

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  26. jinksys

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  27. L

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  28. F

    Basis and dimension: Need help finding my mistakes

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  29. M

    2D surfaces in the third dimension?

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  30. K

    Does the 4th dimension mean our universe is definite?

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  31. G

    Jacobian and the dimension of a variety

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  32. M

    Exploring Time as a Curled-up Dimension

    So, I was just thinking what if the time dimension was a tiny curled up dimension like one of these supposed extra 8 dimensions from string theory ? If it allowed just enough wiggle room for objects to move the idea has some features that appeal to me. - time is symmetrical ; no past , no...
  33. C

    Tesseract with Time as Fourth Dimension

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  34. R

    Dimension for Vector Space involving Planes

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  35. I

    One Dimension Kinematics Question

    Homework Statement A fugitive tries to hop a freight train traveling at a constant speed of 6.0 m/s. Just as an empty box car passes him, the fugitive starts from rest and accelerates at a=4.0 m/s2 to his maximum speed of 8.0 m/s. (a) How long does it take him to catch up to the empty box...
  36. S

    Higher Dimension Integrals: Solving for n in Exponential Functions

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  37. T

    I have some trouble with the fourth spatial dimension

    I suppose it might be appropriate here. I'm not really looking for an answer in pure maths (Topology is waaaaaaaaaaay beyond me), but rather in concept. Now, I can appreciate the properties of a tesseract, how it relates to and derives from the cube (I think of it as either putting 8 cubes...
  38. S

    Null space vs Col space dimension?

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  39. T

    Is Time the Fourth Dimension in Euclidean Space?

    Is it the additional dimension, Time, added to the Euclidean (x,y,z) 3-D space? Or does it even make sense to ask this question? And instead consider 4D space-time (Minkowski Space) as a manifold in which 3D is embedded? I'm confused. :(
  40. M

    Issue with the definition of the Hausdorff dimension

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  41. A

    Basis and dimension of the solution space

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  42. R

    Examining the Dimension of a Photon Impacting an Electron

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  43. Z

    Entanglement possible through a higher dimension.

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  44. P

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  45. B

    Does String Theory Imply the Existence of a 0th Dimension?

    Is string theory the replacement of the zeroth dimension? Or could it be thought of as such?
  46. G

    Tarzan and Jane(collision in one dimension problem)

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  47. B

    Finding a basis and dimension of a subspace

    Homework Statement Let S={v1=[1,0,0,0],v2=[4,0,0,0],v3=[0,1,0,0],v4=[2,-1,0,0],v5=[0,0,1,0]} Let W=spanS. Find a basis for W. What is dim(W)? Homework Equations The Attempt at a Solution i know that a basis is composed of linearly independent sets. This particular problem's...
  48. R

    How to describe the 4th dimension?

    what is 4th dimension? is it just 1 dimensional space? how come time is described as the 4th dimensional path? thanks..
  49. D

    Why is the dimension of the vector space , 0?

    In other words, why is dim[{0}]=0. My math professor explained that since the 0 vector is just a POINT in R2 that the zero subspace doesn't have a basis and therefore has dimension zero. This is not satisfactory. For example, I know R2 has a dimension 2, P_n has dimension n+1, M_(2,2) has...
  50. P

    Elastic and Inelastic Collisions in One Dimension - Need Help, due tonight

    I have a lab report due tonight. We were working on these in lab and I'm not sure how to tell which one was elastic or inelastic. We had three cases. We had to use the conservation of momentum and kinetic energy formulas.A ) Cart A and B have equal masses. Car B is at rest. Car A strikes car B...
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