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. 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.
  2. 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 :/
  3. B

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

    dimension of sl(2,R) = 1*(2*2-1) = 3, is isomorphic to so(2,1) : 2+1 = 3 dimension of sl(2,C) = 2*(2*2-1) = 6, is isomorphic to so(3,1) : 3+2+1 = 6 dimension of sl(2,H) = 15, is isomorphic to so(5,1) : 5+4+3+2+1 = 15 dimension of sl(2,O) = 45, is isomorphic to so(9,1) : 9+8+7+6+5+4+3+2+1 = 45...
  4. N

    Understanding the Dimension of a Variety in Algebraic Geometry

    This should be a simple question for anyone familiar with basic algebraic geometry, but the concept is getting the best of me. Basically I am unclear on the concept of dimension of a variety. I'm still stuck in the algebraic mindset that dimension is the (minimum) number of elements needed to...
  5. W

    Motion in 2 dimension (undergrad level)

    Homework Statement In an action-adventure film, the hero is supposed to throw a grenade from his car, which is going 79.0 km/h , to his enemy's car, which is going 116 km/h . The enemy's car is 16.1m in front of the hero's when he let's go of the grenade. If the hero throws the grenade so...
  6. S

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

    I read Stephen Hawking new book and did a little research on M-theory. Apparently in the 11th dimension there are different brains that crash into each other. 2 of these brains crashing supposedly rendered our big bang, and explains why stuff came out in blobs (not homogeneous). But how...
  7. G

    CAPA problem - Kinematics in 1 Dimension

    Homework Statement A car accelerates at 2.10 m/s2 along a straight road. It passes two marks that are 29.6 m apart at times t=4.10 s and t=4.90 s. What was the car's velocity at t=0? I'm assuming the car has constant acceleration of 2.10 m/s2 Given \Deltax = 29.6 m a = 2.10 m/s2 vi = ...
  8. S

    Kinematics in one dimension - hot air balloon and bullet

    Homework Statement A hot air balloon is ascending straight up at a constant speed of 8.10 m/s. When the balloon is 13.0 m above the ground, a gun fires a pellet straight up from ground level with an initial speed of 28.0 m/s. Along the paths of the balloon and the pellet, there are two...
  9. W

    Dirac equation in one dimension

    i am now studying dirac equation and klein paradox if we confine to one dimension, we only need one alpha matrix, not three so in lower dimensions, maybe the dirac spinor is not of four components but fewer? i am curious about this question because it seems that as for the Klein...
  10. R

    Is 4th Dimension Time? Exploring the Possibilities

    So do we know for sure what the 4th spatial dimension is? I have read in many places that it is time. I would say this makes sense because we are perceiving time in 3 dimensional slices in our dimension. The same way a 2D being would perceive 2D slivers of a 3D object in its own dimension.
  11. 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...
  12. I

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

    Suppose I have a basis for a subspace V in \mathbb{R}^{4}: \mathbf{v_{1}}=[1, 3, 5, 7]^{T} \mathbf{v_{2}}=[2, 4, 6, 8]^{T} \mathbf{v_{3}}=[3, 3, 4, 4]^{T} V has dimension 3, but is in \mathbb{R}^{4}. How would one switch basis for this subspace, when you can't use an invertible...
  13. M

    Time as a fourth dimension

    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...
  14. E

    Find the maximum and minimum dimension of a closed loop

    Dear all, Is there a method to find the maximum and minimum dimension of an irregular closed loop? This is a problem when we want to define the full-width - half maximum of a image. The level contour of this image at its half maximum can be an irregular closed loop. Any reference or...
  15. sbrothy

    Fractal dimension of the universe = 2?

    (Maybe this should go under General Math or maybe even Topology but since it's about the dimension of the universe I'll put it here. Feel free to move it.) I've had too much coffee and on one my frequent wiki binges - reading about life, the universe and everything - I've come up with a...
  16. A

    The possibility of Perception being a dimension?

    The possibility of "Perception" being a dimension? Is it possible that perception itself could be a dimension? Now think deeper, not personal perception, but instead the idea of perception in general. Because quiet simply, there are lots of things that exist that we cannot perceive. Just as...
  17. jinksys

    Lin Alg - Find the basis and dimension

    Find the basis and dimension of the following homogeneous system: A = |1 0 2| |x1| |2 1 3| |x2| = [0,0,0] |3 1 2| |x3| My attempt: Solving the coefficient matrix for RREF, I get the identify matrix. So, x1=x2=x3=0 and the only solution is a trivial one. Does that mean...
  18. L

    Linear Momentum (One Dimension) Quiz Question

    Hello forum, today in my gr. 12 physics class I had a interesting question which I could not prove at all for the life of me. It is a really simple question, just with a little twist I suppose. As I was driving home from class I was thinking about what I was missing and thought up a solution...
  19. F

    Basis and dimension: Need help finding my mistakes

    Homework Statement He made some notes, but I'm still confused. The Attempt at a Solution
  20. M

    2D surfaces in the third dimension?

    Hello, I just need to know whether or not surfaces with zero size in the third dimension, 6x8x0, is considered two-dimensional. The surface is there all the time. It has a location in the third dimension, so wouldn't it be a 3D object? I am not sure whether I should call a flat surface (as...
  21. K

    Does the 4th dimension mean our universe is definite?

    I've recently been learning about relativity and quantum physics. I understand it quite well now but i have a thought experiment that doesn't make sense. If i travel into the future through a high speed train and see the world blown up by some epic disaster, doesn't this mean that this was...
  22. G

    Jacobian and the dimension of a variety

    I have the following problem: I'm studying a system of polynomial equations in R^n and I'm looking at the surface which is the solution set of this system. I'm mainly interested in the dimension of this surface at a given point. Now, naively, one would try to compute the Jacobian (of the m...
  23. 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...
  24. C

    Tesseract with Time as Fourth Dimension

    Hi forum, I was having some difficulty in understanding how time as a fourth dimension allows a cube of three spatial dimensions to obtain the properties of a tesseract. Suppose we allow a cube to exist for some interval of time. At the start such a cube would have eight vertices as well as...
  25. R

    Dimension for Vector Space involving Planes

    I don't really understand what the dimension of a vector space for planes is. Is it 2 or 3 and why? What's the difference between the dimension of the plane as a surface (dimension of all surfaces is 2) and the dimension of plane in a vector space? Also, Wikipedia says that only planes passing...
  26. 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...
  27. S

    Higher Dimension Integrals: Solving for n in Exponential Functions

    Homework Statement Calculate the following integrals: (a) I(n,\alpha) = \int_{0}^{\infty} e^{-\alpha x^2}x^n dx for n whole integers and n \ge 0 Calculate all results till n=5. Tip: First calculate I^2(0,\alpha) and I(1,\alpha) and then use this to calculate n>1. (b) I(n)=\int_{0}^{\infty}...
  28. 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...
  29. S

    Null space vs Col space dimension?

    I have a question in my linear algebra text that asks: Give integers p and q such that Nul A is a subspace of Rp and Col A is a subspace of Rq. What determines these values? Why are the values of p and q different between the Nul space and Col space? The matrix in question is a 3 x 4...
  30. 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. :(
  31. M

    Issue with the definition of the Hausdorff dimension

    Issue with the definition of the Hausdorff dimension Homework Statement http://mathworld.wolfram.com/HausdorffDimension.html" involves a n-dimensional Hausdorff measure of 0. I'm having trouble understanding cases that would give such a value. Homework Equations the Hausdorff...
  32. A

    Basis and dimension of the solution space

    Homework Statement Find (a) a basis for and (b) the dimension of the solution space of the homogeneous system of equations. x - 2y + z = 0 y - z + w = 0 x - y + w = 0 Homework Equations The Attempt at a Solution (a) [1 -2 1 0] => [1 0 -1 2] [0 1 -1 1] => [0 1...
  33. R

    Examining the Dimension of a Photon Impacting an Electron

    To remove an electron from a H-like atom, the action of a photon is required. But what is the dimension of a photon, that hits the electron? If it is considered a wave, then is it possible to have just energy without mass, and it is known that the extra energy of the photon is imparted to the...
  34. Z

    Entanglement possible through a higher dimension.

    Dear everybody... Could it be true... that Entanglement is made possible because the forces between the 2 electrons are being folded through a higher dimension, so there would be an explanation for the simultaneous reaction of the 2 electrons even if there`s a astronomical distance between...
  35. P

    Computing Correlation Dimension of Attractor | Hi, Explain?

    Hi, I read that the dimension of an attractor is not necessarily an integer. I then tried to compute the dimension of the attractor defined by the system I’m studying but I found two definitions for the dimension: -The capacity dimension; -The...
  36. 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?
  37. G

    Tarzan and Jane(collision in one dimension problem)

    Homework Statement Tarzan is in the path of a pack of stampeding elephants when Jane swings into the rescue on a rope vine, hauling him off to safety. The length of the vine is 27 m, and Jane starts her swing with the rope horizontal. If Jane's mass is 56 kg, and Tarzan's mass is 91 kg, to...
  38. 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...
  39. 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..
  40. 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...
  41. 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...
  42. A

    CW Complex: top dimension cells

    1. The problem statement Given an n-dimensional CW complex on a space X, show that each n-cell is open in X. Definition of CW Complex 1. The cells form a disjoint union of X 2. For each k-cell e, there is a relative homeomorphism : \phi: (D^k, S^{k-1}) \rightarrow (e \cup X^{k-1}...
  43. D

    Time as a fractional spatial dimension

    I've been day dreaming about this idea for a month or two. There are many hypothesises that describe the universe in some higher number of dimensions. As far as I know, this is always an integer number of dimensions. I've been wondering if it is possible for the universe to have a...
  44. G

    Did I enter a parallel universe during my coma?

    New here. I just needed to tell my story somewhere. I'll warn you its not very interesting. Its just a strange thing that happened in my life and I cannot explain. I was detoxing of benzodiazepines at a medical detox center. I had a seizure. As a result I went into a coma or coma like...
  45. C

    Subspace matrix, dimension and basis

    So an example was the matrix: A = \left(\begin{array}{cccc} a&a+b\\ b&0\\ \end{array} \right) is a subspace of M2x2. and is the linear combination a*\left(\begin{array}{cccc} 1&1\\ 0&0 \end{array} \right) + b*\left(\begin{array}{cccc} 0&1\\ 1&0 \end{array} \right) Meaning it has...
  46. L

    Basis and Dimension of matrices

    If S is subspace of R6x6 consisting of all lower triangular matrices, what is the dimension of S? Does anyone know the properties about dimension of lower triangular matrices?
  47. A

    Imagining the 10th dimension by rob bryanton

    I need to ask a scientific community here if there are any sort of basis for the video and the book mentioned above in the title? once i got to the fifth dimension it all turned to non sense that does not relate in any way to string or M theory. Supposing I am right about what i said...
  48. M

    Gravitons moving out of our dimension

    Is my understanding correct. Gravity is a week force because gravitons escape out of our dimension into other dimensions. If so would gravitons not also be able to re-enter our dimesion. Therefore cancelling out this theory
  49. D

    Dimension of subspace

    C[-pi, pi] Span(1, cos (2x), cos2 (x)) Doing the Wronskian here is pain so what other method would be more appropriate?
  50. S

    Dimension of intersection of U and V

    Homework Statement Prove true: For any subspaces U,V of R^n dim(U intersect V) <= min(dim(U), dim(V)) Homework Equations Min(a,b) = the minimum value of A and B The Attempt at a Solution I know this statement is true however I can't quite figure out where to start on how to...
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