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.

View More On Wikipedia.org
Recent contents Top users
  1. 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...
  2. D

    Dimension of a multivariate polynomial space

    Consider the space of all polynomials in n variables of degree at most d. The dimension of that space is C(n+d,d). How do I calculate the dimension of that same space when I restrict the domain of the polynomials to the unit ball? In that case all the polynomials (sum(i=1..n) x_i^2)^p with p a...
  3. M

    Vector Spaces of Infinite Dimension

    I was hoping you guys could help me in understanding some vector spaces of infinite dimension. My professor briefloy touched n them (class on linear algebra), but moved on rather quickly since they are not our primary focus. He gave me the example of the closed unit interval where f(x) is...
  4. sankalpmittal

    A Numerical on motion in one dimension .

    Homework Statement A boy throws a ball with v velocity upwards as well as downwards . What will be the displacement covered by the two balls if air friction and external force or viscosity is neglected ? Homework Equations According to me the equations of motions are relevant here ...
  5. 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)?
  6. M

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

    With normal vectors i usually check there is the correct number of vectors i.e 3 for R3 2 for R2 etc and then just check for linear independence but reducing the matrix that results from c1v1+c2v2+..cnvn=0 and determining of unique solution or infinite solutions. There are the right number of...
  7. Y

    Dimension alignment in a station in curve

    I have a question regarding constructing subway platforms in curved line. As we know, in a curve, wagons get out of linear alignment and become closer in inner radius and separate from each other in outer radius. I want to know the relation between dimension of car, dimension of structure...
  8. 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...
  9. S

    What is the Dimension of a Symmetric 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^{...
  10. Z

    Inverse function in one dimension

    can a function in ONE dimension have NO inverse ?? i mean if given the inverse function f^{-1} (x) = g(x) + \sum_{k=-N}^{k=N}c_{k}exp(ixlogk) the first function g(x) is an smooth function , the last Fourier series is a 'noise correction' t o this function g , N is a big but finite...
  11. atyy

    What is the dimension of LQG (Rovelli's TQFT)?

    In http://arxiv.org/abs/1010.1939 Rovelli describes his present model as a generalized TQFT. The current quantum groups stuff is also strongly TQFT inspired (as spin foams in general are). This makes me wonder whether LQG should not in fact be "higher dimensional". The reason is that I've...
  12. M

    Mathematica Plot 3D matrix as 2D plot and 3rd dimension as color in Mathematica

    Hi! It's been like two days since I have tried to make this work, still I got nothing. Searched Google etc. but no help there. I have a three dimensional matrix in form of {{a,b,c}, {d,e,f}, {g,h,i}, ... etc. } with a total of 51 elements, i.e. 51x3 matrix. What I want is to plot it as...
  13. M

    How to Calculate the Box Counting Dimension in Chaos Theory?

    Box counting dimension! please help! Hi all, I am working on a problem from Chaos theory, I have to find the box counting dimension of the set {0}U{n^-p} where n is an integer and p>0. I started this problem by considering p=1. So, the set looks like {0,1,1/2,1/3,...}. If I take intervals...
  14. P

    Please help to find the dimension that will maximize ?

    Please help to find the dimension that will maximize...? The Park Service is building shelters for hikers along the Appalachian Trail. Each shelter has a back, a top, and two sides. Find the dimensions that will maximize the volume while using 384 square feet of wood. length (across the...
  15. D

    Dimension of electric charge, hypercharge and isospin

    For SU(2) the three represented gauge fields are A_\mu^1, A_\mu^2 and A_\mu^3 and for U(1) the gauge field is B_\mu. The A_\mu^3 and B_\mu are electrically neutral. The photon \gamma and Z particle are combinations of these. My interest is the dimensions of the following parameters...
  16. M

    Is Time Truly the Fourth Dimension in Spacetime?

    "Spacetime has No Time Dimension" http://www.dailygalaxy.com/my_weblog/2011/04/spacetime-has-no-time-dimension-new-theory-claims-that-time-is-not-the-4th-dimension.html#more
  17. B

    Question concerning fractal dimension Hausdorff Measures.

    Homework Statement I'm currently trying to work my way through Frank Morgan's Geometric Measure Theory and I had a quick question regarding his definition of Hausdorff measure. Which is: Homework Equations $\mathscr{H}^m(A) = \lim_{\delta \rightarrow 0} \inf \left\{ \sum \alpha_m \left(...
  18. Pythagorean

    Early Universe had just one Spatial Dimension?

    Soft Article: http://www.sciencedaily.com/releases/2011/04/110420152059.htm Journal Article: http://prl.aps.org/abstract/PRL/v106/i10/e101101
  19. Telemachus

    Rigid body kinetics in 3 dimension space

    I need some help with the equation of moments for this exercise: Each wheel of an automobile has a mass of 22 kg, a diameter of 575 mm, and a radius of gyration of 225 mm. The automobile travels around an unbanked curve of radius 150 m at a speed of 95 km/h. Knowing that the transverse distance...
  20. S

    How Do You Calculate the Angle to Reach a Moving Target in a River?

    I was given this question as a study question but no solution provided. I am unsure how to solve the question. Homework Statement A man is swimming 3m/s down a river. The man is 50m from the edge and 1000m from the top of the river. If I am in a boat at the top corner of the river...
  21. D

    Dimension of The Intersection of Subspaces

    Homework Statement If V and W are 2-dimensional subspaces of \mathbb{R}^{4}, what are the possible dimensions of the subspace V \cap W? (A). 1 only (B) 2 only (C) 0 and 1 only (D) 0, 1, 2 only (E) 0,1,2,3, and 4 Homework Equations dim(V + W) = dim V + dim W - dim(V \cap W) dim (V + W) \leq...
  22. R

    One dimension interference of waves

    Homework Statement Two speakers are placed on the x axis, they produce monotonic sound waves of the same frequency. There is a microphone also on the x axis, but far away from the speakers. It detects a sound minimum when the second speaker is 20 cm behind the first. Then it detects a maximum...
  23. A

    Understanding Hausdorff Dimension of 1-d Line

    Hi, I had a question about understanding some basic thing about the Hausdorff dimension. Specifically, I'm trying to understand why the two dimensional Hausdorff dimension of a 1-d line is zero. In terms of the two dimensional Lebesgue measure, I can see that I can cover the line by a...
  24. P

    Dimension Analysis: Solving x=ut- y^{2} * z^{2} / V

    Homework Statement x=ut- y^{2} * z^{2} / V x,y,z is length u is speed t is time V is volume The Attempt at a Solution m = m/s * s - (m^{2} * m^{2} / m^{3}) m=m- m^{4}/m^{3} m=m-m m=0 there is inconsistent? is this correct?
  25. D

    Dimension analysis problem for Vibration Experiment

    1.I am busy with an assignment based on a Vibration experiment in a Mechanical Engineering degree program The procedure is documented in the lab handout and one part is to compare the measured natural frequency to the calculated natural frequency The formula given in the handout for natural...
  26. O

    What is the Dimension of a 3-D Rotation Matrix?

    I have a similar question about rotation matrices. I'm trying to understand the dimension of the matrix given below which is a 3-D rotation. I think that its dimension is 3 but unsure. Any help appreciated. Thanks, John [(cosx sin x 0), (-sinx cosx 0), (0 0 1)] with ( ) = row,
  27. V

    Particle Motion in One Dimension

    Two bodies begin a free fall from rest from the same height. If one starts 1.0 s after the other, how long after the first body begins to fall will the two bodies be 10 m apart? x = x + vt +1/2at^2 I'm having trouble visualising a solution.
  28. F

    What are the Four Dimensions in Our Space-Time Continuum?

    Is the 4th dimension time? 0D = dot 1D = line 2D = plane 3D = sphere 4D = time Am I correct?
  29. M

    Finding basis of spaces and dimension

    Homework Statement Find a basis for each of the spaces and determine its dimension: The space of all matrices A=[a b, c d] (2x2 matrix) in R^(2x2) such that a+d=0 Homework Equations The Attempt at a Solution So I jumped at this question without knowing too much about spaces and...
  30. M

    Linear algebra: subspaces, linear independence, dimension

    Homework Statement 1. Consider three linearly independent vectors v1, v2, v3 in Rn. Are the vectors v1, v1+v2, v1+v2+v3 linearly independent as well? 2. Consider a subspace V of Rn. Is the orthogonal complement of V a subspace of Rn as well? 3. Consider the line L spanned by [1 2...
  31. C

    Calculating Velocity and Distance in One-Dimensional Motion

    Homework Statement The mass of a van with a driver is 2000 kg . When the van accelerates, the velocity increases with a uniform acceleration of 3.0 m/s2. Homework Equations a) The van starts at rest. Find the velocity after 4.0 s. b) How far does the van travel in the first 4.0 s? c)...
  32. S

    Periodic table of shapes to give a new dimension to maths

    Mathematicians are creating their own version of the periodic table that will provide a vast directory of all the possible shapes in the universe across three, four and five dimensions, linking shapes together in the same way as the periodic table links groups of chemical elements. The...
  33. C

    Proving a property of the dimension of eigenspaces in a finite dimensional space

    Homework Statement Prove that if A: V - >V is a linear map, dim V = n, and h1,...,hk (where 1,...,k are subscripts) are pairwise different eigenvalues of A such that their geometric multiplicities sum to n, then A does not have any other eigenvalues. Homework Equations Note sure if this is...
  34. G

    Motion in two dimension problem. Range and Height.

    Homework Statement A rocket is launched from level ground with a speed of 30.0 m/s at an angle of 40 degrees above the horizontal. Find: a) Maximum height reached by the rocket. b) The time rocket is in the air. c) the horizontal distance where the rocket lands. Homework Equations...
  35. V

    Tension Between Objects in 1 Dimension

    Homework Statement Here is the problem verbatim (values have been slightly changed, also assume a frictionless environment): "A 4.3 kg block A and 6.0 kg block are connected by a string of negligable mass. Force FA = (15 N) acts on block A; force FB = (24 N) acts on block B. What is...
  36. Z

    Linear algebra - dimension and intersection

    Homework Statement let V be a finite dimensional vector space of dimension n. For W \leq V define the codimension of W in V to be codim(W) = dim(V) - dim(W). Let W_i, 1 \leq i \leq r be subspaces of V and S = \cap_{i=1}^{r}W_i. Prove: codim(S) \leq \sum_{i=1}^{r} codim(W_i)Homework...
  37. C

    Motion in one dimension, Kinematics equations

    Homework Statement Two spacecraft are 13,500m apart and moving directly toward each other. The first spacecraft has in initial velocity of 525 m/s and accelerates at a constant -15.5 m/s^2. They want to dock, which means they have to arrive at the same position at the same time with...
  38. D

    Dimension of A(Y) and Height of p in Ring Let A = k[x,y,z]

    Let A = k[x,y,z] and Y = \{(t,t^2,t^3)|t \in k\}, which is irreducible. It corresponds to the prime ideal p=(y-x^2,z-x^3). A(Y) is generated by x,y,z of degree 1 as a k-algebra in its graded ring structure. Each group corresponding to the degree d is spanned by the linearly independent...
  39. A

    What is the dimension of the impulse response of an electric circuit?

    Hi all The impulse response h(t) of an electric circuit (maybe in some special cases) is the derivative of the step response s(t) of the same circuit. right? So does it mean they have different dimension, namely if the dimension of s(t) is X, then the dimension of h(t)=ds/dt is x over...
  40. C

    Dimension of vector space (proof)

    Hi, I'm having trouble understanding a proof of the following theorem which allows it to be shown that all bases for a vector space have the same number of vectors, and that number is the dimension of the vector space (as you probably already know): Homework Statement Theorem: If S =...
  41. D

    Dimension of the image of a linear transformation dependent on basis?

    First of all I would like to wish a happy new year to all of you, who have helped us understand college math and physics. I really appreciate it. Homework Statement Determine the dimension of the image of a linear transformations f^{\circ n}, where n\in\mathbb{N} and...
  42. S

    Exploring 4th Dimension in General Relativity

    hi, what do you think that in general relativity, since space is curved, the curvature could be interpreted as a 4th space dimension "hyperspace"? in other words, the 4th space dimension would take the place of time, for a coordinate observer?
  43. J

    Time dimension extent of a 4-dimension object

    In three dimensions physical objects have an extent ( non-zero length ) along all three dimensions. If it is accepted that the four dimensions of general relativity correspond to physical reality, does that imply that physical objects also have a non-zero ( non-infinitesimal ) extent along...
  44. T

    Dimension of 7x7 Matrices w/ Zero Trace: 48

    1. Homework Statement Find the dimension of the set of 7x7 matrices with zero trace Relevant Equations The dimension of a standard basis matrix n x n is n^2 Zero trace = sum of diagonal elements = 0 Attempt at Solution I started with dim(M) = n^2 where M is an nxn matrix. Then I...
  45. P

    Solve for Initial Velocity: Football Field Goal Problem | No Air Resistance

    Homework Statement A football player kicks a field goal from a distance of 45m from the goalpost. The football is launched at 35 degrees above horizontal. What initial velocity is required so that the football just clears the goalpost crossbar that is 3.1m above the ground? ignore air...
  46. D

    Dimension of Space: How Do We Know It's 3?

    How do we know that the (effective) dimensionality of "space" is three in a typical physical theory?
  47. S

    Linear Algebra Dimension Proof

    Homework Statement Let V\subset\,R^n be a subpace with dim(V) = D. Prove that any S vectors in V are linearly dependent if S > D. Homework Equations Rank-nullity theorem? The Attempt at a Solution dim(V) = D implies that there are D vectors in a basis for V. If S > D then there...
  48. R

    Extra Dimensions: Exploring Beyond 3D

    I just want to clear this up: why can't 5th till 11th dimension be represented in 3 dimensions no matter how small this extra dimensions are? One analogy for an extra dimension is a long rope that represents one dimension but this rope is composed of thinner twisting ropes which now represent...
  49. Z

    How would gravity or any 1/r^2 force act in one dimension?

    From what I know, in 2 dimensions gravity works as 1 / R, 3 dimensions is 1 / R^2, and in 4 dimensions 1 / R^3 But what about one dimension? Is it just proportional to R? So the formula would be F = GMMR? Please explain in detail, how gravity would behave in one dimension and how...
  50. G

    Dimension of a Set: Definition & Explanation

    I'm familiar with the notion of the dimension of a vector space. Sometime earlier though, I ran into something asking for the dimension of a set of matrices. In general context, what is meant by the dimension of a set?
Back
Top