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. H

    Thermal expansion of each dimension of a solid

    Hi, I am trying to work out how much each dimension of a solid (for instance an annular disc) made out of steel changes assuming that the solid is heated uniformly and is not constrained at any of its boundaries. Am I right in saying that, the linear expansion equation L = L_0 (1+ α ΔT) can be...
  2. binbagsss

    Modular Forms, Dimension, Valence Formula

    Homework Statement What is the dimension of ##M_{24}##? Homework Equations attached The Attempt at a Solution [/B] I am confused what the (mod 12) is referring to- is it referring to the ##[k/12]## where the square brackets denote an equivalent class and the ## k \equiv 2## / ##k \notequiv...
  3. T

    I Dimension using box counting technique

    On an exam we just took, we were asked to find the dimension of a set using the box counting technique. So choose an epsilon, and cover your object in boxes of side length epsilon, and count the minimum number of boxes required to cover the object. Then use a smaller epsilon and and count the...
  4. L

    Special Relativity -- two ships moving in one dimension

    Homework Statement You approach an enemy ship at a speed of 0.5c measured by you, and the ship fires a missile toward your ship at a speed of 0.7c relative to the enemy ship. What speed of the missile do you measure, and how much time do you have measured by you and the enemy ship before the...
  5. SakoNova

    I Could a Negative Dimension exist

    Instead of the 10,11, and 26 dimensions proposed by various theories, could there be, let's say 10 positive dimensions, existing of 9 of space and one of time, but also 1 negative dimension? This is just a random theory. I have no idea what a negative dimension would hold, but I think it would...
  6. Kaura

    Higher Dimension and Randomness

    I often here claims that higher dimensions such as the 5th and 6th dimensions deal with different possible realities, be it branching off or from different start conditions. I find this confusing and would like to have it cleared up how it is possible for different so called realities to occur...
  7. C

    I Bound states of a periodic potential well in one dimension

    Hi, I'm trying to understand the bound states of a periodic potential well in one dimension, as the title suggests. Suppose I have the following potential, V(x) = -A*(cos(w*x)-1). I'm trying to figure out what sort of bound energy eigenstates you'd expect for a potential like this. Specifically...
  8. Docdan6

    Classical behavior, 3 dimension wave function and reflection

    Homework Statement I'm a pharmacologist and I have a modern physics course to do. This is not my field and I'm completely lost... We were given this problem to do. Thanks a lot in advance. Consider a potential where U(x) = 0 for x ≤ 0 U(x) = -3E for x > 0 Consider a particle of energy E...
  9. Luca_Mantani

    A Computation of anomalous dimension in MS scheme

    Hi, I am computing the anomalous dimension of a mass operator in the MSbar scheme, but i have a doubt. The following is the solution of an exercise given by a professor but i don't understand a passage. I have computed the counterterm ##\delta## and i have the formula $$\gamma=-\mu...
  10. R

    Why does translation work with the extra dimension (Homogeneous coordinates.)

    Homework Statement hi I have read a lot on homogenous coordinates and I feel like I now have a solid foundation. However none of the videos or books I have read give an explicit reason as to why translation with the extra dimension works(i.e. it does not result in scaling). Here 's what I...
  11. Kara386

    Dimension of a Lie algebra

    Homework Statement Show the (real) dimension of su(n) is ##n^2-1##. Homework EquationsThe Attempt at a Solution ##su(n) = \{ A \in M_n(\mathbb{C}) | A+A^T = 0, tr(A) =0 \}## Maybe the solution is obvious, because I can't find a thing online about how to do this. But I can't see how to do it! I...
  12. Matejxx1

    Find the basis of a kernel and the dimension of the image

    Homework Statement Let ##n>1\in\, \mathbb{N}##. A map ##A:\mathbb{R}_{n}[x]\to\mathbb{R}_{n}[x]## is given with the rule ##(Ap)(x)=(x^n+1)p(1)+p^{'''}(x)## a)Proof that this map is linear b)Find some basis of the kernel b)Find the dimension of the image Homework Equations ##\mathbb{R}_{n}[x]##...
  13. S

    B Photography in 4D World: Time & Space

    If time is a dimension, what would be the dimension of a photograph in such a space?
  14. A

    Draw the third view and the points on each view

    Homework Statement Draw the third view and the points on each view. In the first picture you have what the exercise has given us and in the second what I have drawn. I have turned the page so that the first view is the one on the left, the horizontal one is the one on the right. Is that OK...
  15. M

    MHB Determining Basis and Dimension for Vector Subspaces $U_1$ and $U_2$

    Hey! :o Let $U_1,U_2$ be vector subspaces of $\mathbb{R}^4$ that are defined as $$U_1=\begin{bmatrix} \begin{pmatrix} 3\\ 2\\ 2\\ 1 \end{pmatrix}, \begin{pmatrix} 3\\ 3\\ 2\\ 1 \end{pmatrix}, \begin{pmatrix} 2\\ 1\\ 2\\ 1 \end{pmatrix} \end{bmatrix}, \ \ U_2=\begin{bmatrix}...
  16. T

    Is gravity a function of the 4th dimension?

    I recently came across an example of a fictional 2 dimension being and how that being would experience 3 dimensional interaction. In this is example the 2d being was on a pool table and would only see flashes of the pool balls as they interact with the 2d plane. Is gravity similar; however, a...
  17. binbagsss

    A Modular forms, dimension and basis confusion, weight mod 1

    Hi, Excuse me this is probably a really stupid question but I ask because I thought that the definition of the dimension of a space is the number of elements in the basis. Now I have a theorem that tells me that ## dim M_{k} = [k/12] + 1 if k\neq 2 (mod 12) =[k/12] if k=2 (mod 12) ## for ## k...
  18. Julie1703

    B The dimension of time in String Theory

    I was doing some reading on String Theory...I'm not a scientist, but i enjoy it (so forgive me if my question is stupid) and i was wondering: Being time a dimension, could it be the dimension the strings exist in? And interaction between all particles and all forces determine how they vibrate...
  19. ChrisisC

    B Can a point-like particle really exist in a 0-dimensional universe?

    How is it possible that a point like particle is 0 dimensional? Could it only exist within pure mathematics? or actually exist physically in our universe?
  20. ChrisisC

    B Flatlanders in the 3rd dimension "Hyperspace"

    Is the 4th dimension right under our noses? In the book "Hyperspace" written by Machio Kaku (fantastic book, check it out), Kaku uses what he calls "flatlanders" to depict how, us, 3 dimensional people, could actually be involved in the 4th SPACIAL dimension (4th dimension is not used in the...
  21. haruspex

    Insights Can Angles be Assigned a Dimension? - Comments

    haruspex submitted a new PF Insights post Can Angles be Assigned a Dimension? Continue reading the Original PF Insights Post.
  22. M

    What Is the Dimension of Eigenspaces for Given Characteristic Polynomial?

    Homework Statement For c not equal to ±1, what is the dimension of the eigenspaces of A The characteristic polynomial of A is (x-1)(x+1)(x-c) The Attempt at a Solution each term in the characteristic polynomial has a multiplicity of 1 so does this mean that the dimension of the eigenspaces...
  23. Mr Davis 97

    Dimension and basis for a subspace

    Homework Statement ##\mathbb{H} = \{(a,b,c) : a - 3b + c = 0,~b - 2c = 0,~2b - c = 0 \}## Homework EquationsThe Attempt at a Solution This definition of a subspace gives us the vector ##(3b - c,~2c,~2b) = b(3,0,2) + c(-1,2,0)##. This seems to suggest that a basis is {(3, 0, 2), (-1, 2 0)}, and...
  24. ShayanJ

    A For finite dimension vector spaces, all norms are equivalent

    I searched for a proof of the statement in the title and found this document. But it just proves that for two norms ## \rho(x) ## and ## ||x|| ##, we have ## m\rho(x)\leq ||x|| \leq M \rho(x) ## for some m and M. But how does it imply that the two norms are equivalent? Thanks
  25. T

    A Does spacetime lose its determinism in third dimension?

    In a lecture from a course in QM, it was mentioned that Shroedinger's equation is deterministic in one and two dimensions. But in third dimension it gives unstable solutions, loosing it's determinism. It was mentioned that "in space of D dimensions Gauss theorem leads to the conclusion that...
  26. T

    MHB Help finding length of perpendiculars in a box of known dimension

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  27. M

    B Is Time a Dimension or a Measure of Movement?

    Because it takes time to get from one point to another?
  28. P

    I Can one determine the velocity of a photon in the fourth dimension using limits?

    Can one shed light on the velocity of the photon through the fourth dimension x4 using limits? To begin with, please study the mathematics from Brian Greene’s book An Elegant Universe. The upshot is that the faster an object moves through space, the slower it moves through the fourth...
  29. A

    I What is the velocity of the photon through the fourth dimension x4?

    What is the velocity of the photon through the fourth dimension x4? Photons are real, physical entities. The fourth dimension is a real, physical entity. Therefore, photons must have a relationship with the fourth dimension. They must have some velocity relative to it. What is the velocity...
  30. P

    I Can there be a bounded space w/o a boundary w/o embedding?

    Can there be a bounded space without a boundary without embedding in a higher spatial dimension? This seems to be the kind of question I get stuck on when the big bang comes up. Thanks
  31. JuanC97

    I Dimension of the group O(n,R) - How to calc?

    Hi, I want to find the number of parameters needed to define an orthogonal transformation in Rn. As I suppose, this equals the dimension of the orthogonal group O(n,R) - but, correct me if I'm wrong. I haven't been able to figure out how to do this yet. If it helps, I know that an orthogonal...
  32. F

    Finding the dimension of S

    Homework Statement Let S denote (x,y,z) in R3 which satisfies the following inequalities: -2x+y+z <= 4 x-2y+z <= 1 2x+2y-z <= 5 x >=1 y >=2 z >= 3 Homework Equations How to find the dimension of the set S ? The Attempt at a Solution I have tried to transform the inequalities into matrix form...
  33. G

    Kinematics-Motion in one dimension

    Homework Statement Two points P and Q move in a straight line AB.The point P starts from A in the direction AB with velocity and acceleration http://latex.artofproblemsolving.com/b/b/2/bb2c93730dbb48558bb3c4738c956c4e8f816437.png.At the same instant of time Q starts from B in the direction of...
  34. Chevreuil

    Spring+mass to do known job. How dimension? Impulse? Energy?

    Hello there, I'm working on a design project where I have come upon a mechanical problem that I'm having trouble with. Basically I'm making a kind of specialized stapler (at least I think that's a good translation...), and I want it to clamp the staplers using a mass accelerated by a spring...
  35. F

    A Symplectic Majorana Spinors in 5 Dimension

    I need to know if the Symplectic Majorana spinors in 5 dimension have any advantage with respect to the Dirac spinors in 5 dimension, since they have the same number of components. For example if the Symplectic Majorana spinors have a manifested symmetry that the Dirac spinors don't have, or if...
  36. KT KIM

    I Unit and dimension about De broglie wave

    From the de Broglie wave formula we know, Rhamda=h/p In actual examples of course the answer would be 'something [meters]' I am having hardtime to understand how unit of h/mv [J*s]/[kg]*[m/s] turn into wavelength unit [m] I studied the Mass-Energy relation part earlier, But still can't get...
  37. Nasbah BM

    I Why we use Dirac delta function? (in 1 Dimension & 3 Dimesions)

    I want to understand why and where exactly we use dirac delta function? what is its exact use?
  38. R

    What Does a Constant Velocity Graph Look Like?

    Homework Statement Identify the motion that the different sections of the graph describe Homework EquationsThe Attempt at a Solution t=0-2 is accelerating t=2-4 is stationary t=4-5 accelerating t=5-9 decelerating/slowing down Options which are given: Constant speed, moving back to the...
  39. toforfiltum

    B About dimension of vector space

    I'm confused about this. I know that if the dimension of the vector space is say, 2, then there will be 2 elements, right? eg. ## \left( \begin{array}{cc} 1 & 0\\ 0 & -1 \end{array} \right)## What I want to know is if the dimension of vector space is still two if the matrix is like this...
  40. G

    MHB Dimension of $m \times n$ Matrices: Finding Basis

    What's the dimension of the space of $2 \times 2$ matrices? What's the dimension of the space of $m \times n$ matrices? I know that matrices of size $m \times n$ with components in field $K$ form a vector space over $K$. To find the dimension, I would have to find basis. This I'm not quite sure...
  41. E

    Free Particle moving in one dimension problem

    Homework Statement 5) A free particle moving in one dimension is in the state Ψ(x) = ∫ isin(ak)e(−(ak)2/2)e(ikx) dk a) What values of momentum will not be found? b) If the momentum of the particle in this state is measured, in which momentum state is the particle most likely to be found? c)...
  42. G

    MHB Dimension & Subspace of $\mathbb{R}^3$

    Check whether the following are subspaces of $\mathbb{R}^3$ and if they're find their dimension. (a) x = 0, (b) x+y = 0, (c) x+y+z = 0, (d) x = y, (e) x = y= z, and (f) x = y or x = z. (a) Let $S = \left\{(x, y, z) \in \mathbb{R}^3:x = 0 \right\}$. I want to check whether $S$ is subspace of...
  43. jk22

    How does Coulomb's law change in 3+1 dimensional spacetime?

    The power law of Coulomb depends on the dimension treated . It is $$1/r^{n-1} $$ where n is the dimension. In n=3 we get the inverse square law. How does this go into considering now spacetime 3+1 dimensional ? Would it modify the law and how ?
  44. G

    Linear algebra: Prove that the set is a subspace

    Homework Statement Let U is the set of all commuting matrices with matrix A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...
  45. Y

    R^(1/2) Question, half dimension space meaning?

    Homework Statement I know some people use R to mean all numbers in only one dimension which span a line. R^2 meaning numbers with two dimensions that span a plane. R^3 meaning three dimensional numbers that span three dimensional space. I'm not really sure if I worded that correctly but I think...
  46. P

    Difference between direction and dimension

    Hello, Can you tell me what is the difference between direction and dimension?
  47. G

    Quantum potential problem -- Particle confined in 1 dimension

    Homework Statement A particle with mass m and electric charge e is confined to move in one dimension along the x -axis. It experiences the following potential: V(x) = infinity when x<0, V(x) = -e^2/4*pi*ε*x when x≥0 For the region x ≥ 0 , by substituting in the Schrödinger equation, show that...
  48. Ez4u2cit

    What dimension does space-time curve in?

    Is not space curvature the curving or projecting into a higher dimension? Like a curved sheet of paper perceived by a two dimensional creature? The mystery seems to reside in our ape brains being unable to perceive (but not conceptualize) higher dimensions than three or relativistic, quantized time.
  49. N

    Spivak & Dimension of Manifold

    1. Homework Statement I'm taking a swing at Spivak's Differential Geometry, and a question that Spivak asks his reader to show is that if ##x\in M## for ##M## a manifold and there is a neighborhood (Note that Spivak requires neighborhoods to be sets which contain an open set containing the...
  50. S

    Why Do the Terms in the Lagrangian Action Have the Same Dimension?

    Dear all, If we consider the lagrangian to have both geometric parts (Ricci scalar) and also a field, the action would take the form below: \begin{equation} S=\frac{1}{2\kappa}\int{\sqrt{-g} (\ R + \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi -V(\phi)\ )} \end{equation} which are...
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