What is Dimensions: Definition and 1000 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. FallenApple

    A Markov Chain as a function of dimensions

    Here is an animation I created in R. I built this Markov chain of order 50 by correlating the information in one of the coordinates while randomly varying the rest. Is there an explanation for the clustering and flattening out over increasing dimensions of the vector space? Is it due to the...
  2. M

    How does one dimensional light warp multiple dimensions?

    If light is one dimensional, yet has gravity, and gravity is the warpage of spacetime, and spacetime has four dimensions, then how does a one dimensional wave/particle warp multiple dimensions?
  3. K

    I Testing string/m-theory extra dimensions prediction

    String theory and M-theory and Kaluza Klein theories predicts additional extra dimensions at least 2 recent papers proposed tests these predictions of extra dimensions String theory phenomenology and quantum many body systems Sergio Gutiérrez, Abel Camacho, Héctor Hernández (Submitted on 24...
  4. DaveC426913

    B Knot theory: closed loops in n dimensions

    I'm not a math machine, but I dabble in dimensional stuff. I think this falls under knot theory. I have built several prototypes of a tesseract. Each of them sits in a little case in my office. One of them is made from truncated cubes, held together with elastic cord: In theory, the...
  5. Ken Gallock

    Clifford algebra in higher dimensions

    Homework Statement Consider gamma matrices ##\gamma^0, \gamma^1, \gamma^2, \gamma^3## in 4-dimension. These gamma matrices satisfy the anti-commutation relation $$ \{ \gamma^\mu , \gamma^\nu \}=2\eta^{\mu \nu}.~~~(\eta^{\mu\nu}=diag(+1,-1,-1,-1)) $$ Define ##\Gamma^{0\pm}, \Gamma^{1\pm}## as...
  6. Telemachus

    I Sturm-Liouville theory in multiple dimensions

    Hi there. It is well known from the theory of differential equation the Sturm-Liouville theory, concerning with the eigenvalue problem: ##\displaystyle \frac{d}{dx} \left ( p(x) \frac{dy_n}{dx} \right ) - q(x) y_n(x)=\lambda_n w(x) y_n(x)## with prescribed boundary condtitions for ##a\leq x...
  7. T

    I Origin of 3 spatial dimensions

    A recent post asked "why does physical space need to be 3-dimensional?" However, the thread became infected with personal insults and was (properly) closed. I felt sorry that PeterDonis, whose posts I respect, was subjected to this abuse ! I have read several posts which start by assuming that...
  8. AuntyMatter

    I Black holes and extra dimensions

    We often hear about string theory requiring extra dimensions over and above our 4D spacetime. Does this shed any light on the singularity which is supposed to exist at the centre of a black hole? Does our normal spacetime simply join the other dimensions in their hidden (curled-up) form? This...
  9. K

    I Signatures of extra dimensions in gravitational waves

    Signatures of extra dimensions in gravitational waves David Andriot, Gustavo Lucena Gómez (Submitted on 24 Apr 2017) Considering gravitational waves propagating on the most general 4+N-dimensional space-time, we investigate the effects due to the N extra dimensions on the four-dimensional waves...
  10. Phys pilot

    Showing the invariance of Levi-Civita symbol in 4 dimensions

    Homework Statement i am showing the invariance of the Levi-Civita symbol in 4 dimensions The Attempt at a Solution $$\varepsilon_{ijkl}'=R_{im}R_{jn}R_{kp}R_{lt}\varepsilon_{mnpt}$$...
  11. B

    I Hamiltonian for spin-1/2 particle in B-field: units issue

    Take a spin-1/2 particle of mass ##m## and charge ##e## and place it in a magnetic field in the ##z## direction so that ##\mathbf B=B\mathbf e_z##. The corresponding Hamiltonian is $$\hat H=\frac{eB}{mc}\hat S_z.$$ This must have units of joules overall, and since the eigenvalues of ##\hat S_z##...
  12. Phys pilot

    I Levi-Civita properties in 4 dimensions

    first of all english is not my mother tongue sorry. I want to ask if you can help me with some of the properties of the levi-civita symbol. I am showing that $$\epsilon_{ijkl}\epsilon_{ijmn}=2!(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$ so i have this...
  13. Phys pilot

    Showing Levi-Civita properties in 4 dimensions

    first of all english is not my mother tongue sorry. I want to ask if you can help me with some of the properties of the levi-civita symbol. I am showing that $$\epsilon_{ijkl}\epsilon_{ijmn}=2!(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$ so i have this...
  14. ScrollPortals

    What's so special about higher dimensions?

    What's so special about higher dimensions? I did some youtube research and didn't find a lot of information. All I found is that beyond 5 dimensions all dimensions have only 3 platonic solids. I've got this simulation I'm working on. I don't want to go into detail for obvious reasons but so...
  15. J

    I Lorentz transformation in 2 dimensions

    Hi folks, This is the Lorentz transformation in 1D, x axis: I want to get the second term of the time t equation, I mean vx/c2, in two dimensions, I mean for a point in the XY plane. I know this term arises because if we want to syncronize a point B with the origin what we do is sending a...
  16. T

    B Are All Points on a Sphere Considered Dimensions?

    I don't have a degree in Physics but studied it in secondary school and I was wondering , Can each point on a sphere be classed as another dimension in relative comparison to like a 1 dimension plane ,two dimension plane ect. ?
  17. 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...
  18. S

    I ##SU(2)## generators in ##1##, ##2## and ##3## dimensions

    The ##{\bf su}(2)## Lie algebra in a representation ##\bf R## is defined by ##[T^{a}_{\bf R},T^{b}_{\bf R}]=i\epsilon^{abc}T^{c}_{\bf R},## where ##T^{a}_{\bf R}## are the ##3## generators of the algebra. In ##2## dimensions, these generators are the Pauli matrices ##T^{1}_{\bf 1} =...
  19. I

    Elastic Collision in two dimensions question

    Homework Statement [/B]A 2.0 kg ball moving with a speed of 3.0 m/s hits, elastically, an identical stationary ball as shown (The first ball is traveling to the right). If the first ball moves away with angle 30° to the original path, determine the speed of the first ball after the collision...
  20. M

    I Double slit experiment in 3 dimensions?

    Has anyone performed the double slit experiment in three dimensions? By this I mean, having double slits and a detection screen on three sides? This would imply there are 6 slits and 3 detection screens. What would we observe? Would we see the electron exhibiting wave like behavior (ie...
  21. D

    I Are Strings the Key to Understanding the Dimensions of Our Universe?

    If strings are one dimensional objects and they make up every particle that we know of then there is no second or third dimension. There are only one dimensional objects that seem like they are three dimensional objects because they spin as they move through space.
  22. sampsont

    I Angle between two vectors with many dimensions

    The angle between two ℝ2 or ℝ3 vectors makes sense. I have a vector like [0 0.707 1 0.707 0 -.707 -1 -.707]T (Actually my vector is A/D conversions of a sine wave from the wall power outlet. The sample rate is 4800 so there are 80 samples, separated by 1/Fs for a 60Hz sine wave. If I...
  23. D

    I Physical Dimensions of electron shells

    I want to model atoms of the periodic table using OpenGL (API for 3D graphics). I was told by a physics teacher one time that this cannot be done because it's not solvable. Can you guys confirm? Apparently only the shells of the hydrogen atom has been solved meaning that I can only model the...
  24. FallenApple

    I How does the Holographic Principle relate to 11 dimensions?

    So the Holographic Principle says that our 3 dimensonal world is encoded in the 2d surface of the cosmological horizon. What about the remaining dimensions? Are they there in the regular 3space or are they also encoded?
  25. B

    B Speed of light and higher dimensions

    will there be any effect on the speed of light ,when it travels from higher dimensions to three dimensions of space ?
  26. S

    B Perceiving Higher Dimensions: Understanding the Unknown

    Why can't we perceive higher dimensions?
  27. Kara386

    Dimensions of interaction/radiation length

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  28. A

    I Dimensions of the pion decay constant

    The pion decay constant f_{\pi} is defined by <0|\overline{d}\gamma^{0}\gamma^{5}u|\pi^{+}>=f_{\pi}m_{\pi} where I have set the momentum \boldsymbol{p}=0 (and used the temporal component of the axial vector). Now, at the right-hand-side the dimension is two (the decay constant and the mass are...
  29. P

    Calculating the dimensions of a lever and a column

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  30. H

    Coulomb's Law in Two Dimensions

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  31. S

    I Gaussian integral in two dimensions

    I am trying to evaluate the following integral. ##\displaystyle{\int_{-\infty}^{\infty}f(x,y)\ \exp(-(x^{2}+y^{2})/2\alpha)}\ dx\ dy=1## How do you do the integral above?
  32. 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.
  33. toforfiltum

    Find ratio between dimensions of can w/ largest capacity

    Homework Statement A cylindrical metal can is to be manufactured from a fixed amount of sheet metal. Use the method of Lagrange multipliers to determine the ratio between the dimensions of the can with the largest capacity. Homework EquationsThe Attempt at a Solution $$V(r,h)=\pi r^2h$$ $$2...
  34. BrainMan

    Dynamics in two dimensions problem

    Homework Statement Homework Equations xf = x0 + v0cos(θ)t + (1/2)axt2 yf = y0 + v0sinθt - (1/2)gt2 F = ma The Attempt at a Solution [/B] I first tried to find the acceleration in the x direction so I could solve for the force xf = x0 + v0cos(θ)t + (1/2)axt2 In order to find the time I used...
  35. S

    A Spinors in dimensions other than 4

    The Dirac equation describes the behaviour of non-interacting spin-##1/2## fermions in a quantum-field-theoretic framework and is given by ##i\gamma^{\mu}\partial_{\mu}\psi=-m\psi,## where ##\gamma^{\mu}## are the so-called gamma matrices which obey the Clifford algebra...
  36. F

    I Constructing dimensions out of a graph structure?

    I have a question to anyone experienced with graphs and topology. The question is relevant for this topic: https://www.physicsforums.com/threads/a-graph-based-model-of-physics-without-dimensions.887694/ Is it possible to construct an arbitrarily large graph (a set of vertices, a set of edges)...
  37. Nader AbdlGhani

    I Why Planck's Constant Has Dimensions and a Unit?

    Despite being a constant, It has both dimensions and a unit, can someone kindly explain why ?
  38. M

    I What would a universe look like if it had 2 time dimensions?

    We live in a world with 3 space dimension and 1 time dimension, and i saw a video how a universe would look like is their where n-space dimension but what would happen if their where 2 time dimensions?
  39. K

    Places to learn kinematics in one and two dimensions W/Calc

    Im not sure where to find a good place to learn kinematics in one and two dimensions. Using calculus
  40. P

    B Visualizing Spacetime: 3 Dimensions Not Enough?

    I've tried to imagine and draw the effect of a large object would have on space time in 3d but it just doesn't seem right. Would I be right to assume that maybe spacetime has more than 3 dimensions?
  41. S

    Electric field in an arbitrary number of dimensions

    I am looking to use Gauss's law to find the electric field in ##1+1## dimensional spacetime: ##\int \vec{E}\cdot d\vec{A}=\frac{Q}{\epsilon_{0}}## Now, for a point charge in ##1+1## dimensional spacetime, the Gaussian surface is the two endpoints (a distance ##r## away from the point charge)...
  42. C

    B Photon Dimensions: Wave or Particle?

    I know photon is considered a point particle, so it means it has no physical dimensions, if I'm not wrong. Plus, it has a probability wave function associated to it. It also interacts as a wave which consists of an electrical and magnetic field perpendicular to each other and delayed by a...
  43. A

    B I can't understand more than 3 spatial dimensions

    Dear Physics enthusiasts, I'm just a curious guy, I don't have any fancy credential, and in fact I don't have formal education at all (for 'legal' purposes I'm just "literate"). I'm being honest about that right away so if that's a problem (what seems to be the case for a lot of...
  44. Guaicai

    Questions about the Dimensions Analysts changing the equation

    The first equation of the left hand side is the Schrodinger Equation of the Harmonic Oscillator.The second equation of the left hand side is the new equation doing by the Dimensions Analysts. But I don't understand how the new equation come from according with Dimension Analysts of the right...
  45. M

    B What dimensions aren't geometry based?

    I.e. x,y,z
  46. M

    Products of gamma matrices in n dimensions

    Hello, i have here some identities for gamma matrices in n dimensions to prove and don't know how to do this. My problem is that I am not very familiar with the ⊗ in the equations. I think it should be the Kronecker-product. If someone could give me a explanation of how to work with this stuff...
  47. RyeLie

    String Theory in 7 dimensions?

    String Theory implies that the universe has 11 dimensions. If the 3 Dimensions represents space and the other one is time, what about the 7 dimensions??
  48. levadny

    Dynamic Arrays (One, two, three dimensions)

    Dear Friends! My post about dinamic arrays. For example little bit code. // size int const X = 5; int const Y = 4; int const Z = 3; // one dimension array printf("\nOne dimension\n"); // array of pointers int * Arr; // create array Arr = new int [X]...
  49. P

    B Influx vs. radius equation, mistake in the dimensions?

    The video:
  50. pellman

    I SU(2) and su(2) have different dimensions?

    The Lie group SU(2) is the set of unitary 2x2 matrices with determinant 1. These matrices can be written a b -b* a* Thus, as a manifold, we can think of a coordinate chart consisting of the four real numbers making up the two complex numbers a and b. It is a...
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