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.

View More On Wikipedia.org
Recent contents Top users
  1. V

    Acceleration in Two Dimensions

    Homework Statement A car initially traveling due north goes around a semicircle having a radius of 500 m at a constant speed of 20 m/s. A) How long does this take? B) What is the magnitiude and direction of the average acceleration? Homework Equations a = Δvx/Δt(xhat)+Δvy/Δt(yhat)...
  2. A

    Components of Einstein's Equations in 4 dimensions

    In this excerpt from the notes of Sean M. Carrol, he says: "Einstein’s equations may be thought of as second-order differential equations for the metric tensor field gμν. There are ten independent equations (since both sides are symmetric two-index tensors), which seems to be exactly right...
  3. Matt Benesi

    Rotation from axis to axis in 4 dimensions

    I can easily visualize the 3 dimensional way to do so, and already have. I'd like to rotate from the (-1,-1,-1,-1) -- (1,1,1,1) line to the x-axis (-1,0,0,0) -- (1,0,0,0). To do so in 3 dimensions (-1,-1,-1) -- (1,1,1) line to the x-axis (-1,0,0) -- (1,0,0) I rotated around the z...
  4. P

    Kinematics-acceleration in two dimensions

    Homework Statement A tennis player standing 8.0 meters from the net hits a ball 1.5 meters above the ground toward her opponent. The ball leaves her racquet with a speed of 25.0m/s at an angle of 14.0 degrees above the horizontal. The net is 1.0 meters high. The baseline is 12 meters back...
  5. D

    The logic of the 11 dimensions in M theory

    According to M-theory as I understand, the 7 additional spatial dimensions to our familiar 3 are curled up all around us but they are too small for us to see. However I have difficulty in understanding how this constitutes a "dimension" because dimensions allow additional degrees of freedom...
  6. J

    Dimensions of k in Nusselt Number

    Homework Statement A common dimensionless group used in heat transfer calculations is defined as: Nu=\frac{hD}{k} where h is the heat transfer coefficient, and D is the diameter of the pipe in which heat transfer takes place. Please determine the dimensions of the quantity k in...
  7. Markus Hanke

    Metric of Manifold with Curled up Dimensions

    Would someone here be able to write down for me an example of a metric on a manifold with both macroscopic dimensions, and microscopic "curled up" dimensions with some radius R ? Number of dimensions and coordinates used don't matter. Not going anywhere with this, I am just curious as to how...
  8. P

    What Are the Dimensions for Force and Distance in This Problem?

    Homework Statement Let P represent a force and x, y, and z represent distances. Determine the dimensions for each of the quantities listed below. I attached the problem I need help with. (it's a small picture) So I'm a bit confused since the it involves both dx and dy. The dimensions for P...
  9. W

    Equation z = ct + d, z is measured in meters and t in seconds; dimensions of D?

    Homework Statement In the equation z = ct + d, z is measured in meters and t is measured in seconds. What are the dimensions (units) of d? answer options are... s/m, m/s, m, s, m*sHomework Equations The Attempt at a Solution plugged in m for z, s for t, and m/s for c. solved for D and got 0.
  10. P

    Kinematics in 2 dimensions; position vector problem.

    Homework Statement "A particle moves in the horizontal plane that contains the perpendicular unit vectors i and j. Initially it is at the origin and has velocity 18ims^-1. After accelerating for 10 seconds its velocity is (30i + 8j)ms^-1. Assume that the acceleration of the particle is...
  11. A

    Spatial dimensions inside a black hole

    hello all I am so glad to have found this forum. I've always had an interest in astrophysics, cosmology, SR/GR, etc, and no place to ask questions. I'm an engineer and was once a member of Mensa (I only left the organization because I thought other members were crazy. Sorry). So although I'm...
  12. haael

    Are anyons possible in higher dimensions?

    Weinberg wrote that in 3D and higher spaces all particles must be bosons or fermions. The proof used the fact that particles are really indistinguishable i.e. we can't "mark" any particle and the mathematical replacement of two particles of the same type should not change any physical...
  13. P

    The space between a unit 'sphere' in n dimensions within an n-dimensional cube

    If we have a unit circle within a square s.t. the square touches the circle in 4 places then the biggest gap we can find is just √2 - 1. Doing a similar thing with a sphere in a cube we get √3 - 1 I've heard the n-dimensional analogue is √n - 1. Which is crazy as it means the gap is bigger...
  14. R

    Problem with Understanding Dimensions

    Here's what i know about Dimensions... Infinite stacks of a 1D world makes a 2D world, And Infinite stacks of a 2D world makes a 3D world and Infinite stacks of 3D world makes a 4D world and so on... So if we humans are 3D creatures, then we perceive 4th D as time, Same goes for 4D creatures...
  15. H

    Does the D'Alembertian have the same dimensions

    Homework Statement I will get to it. Homework Equations None. The Attempt at a Solution Does the D'Alembertian have the same dimensions 1/length as the Laplacian operator except the D'Alembertian takes into consideration four dimensions?
  16. S

    Diffusion Equation on a plate - 2 dimensions

    Homework Statement The edges of a thin plate are held at the temperature described below. Determine the steady-state temperature distribution in the plate. Assume the large flat surfaces to be insulated. If the plate is lying along the x-y plane, then one corner would be at the origin...
  17. I

    Dimensions of klein-gordon field

    Does anyone know how to write the classical solution to the Klein-Gordon equation in NON natural units? Where do all the c's and h's go?
  18. C

    Where will you find mathematics of n spacial dimensions?

    For example, finding the properties of an n dimensional figure. Is this called something in math, or do I just refer to it as 'geometry in multiple dimensions'? What subjects can I find this topic under?
  19. bcrowell

    Exploring Torsion in 2 Dimensions: A Differential Geometry Perspective

    This is really more of a differential geometry question than a GR question. Is it possible to have torsion in two dimensions? I've seen statements implying that torsion is only of interest in 3 or more dimensions, but I don't see why. My understanding is that in two dimensions, you could not...
  20. S

    Tool for measuring inner dimensions of a tube from an offset position

    Hi, I'm trying to measure how far a rod that passes through a tube is offset from the center. Assume that the axis of the tube runs parallel to that of the rod. I can only access the interior of the tube from above (it hangs vertically), and the access hole I have is smaller than the radius...
  21. K

    Product Groups and their dimensions

    My understanding was that the product of two groups A and B will yield a group C for which the dimension of C is dim(A)*dim(B). Now however, the author I'm reading defines the group product multiplication as: (a1, b1) * (a2, b2) = (a1*a2, b1*b2), for a1,a2 in A and b1, b2 in B. Does this...
  22. MathematicalPhysicist

    Empirical test for extra curled dimensions.

    How would you test empirically for the existence of extra curled dimensions of space?
  23. J

    Other dimensions have been proven .

    Other dimensions have been proven... Since other dimensions have been proven both mathmatically and at cern, I do not consider the topic of warping out of our current space/time dimension out of the realm of possibility especially if it is done on an atomic or quantum level initially. A carrier...
  24. H

    What are the dimensions of an integral in terms of energy and time?

    Homework Statement Homework Equations The Attempt at a Solution Just a question on dimensional analysis here. I believe that when an integral is taken with respect to time for instance, dt appears as a dummy variable yes? Imagine we had \int E\ dt Does this have dimensions of energy times...
  25. A

    Wess Zumino model in two dimensions

    Homework Statement Hi! I need some help to describe a Wess Zumino model in two dimensions: spinors are real (because of the Majorana condition \theta=\theta^{\ast}) and have two components; the superfield is: \phi \left( x,\theta\right)=A\left( x\right) +...
  26. S

    Dimensions of Universe: Exploring the Past and Speculating the Future

    In the distant past at bigbang the universe was very hot and dense. Can i say it was dimensionless then? If so did it come to this 3 dimension through 1 and 2?a little more speculation... is/will it go to higher dimensions?
  27. J

    How accurate is this video about higher dimensions?

    I thought it was pretty straightforward until 4:26, until they started making some questionable claims. I'm no expert so I thought I'd ask here. http://www.youtube.com/watch?v=0ca4miMMaCE
  28. A

    Mass dimension of a scalar field in two dimensions?

    Which is the mass dimension of a scalar filed in 2 dimensions? In 4 dim I know that a scalar field has mass dimension 1, by imposing that the action has dim 0: S=\int d^4 x \partial_{\mu} A \partial^{\mu} A where \left[S\right]=0 \left[d^4 x \right] =-4 \left[ \partial_{\mu} \right]=1...
  29. T

    The dimensions of elementary particles in quantum physics and string theory

    I noticed that in quantum physics, an elementary particle has no dimensions, and is point like, but in string theory has one dimension. Why is this?
  30. K

    Quantum tunneling and extra dimensions

    Hello. Okay, so there are some theories like string theory that say that there are extra dimensions of space that we cannot see. Extra dimensions if they exist would allow more degrees of freedom in movement. This website demonstrate how barriers insurmountable in a lower dimension would be...
  31. J

    Dimensions of a room to be electrically heated

    A room 7m x 6m x 3m high is to be electically heated to maintain the inside temperature at 20 C, when it is 0 C outside. The room has 2 windows each 1.2m x 1.8m and a sliding glass door 2.4m x 2.1m. The transmission coefficients u in Wm^2/C are given below. Floor: concrete slab (100mm thick)...
  32. J

    Dimensions - Disc Brake Surface Area

    Dimensions -- Disc Brake Surface Area During frequent braking under race conditions the disk brake rotors on the car described above reach a temperature of 500C. These disk brakes rely on forced convection to cool them. The dimensions of each disk rotor are: outer radius 130mm; inner radius...
  33. Demystifier

    String theory in arbitrary number of dimensions

    A long time ago, physicists thought that only a small class of quantum field theories (QFT's) makes physical sense - those which are renormalizable. But then gradually it became accepted (Weinberg was the most influential figure in that regard) that QFT does not really need to be renormalizable...
  34. H

    The dimensions of something please

    I read about this expression for the Coriolis force \frac{\omega c}{\sqrt{G}} Would I be right in saying this has dimensions of force? Thank you!
  35. P

    Coordinate transformation of a tensor in 2 dimensions

    Homework Statement Given a symmetric tensor T_{\mu\nu} on the flat Euclidean plane (g_{\mu\nu}=\delta_{\mu\nu}), we want to change to complex coordinates z=x+iy, \,\overline{z}=x-iy. Show, that the components of the tensor in this basis are given by...
  36. P

    Converting real life dimensions of a canvas into video file dimensions.

    Hey people, 1. The problem statement, I'm doing a Uni Art assignment, and I'm going to be projecting a video onto a canvas (which I don't know the exact dimensions of yet). Video file dimensions are stated as 1080 x 980 and so forth, whereas the real life canvas will be something like 50...
  37. G

    Dimensions and Degrees of Freedom

    I had three questions that had to do with spacetime, digital physics, and the holographic principle: 1. Why are there dimensions in space? 2. Why isn't there just some sort of information matrix as suggested in digital physics? 3. How do these two ideas work with the holographic principle?
  38. C

    Length of wire in a coil of known dimensions?

    suppose that i have a wire with a known thickness ... and i want to wind it around a solid cylinder with known dimensions, and in the end make a coil with the same length as the solid cylinder how do i relate between the length of wire required and the diameter of this coil?
  39. A

    So, no prefixes...Do planes in higher dimensions satisfy Euclid's definition?

    A problem in Linear Algebra by Jim Hefferson: Euclid describes a plane as \a surface which lies evenly with the straight lines on itself". Commentators (e.g., Heron) have interpreted this to mean \(A plane surface is) such that, if a straight line pass through two points on it, the line...
  40. Vorde

    Half Question/Half-Challenge: Dimensions of of the Earth-Moon System

    I've been considering this problem for about a year now (whenever I remember about it, that is), and I've come to the conclusion that I can't figure out a way to do what I want to see is possible, and so I've decided to ask it here to see if anyone else can. The original question I...
  41. E

    Fermion Current Commutators in 2 dimensions

    Homework Statement Given the current: J^{\epsilon}_{0} (t,x) = \overline{\psi_{L}}(t,x + \epsilon) \gamma^{0} \psi_{L}(t,x - \epsilon) = \psi_{L}^{\dagger} (x + \epsilon) \psi_{L}(x - \epsilon) with \psi_{L} = \frac{1}{2} (1 - \gamma^{5}) \psi_{D}. Use the canonical equal time...
  42. J

    Statisics - linearity and best-fit in 3 dimensions

    Howdy folks I've gotten a number of answers to this in various fora, some contradictory. I need to do 3 things to a set of datapoints in 3space (X,y, and z real values). 1)Test for linearity (pearson's R?). 2)If passed, find line of best fit (SSE?) 3)See if line of best fir is...
  43. S

    Finding the equation of a straight line in 3 dimensions.

    Homework Statement Prove that the shortest path between two points in three dimensions is a straight line. Write the path in the parametric form: x=x(u) y=y(u) z=z(u) and then use the 3 Euler-Lagrange equations corresponding to ∂f/∂x=(d/du)∂f/∂y'. Homework Equations Stated them...
  44. P

    Solved: Dimensions of b/a in Pressure Equation

    Homework Statement Assuming that Pressure (P) of a particle is given by P = b - t^2 / ax where t = time, x = position Find the dimension of b/a. Homework Equations The Attempt at a Solution Using dimensional homogenity, I know that b represents time interval.
  45. S

    Einstein: Balancing 3D Positions & Time in E=mc2

    If Eisenstein taught us the space time is one and the same thing, then how did he mathematicaly prove that. How did he put dimentions into his E=mc2 formula. The first 3 dimensions are strictly positions, positions in 3D require 3 numbers, and time is linear movement in one direction, the...
  46. N

    Mathematica Mathematica: NDSolve in 3 dimensions

    Hi I am trying to solve Newtons equation for a particle in (x, y, z) using NDsolve. Here I what I have so far: sol = NDSolve[{ x''[t] == acceleration[x'[t], y[t], z[t]], y''[t] == acceleration[y'[t], x[t], z[t]], z''[t] == 0, x[0] == 2, x'[0] == 0, y[0] == 0, y'[0] ==...
  47. D

    Question regarding possible extra time dimensions

    Hello, I am an amateur physics enthusiast and would like to pose a question to you. It is my understanding that for string theory to make sense there should be 11 dimensions. Three extended spatial dimensions and one time dimension that we perceive, as well as seven hidden dimensions which...
  48. R

    Exploring Vector Spaces and Dimensions: A Homework Challenge

    Homework Statement Homework Equations From my notes I'm aware of the following equation: dim(A + B) = dimA + dimB − dim(A ∩ B).The Attempt at a Solution I'm assuming part of the solution involves the equation above and rearranging it but I'm not sure how I would determine dim(A + B). I also...
  49. H

    My Professor frequently uses the term higher dimensions. Could someone

    My Professor frequently uses the term higher dimensions. Could someone tell me exactly what is a dimension. I think of it as some parameter which can vary. Also, I only know of 4 dimensions: time, length, width and height. Could someone give me more examples of dimensions? And lastly could...
  50. Math Amateur

    Finite Reflection Groups in Two Dimensions - R2

    I am seeking to understand reflection groups and am reading Grove and Benson: Finite Reflection Groups On page 6 (see attachment - pages 5 -6 Grove and Benson) we find the following statement: "It is easy to verify (Exercise 2.1) that the vector x_1 = (cos \ \theta /2, sin \ \theta /2 )...
Back
Top