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.

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  1. B

    Info on wave mechanics prob in one dimension need

    Hi there, i need help in a couple of questions that I'm just stumped one of them : A) use induction to show that [ x (hat)^n, p(hat) sub "x" ] = i (hbar)n x(hat)^(n-1) - so far I've figured out this equation is in relation to solve the above eq, but I'm not entirely sure how to connect the...
  2. C

    Finding a Basis and Dimension of Set W in R^4

    The set W={(a, 2a, b, 0)| a, b eR} is a subspace of R^4. Find a basis for W and find dimW
  3. B

    Exploring Subspaces and Dimension in Z_2^3

    Hi, I'm wondering how I would decide how many "subspaces of each dimension Z_2^3 has." The answer is: 1 subspace with dim = 0, 7 with dim = 1, 7 with dim = 2, 1 with dim = 3. I'm looking for subsets of Z_2^3 which are closed under addition and scalar multiplication. An arbitrary vector in...
  4. M

    Is Time Truly the Fourth Dimension or Something Else Entirely?

    I'm just curious to see what other people in this group have to say about this theory I have. I used to think that time was the fourth dimension. It seemed pretty logical due to the way dimensions are built up. The zero dimension is a point, the first dimension is an infinite number of...
  5. M

    How Can You Conduct a Simple Single Dimension Motion Lab Under Time Constraints?

    I need to create a lab today for a quarterly but I don't have time to acctually experiment, what is something I could easily BS and get a good grade?
  6. Greg Bernhardt

    Motion in One Dimension tutorial

    Author: Dr. Donald Luttermoser of East Tennessee State University III. Motion in One Dimension A. Displacement. 1. The displacement of an object is defined as the change in its position. 2. It is given by the difference between its final and initial coordinates: x  xf − xi = displacement...
  7. P

    What Is the Dimension of the Vector Subspace Spanned by e^x and e^2x?

    The functions e^x and e^2x I have to find the dimension of the vector subspace spanned by this set. Im not sure where to start, I do know how to solve other problems asking the same question just different function. Any help would be greatly appreciated. thanks a.p
  8. J

    Motorboat Downstream Velocity: Point A to 6km Away

    A Motorboat Going Downstream Overcame A Raft At Point 'a' . 60 Minutes Later It Turned Back And After Some Time Passed The Raft At A Distance 6 Kilometer From Point 'a'.find The Flow Velocity Assuming The Duty Of Engine To Be Constant
  9. C

    One Dimension Free-Fall Problem

    5. Sue is watching Hugh 7.2 meters below her when she sees him throw a ball up to hit her. She pulls in her head, but Hugh purposely threw the ball hard enough to hit her on his way down 1 second after it passes her on the way up. Explain how hugh figured this out... I used a sort of system...
  10. P

    Good problem on motion in one dimension

    PLEASE SOLVE THIS WITHOUT USING THE CALCULUS a man starts walking from a point P. after t seconds another man starts from the same point. they reach the nearer end af a bridge such that the time interval between them is T seconds. the length of the bridge is L meters. they reach the other...
  11. J

    Speed of Motorboat Going Downstream at Point 'a': 6 km/hr

    A Motorboat Going Downstream Overcame A Raft At Point 'a' . 60 Minutes Later It Turned Back And After Some Time Passed The Raft At A Distance 6 Kilometer From Point 'a'.find The Flow Velocity Assuming The Duty Of Engine To Be Constant
  12. B

    Find the Shortest Distance Between Two Lines: L1 and L2

    Hi, Can someone help me with this: What is the shortest distance between the two lines: L1: r= (1,0,0) +t( 2,3,4) L2 u= (2,1,0) +s(1,2,0) Thank you very much for your help B
  13. M

    How Far Will You Go Before Stopping? Can You Avoid Hitting the Dog?

    You are driving your new sports car at a velocity of 90km/hr, when you suddenly see a dog step into the road 50m ahead. You hit the brakes hard to get maximum deceleration of 7.5m/s^2. How far will you go before stopping? Can you avoid hitting the dog? I got 540m, which I know is wrong. V=0...
  14. T

    Complex Plane Dimension Number

    What dimension between space-time and 11 Dimensions is allocated to the complex plane 'visualized' and used in complex number theory? Complex numbers are used in every branch of maths and physics, based on an imaginary complex plane, z = x +iy where y is an imaginary axis and i^2=-1. - It's...
  15. S

    Calculating Acceleration and Displacement in 1D Motion

    :confused: :confused: Help! I need to know how to work this problem! A motorcyclist moving with an initial velocity of 8.0 m/s undergoes a constant acceleration for 3.0s, at which time his velocity is 17.0 m/s. What is the acceleration, and how far does he travel in the 3.0 s interval?
  16. B

    Solving Dimension Matching Problem with Brad - 65 Characters

    Hi, I have a problem about dimension matchig. the dimensional relation between the period τ , the pendul length l, and the acceleration of the gravity g takes the form: [ τ ]=[l^r] [g^s] Use the fact that the dimendion of τ is [T], that of l is [L], and that of g is [L/T^2] to...
  17. S

    Confused about fourth spatial dimension

    lets consider a sphere and let there be a two dimensional man and for him he is living on a two dimensional place.we r observing it from third dimension. now this two dimensional man be fred. one day fred decides to make a circle using a rope.now start imagining he is on a sphere and he tkaes...
  18. S

    Is Time the Same for All Observers in Different Dimensions?

    Is ten minute for an observer in different dimension, with the same speed relatively, same for every observer?
  19. C

    General Questions: Motion in One Dimension

    1. Can you think of physical phenomena involving the Earth in which the Earth cannot be treated as a particle? I would say the rotation of the earth. Would there be anything else? A particle is something in which rotational and vibrational considerations are disregarded. What would be a...
  20. A

    Kinematics - One Dimension Problem

    The problem is stated as follows: A typical automobile has a maximum deceleration of about 7m/s^2; the typical reaction time to engage the brakes is 0.50 s. A school board sets the speed limit in a school zone to meet the condition that all cars should be able to stop in a distance of 4...
  21. S

    Understanding Dimensions: The Interplay of Mathematics and Philosophy

    yes I have read the wikipedia definition and followed my nose around to see how it gets interpreted in various models and theories but it doesn't say why it is that when 3 come together it forms a sphere and creates an instant of time ? how can one dimension exist independently of others and...
  22. E

    Help Me Relative Motion In One Dimension.

    Help Me...Relative Motion In One Dimension. What Is the relative motion in one dimention? Drive the Acceleration equation.
  23. C

    Calculating Acceleration from Change in Velocity and Distance

    An electron with initial velocity v_{x}_{o} = 1.0 \times 10^{4} meters/sec enters a region where it is electrically accelerated. It emerges with a velocity v_{x} = 4.0 \times 10^{6} meters/sec. What was its acceleration, assumed constant? The accelerated region is 1 cm. So acceleration is...
  24. C

    Three dimension space and one dimension time means

    Can someone tell what are dimensions and what does the line "Three diemsion space and one dimension time means". please. :confused:
  25. E

    Linear Transformation: Kernel, Dependence, Dimension, Basis

    This is probably a simple question, but just to be sure: if the kernel of linear transformation is {0}, then the set is linearly dependent since 0-vector is LD, right? So dimension is 0, right? Then what's the basis of kernel? No basis? thanks in advance.
  26. B

    Inelastic Collisions in One Dimension

    Here's my problem: Car A (mass 970 kg) is stopped at a traffic light when it is rear-ended by car B (mass 1600 kg). Both cars then slide with locked wheels until the frictional force from the slick road (with a low mk of 0.23) stops them, at distances dA = 5.8 m and dB = 3.6 m. What are the...
  27. D

    Funny definition of dimension of a topological space

    I'm reading this book (Hartshorne), and it uses a funny definition of topological dimension, which I'm having a hard time convincing myself is the usual one. The definition is as follows: dim X is the supremum of natural numbers such that there exists a chain Z_0\subset Z_1\subset \dotsb...
  28. S

    What Are the Different Definitions of Dimension in Mathematics?

    Dimension, what does this mean?! and Has anyone heared about 2-d matter?
  29. S

    What is the Definition and Significance of Dimension in Mathematics?

    Has the Dimension a mathematical defination?
  30. B

    Why Don't the Vectors (1,2,-5), (3,8,-19), and (1,-2,3) Span R^3?

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  31. S

    Why Can't I Use V = Vo + at to Calculate Time for a Car Falling Off a Cliff?

    ok I've just cracked the ****s, for this question: A car rolls gentley(Vo=0 m/s) off a vertical cliff. How long does it take for it to reach 100km/hr. why can't i use V = Vo + at where: V= 100 Vo = 0 a = 9.8(gravity) t = ? the only equations I've been given to do this are: V...
  32. F

    Why are dimension > 4 operators non-renormalizable?

    Hi everyone--I'm curious why terms in the Lagrangian with mass dimension greater than four are "nonrenormalizable." I understand that the action must be dimensionless, hence the Lagrangian [density] has mass dimension 4. However, in effective field theories, we can end up with terms of...
  33. D

    Nontrivial Subspace with Equal Dimensions: Linear Algebra Example

    Hi, I'm just learning for my linear algebra exam and I wonder if somebody could give me an example of a nontrivial subspace which has as many dimensions as the original space. Thanks a lot
  34. D

    Can a Nontrivial Subspace Have the Same Dimension as Its Original Space?

    Hi, I'm just learning for my linear algebra exam and I wonder if somebody could give me an example of a nontrivial subspace which has as many dimensions as the original space. Thanks a lot
  35. G

    Are Other Dimension Universes Plausible?

    I mean is it somewhat science or is it purely sci-fi?
  36. M

    Dimension of Soln Space of Heat Equation: Is It Infinite?

    What is the dimension of soln space of the heat equation: \frac{\partial U }{\partial t}=a^2\frac{\partial^2 U}{\partial x^2} U(0,t) = U(L,t) = 0 U(x,0)= f(x) Is it infinite , if so why?
  37. marcus

    The running of the spacetime dimension (AJL)

    we are familiar with the "running" of coupling constants, as for instance in QCD or even in QED where alpha (the electrodynamic coupling) is about 1/137 at long distances and low energies but increases to about 1/128 as energies increase and distances diminish. Who would have imagined that...
  38. D

    Kinematics in one dimension (easy problem)

    Im new to this and i know this must be a very simple problem. I can't seem to find any formulas for distance or a way of doing this problem. Most of the help sites give you the distance and want you to find something else. Could someone help me understand what i need to do to solve this. Thank...
  39. S

    Proving "Quantitized Dimension" Theory: Answers Needed

    Recently I was thinking about these statement: 1. Dimension is as if a 3-D graph, which every coordinate is the smallest unit for dimension. 2. Volume of an object is a whole number multiply the volume of the dimension unit. 3. Distance is a whole number multiply the length of the dimension...
  40. M

    Relativistic velocity in the time dimension

    I would be interested to hear what others think of the general concept of relativistic velocity in the time dimension, analogous to spatial velocity. To get going, consider the Minkowski velocity 4-vector: \gamma(c,v_1, v_2, v_3) The time component of this 4-velocity is \gamma c. With...
  41. S

    What is a black hole in 4th dimension looks like?

    If there is a black hole in the 4th dimension, how will it looks like in the observer in 3rd dimension?
  42. JamesU

    Exploring the 3rd Dimension: A Deeper Understanding

    While we're in a 3-D world, we can only see two dimensions at a time. By seeing all three dimensions, we would be able to see every face of every object in our world. But a lot of people, when I explain that to them, don't accept it and say it's false. Can anyone come up with a good explenation...
  43. T

    Thoughts on visualizing the 4th dimension

    Just an idea: It occurred to me that people can't see the fourth dimension because, in reality, we can't even see the third. Think about it. When you're looking at something, each of your eyes registers a 2D image, and the sense of depth (the third dimension) is inferred from the two 2D...
  44. E

    Understanding Multiplicity and Dimension in Eigenvectors

    Hi all, I have a homework problem that I would like someone to check: this relates to the eigenvectors: in the problem we are given characteristic polynomial, where I put x instead of lambda: p(x) = x^2*(x+5)^3*(x -7)^5 Also given A is a square matrix, and then these questions (my answers)...
  45. JamesU

    Can we comprehend the 4th dimension.

    If we exclude time as the fourth dimension, is it possible that over many years of trial and study, that our thee-dimensional brains can comprehend a higher dimension? :bugeye:
  46. S

    Can Dimension and Time be quantized?

    Can Dimension and Time be quantized? For example, distance and time is a whole number multiple a smallest unit. p/s: I don't have any facts to support my question, so this is possibly one of my illogical thought :smile:
  47. E

    How many 3D cubes can fit inside a hypercube in R^4?

    let suppose we have an hypercube in R^4 then m y question is how many 3-dimensional cubes could we put inside our hypercube?...
  48. A

    Volumes in the 4th spatial dimension

    How would you calculate the volume of a 4-dimensional object? Like a hypercube, hypersphere, etc...
  49. G

    8 Dimension creatures roaming the universe?

    Does this mean there are 8 Dimension creatures roaming the universe?
  50. N

    What do we call the dimension measured with time?

    I am told time is a dimension, indeed the 4th dimension but then I am also told time is effected by other things such as gravity These two seem to be at odds with each other. the three classical dimensions are length breadth and width. yet no one I know of argues that gravity or any...
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