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. J

    Time as a Dimension: Is Time Expanding?

    This may be a dumb question, but if the universe is expanding and time can be treated as just another dimension, does this mean that time is 'expanding'? And if so, what the heck does that mean?
  2. D

    Fortran How many bytes in DIMENSION? - A Fortran Question

    How many bytes in DIMENSION?? - A Fortran Question I'm looking at some Fortran code. Some variables are of type dimension. How do I tell if these variables are integers or floating points numbers? How do I know if they are signed or unsigned? How do I tell the number of bytes occupied by...
  3. J

    Vector space dimension of little-l 2

    Homework Statement I'm trying to understand why \ell_2^\infty as a vector space over \mathbb{C}, has uncountable dimension. Homework Equations The Attempt at a Solution Firstly, I'm not really clear on the meaning of basis in infinite dimensions. Is it still true that any element...
  4. L

    Football player kinematics in one dimension

    Homework Statement A football player, starting from rest at the line of scrimmage, accelerates along a straight line for a time of 3 seconds. Then during a negligible amount of time, he changes the magnitude of his acceleration to a value of 1.1 m/secondssquared. With this acceleration, he...
  5. M

    Kinematics in two Dimension- projectile motion

    I am really having trouble solving this problem. Can someone help ASAP?? An airplane with a speed of 87.9 m/s is climbing upward at an angle of 59.8 ° with respect to the horizontal. When the plane's altitude is 652 m, the pilot releases a package. (a) Calculate the distance along the ground...
  6. B

    Two dimension relative morion help

    Two dimension relative morion help! Homework Statement After flying for 15 min in a wind blowing 42 km/h at an angle of 18° south of east, an airplane pilot is over a town that is 60 km due north of the starting point. What is the speed of the airplane relative to the air? Homework...
  7. Robert100

    How is 3-space curved without a fourth spatial dimension?

    When I was studying physics in college I learned two contradictory ways to understand Einstein's theory of general relativity (GR). The most common way was to visualize our three-dimensional space as being distorted by mass. Anything with mass created a distortion in spacetime. Just as a...
  8. K

    Rank, Dimension, Subsapce, Column Space

    1) True or False? If true, prove it. If false, prove that it is false or give a counterexample. 1a) If A is m x n, then A and (A^T)(A) have the same rank. 1b) Let A be m x n and X E R^n. If X E null [(A^T)(A)], then AX is in both col(A) and null(A^T). [I believe it's true that AX is in...
  9. G

    Exploring the 6th Dimension: A 10th Grader's Perspective

    Ok, so I'm new to this message board and I'm only in 10th grade so please bere with me. So being from a small town I've decided to take on a big project and write a research paper on the Sixth Dimension. Here is my understanding of the dimensions so far: Zeroth Dimension= a point, consisting of...
  10. A

    Vector Spaces, Dimension of Subspace

    Find the dimension of the subspace spanned by the vectors u, v, w in each of the following cases: i) u = (1,-1,2)^T v = (0,-1,1)^T w = (3,-2, 5)^T ii) u = (0,1,1)^T v = (1,0,1)^T w = (1,1,0)^T Right, how do I go about this, do I have to find the subspace first then do the dimension. Can...
  11. S

    Why is Planck time measured differently from other dimensions?

    Hi, I'm new here. I've loved all the qualitative aspects of physics for a while now and I'm just starting to get into some of the more quantitative aspects of physics, the actual math. I'm such a complete amateur at all this and I'm only in high school so please excuse my question if it seems...
  12. C

    Understanding Dimensions to the Basics of Dimensional Concepts

    What exactly is a "demension"? Correct my spelling if it's spelled wrong. Thanks
  13. A

    Energy as Dimension: Exploring the Possibilities

    What would happen if we consider Energy as Dimension ? on other way can we consider Energy as Dimension ??
  14. wolram

    Mathematicians Map 248-Dimension Structure of E8

    I am not sure if this is of any interest. http://www.newscientist.com/article/dn11410-mathematicians-finally-map-248dimension-structure.html Tools Enlarge image The E8 root system consists of 240 vectors in an eight-dimensional space. These vectors are the corners...
  15. P

    A completely different look at the 4th dimension

    I have an idea that completely renivents the idea of the 4th dimension that has absolutely nothing to do with time. BUt however, it will be easier to first think about the lower dimensions. imagine a perfectly 2 dimensional universe which can be represented with a xy plane where -10 meters <...
  16. D

    Need Help about Lebesgue Covering Dimension

    guys I'm currently working on a paper that contains of topological dimension. as i dashed around the internet i still haven't figured out the way to define the dimensions. and if it's not much of a burden for you guys, could you explain to me in simple words. thanks guys 1.Cantor's proof that...
  17. J

    Understanding Span, Basis, and Dimension to Linear Algebra Concepts

    I need some help to understand the concepts of span, basis, and dimension. 1-How do you determine if a set of vectors [in matrix form] span a region? -Do you set the given matrix set to arbitary numbers and see if there is a unique, infinite, or no solution? ie, set a 4X4...
  18. F

    Kinematics in 1 Dimension: Solving Basic Problems

    Hello! Could you please check for errors in the following HW problems? Thank you! A. Homework Statement A rolling ball moves from x1=3.4 cm to x2=-4.2 cm during the time from t1=3.0 s to t2=6.1 s. What is the average velocity? Homework Equations and The Attempt at a Solution Average...
  19. G

    Dimension of angular displacement, velocity and acceleration

    Homework Statement I have a homework question that I do not understand. I have searched everwhere for the answer. I think I did not find the answer because I am do not understnad waht the question is asking for. Some help would be greatly appreciated. The question is: What are the...
  20. J

    Time a fourth dimension? is a boost a rotation?

    is a boost a rotation of a four vector? if the boost rotates an object more into the time dimension, ywo things happen: it's velocity increases. it's length contract in the direction of its motion. does this make sense? thanks!
  21. D

    The Fourth Dimension: Introduction to a Book

    http://sprott.physics.wisc.edu/pickover/fourth.html I just read this, and it makes no sense. In my opinion of the fourth dimension all that is written here is garbage. What do you get out of this? It is an introduction to a book.
  22. K

    Wave packet propagating in one dimension

    I am working on a problem that goes "Show that for a wave packet propagating in one dimension, for a free particle Hamiltonian" m d<x^2>/dt = <xp> + <px> What I think I want to do. Use d<A>/dt = i/hbar * <[H,A]> Which leads me to d<x^2>/dt = i/hbar * <[H,x^2]> For the free...
  23. C

    One and Two Dimension collisions

    My question is not so much what to do it is just i have problems solving for a certain equation. A 5.00kg ball, moving to the right at a velocity of 2m/s on a frictionless table, collides head-on with a stationary 7.50kg ball. Find the final velocities of the balls if the collision is (a)...
  24. M

    Exploring the Possibility of a 4th Dimension

    Does a dimention other than the third dimention possibly eixist? For some thing to exist doesn't it need thickness width and length?.People say time is the 4<sup>th</sup> dimention is there any kind of proff for it:confused: .
  25. S

    What is the role of time as the fourth dimension in scientific theories?

    i came across this article that says time is considered the 4 dimension(i think) and check up on wikipedia but doesn't know what they are talking about.so does anyone know where i can read up on this or better explain to me? :rolleyes:
  26. S

    What is the Dimension and Linear Independence of Subspaces?

    I'm having trouble understanding bases outside of R^n; I've got a handle on those I think. Here are some examples: 13. In C(-Pi,Pi), find the dimension of the subspace spanned by 1, cos(2x), and cos^2(x). 14. Find the dimension of the subspace of P3 spanned by x, x-1 , x^2+1, x^2-1...
  27. A

    A few problems on MOTION IN ONE DIMENSION

    A few problems on "MOTION IN ONE DIMENSION" INTRODUCTION: Here are a problem on "Motion in one-dimension". THE PROBLEMS: (1) "Car A is just behind Car B.Both are 5m long.The velocity of A is 15 m/s while that of B is 10 m/s. Find the road distnace covered by Car A to cross Car B." WHAT MY...
  28. C

    How Can I Dimension an Electromagnet for Maximum Strength?

    Hi, I like to research the possibility to build an electromagnet that is quit big and has a lot of strength. As I really don't know much about magnetism, I don't see how to start with this. Anyone that please can help me with some formulas? I like to make a sort of excel or MATLAB...
  29. O

    Kinematics problem (one dimension)

    Hello, I am a tenth-grader physics student, and I have a mechanics-realted problem you may think is quite trivial, but I have been unsuccessfully trying to solve it for a whole day, so I would thank you if you answered. Please consider the attached graph; the question relates to it: Two...
  30. E

    What dimension does process measure

    When someone says their transistors are a "90nm process" (or similar) I always assumed they meant the minimum gate length their fabrication process could produce. But then while reading this: http://www.intel.com/technology/silicon/65nm_technology.htm I saw this: "Intel's 65nm...
  31. B

    Exploring Probability Spheres in Fractional Dimension

    I have heard of such idea: A sphere of fractional dimension 0<s<1 is understood as a probability sphere with probability s to have an electron at a certain position for example the volume of the sphere S^{n-1} in \Re^n has volume Vol(S^{n-1})= (2\Pi^{n/2})/(\Gamma(n/2) and we can...
  32. T

    Projectile motion in 1 dimension problem

    Hi, having trouble with the following problem: A stone is thrown vertically upward from a bridge 30.0m high at an initial velocity of 15.0 m/s . How long will it take for the stone to hit water? I set it up first by writing my givens down: initial velocity = 15.0 m/s. distance (x)...
  33. D

    There's a question of dimension

    As known,we can use 3 basic dimensions to represent other dimensions,but how to represent I with L,M,T?
  34. G

    Is the Universe Expanding in the Time Dimension?

    A thought has been bother me for a while. Is the universe expanding in the time dimension? Let me explain. The future does not exist yet, the future is a concept, an artifact of the human mind. The past (from the time perspective) don't exist anymore, yet we can see the evidence of its passage...
  35. T

    Dimension of V in 2x2 Hermetic Matrix

    V is the vector plane of all hermetic matrix in order of 2x2. what is the dimension of V?
  36. P

    Elementary physics in 1 dimension

    The driver of a car wishes to pass a truck that is traveling at a constant speed of 19.0 m/s. Initially, the car is also traveling at a speed 19.0 m/s and its front bumper is a distance 23.7 m behind the truck's rear bumper. The car begins accelerating at a constant acceleration 0.550 m/s^2...
  37. J

    What is the Cardinality and Dimension of \mathbb{Z}^{3}_{7}?

    Find the cardinality and dimension of the vector space \mathbb{Z}^{3}_{7} over \mathbb{Z}_{7}. \mathbb{Z}^{3}_{7} = \{ (a,b,c) \; | \; a,b,c \in \mathbb{Z}_{7} \}. Then since \mathbb{Z}_{7} is a field 1 \cdot a = a \; \forall \; a, so B = \{ (1,0,0), (0,1,0) , (0,0,1) \} is a basis of...
  38. R

    Motion in one - dimension question

    An indestructible bullet 2.00 cm long is fired straight through a board that is 10.0 cm thick. The bullet strikes the board with a speed of 420 m/s and emerges with a speed of 280 m/s. (To simplify, assume that the bullet accelerates only while the front tip is in contact with the wood.) the...
  39. Y

    Time as Dimension: Unveiling the Mystery

    I just do not understand why it is so essential for time to be a dimension. I'm by no means a specialist in this field, and I don't really know the core workings of relativity, but why is it that time must be, or at least is almost always referred to as a dimension? The only way I seem to be...
  40. N

    What is the dimension of the spacetime interval?

    Just what the title says. In the book Spacetime Physics, by Taylor and Wheeler, the time coordinate is measured in metres of light-travel time, but that's just a roundabout way of saying that they are using the second...or am I missing the point. :blushing:
  41. V

    If Time Is The Fourth Dimension

    does that mean that the past and the future exist somewhere? It makes it seem like everything we're experiencing is one big solid object in some higher dimension we can't visualize, naturally, just as an ever widening and then shrinking disk seen by a two dimensional person is really a 3...
  42. S

    Calculating Subspace Dimension in R4: U = [a b c d] | a+b=c+d

    Find a basis and calculate the dimension of the following subspaces of R4 U = [a b c d] | a+b=c+d in R now a basis is supposed to be linear independant but that conditon on a,b,c,d makes this a bit strange can i just say d = a+b - c and then pick a basis as (1,0,0,1),(0,1,0,1),(0,0,1,-1)...
  43. B

    Calculating Fractal Dimension for the Lorenz Strange Attractor

    Hi. How can I "experimentally" (by way of computer simulation) calculate an approximate value for the dimension of a fractal object? The object in question is the Lorenz strange attractor, which has a dimension between 2 and 3. Also, I know there is a number of different ways to define...
  44. X

    So, What is the Dimension of the Subspace of Matrices with Trace 0?

    Hey Everyone, I have this question on my assignment that's confusing me, I think I have the logic right, but for some reason I am getting the wrong answer.. Anyways the question states: If S is the subspace of M5 (R) consisting of all matrices with trace 0, then dim(S) = ? Now, I know...
  45. mattmns

    How do I determine the dimension of a space in linear algebra?

    There are these questions in the book that ask us to find the Dimension of a particular space. Do I just find a basis for the space, and then the number of elements in that basis is the dimension for the space? Or is there some trick to finding the dimension? Thanks!----------- For example...
  46. W

    Dimension Question: How Does Math Shrink 3-D Shapes?

    I remember when I was in school my professor mathematically shrank a cube to a point and when he did so it was a sphere. So I was wondering does this mean that a sphere is the best way to think 3-D. Then I also wondered what would a square shrink to, I would guess a circle but I do not...
  47. W

    What are the ten dimensions of string theory?

    Hey, I have not taken any string theory courses yet, but I had heard that String theory says there are probably 8? dimensions. I was wondering what the other 4 dimensions are and how they would be described. I know of: 0d - Spatial Dimension (a dot) 1d - Spatial Dimension (Can be viewed as...
  48. O

    How far does a photon move in the time dimension when it travels for one second?

    How far does a photon move in the time dimension when it travels for one second? How far does a photon move in the spatial dimension when it travels for one second? How far does a golf ball move in the time dimension when it stays stationary on a tee for one seceon? How far does a golf ball...
  49. L

    Could the Big Bang Be a Pinched-Off Bubble from a Greater Dimension?

    It is possible that the "big bang," should it ever have occurred, was actually a bubble from a greater dimension, pinched off. It has been postulated that as matter is gathered into a black hole, it condenses and forms what could be equated to a water droplet in space-time. When that drop...
  50. T

    What is each dimension in string theory?

    i'm currently reading Hyperspace by michio kaku and it says that there are 10 dimensions in superstring theory, what are each of the dimensions?
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