Dimensions Definition and 1000 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. PcumP_Ravenclaw

    Understanding fractional and higher dimensions

    Halo, I was reading about geometry from Tim Gowers book titled "A very brief introduction to mathematics". I came across fractional dimensions and the 4th dimension. The koch snowflake has dimension 1.2 yet he could comfortably drawn it on a 2d page (or is it complete?). Has not he just...
  2. 24forChromium

    How do you add angular momentum in different dimensions?

    Say a ring is spining around the z-axis, an angular impulse is then applied to it in the x-axis, what is the resultant motion qualitatively and quantitatively? How can it be calculated? (You can make up the quantity of z-angular momentum and x-angular impulse)
  3. I

    'forces in two dimensions' problem

    Homework Statement hello guys, that's the problem, when i tried to solve it using soh cah toh and Pythagoras law it resulted in answers e' and f', which make sense to me because both result it 54.6N and both relevant in directions so i am a bit confused.[/B] Suppose that a force with a...
  4. jfizzix

    Curvature implying Closedness in N dimensions

    A two-dimensional surface with everywhere positive curvature is a closed surface with no boundary (isomprphic to a sphere). Is this true for higher dimensional surfaces as well? Would a three-dimensional surface, with everywhere positive curvature be a closed hypersurface isomorphic to a...
  5. jk22

    How to make a 2-sphere in 4 dimensions

    I want a closed two dimensional manifold embedded in four dimensions. Is the following way a good one : Parametrize a 3 sphere with $$\theta,\phi,\chi$$ Put $$\chi=f(\theta,\phi)$$ Does it make the manifold closed ?
  6. Ien Cleary

    Why does imaginary time behave like space?

    I know what imaginary numbers are, but I'm struggling to understand why the Lorentz transformation makes a time-like dimension space-like. I suppose what I'm really asking is what is the difference between time-like and space-like. I've read that it has something to do with special relativity...
  7. N

    String theory, string dimensions?

    Strings are said to be one dimensional, due to the math. And I understand that there are problems in the math when they put the strings in 3 or 2 dimensions. According to string theory, lengths smaller than Planck length have no physical significance. Could it be that strings are 3...
  8. S

    Why chiral fermions don't exist in odd dimensions?

    In four dimensions, left and right chiral fermion can be written as \psi_L= \begin{pmatrix} \psi_+\\ 0 \end{pmatrix},\qquad \psi_R= \begin{pmatrix} 0\\ \psi_- \end{pmatrix}, respectively, where \psi_+ and \psi_- are some two components spinors(Weyl spinors?). In this representation, the...
  9. alexmahone

    MHB Is there a correct way to include dimensions in economics equations?

    A typical demand function is y=5000-p where y is quantity demanded and p is price. But this equation isn't dimensionally correct. What am I missing?
  10. B

    Why is it necessary to have 11 dimensions in M Theory?

    I do apologize if this is the wrong place to post this. Why is it necessary to have 11 dimensions in M Theory? I've been at it for days emailing universities, Twitter, etc. No one has attempted to give me an answer and it's frustrating me. I just want to know what the purpose of each of the...
  11. Orodruin

    Discussion on tensor dimensions

    <<Mentor note: This thread has been split from this thread due to going a bit off-topic.>> I would actually disagree with this. Any tensor has well defined units, but its components may not have the same units as the tensor basis may consist of basis tensors with different units. For example...
  12. S

    Exploring the Possibility of Larger Dimensions in String Theory

    I hope I'm asking in the right forum. I'm trying to understand the subject of more dimensions a little better. Does the dimensions predicted by string theory have to be small? Can our known dimensions predicted by string theory actually be inside much larger dimensions?
  13. M

    Calculating Velocity in Two Dimensions: A Real-Life Swimming Challenge

    Question: In an anniversary celebration of Marilyn Bell's 1954 feat- she was the first person to swim across Lake Ontario- a swimmer set out from the shores of New York and maintained a velocity of 4m/s (N). As the swimmer approached the Ontario shore, she encountered a cross-current of 2 m/s...
  14. H

    Why can't we perceive extra dimensions?

    String theorists postulate that our world has 9 dimensions and these extra dimensions are very twisted up such that they are too small to be seen. Question 1: Why don't we postulate, instead, that the reason we can't see these extra dimensions is because we are not genetically wired to do so...
  15. Caneholder123

    How Does Keplerian Motion Work in 4 Dimensions?

    This is the link to the relevant paper. I have to show that vector (t', x', y', z') lies on the sphere. But for that to be, V^2 has to be 1 according to the equation in the introductory part of the section 2. That, by definition, means that E=-m/2. What does that mean, and why is this solution...
  16. S

    Meaning of representations of groups in different dimensions

    Problem This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions. Relevant Example Take SO(3) for example; it's the...
  17. amjad-sh

    Closure relation in infinite dimensions

    The closure relation in infinite dimension is : ∫|x><x|dx =I (identity operator),but if we apply the limit definition of the integral the result is not logic or intuitive. The limit definition of the integral is a∫b f(x)dx=lim(n-->∞) [i=1]∑[i=∞]f(ci)Δxi, where Δxi=(b-a)/n (n--.>∞) and...
  18. Avatrin

    Free particle in three dimensions (angular momentum)

    Homework Statement A particle with mass m can move freely in three dimensions. Explain why the stationary states of the particle are determinate states for angular momentum (L_z and L^2) Homework Equations L^2 = L_x^2 + L_y^2 + L_z^2 L = r \times p \hat{H} =...
  19. Saffat Rafsan

    How to imagine higher dimensions?

    In the links below Carl Sagan and TED-Ed described about higher dimension: and here's a description of Brian Greene: Carl Sagan and TED-Ed explained, we can not see the higher dimensions because we are limited to perceive only three dimensions. They didn't say a dimension can be small or...
  20. K

    Optimization Problem: Minimizing rectangle dimensions

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  21. kostoglotov

    Newton's Method generalized to 3 dimensions

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  22. K

    Extra Dimensions: Lorentz Covariance, String Theory, Symmetries & More

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  23. S

    Showing equality of dimensions

    Homework Statement [/B] Let α ∈ ℂ be a complex number. Let V = ℚ(α) be the rational vector space spanned by powers of α. That is ℚ(α) = <1,α,α2,...>. 1) If P(t) is a polynomial of degree n such that P(α) = 0, show that dimℚℚ(α) is at most n.Homework EquationsThe Attempt at a Solution...
  24. S

    MHB Is 1, α, α², ..., αⁿ⁻¹ a Basis for ℚ(α)?

    Let α ∈ ℂ be a complex number. Let V = ℚ(α) be the rational vector space spanned by powers of α. That is ℚ(α) = <1,α,α2,...>. 1) If P(t) is a polynomial of degree n such that P(α) = 0, show that dimℚℚ(α) is at most n.Here is my attempt to solve this question. Please give me some...
  25. K

    What is the Velocity of a Projectile Using Kinematic Equations?

    Homework Statement I need to find velocity v from the drawing. Homework Equations Equations of kinematics The Attempt at a Solution I found that v=12.134 m/s . I need to know is it correct. [/B]
  26. George Zucas

    What is an I55 beam and how does it compare to other common beam types?

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  27. L

    Equilibrium in Two Dimensions: What Force Does a Hanging Boy Exert on Each Tree?

    Homework Statement a boy on whom the force of gravity is 400 N hangs on to the middle of a rope stretched between two trees. The rope sags in such a way that it makes an angle of 170 degrees at the boys hands. what force does the rope exert on each tree? Homework Equations F=ma The Attempt at...
  28. D

    Dimensions of Parallel Plate Capacitors

    Homework Statement The electric field between two circular plates of a capacitor is changing at a rate of 1.5 x 10^6 V/m per second. If the displacement current at this instant is Id = 0.80 x 10^-8A, find the dimensions of the plate. Homework Equations Id = ΔQ/Δt = ε0(ΔΦE/Δt) ΦE = EA Q = CV...
  29. RJLiberator

    Arc Length in Three Dimensions Question

    Homework Statement Find the arc length of r(t)= <tsin(t), tcost(t), 3t> from 0 to t to 2pi (inclusive) Homework Equations Integral from 0 to 2pi of the magnitude of r'(t) dt The Attempt at a Solution 1. Must find the derivative of the function. Using the product rule a few times, the...
  30. RJLiberator

    Calculus based kinematics (two dimensions) question

    Homework Statement A soccer ball is kicked from the ground at an angle of 30◦ with speed 30m/s. Determine the duration of the flight, range and maximal height of the trajectory. Homework EquationsThe Attempt at a Solution 1. Break down the velocity vector into x and y components: 30cos(30) =...
  31. ChrisVer

    FRW Metric in d Dimensions: Can I Expand?

    I was wondering if I can expand the FRW metric in d spatial dimensions, like: g_{\mu \nu}^{frw} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & - \frac{a^2(t)}{1-kr^2} & 0 & 0 \\ 0 & 0 & - a^2(t) r^2 & 0 \\ 0 & 0 & 0 & -a^2 (t) r^2 \sin^2 \theta \end{pmatrix} \rightarrow g_{MN} = \begin{pmatrix} g_{\mu...
  32. T

    Motion in Two and Three Dimensions

    Homework Statement A frightened rabbit runs onto a large area of level ice that offers no resistance to sliding, with an initial velocity of 6m/s toward the east. As the rabbit slides across the ice, the force of the wind cases it to have a constant acceleration of 1.4 m/s^2, directed due...
  33. psuedoben

    Could multiple dimensions explain quantum leap?

    please point out the flaws/add to my reasoning here: it is known that energy is quantized, but could that be for just the 3 dimensions we're able to experience? Is it illogically to propose that energy might not be quantized when looking at all of the dimensions together? this could explain...
  34. RJLiberator

    How to compute the integral of r(t) in three dimensions?

    Homework Statement r(t) = <cos(t), 2t, sin(t)> from 0 to 2pi (included). Compute the integral from 0 to 2pi of r(t). Homework EquationsThe Attempt at a Solution [/B] We haven't learned this in class yet, so I am getting a head. My idea is to simply put r(t) in and integrate term by term so...
  35. M

    One or more couldn't be resolved in descending dimensions

    In the process of descending from a higher dimension to a lower one, there must be more than one factor that could not be solved (or descend.) My little own experiment.. http://mr-none.meximas.com/public_html/pic/1.JPG Steps: 1, wrap a plastic bag around a basketball. 2, draw a "triangle" with...
  36. W

    Variables & Dimensions (total noob)

    So, I have some questions regarding 'dimensions' and 'parameters' in physics. (there are differences in the meaning between physics and mathematics, as far as I understand it) ---- As far as I understand it, we live in a 4-dimensional world, consisting of 3 space and 1 time dimensions. Yet we...
  37. Einstein Devotee

    Time dilation why or how, Special Relativity causes

    I understand the theory of special relativity and the mathematics which support it. I even understand that the time dilation has been proven. Therefore I am going to ask a question which on first blush may appear that I disagree with it but that is not the case. The question I can not seem to...
  38. K

    Relative position and velocity question (Two dimensions)

    Homework Statement A freight train is moving at a constant speed of 10 m/s. A man standing on a flatcar throws a ball into the air and catches it as it falls. Relative to the flatcar, the initial vellocity of the ball is 15 m/s straight up. a. What are the magnitude and direction of the...
  39. AlphaA

    A question about Motion in One & Two Dimensions

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > A balloon going upwards with a velocity of 12 m/s, is at a height of 65m from the Earth at any instant. Exactly at this instant a packet drops from it. How much time will the packet take in...
  40. TheDemx27

    Graphing Functions in n Dimensions, Parametric Equations

    So I was watching this video on Khan Academy, and it talks about graphing functions that have values in multiple dimensions. It shows how to represent a linear function in 3 dimensions with a set of vectors, L ={p1 + t(p1-p2)|t∈R} where p1 and p2 are vectors that lie on the line you want to...
  41. nomadreid

    Mismatched dimensions in a tensor product with CNOT

    I am working through an explanation in Nielson and Chuang's Quantum Computation book where they apply a CNOT gate to a state α|0>|00> + β|1>|00>. (The notation here is |0> = the column vector (1,0) and |1>=(0,1), while |00> = |0>|0>, and |a>|b>=|a>⊗|b>, ⊗ being the tensor (outer) product. I am...
  42. bcrowell

    Gorini vs. Jin: Superluminal Frames in 1+1 Dimensions

    Gorini 1971 (reference below) proves a no-go theorem for attempts to extend the Lorentz group to superluminal velocities in n+1 dimensions, for n\ge3. His assumptions are stated in abstruse notation, but he gives the following translation into plainer language: > Axiom: [...] time has a...
  43. M

    Matlab - Index exceeds matrix dimensions when writing .txt

    Homework Statement Index exceeds matrix dimensions Homework EquationsThe Attempt at a Solution I am trying to output into a text file something like this: Name ID scE ccE scC ccC Fake Subject 1 1 3 4 5 2 Fake...
  44. Anique

    Are These Dimensions of Physical Quantities Correct?

    Hello there, I have a confusion between dimensions and units! All of others tell me its not unit but some use units! Please verify the following Quantity Dimension Length L Mass M Time T Temperature Θ Amount of...
  45. CaptDude

    Dark Matter and extra dimensions

    I have considered the possibility that dark matter is so hard to find because it exists in extra dimensions. Before posting here I googled this, and found that some scientists postulate this exact possibility, while others consider that extra dimensions themselves cause the effects that are...
  46. A

    Moment in different dimensions in 2 D

    Homework Statement This isn't exactly a HW question, it's a question in my statics midterm today, he wants the reactions at the supports B and C, which is easy but the problem is this 60 N.m moment, all my friends were like :" let's just sum the moments around B ( due to the 50 N and the...
  47. M

    Piezoelectric and minimum dimensions of an actuator beam?

    Homework Statement We want to build a micro-positioning system that produces a maximum travel of 5 μm by using a piezoelectric linear actuator beam with a square cross section (a^2) and length (l) as shown below. A piezoelectric material with piezoelectric coefficients d13 = -2.0×10-8, d33...
  48. W

    MHB Unravel the Mystery: 5 Different Rooms, 5 Different Dimensions

    Have fun: Art, Ben, Cam, Dan and Ern each added a different kind of room to his house. Each added room can have a width of 10, 11, 12, 13 or 14 feet, and a length (not necessarily respectively) of 15, 16, 17, 18 or 19 feet. From the clues below, determine the kind of room each man added as...
  49. D

    Scaling / engineering dimensions for these interacting fields?

    Hi guys, So I have the following Lagrangian: \mathcal{L}=\bar{\psi}(i\partial_{\mu}\gamma^{\mu}-m)\psi+\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)-\frac{M^{2}}{2}\phi^{2}-\lambda\bar{\psi}\psi\phi where m is the is the mass of the spin-half Dirac field and M is the mass of the spin-0...
  50. T

    Can time have more than one dimension?

    hi guys, i am new here, not even sure this is right place to put this questions, I always understood time as a 4 dimension, just a diferent reference when compared with x,y and z. but is it possible that we r trying to see it in a different way? is it possible that time itself have its own x y...
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