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.
It is assumed that gravitation in n dimensions would follow an approximate 1/d^(n-1) law. In our 3D world the attraction of a uniformly dense sphere is the same as if all the mass were concentrated at its center. I have read for n>3 this is not so. I want to find out what the result would...
If we accept the concept that the universe began as a zero-dimensional point, wouldn't that imply that our 3+1 dimensions, or 9+1 / 10+1 dimensions of string theory, emerged at the moment of the Big Bang or some presumably Planck-order time after? Are there any theories on this?
When going from 3 to 2 dimensions, I am unsure about how the number of polarizations will be affected.
I know the following though:
The 1/8 factor becomes a 1/4 since we are now integrating over the positive quadrant in 2d rather than the positive octant in 3d.
The ##4\pi n^2## becomes a ##2\pi...
I don't believe this question will interest anyone else, but remains worthy of notice because the solution is a formula that depends on the 36th root of a number.
You unexpectedly find yourself in a space with four Euclidean dimensions. Fortunately your molecules have been rearrange into a four...
What is the velocity of a photon through the four dimensions?
What is the velocity of a photon through the three spatial dimensions x1, x2, x3?
What is the velocity of a photon through the fourth dimension x4?
A 4D planet has no axis of rotation. Nothing special about planets, all freely rotating 4D bodies have two perpendicular planes of rigid rotation. (Clifford proved this in the 19th century.)
Now there is nothing stopping us from thinking of planes of rotation here in 3D Universe. It's the...
The object takes a step [x, y] in 2 dimensional space. This is represented the change in coordinate ##x \vec e_x + y \vec e_y## where ##e_x## and ##e_y## are basis vectors in this space.
Suppose we define a non-linear / parametric transformation of this ##\vec e_x## and ##\vec e_y## basis...
I have a volume of 0.04 cubic centimeters. I need to convert into something I can work with in my shop. Tolerance is not important.
To get dimensions of a cube with this volume, I calculated the cube-root which came out to 0.341. So, my cube would be (rounded) 3mm on a side, correct?
Next...
1.Does the Maxwell Boltzmann distribution change depending on the shape of the container? Pressure and the volume is constant. How is the Distribution affected whether the gas is in: a,sphere b,cube c,cuboid?
Why does/doesn’t the distribution change depending on the shape of the container...
Hi, just wondering if the dimension of velocity is m/s that can be described as what distance is passed in a specific time, then how can I describe volt which is kg•m²÷s³÷A? Mass in the area that moved...
I can't even imagine.
Thanks.
Let's say we have four 3D spaces: (x, y, z) , (x, y, iz) , (x, iy, iz) and (ix, iy, iz), with i being the imaginary unit. Now, let's say we have a donut in each of these spaces. Geometrically, the donuts are different objects, have different equations and different properties (I think) but would...
In string theory, the universe can have 9-10 spatial dimensions and the reason why we experience 3 is because those higher dimensions compactify. Under right conditions the extra dimensions can decompactify into the macroscopic dimensions we see. Why do some dimensions curl up or expand? Is...
Let ##a, b##, and ##c## be real numbers such that ##a## and ##c## are positive and ##ac > b^2##. Evaluate the double integral $$\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-ax^2 - 2bxy - cy^2}\, dx\, dy$$
You have a design problem statement, which gives you numerical specifications for performance, spatial constraints etc, and you have finally come up with some mechanism ideas you synthesised into a machine concept.
You’ve sketched your design’s parts and the final assembly.
So: How do you now...
This is really a conceptual question about time dimensions bit it came up in the context of astronomy so I'll put it here.
I was gazing up at the sunset sky the other night and saw a very bright (twinkling) star. I was about to make a note about its location so I could look it up later. I...
Some BSM theories, often but not exclusively string theory inspired, have an extra time dimension, as well as extra spatial dimensions.
I'm trying to make sense of what it even means to have more than one time dimension, ideally, with a concrete example that illustrates how the second time...
In the paper
C. S. Lent and P. D. Tougaw, "A device architecture for computing with quantum dots," in Proceedings of the IEEE, vol. 85, no. 4, pp. 541-557, April 1997, doi: 10.1109/5.573
about quantum dots, it is stated that the basis vectors in the space of quantum states for a single cell...
I'm interested in describing a 6-dimensional space of which three are compactified to small circles. Globally this space looks 3-dimensional, like a 2-dimensional cylinder looks 1-dimensional globally.
Kaluza and Klein did a similar thing in the context of 4-dimensional spacetime. They extended...
We are in phase space of ##6N## dimensions. Each point ##\mathbf r## in this space has ##6N## coordinates.
Pathria writes "Consider an arbitrary "volume" ##\omega## in the relevant region of the phase space and let the "surface" enclosing this volume be denoted by ##\sigma## then the net rate...
My high school physics days are long ago ;) This is not homework, well, other than it is work, at home.
For a real application: Very space constrained "workshop", got a bench drill press, and want to build a table on wheels for it, to be able to move it into a corner when not needed.
Those...
https://arxiv.org/abs/hep-th/0303031#:~:text=In%20gravitational%20theories%20with%20extra,of%20our%20four%2Ddimensional%20world.
This paper seems to suggest that if there are extra dimensions , then they are unstable and can expand turning our 4d universe into a higher dimensional space.
I read Brian Greenes book "The Elegant Universe".
In it there is one chapter talking about extra dimensions. The first idea was proposed by Kaluza and Klein. They suggested that the extra fifth dimension exists as a circle, is it right. Is this called a KK circle?
Now Greene also mentioned that...
Is there anything we know of in this physical world that is less than 3 spatial and 1 time dimension? Or is that pretty much everything, no more, no less?
If a linear space ##V## is finite dimensional then ##S##, a subspace of ##V##, is also finite-dimensional and ##dim ~S \leq dim~V##.
Proof: Let's assume that ##A = \{u_1, u_2, \cdots u_n\}## be a basis for ##V##. Well, then any element ##x## of ##V## can be represented as
$$
x =...
... by a factor of 10, then would it operate exactly the same way as if it did not contract?
I ask this because there could be an observer at a sufficient speed for which our solar system contracts by a factor of 10. Wouldn't our solar system and its contents have to behave differently for the...
An amusement park wants to place a new rectangular billboard to inform visitors of their new attractions. Suppose the length of the billboard to be placed is 4m longer than its width and the area is 96cm^2. What will be the length and the width of the billboard?
I had a few questions about this paper by Nima Arkani-Hamed, Georgi Dvali and Savas Dimopoulos (https://arxiv.org/abs/hep-th/9907209) which is closely related to the concept of branes (https://en.wikipedia.org/wiki/Brane)
1. The authors of the paper mention that the branes in the model could...
I'll try to phrase this as clearly as possible but my use of terminology might need to be refined. That may be what ultimately comes of this thread, but hopefully the question as I phrase it will make enough sense. I'm not necessarily asking that a proof be provided, rather, I am interested to...
I am planning to build my own laptop stand with an acrylic sheet. The final product should look like the one in the attached image (bottom right)
Material: most laptop stands are made with 6mm acrylic thickness, hence this is what I will use. Also I already have this material at home.
I am...
I got the answer right and here's my work, but I want to know how to get the final direction correct [E 76 S]:
Dtx = 2.78 [W] + (6.25)(sin40)
Dtx= 1.237
Dty= (6.25)(cos40)
Dty= 4.95
Dt= 4.95 (pythagorean thereom involving 1.237 and 4.95)
Dangle = tan^ (4.95/1.237) = 76
Therefore, it is...
I am studying polymer quantum mechanics. In it, they say that the momentum, ##p## eigenvalue, has the dimensions of ##(mass)^{-1}## and similarly ##\omega## has the dimensions of ##mass##. How it is possible, please someone explain that to me. Even a little hint would work.
I don't get it...
Heloo guys, diego here,
I was wondering if there exist multiple space dimensions, and we can only see in the 3th dimension, it is entirely possible to have multiple time dimensions.
And another thing, we all learn(simplified version coming) 1 dimension is basically a point, 2nd dimension is a...
hi guys
my analytical mechanics professor asked a question the other day about, how come the generalized forces##Q_{\alpha}## doesn't need to have a dimension of force, and the generalized coordinated ##q_{\alpha}##as well doesn't need to have a dimension of length, but the generalized work...
I am studying the real scalar field theory in ##d## spacetime dimensions as beautifully presented by M. Srednicki QFT's draft book, chapter 18 (actually, for the sake of simplicity, let us include polynomial interactions of degree less than or equal to 6 only)
\begin{equation*}
\mathcal{L}...
There are several models of brane cosmology (https://en.wikipedia.org/wiki/Brane_cosmology) and several physicists working in this field (e.g Lisa Randall and Raman Sundrum), but as you will notice, apparently they are all directly related to string theory. This has several consequences, for...
I have a 2-dimensionsal smooth function ##f(x,y,t)##. There may be multiple traveling waves across the domain. None of them are precisely traveling waves (the shape of the wave changes as it travels). Here is how one of these waves would look in 1-dimension:
I want to find the speed of these...
I have been trying to imagine extra dimensions for a long time now. I came across this article and it is certainly the most vivid pictures I have ever seen painted of the extra dimensions:
https://phys.org/news/2014-12-universe-dimensions.html
For example, it describes "In the sixth, we would...
String Theory and related theories like M Theory have strong constraints in the number of dimensions where they can be formulated (for example, in the case of M theory, it is only allowed in 11D or in the case of bosonic string theory is only allowed in 26D.
Since string theory and related...
Hello. Questions: How can special relativity describe four dimensions when we only see three spatial dimensions and we experience time?Why time is the fourth dimension and not another physical quantity? Is this only a generalisation that was needed to describe more physical phenomena and the...
Wich The Program (Software) Using To Draw Statics And Dynamics Mechanichal Same This 100% When Create This below Which Program Using ?
And set Dimensions
Thanks For All