Dimension Definition and 908 Threads

Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.

I always thought that one independent equation cuts down the dimension by 1, so if we had two planes, say x - y - z = 1 and x + y + z = 1, then because these are two independent equations, the dimension of the intersection should be 1 because each plane is cutting down the dimension by 1. Using...

At least according to Tim Anderson Ph.D who wrote the paper in Physics Review. https://news.knowledia.com/US/en/articles/a-5th-dimension-may-explain-quantum-theory-the-infinite-universe-medium-6f1d6fd371e068a07f357b9babe9ab2eec06d034 What do you make of this? "The paper simply presents...

It's just think that if we measure 1 second with a clock we should be able to "see" a 300'000km long piece of something in space or not ? Or does the time extension only has to be understood as a set of numbers indicating timelaps, so that there is no "geometry" of time ?

Suppose the universe were described by internal geometry by a ball, i.e. the metric where : $$diag(1,r^2,r^2 sin(\theta)^2)$$ Now if we go to exterior geometry and suppose there existed a 4th timelike dimension the manifold were for example modelized by : $$\left(\begin{array}{c}...

The perimeter of a circle increases by radius, the surface area of a ball increase by radius(which is height which is the third dimension if the ball is a planet like the Earth), and the universe is expanding by time, can we say that the fourth dimension is time by this ?

To form a 2-torus, a narrow tube can be bent into a loop and joined end to end: But instead of forming this loop in our three-dimensional space, the loop can also be formed in a direction perpendicular to three-dimensional space, moving it into the fourth dimension of space. What's the name of...

My thoughts are: a) it should just be N^2 b) just N since they're identical c) due to Pauli exclusion would it be N^2 - N since they have to be different states?

adding all the torques around the red circle position (taking clockside direction as positive ): -M*g*L*sin(theta)-k*x*y=I *w (considering that the suspension bar is of negligible mass as the problem indicates )here "x" is the normal "x" of hooke's law (I don't know exactly what it is for a...

A topless square box is made by cutting little squares out of the four corners of a square sheet of metal 12 inches on a side, and then folding up the resulting flaps. What is the largest side area which can be made in this way? What information I have so far is that since the side of the...

This is wild. I was always fascinated with the Mandelbrot set, as well as the bifurcation diagram. I had no idea the Mandelbrot diagram was a different visualization of the bifurcation diagram. Question: is this video accurate? I always question the veracity of YouTube science videos.

Reading old text about the limit of large N_c, there are some remarks telling that for finite N we have a colour tube and that it becomes a string in the limit. It sounds as the extra dimension of M-Theory, and I wonder if the math of both calculations are related somehow. Is the transition to...

As I understand it, dimension is a way of describing direction, with the first three spatial dimensions being straight lines which extend infinitely in one direction, perpendicular to each other. In string theories, several additional dimensions are required, sometimes up to nine or 10, I...

Hi, if I have a equation like (just a random eq.) p_dot = S(omega)*p. where p = [x, y, z] is the original states, omega = [p, q, r] and S - skew symmetric. How does the equation appear if i only want a system to have the state z? do I get z_dot = -q*x + p*y. Or is the symmetric not valid so I...

Imagine we draw a two dimensional finite plane with coordinate axes; for simplicity, let's make it a square. Now, suppose we add a third dimension that represents the possible distances between any two points on the square. Now we have a three dimensional space. What shape will that space have...

The farthest distance of two places in an area is 200 km. If someone wants to make a map of that area on a 1 m × 1 m paper, the possible scale to make it is ... a. 1 : 210 b. 1 : 2.100 c. 1 : 21.000 d. 1 : 210.000 Can you help? The 200 and 210 makes me think that the distance on map won't be an...

Hello, It has been a long time since I first looked at this, so thought I might ask for some help in clarifying this problem: Is an equation of the form --> Velocity = (Distance) * (Trigonometric function) a valid one in physics? If so, what is the relationship of trigonometric functions...

If ##\textbf{u}_1,...,\textbf{u}_n## form a basis in a linear space, how does one determine the dimension of the span ##\textbf{u}_1-\textbf{u}_2, \textbf{u}_2-\textbf{u}_3,...,\textbf{u}_n-\textbf{u}_1##? Since ##\textbf{u}_1,...,\textbf{u}_n## form a basis, they're linearly independent. If one...

Summary: Integrating the 1 dimensional MB Distribution in terms of translational kinetic energy up to infinity, does not yield ##\frac{1}{2}k_BT## as it should be. The format for the 3 dimensional Maxwell-Boltzmann Distribution is ##A\cdot e^{-\frac{E}{k_BT}} \cdot g(E)## in which ##A## can be...

Problem Statement: Trying to understand the principles for the equation e=mc2. Relevant Equations: E=mc2 So just started at a physics A level and so far loving it. I understand this question has indeed been asked before, however for different reasons. I understand the purpose of e=mc2 and I...

as calculations are technically difficult in curved spaces, I wonder if we would obtain the same results by adding one additional (virtual) dimension in order to embed the space in a higher order Euclidean volume, just to facilitate the treatments? (for example embed a 3D hypersphere in a 4D...

If we set a dimension for the unobservable, we may stumble on a unifying theory for the large and small. 3D + Time + Waves When I say Waves, I'm talking about the waves a particle becomes when it is unobserved and going through the double slit. If waves only exist as math, observation pulls...

Motivated by some apparently intractible unsolved problems, I see cosmology as a beautiful mathematical description of a strongly flawed paradigm. This forum wisely does not allow laying out alternate paradigms, so I try to ask questions to guide my immature understanding , in the spirit of...

Suppose we use fractional derivatives (https://en.m.wikipedia.org/wiki/Fractional_calculus) in GR, hence we have a local group symmetry ##SO(3-\epsilon,1+\epsilon)## does any reference exist about an equation for ##\epsilon## ?, since it could depend on coordinates too.

Homework Statement Problem given to me for an assignment in a math course. Haven't learned about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below. [/B] Homework Equations Unsure of relevant equations The Attempt at...

<Moderator's note: Moved from a technical forum and thus no template.> I am at the beginners level of linear algebra and having problem of the intersection of matrices. Your kind help is much appreciated for the following question Let\quad M1=\begin{Bmatrix} x & -x \\ y & z \end{Bmatrix},\quad...

In several places, for example https://xxx.lanl.gov/pdf/chao-dyn/9406003v1, it is claimed that the Riemann zeta function is a fractal under the assumption of a positive result for the Riemann Hypothesis, because (1) the Voronin Universality Theorem, and (2) if the RH is true, then the zeta...

Let $$\left[\begin{array}{rrrrrrr} 1 & 0 & -1 & 0 & 1 & 0 & 3\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 4 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \end{array}\right]$$ Find a basis for the null space of A, the dimension of the null space of A, and...

Let ##(x_1,x_2,x_3)=\vec{r}(\theta,\phi)## the parametrization of a usual sphere. If we consider a projection in two dimension ##(a,b)=\vec{f}(x_1,x_2,x_3)## Then I don't understand how to use the metric, since it is ##g_{ij}=\langle \frac{\partial\vec{f}}{\partial...

In looking at the definition of a Hausdorff dimension of a space S = inf{d>0: inf{Σirid: there is a cover of S by balls with non-zero radii} =0} where i ranges over a countable set, it would appear that it would be acceptable to take the index set to be finite, but I am not sure how you would...

I do not understand the principle, to me it would make more sense if there was no line, there is nothing. The best way I can think of it would be, the first dimension is the observer, or nothing. I really don't know, its just this way makes more sense in my head, though I am lacking any...

Is time its own dimension or is it a constant that remains through all dimensions? Also, are there multiple dimensions of time, and how do we know the answers to these questions?

I have a vector ##\textbf{v} \in \mathbb{R}^{3N}## and a function ##\textbf{Ψ} : \mathbb{R}^{3N} \longrightarrow \mathbb{R}^p## such that ##\textbf{Ψ}(\textbf{v})=0##. Why the set ##T=\{ \textbf{x} \in \mathbb{R}^{3N} \ | \ \textbf{Ψ}(\textbf{x})=0 \}## has dimension ##n=3N-p##?

I have been learning about the idea of "dimension" of a shape in terms of how its "mass" scales down when we cut it into self-similar parts. For example, However, the term "dimension" is closely linked with the idea of degree of freedom. So my question is, is there any sense in which the...

I am trying to distinguish two different plant species based on their chemical signature using statistical signal classification. This is what the average curves of two species look like with their standard deviation bounds: These spectra are made up of ~1400 points. So it was suggested to...

The question is not to distinguish space from time, but in general, what distinguishes a spatial dimension from other types of dimensions? For example, Hilbert space has an infinite number of dimensions, but they are not spatial; string theories add extra spatial dimensions. Is there a...

Homework Statement A turtle and a rabbit engage in a footrace over a distance of 4.00km. The rabbit runs 0.500km and then stops for a 90min nap. Upon awakening, he remembers the race and runs twice as fast. Finishing the course in a total time of 1.75h, the rabbit wins the race. A) Calculate...

This question is inspired by one question, which was about representations that can be realized homologically by an action on a graph (i.e., a 1-dimensional complex). Many interesting integral representations of groups arise via homology from a group acting on a simplicial complex that is...

I am developing a FORTRAN code (.f90) which "ll calculate some matrix in some time interval (dt1=0.001) and these matrices have to be integrated in some time steps (dt=0.1). Though I am experience in FORTRAN 77, new to FORTRAN 90. I am unable to make dimension of matrix real (I think that is the...

Hello everyone .I cannot understand what Q14 is asking about. May I know how to solve Q14 shown in the pic？

Homework Statement Find the dimension of the subspace of all vectors in ##\mathbb{R}^3## whose first and third entries are equal. Homework EquationsThe Attempt at a Solution So I arrived at two solutions and I'm not entirely sure which is the valid one. #1 Let ##H \text{ be a subspace of }...

If we do the thougt experiment to delete one of the three spacedimentions we end up with a flat-land universe, which is fully possible to imagine (while probably not existing) But if we delete the time dimesion it becomes inpossible to imagin. Whithout time there is no existence. Could that mean...

In a holographic universe model, could our 3D universe be encoded in 3D and still be a holographic universe, instead of 3D information encoded in 2D space? Or is the standard model (non-holographic) of the universe already 3d information encoded into 3d space...

Homework Statement [/B] In an elastic head-on collision, a 0.60 kg cart moving at 5.0 m/s [W] collides with a 0.80 kg cart moving at 2.0 m/s [E]. The collision is cushioned by a spring (k=1200 N/m). a) Find the velocity of each cart after the collision b) Find the maximum compression of the...

I am trying to understand how to define a metric for a positively curved two-dimensional space.I am reading a book and in there it says, On the surface of a sphere, we can set up a polar coordinate system by picking a pair of antipodal points to be the “north pole” and “south pole” and by...

Homework Statement Homework EquationsThe Attempt at a Solution Dimension of length using h,G,c [h] = [F r] ##[G] =[ \frac { Fr^2}{m^2} ] \\ [\frac { hG}c] = [L] ## So, the answer is option (b). Is this correct?

Hi, I have heard (or imagined) that a wavefunction, where Psi is on the y-axis and the positions x is naturally on the x-axis, is really a one-dimensional system in Physics (not in mathematics), because the signal or the oscillation of the wavefunction is not really a dimension, and only the...

Homework Statement Ball A is dropped from the top of a building of height h at the same instant that ball B is thrown vertically upward from the ground. At what height will the balls collide if the collision occurs when the balls are moving in the same direction and the speed of A is 4 times...

Homework Statement Given the six vectors below: 1. Find the largest number of linearly independent vectors among these. Be sure to carefully describe how you would go about doing so before you start the computation. 2 .Let the 6 vectors form the columns of a matrix A. Find the dimension of...

Can the wave function in four dimensions be expressed as e^i(kx+ky+kz-wt)?