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1

I think Gennaro covered, this, I'll give a layman's explanation. In spacetime there are four general dimensions, three of space and one of time. Why is it that other dimensioned qualities seem to be rarely considered as part of spacetime? For example, why isn't speed part of spacetime, forming a five-dimension spacetime-speed manifold? Objects in ...


2

We need to clarify what we mean by dimensions of a mechanical system. They refer, in the standard terminology, to the number of different degrees of freedom we need to describe the kinematics of a point particle (in this case). To this extend, given any reference frame $S$, an event in the space-time is identified by its position $(x,y,z)$ and the time $t$ ...


-3

A spacial dimension is a direction in which a body can move. This line of direction must lay 90° to other dimensions. The 3 dimensions commonly known to us are forward-back, up-down and front-back. A moving body in our universe used one or more of these directions when it moves. So far we have not detected any 4th spacial direction/dimension in our universe. ...


1

In 2D a circunference is $$(x-x_c)^2 + (y-y_c)^2 = r^2$$ A circle is: $$(x-x_c)^2 + (y-y_c)^2 \leq r^2$$ In 3D sphere is: $$(x-x_c)^2 + (y-y_c)^2 + (z-z_c)^2 = r^2$$ A ball is: $$(x-x_c)^2 + (y-y_c)^2 + (z-z_c)^2 \leq r^2$$ You can even have a "1D ball/circle": $$(x-x_c)^2 \leq r^2$$ which is just a line segment. A 1D circunference/sphere is ...


8

There is not a good mathematical or physical answer to what a circle becomes in 3D except to say it stays a circle. The statement that a circle becomes a sphere is an analogy: As CuriousOne points out, one is extracting the important thing about a circle, that it is all points at a given distance from the center, and carrying that to 3D to get a sphere. ...


2

Generally and fundamentally, the answer to your question is no. Photography / 2D imaging is the prototypical example of the mathematical notion of projection, which, by its definition, destroys information by extracting "class" information from sets of objects: forgetting about fine differences and reporting only the class. Here, for example, if $z$ is the ...


2

I know some derivations in which one can track the emergence of the concrete value, without having to buy that the second order contribution in the Euler-MacLaurin formula (see other answer) is $-\frac{1}{2!}$ times the second Bernoulli number $B_2$. The limit $\lim_{z\to 1}$ of the sum $0+1\,z^1+2\,z^2+3\,z^3+\dots$ diverges, because of the pole in ...


-1

We have some theories witch say we have 10 or 11 or 26 dimensions For Example according to String Theory we have 10 Dimensions But due to M Theory we have 11 Dimensions


1

Why can't we experience them like the first three dimensions? The usual explanation is that these additional dimensions, if they exist, are tightly curled up or compacted. Humans can't move around in them like we can move through the three "normal" spatial dimensions we are familiar with. Why are we not able to visualize Dimensions beyond 3 Mostly ...


1

Our rigorous definition of "dimension" comes from linear algebra. This is going to be a quick run-through of the mathematical way of describing dimensions, and then its physical significance. The first concept needed is a vector space. A vector space can effectively be thought of as a collection of points that satisfy a few particular (and very useful) ...


3

In a geometrical context, the dimension (at a point) roughly speaking is the number of coordinates you need to identify any point in a fixed neighbourhood. Note that in a geometrical context, the least you need is a topology, in order to be able to speak of continuity, neigbourhoods, etc. This intuitive definition used to work quite well, but at a certain ...


6

I think this depends a lot on what you are doing and how you look at whatever you are looking at. Speaking of which, how many dimensions does the content displayed by your computer monitor have? Two, I guess, could be one answer. It's not three dimensional and it certainly is not one strip of pixels. Let me quote from Carl's great answer that I like to ...


28

In this context, I usually explain it (non-mathematically) by saying that the number of dimensions is the number of values you need to specify where an event occurs. For most people this involves space and time (but for particle physicists it might involve more values ;). Anyway, certainly even people before Einstein would need to specify the time as well ...


35

Coming from a math perspective, I would define a dimension as "any property which is orthogonal to all other properties." "Orthogonal" here means you cannot get to one property by applying scalar operations on another. For example, the x-axis dimension can never become a y-axis value, and similarly for time vs. spatial dimensions. For that matter, ...



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