Moderator: Alyrium Denryle
linkyNature.com is reporting that there may be evidence of 6 dimensions.
Galaxies seem to behave as there were more matter in them than is actually visible.
'One explanation, they say, is that three extra dimensions, in addition to the three spatial ones to which we are accustomed, are altering the effects of gravity over very short distances of about a nanometre.'"
It takes a special kind of mind, and a LOT of well-written explanation materials. For a few unusual reasons, I started tinkering with ideas of multidimensionality at a young age, so I think I "expanded" my mind. Try reading Hawking's books on the subject, he explains them pretty well.Superboy wrote:I've never quite been able to grasp the existence of other dimensions in this sense. Is there an easy way to explain it, or is it just something that's really hard to wrap your head around?
It depends if you want just mathematical concept or imagine it somehow. I don't think there is a way to imagine more than 3 dimensions in visual way , only in some symbolic way. ( but that's maybe my limited mind ).Superboy wrote:I've never quite been able to grasp the existence of other dimensions in this sense. Is there an easy way to explain it, or is it just something that's really hard to wrap your head around?
Its Son of the Suns way of giving you a cookie for your wit in the Scientific Paper thread.FireNexus wrote:Addendum: When did my user title become "cookie" and why?
It's pretty easy to explain, but practically impossible to visualize.Superboy wrote:I've never quite been able to grasp the existence of other dimensions in this sense. Is there an easy way to explain it, or is it just something that's really hard to wrap your head around?
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One can define a new orthogonal axis, true enough, but your statement is somewhat misleading--just because there is another coordinate does not mean that it is orthogonal to the rest, even assuming that there is some sort of inner product that lets one compute distances and angles between vectors.Surlethe wrote:Mathematically, we can deal in many, many dimensions; it's just as easy as defining a new axis into existence at right angles to the others. For example, let v = <v_1,v_2,v_3,...,v_n>. Then v = <v_1,v_2,v_3,...,v_n>, where each component is at right angles to each other one.
Ironically, sometimes introducing more dimensions can actually help in visualizing a hyperdimensional object, or at least aid in understanding. As a simple example, suppose one has the equation a²+b²+(a-b)² = c²+d² over the reals. One can interpret it as a three-dimensional surface in four-dimensional (Euclidean) space, but it might not be immediately obvious as to what kind of geometry it has. But if there is a particular (a,b,c,d) that satisfies this equation, then both sides of it are equal to some particular positive number, say e². One has introduced another axis, creatively named e. Concentrating on what happens to the surface as one varies e, one in fact has two independent equations for any fixed e: a²+b²+(a-b)² = e², c²+d² = e². But the former is an ellipse in (a,b)-plane and the latter is a circle in (c,d)-plane. Effectively, the hypersurface has been "split" into two a pair of ordinary three-dimensional objects that are "connected" through e: (a,b,e)-space and (c,d,e)-space, both of which form double-cones (one elliptic, the other circular). Instead of trying to imagnie the complete object in four dimensions, one can concentrate on two separate surfaces in those two 3-spaces and consciously think of them as connected. Another possible method is to take a plane, say (c,d), and think of each point as 'leading to' a unique plane (a_c,b_d) in which there is an ellipse. This is easier to generalize to higher number of dimensions, but less visual.Surlethe wrote:Now, the hard part is visualizing a vector like that.
It really is very simple to 'get', but truly understanding it would have to require lots of maths. Try reading Flatland, a classic introductory booklet (but if you haven't got much time skip to Part II).Superboy wrote:I've never quite been able to grasp the existence of other dimensions in this sense. Is there an easy way to explain it, or is it just something that's really hard to wrap your head around?
But aren't the components of a vector, by definition, orthogonal?Kuroneko wrote:One can define a new orthogonal axis, true enough, but your statement is somewhat misleading--just because there is another coordinate does not mean that it is orthogonal to the rest, even assuming that there is some sort of inner product that lets one compute distances and angles between vectors.Surlethe wrote:Mathematically, we can deal in many, many dimensions; it's just as easy as defining a new axis into existence at right angles to the others. For example, let v = <v_1,v_2,v_3,...,v_n>. Then v = <v_1,v_2,v_3,...,v_n>, where each component is at right angles to each other one.
Definition of vector spaces demands just vector addition and scalar multiplication to be defined. And a bit of nitpicking : Components of the vector are scalars ( numbers for example ) so the term orthogonal doesn't apply to them.Surlethe wrote:But aren't the components of a vector, by definition, orthogonal?Kuroneko wrote:One can define a new orthogonal axis, true enough, but your statement is somewhat misleading--just because there is another coordinate does not mean that it is orthogonal to the rest, even assuming that there is some sort of inner product that lets one compute distances and angles between vectors.Surlethe wrote:Mathematically, we can deal in many, many dimensions; it's just as easy as defining a new axis into existence at right angles to the others. For example, let v = <v_1,v_2,v_3,...,v_n>. Then v = <v_1,v_2,v_3,...,v_n>, where each component is at right angles to each other one.
OK. Gotcha. I was thinking of components like v = ai + bj + ck + ... = <a, b, c, ...>. And I'm currently (mathematically) still living in Euclidean space, so non-orthogonal axes didn't even occur to me.anybody_mcc wrote:Definition of vector spaces demands just vector addition and scalar multiplication to be defined. And a bit of nitpicking : Components of the vector are scalars ( numbers for example ) so the term orthogonal doesn't apply to them.Surlethe wrote:But aren't the components of a vector, by definition, orthogonal?Kuroneko wrote: One can define a new orthogonal axis, true enough, but your statement is somewhat misleading--just because there is another coordinate does not mean that it is orthogonal to the rest, even assuming that there is some sort of inner product that lets one compute distances and angles between vectors.
System of axises ( basis of that vector space ) doesn't have to be orthogonal. But for Euclidean ( real with inner product ) space orthogonal , even orthonormal , basis exists , but not every basis is orthogonal.
No. There is nothing in the term 'vector' that requires them (or, rather, the corresponding members of the basis) to make any angle, much less to make a right angle. For example, polynomials make a vector space, and so do polynomials of degree less than n, which can correspond directly with your equation v = <v_1,v_2,...,v_n> (say the basis is (1,x,x^2,...,x^{n-1})). But let's assume that there is a larger structure (an inner product <u,v> = |u||v|cos θ that lets us compute the angles).Surlethe wrote:But aren't the components of a vector, by definition, orthogonal?
Even in Euclidean space, coordinates can be non-orthogonal. For example, if the Euclidean metric for two-dimensional space is ds² = dx²+dy², which is just the Pythagorean theorem. If one defines the metric to be ds² = 2[du²-2dudv] + dv² instead, then one is still dealing with Euclidean (flat) space, just in different coordinates (they can be related by x = u+v, y = u, to be precise). None of this is a truly fatal flaw to your statement; it's only that one should simply define them to be orthogonal rather than assume that simply having a vector makes them orthogonal.Surlethe wrote:OK. Gotcha. I was thinking of components like v = ai + bj + ck + ... = <a, b, c, ...>. And I'm currently (mathematically) still living in Euclidean space, so non-orthogonal axes didn't even occur to me.
