What is dimension? And how can there be more than 3 of them ?

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DrRocket

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<p>I have noticed occasional confusion in some of the threads with the notion of dimensions and with manifolds.&nbsp; I won't talk about manifolds here, since one first needs to understand Euclidean spaces and dimensions to discuss them, and since they are a bit more complicated.&nbsp; I may tackle manifolds in another thread -- later.</p><p>But let's talk about dimension.&nbsp; Dimension is just a another way of saying "degree of freedom".&nbsp; You walk down the street and are free to go straight ahead, backwards,&nbsp;left or right.&nbsp; Basically you can describe your position on a map with two coordinates. So the surface of the earth on which you walk is basically a two-dimensional plane (we are presently not advanced enough to graduate from the flat earth society).&nbsp; You come to a building and climb&nbsp; a set of stairs, so you have found a third dimension.&nbsp; And surely everyone has been exposed to the idea that in order to call a meeting you need to specify the place (3 dimensions now) and time ( a fourth dimension).</p><p>So to go to a meeting you need the coordinates in a 4-dimensional space (note that the word "space" is used here as it is used by a mathematician or physicist and may have nothing to do with the usual 3 spatial dimensions).&nbsp; But suppose that you are not attending a business meeting but rather are attending the theater.&nbsp; To see the show you need to know not only the place (3 dimensions) and time (a 4th dimension) but also the ticket price (a fifth dimension).</p><p>If you have ever taken a thermodynamics course you have been exposed to the notion of state variables.&nbsp; Or you may have seen them in a course on mechanics and dynamical systems.&nbsp; Let's look at the the latter.&nbsp; To specify the state of a particle you need to know the position (3 variables) and the momentum (3 more variables).&nbsp; So the state space, or phase space, for a particle is 6-dimensional.&nbsp; Now suppose that you have N particles.&nbsp; The description of each requires 6 variables and there are N of them so the dimension of the phase space for the system is of dimension 6N.&nbsp; If you happen to be a control engineer, then the state space for control systems is another very similar example (this is basically where Rudolph Kalman got the idea for state space analysis).</p><p>Here's another way to look at it.&nbsp; If you consider the set of all real valued functions defined on a single point, then any such function is described by a single number and the set of all such functions is 1-dimensional.&nbsp; If you consider the set of all real-valued functions defined on two points, say 0 and 1 then a function is described by two numbers and the set of such functions is 2-dimensional.&nbsp; It you consider the set of all real-valued functions defined on the integers 1,2,3,...,N then that set is N-dimensional.&nbsp; And if you consider the set of all real-valued functions defined on all&nbsp;positive integers (or all integers for that matter)&nbsp;then that set is infinite dimensional.</p><p>Now let's go one step further.&nbsp; Consider the set of functions defined on the&nbsp; integers such that the infinite sum of the values when squared is summable as an infinite series.&nbsp; That set of functions is also infinite dimensional.&nbsp; But you can also use the value of such functions as a coefficient for&nbsp;the function that takes x to exp(-ix) where i is the square root of negative 1.&nbsp; If you do that you capture the theory of Fourier series, for functions that are periodic and that are are square integrable over the unit interval from -pi to pi.&nbsp; So you see that the notion of an infinite dimensional space can have some practical applications.</p><p>So you see, in reality&nbsp; you deal with higher dimensional constructions all the time.&nbsp; There is nothing strange about it, once you understand what a dimension really is.&nbsp; </p><p>If you play a similar game on a finite set of integers, you get the theory of the Fast Fourier Transform.&nbsp; Another application of higher-dimensional spaces.</p> <div class="Discussion_UserSignature"> </div>
 
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jbachmurski

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Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>but also the ticket price (a fifth dimension)Posted by DrRocket</DIV><br /><br /><p>&ldquo;ticket price (a fifth dimension)&rdquo;? I&rsquo;m not sure that that qualifies as a dimension. I suppose if its connected and perpendicular to all preceding dimensions&hellip; but I can see where that might lead there being millions of dimensions pertaining to any given event.</p>
 
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DrRocket

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>&ldquo;ticket price (a fifth dimension)&rdquo;? I&rsquo;m not sure that that qualifies as a dimension. I suppose if its connected and perpendicular to all preceding dimensions&hellip; but I can see where that might lead there being millions of dimensions pertaining to any given event. <br />Posted by jbachmurski</DIV></p><p>It is an additional independent parameter necessary to get you into the show.&nbsp; And your perceptions are correct there could be indeed be millions of dimensions pertaining to any event.</p><p>In fact there can be infinitely many dimensions quite easily.&nbsp; I talked about the set of all functions defined on the integers being infinite dimensional.&nbsp; You can also consider&nbsp;the set of real-valued (oir complex-valued) functions defined on a interval.&nbsp; That is also infinite dimensional.&nbsp; If you look at just the continuous functions that too is infinite-dimensional.&nbsp; Now consider the solution to some partial differential equation, say the heat equation on a region in the plane bounded by a closed curve, say a circle.&nbsp; The solutions to that PDE are dependent on the boundary conditions on the circle and those boundary conditionis are defined by a continuous function on the circle.&nbsp; So the set of boundary conditions has an infinite-dimensional structure and the solution to the heat equation, the "event" depends on this infinite dimensional space.&nbsp; </p><p>The notion of perpendicularity, or orthogonality, &nbsp;requires another concept, that of an inner product.&nbsp; I deliberately avoided getting any more complicated than simply discussing dimensions.&nbsp; With an inner product there are many more things that one can do, and it works in some infinite dimensional spaces as well, primarily in Hilbert spaces.&nbsp; In addition, once you a have a notion of inner productd you can start to talk about geometry, but that will be another subject.&nbsp; </p><p>There is no meaning to connnectedness of one dimension to another.&nbsp; There is a topological meaning to the tern "connected" and all&nbsp;real and complex&nbsp;topological vector spaces are connected.&nbsp; </p> <div class="Discussion_UserSignature"> </div>
 
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emperor_of_localgroup

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'> If you have ever taken a thermodynamics course you have been exposed to the notion of state variables.&nbsp; Or you may have seen them in a course on mechanics and dynamical systems.&nbsp; Let's look at the the latter.&nbsp; To specify the state of a particle you need to know the position (3 variables) and the momentum (3 more variables).&nbsp; So the state space, or phase space, for a particle is 6-dimensional.&nbsp; Now suppose that you have N particles.&nbsp; The description of each requires 6 variables and there are N of them so the dimension of the phase space for the system is of dimension 6N.&nbsp; If you happen to be a control engineer, then the state space for control systems is another very similar example (this is basically where Rudolph Kalman got the idea for state space analysis).......&nbsp;&nbsp;&nbsp; It you consider the set of all real-valued functions defined on the integers 1,2,3,...,N then that set is N-dimensional.&nbsp; And if you consider the set of all real-valued functions defined on all&nbsp;positive integers (or all integers for that matter)&nbsp;then that set is infinite dimensional.<br /> Posted by DrRocket</DIV></p><p><font size="2">I agree with what you have written, they are totally correct and accepted, but the bottom line is 'Mother of all Dimensions is 3'. Add time to it, we get 4, even though I feel&nbsp; time is a&nbsp; 'step-dimension' as&nbsp; in 'step son', or 'step dad' - a non-physical dimension. Most&nbsp; other&nbsp; dimensions originate from those 3 basic dimensions, such as 3 components of momentum, acceleration etc.</font></p><p><font size="2">We can call the states in state-space or the variables of a multi-variable function as dimensions because they represent 'degrees of freedom', but in a deeper sense they are not quite dimensions. Let me explain what I think,(and I'm not the brightest person anywhere).</font></p><p><font size="2">&nbsp;Basis of our math, specially calculus,&nbsp; is change. Spatial change (in 3D) is the first thing we experience everyday. The math we developed is actually geometric in nature. We have found that if we apply the same math in state-space or multi-variable functions our computations get easier.&nbsp; We replace x,y,z with say x1, x2, x3, x4......xn and use them in the&nbsp; same mathematical forms as we use for 3D space. When we do that we get the right answers. This may inadvertently lead us to believe we are working in multi-dimensional space. </font></p><p><font size="2">I'm not saying dimensions more than 3 (I mean more than 4) do not exist, but they are probably totally incomprehensible to us.&nbsp; It is very difficult for a 4D person to make a sense about a 5D person, but it is somewhat easier for a 5D person to describe a 4D person. A man can make a good judgement about what it's like to be a cat, but cat's brain would probably fall off if he tries to understand a man.</font></p><p>&nbsp;</p> <div class="Discussion_UserSignature"> <font size="2" color="#ff0000"><strong>Earth is Boring</strong></font> </div>
 
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killium

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<p>I think it depends on what is your definition of "dimension". If you call "dimension" any value something can be described with, then yes it's probably infinite. But if a dimension is a "plane of displacement" then you only need 3 of them to describe (the position) of anything. If you define a dimension as a "tool to find something", then you also need the time.</p><p>&nbsp;</p><p>So the spatial dimensions describes the "where" while the time dimension describe the "when". You can also ask the "why, what, how, how many, how much" etc.... but can those numbers be called "dimensions" ?</p><p>&nbsp;</p><p>Now the big question: If we're striclky talking about spatial dimensions (those needed to answer the "where"), how can there be more than 3 of them ?</p><p>&nbsp;</p> <div class="Discussion_UserSignature"> </div>
 
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derekmcd

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I think it depends on what is your definition of "dimension". If you call "dimension" any value something can be described with, then yes it's probably infinite. But if a dimension is a "plane of displacement" then you only need 3 of them to describe (the position) of anything. If you define a dimension as a "tool to find something", then you also need the time.&nbsp;So the spatial dimensions describes the "where" while the time dimension describe the "when". You can also ask the "why, what, how, how many, how much" etc.... but can those numbers be called "dimensions" ?&nbsp;Now the big question: If we're striclky talking about spatial dimensions (those needed to answer the "where"), how can there be more than 3 of them ?&nbsp; <br /> Posted by killium</DIV></p><p>It's difficult to think about more dimensions that our common experiences describe.&nbsp; Much the same way a 2d stick figure living on a 2d plane would have difficultly physically describing a 3rd spatial dimension.&nbsp; The hidden dimensions of something such as Sting Theory are described as hidden because they are not something we can visually interpret.</p><p>Here's a rotating 4d hypercube.&nbsp; It might be 4d, but it still has to be visuallized in 3d... we are simply not capable to do so otherwise AFAIK.</p><p><br /> <img src="http://sitelife.space.com/ver1.0/Content/images/store/3/5/63f34e3f-48d2-47e7-a5c0-ee03c509fe6f.Medium.gif" alt="" /><br />&nbsp;</p> <div class="Discussion_UserSignature"> <div> </div><br /><div><span style="color:#0000ff" class="Apple-style-span">"If something's hard to do, then it's not worth doing." - Homer Simpson</span></div> </div>
 
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emperor_of_localgroup

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>It's difficult to think about more dimensions that our common experiences describe.&nbsp; Much the same way a 2d stick figure living on a 2d plane would have difficultly physically describing a 3rd spatial dimension.&nbsp; The hidden dimensions of something such as Sting Theory are described as hidden because they are not something we can visually interpret.Here's a rotating 4d hypercube.&nbsp; It might be 4d, but it still has to be visuallized in 3d... we are simply not capable to do so otherwise AFAIK. &nbsp; <br />Posted by derekmcd</DIV><br /><br /><font size="2">Hahaha,</font></p><p><font size="2">It's not 4-dimensional. It is probably a projection of 4D on 3 dimensions.&nbsp; Only thing we can have is projection of higher dimensions on our dimensions. In fact, the animation in your post &nbsp;is a projection of 4D on 2D (2 dimensional computer screen).</font></p><p><font size="2">But I was hoping&nbsp; more people&nbsp; participate &nbsp;&nbsp;in the thread. It's a very interesting subject.</font></p> <div class="Discussion_UserSignature"> <font size="2" color="#ff0000"><strong>Earth is Boring</strong></font> </div>
 
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killium

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>It's difficult to think about more dimensions that our common experiences describe.&nbsp; Much the same way a 2d stick figure living on a 2d plane would have difficultly physically describing a 3rd spatial dimension.&nbsp; The hidden dimensions of something such as Sting Theory are described as hidden because they are not something we can visually interpret.Here's a rotating 4d hypercube.&nbsp; It might be 4d, but it still has to be visuallized in 3d... we are simply not capable to do so otherwise AFAIK. &nbsp; <br />Posted by derekmcd</DIV><br /><br />and yet, you're showing me that 4d hypercube using a 3d representation of it, drawn on a 2d screen! <img src="http://sitelife.space.com/ver1.0/content/scripts/tinymce/plugins/emotions/images/smiley-tongue-out.gif" border="0" alt="Tongue out" title="Tongue out" /></p><p>seriously, it's easy to say a 2D thing would have difficulties imagining a third dimension, problem is, a 2D thing doesn't exist in reality! Even the ink of a drawing have a thickness. Something with 0 thickness, or 0&nbsp;height, or 0 width, can't exist. So this&nbsp;explanation saying that we don't get the 4th dimension cause we're suited for 3 for the same reason 2D thing don't get the third dimension, is flawed, and surely explains nothing to me, in concrete terms.</p><p>&nbsp;</p><p>edit: just to add, even those ants on the balloon can make it that there is a third dimension under thier feet. Isn't what we did ? Aren't we all just ants on a balloon ? <img src="http://sitelife.space.com/ver1.0/content/scripts/tinymce/plugins/emotions/images/smiley-wink.gif" border="0" alt="Wink" title="Wink" /></p><p>&nbsp;</p> <div class="Discussion_UserSignature"> </div>
 
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derekmcd

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Hahaha,It's not 4-dimensional. It is probably a projection of 4D on 3 dimensions.&nbsp; Only thing we can have is projection of higher dimensions on our dimensions. In fact, the animation in your post &nbsp;is a projection of 4D on 2D (2 dimensional computer screen).But I was hoping&nbsp; more people&nbsp; participate &nbsp;&nbsp;in the thread. It's a very interesting subject. <br /> Posted by emperor_of_localgroup</DIV></p><p>What's the point of the laugh?&nbsp; I didn't think I needed to state the obvious.&nbsp; I think it's pretty clear to everyone on the planet that the animation does not physically consist of 4 spatial dimension.</p><p>It's not "probably" a projection... It IS a projection.&nbsp; In fact, it is a perspective geometrical projection used as a visual aid to help conceptualize.&nbsp; Nothing more, nothing less.</p><p>The cosmological balloon analogy is quite the opposite.&nbsp; It is the image of a 3d sphere used to describe 2d space. Even if you try to avoid saying it is a 3d sphere, it is human nature to conceptualize it as such.&nbsp; It's obvious it is not visually 2 dimensional even though it can be mathematically described as a 2-manifold surface. </p><p>You will have to do a better job than this if you want to ridicule me with "haha" posts.&nbsp;</p> <div class="Discussion_UserSignature"> <div> </div><br /><div><span style="color:#0000ff" class="Apple-style-span">"If something's hard to do, then it's not worth doing." - Homer Simpson</span></div> </div>
 
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derekmcd

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>and yet, you're showing me that 4d hypercube using a 3d representation of it, drawn on a 2d screen! seriously, it's easy to say a 2D thing would have difficulties imagining a third dimension, problem is, a 2D thing doesn't exist in reality! Even the ink of a drawing have a thickness. Something with 0 thickness, or 0&nbsp;height, or 0 width, can't exist. So this&nbsp;explanation saying that we don't get the 4th dimension cause we're suited for 3 for the same reason 2D thing don't get the third dimension, is flawed, and surely explains nothing to me, in concrete terms.&nbsp;edit: just to add, even those ants on the balloon can make it that there is a third dimension under thier feet. Isn't what we did ? Aren't we all just ants on a balloon ? &nbsp; <br /> Posted by killium</DIV></p><p>Of the top of my head, images of a shadow are 2 dimensional.&nbsp; Your comments only support the notion the difficulties of visualizing anything but a 3d world.&nbsp; The best descriptions you will find are in mathematics concerning manifolds and topology. </p> <div class="Discussion_UserSignature"> <div> </div><br /><div><span style="color:#0000ff" class="Apple-style-span">"If something's hard to do, then it's not worth doing." - Homer Simpson</span></div> </div>
 
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SpeedFreek

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<p>All the lines that make up the 4D hypercube are the <strong><em>same length</em></strong>. Try to wrap your brain around that and it will probably pop out of your ears... </p><p> <img src="http://sitelife.space.com/ver1.0/content/scripts/tinymce/plugins/emotions/images/smiley-wink.gif" border="0" alt="Wink" title="Wink" /></p> <div class="Discussion_UserSignature"> <p><font color="#ff0000">_______________________________________________<br /></font><font size="2"><em>SpeedFreek</em></font> </p> </div>
 
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killium

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<p>I'm not trying to ridiculise anyone here. I'm asking questions and reading answers. And when those answers don't satisfy my urge to learn, i ask more of them, or ask for clarifications. And if i see a flaw in an answer, i will point it, the goal of this beeing "clearing the flaw", it by either changing the answer, or prooving me there is no flaw in fact and that it's something i didn't understood well. And i'm gladly accepting beeing corrected, in fact, i hope for it, that's how i learn.</p><p>&nbsp;</p><p>About shadows. A shadow doesn't exist. Our mind can infer that this spot receive less light and that it's due to an object hiding the light source, and that the contour of it represents the shape of that object, but the shadow itself, as a seperate component in the universe,&nbsp;doesn't exist in reality.</p><p>&nbsp;</p> <div class="Discussion_UserSignature"> </div>
 
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emperor_of_localgroup

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>What's the point of the laugh?&nbsp;&nbsp; You will have to do a better job than this if you want to ridicule me with "haha" posts.&nbsp; <br />Posted by derekmcd</DIV></p><p><font size="2">Jesus, you guys are too sensitive about intellegence or PC, don't be paranoid, that takes out the spirit of tntellectual information exchange. We are not in any competition here, whatsoever. The animation looked funny to me.</font></p><p><font size="2">Anyway, my other question is why do we always think of extra physical dimensions? Why not an extra non-physical dimension, in addition to 'time'?&nbsp; I think our limitation to know 'everything' comes from lack of a few extra non-physical dimensions. Just my guess. </font></p><p><font size="2">Also note that a 2D 'thing' will glide through a 3D space, but a 3D thing will never fit in a 2D space. Simialrly a 1D thing will be at ease in a 2D space. &nbsp;So our universe can not support 'things' from higher dimensions.&nbsp;<br /></font></p> <div class="Discussion_UserSignature"> <font size="2" color="#ff0000"><strong>Earth is Boring</strong></font> </div>
 
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killium

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>..... Also note that a 2D 'thing' will glide through a 3D space, but a 3D thing will never fit in a 2D space. Simialrly a 1D thing will be at ease in a 2D space. &nbsp;So our universe can not support 'things' from higher dimensions.&nbsp; <br />Posted by emperor_of_localgroup</DIV><br /></p><p>It cannot support things from less dimensions too. As i said (i may be wrong but...), if you're in a 3D universe, anything that exist in that universe ought to be 3D, no more (yes, that's speculation, to which i'm looking for an answer), no less (that one i'm conceptually sure, though it still intuition).</p><p>&nbsp;</p> <div class="Discussion_UserSignature"> </div>
 
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DrRocket

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>It cannot support things from less dimensions too. As i said (i may be wrong but...), if you're in a 3D universe, anything that exist in that universe ought to be 3D, no more (yes, that's speculation, to which i'm looking for an answer), no less (that one i'm conceptually sure, though it still intuition).&nbsp; <br />Posted by killium</DIV></p><p>That rather depends on what you are looking at.&nbsp; Macroscopic objects would almost certainly have to be 3-dimensional in a 3-D world.&nbsp;We think of them as being modeled as a continuum -- the theory of elasticity for instance is a topic in continuum mechanics.&nbsp; A continuum that is of fewer dimensions than 3 would have zero mass in a 3-D world. So macroscopic objects will indeed be 3-dimensional.</p><p>But when you start looking at the atomic and quantum levels things get weird.&nbsp; Elementary particles are currently thought of a being points, 0-dimensional objects.&nbsp; But quantum weirdness also makes it impossible to say exactly where those points are or exactly how fast they are moving.&nbsp; So there is room in current physical theory for 0-dimensional massive objects.</p><p>Speculative physics may include massive things of other dimensions.</p><p>But for everyday life your intuition that objects must be fully 3-dimensional will not lead you astray.<br /></p> <div class="Discussion_UserSignature"> </div>
 
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why06

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Lol as soon as I read this topic I was waiting for someone to pull out the picture of the wikipedia "hypercube" :p <div class="Discussion_UserSignature"> <div>________________________________________ <br /></div><div><ul><li><font color="#008000"><em>your move...</em></font></li></ul></div> </div>
 
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why06

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'> Consider the set of functions defined on the&nbsp; integers such that the infinite sum of the values when squared is summable as an infinite series.&nbsp; That set of functions is also infinite dimensional.&nbsp; But you can also use the value of such functions as a coefficient for&nbsp;the function that takes x to exp(-ix) where i is the square root of negative 1.&nbsp; If you do that you capture the theory of Fourier series, for functions that are periodic and that are are square integrable over the unit interval from -pi to pi.&nbsp; So you see that the notion of an infinite dimensional space can have some practical applications.So you see, in reality&nbsp; you deal with higher dimensional constructions all the time.&nbsp; There is nothing strange about it, once you understand what a dimension really is.&nbsp; If you play a similar game on a finite set of integers, you get the theory of the Fast Fourier Transform.&nbsp; Another application of higher-dimensional spaces. <br /> Posted by DrRocket</DIV></p><p>I think I see where your getting at. Personally I don't know anything about Fourier spaces, but I have learned how to program to a very basic extent. I have also taken a class in Chaos. Dimensions play a very big role in both. In Java programming I have these things called Arrays. Array can not only arange things horizontal and vertically but along an infinite number of dimensions. These Arrays are often use by programs to create intergers and numbers of such vast amounts and complexity that it would be impossible to arrange denote by hand and or a sort of "while loop" (programming jargon). Not only this, but an array can store another array.&nbsp; So that an infinite number of dimensions could simply be one dimension of an Even more encompassing array.</p><p>&nbsp;What Im saying is what if ur universe of infinite dimensons is only really one dimension of an even greater Array. This is really just a mind game though for me. Since I can see no practical application due to my lack of knowledge all I can do is simply suggest ideas on the subject of dimensions themselves. What this could imply? I have no clue. </p> <div class="Discussion_UserSignature"> <div>________________________________________ <br /></div><div><ul><li><font color="#008000"><em>your move...</em></font></li></ul></div> </div>
 
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derekmcd

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Lol as soon as I read this topic I was waiting for someone to pull out the picture of the wikipedia "hypercube" :p <br /> Posted by why06</DIV></p><p>Actually, I pulled it off of someone's avatar from another forum.&nbsp; I forget which forum as it was a year ago or so.&nbsp; This was the first time I had a chance to use it.&nbsp; There's quite a few variations floating around.&nbsp;</p><p>Here's the one I used:</p><p><br /> <img src="http://sitelife.space.com/ver1.0/Content/images/store/7/12/47db644c-6873-4fdc-9c95-59fc411930d3.Medium.gif" alt="" /></p><p>Here's the one from Wiki:</p><p><br /><br /> <img src="http://sitelife.space.com/ver1.0/Content/images/store/7/3/47e391e1-945f-4556-9332-04ec72c435f7.Medium.gif" alt="" /></p><p><br />There different.&nbsp;</p> <div class="Discussion_UserSignature"> <div> </div><br /><div><span style="color:#0000ff" class="Apple-style-span">"If something's hard to do, then it's not worth doing." - Homer Simpson</span></div> </div>
 
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why06

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Actually, I pulled it off of someone's avatar from another forum.&nbsp; I forget which forum as it was a year ago or so.&nbsp; This was the first time I had a chance to use it.&nbsp; There's quite a few variations floating around.&nbsp;Here's the one I used: Here's the one from Wiki: There different.&nbsp; <br /> Posted by derekmcd</DIV></p><p>ah...</p><p>...Well I like the wiki one better, but umm.... thanks for filling me in on the mysteries of ur accuisition of the 4-D hypercube avatar. :/</p> <div class="Discussion_UserSignature"> <div>________________________________________ <br /></div><div><ul><li><font color="#008000"><em>your move...</em></font></li></ul></div> </div>
 
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vandivx

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I think it depends on what is your definition of "dimension". If you call "dimension" any value something can be described with, then yes it's probably infinite. But if a dimension is a "plane of displacement" then you only need 3 of them to describe (the position) of anything. If you define a dimension as a "tool to find something", then you also need the time.&nbsp;So the spatial dimensions describes the "where" while the time dimension describe the "when". You can also ask the "why, what, how, how many, how much" etc.... but can those numbers be called "dimensions" ?&nbsp;Now the big question: If we're striclky talking about spatial dimensions (those needed to answer the "where"), how can there be more than 3 of them ?&nbsp; <br /> Posted by killium</DIV><br />your's is a very sensible view, more than three dimensions in the same sense that we have the three ones are the product of groundless wishes or imagination (there is not an iota of evidence for anything like that)</p><p>I regard the modern quest for extra dimensions as outgrowth of scientist's inability to make any progress in science given the physical reality as it is and hoping blindly that there really may be some higher dimensions to reality using which one would be able to make new progress, it is the 'what if' kind of making science which is no different from science fiction</p><p>&nbsp;</p><p>that is not to say that 3+n dimensional calculations are nonsense, quite the contrary, only one shouldn't be mislead into thinking that such can exist physically, higher than 3D dimensional theories are valid if those extra dimensions are understood to mean extra variables of some kind - typical use is making time the 4th dimension, n-dimensional space when it is understood that it is abstract (that is mathematical) space is fine too, unfortunately modern yahoos of science now play with those extra dimensions as if they were real physical dimensions of the same or similar kind as the three real ones</p><p>&nbsp;</p><p>actually our three dimensions are abstract division of nature in man's minds and do not exist as such out there, nature comes the way it does, 'whole', without being parcelled into dimensions and scientists invented the most efficient way, minimized&nbsp; but sufficient number of abstract coordinate axes to describe it </p><p>&nbsp;</p><p>when it comes to microscopic world of atomic particles, it is a blind alley to speculate about some other spatial dimensions, this time fewer than our three, it is again the inability to make progress in science that makes such flight to zero dimensional reality tempting to some but that's like 'explaining' without really explaining anything (it is invention to suit, to explain away), it is really no wonder that any such explorations into alternate dimensionality never lead to anything this far and I for one don't expect that trend to be broken, I suppose it puts the bacon on the table for some and that may be taken as justification of such pursuits and if others are willing to pay for it it will go on </p> <div class="Discussion_UserSignature"> </div>
 
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emperor_of_localgroup

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I regard the modern quest for extra dimensions as outgrowth of scientist's inability to make any progress in science given the physical reality as it is and hoping blindly that there really may be some higher dimensions to reality using which one would be able to make new progress, .......</p><p>when it comes to microscopic world of atomic particles, it is a blind alley to speculate about some other spatial dimensions, this time fewer than our three, it is again the inability to make progress in science that makes such flight to zero dimensional reality <br />Posted by vandivx</DIV></p><p><font size="2">You are brave to make this post here.&nbsp; But glad more people are paricipating in this thread. I tend to agree with the first paragraph above as I keep reading about new scientific discoveries. This reality is so strange, so mind boggling,&nbsp; that's why we have so many &nbsp;imaginations about it, be it mental imagination or mathematical imagination. &nbsp;What bothers me more , these scientists (I'm sure they know the truth about their theories &nbsp;themselves) could say, "this is one possibility of reality", instead they want average people to believe their findings are "the actual reality". <br /></font></p><p><font size="2">Dimensions higher &nbsp;than 3+1 is also the main foundation of String theory.&nbsp; Their argument is all dimensions above 3+1 are 'curled' up.&nbsp; My question is did they really mathematically find&nbsp; dimensions &nbsp;'curled' up? I wonder what kind of math can predict curling up of dimensions? What I think is they set all dimensions except 3+1 to zero (or set to length below planck's length), and when they did this they found the results agreed with 'things' known in our world.&nbsp; I wonder if this is what they term as 'curled up'?&nbsp;Because no 'reason of curling up'&nbsp; is ever given.</font></p><p>&nbsp;</p> <div class="Discussion_UserSignature"> <font size="2" color="#ff0000"><strong>Earth is Boring</strong></font> </div>
 
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killium

Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>You are brave to make this post here.&nbsp; But glad more people are paricipating in this thread. I tend to agree with the first paragraph above as I keep reading about new scientific discoveries. This reality is so strange, so mind boggling,&nbsp; that's why we have so many &nbsp;imaginations about it, be it mental imagination or mathematical imagination. &nbsp;What bothers me more , these scientists (I'm sure they know the truth about their theories &nbsp;themselves) could say, "this is one possibility of reality", instead they want average people to believe their findings are "the actual reality". Dimensions higher &nbsp;than 3+1 is also the main foundation of String theory.&nbsp; Their argument is all dimensions above 3+1 are 'curled' up.&nbsp; My question is did they really mathematically find&nbsp; dimensions &nbsp;'curled' up? I wonder what kind of math can predict curling up of dimensions? What I think is they set all dimensions except 3+1 to zero (or set to length below planck's length), and when they did this they found the results agreed with 'things' known in our world.&nbsp; I wonder if this is what they term as 'curled up'?&nbsp;Because no 'reason of curling up'&nbsp; is ever given.&nbsp; <br />Posted by emperor_of_localgroup</DIV><br /></p><p>yeah, that one is really special.... curled dimensions....!! again, what is a dimension ? if it's a conceptual "plane of displacement", how can you curl this ?</p><p>&nbsp;</p> <div class="Discussion_UserSignature"> </div>
 
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DrRocket

Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>You are brave to make this post here.&nbsp; But glad more people are paricipating in this thread. I tend to agree with the first paragraph above as I keep reading about new scientific discoveries. This reality is so strange, so mind boggling,&nbsp; that's why we have so many &nbsp;imaginations about it, be it mental imagination or mathematical imagination. &nbsp;What bothers me more , these scientists (I'm sure they know the truth about their theories &nbsp;themselves) could say, "this is one possibility of reality", instead they want average people to believe their findings are "the actual reality". Dimensions higher &nbsp;than 3+1 is also the main foundation of String theory.&nbsp; Their argument is all dimensions above 3+1 are 'curled' up.&nbsp; My question is did they really mathematically find&nbsp; dimensions &nbsp;'curled' up? I wonder what kind of math can predict curling up of dimensions? What I think is they set all dimensions except 3+1 to zero (or set to length below planck's length), and when they did this they found the results agreed with 'things' known in our world.&nbsp; I wonder if this is what they term as 'curled up'?&nbsp;Because no 'reason of curling up'&nbsp; is ever given.&nbsp; <br />Posted by emperor_of_localgroup</DIV></p><p>I don't know if curled-up dimensions are physically real or not, and neither does anyone else.&nbsp; But the idea of how they could exist and not be detected goes something like this.&nbsp; Think of&nbsp;garden hose.&nbsp; It is a three dimensional object.&nbsp; But from far away it looks like a line, which is one-dimensinal.&nbsp; When, in everyday life you think about "where" on a garden hose you really mean a distance taken from one end or the other, a 1-dimensional reference.&nbsp; The idea of curled up dimensions is rather like that, the realization of the dimensions is such that you just don't notice them.&nbsp; </p><p>It is important to recognize that we are not talking about ordinary Euclidean space here, where you can follow an dimensional axis forever along a straight line.&nbsp; We are talking about the space-time manifold, which is only locally Euclidean, but may have curvature.&nbsp; This is analagous to the surface of a balloon which is 2-dimensional, looks like a plane in small patches,&nbsp;but is curved&nbsp;has only a limited spatial extent.</p> <div class="Discussion_UserSignature"> </div>
 
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derekmcd

Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I don't know if curled-up dimensions are physically real or not, and neither does anyone else.&nbsp; But the idea of how they could exist and not be detected goes something like this.&nbsp; Think of&nbsp;garden hose.&nbsp; It is a three dimensional object.&nbsp; But from far away it looks like a line, which is one-dimensinal.&nbsp; When, in everyday life you think about "where" on a garden hose you really mean a distance taken from one end or the other, a 1-dimensional reference.&nbsp; The idea of curled up dimensions is rather like that, the realization of the dimensions is such that you just don't notice them.&nbsp; It is important to recognize that we are not talking about ordinary Euclidean space here, where you can follow an dimensional axis forever along a straight line.&nbsp; We are talking about the space-time manifold, which is only locally Euclidean, but may have curvature.&nbsp; This is analagous to the surface of a balloon which is 2-dimensional, looks like a plane in small patches,&nbsp;but is curved&nbsp;has only a limited spatial extent. <br /> Posted by DrRocket</DIV></p><p>I've seen this analogy applied to electrical/telephone lines stretched across the street.&nbsp; From a distance it appears one dimensional and I can only point out locations on it on the x-axis.&nbsp; A bug on the wire can define both the x and y-axis.&nbsp; The particles running through it can define x, y and z-axis.</p><p>If I'm far enough away from the wire or if it is small enough, it simply isn't detectable.&nbsp;</p> <div class="Discussion_UserSignature"> <div> </div><br /><div><span style="color:#0000ff" class="Apple-style-span">"If something's hard to do, then it's not worth doing." - Homer Simpson</span></div> </div>
 
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why06

Guest
<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>I've seen this analogy applied to electrical/telephone lines stretched across the street.&nbsp; From a distance it appears one dimensional and I can only point out locations on it on the x-axis.&nbsp; A bug on the wire can define both the x and y-axis.&nbsp; The particles running through it can define x, y and z-axis.If I'm far enough away from the wire or if it is small enough, it simply isn't detectable.&nbsp; <br /> Posted by derekmcd</DIV></p><p>I've always had a problem with the way on programs about string theory when they talk about extra dimensions they always fail to explaint the reason why. I've always wondered why in the hell dimensions infinitely more vast then ours would be curled up in an incredibly small space. In my opinion its complete BS. These are mathemiticians devising these theories not scientist. Math can help to calculate phenomenon, but it can not replace science. </p><p>Einstien changed our view of spac-time. it shattered ideas of Euclidean space. However that doesn't mean that everytime someone is trying to figure something out they need to try and redefine the universe. In my oppinion the society of particle physicist have given the mathemiticians to much freedom. Mathematics will take us down the path towards the infitismal and the infinite so that even by the time these M theories, A Theories, and D Theories reach a conclusion the mathematics will be to complex for the <em>real </em>scientist to find any use of the equations. They are losing their connection to reality. </p><p><strong><em>&nbsp;</em></strong></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><strong><em>Whats the point of writing the universe in an equation if the equation is 1000 characters long, infinitely complex, and can no longer be applied to reality? Where is the beauty in that. What happened to parsimony? What happened to always taking the easiest approach? Where have all the scientist gone?</em></strong></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p> <div class="Discussion_UserSignature"> <div>________________________________________ <br /></div><div><ul><li><font color="#008000"><em>your move...</em></font></li></ul></div> </div>
 
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