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

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#### SHU

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Vectors are in fact points and really don't have a dimension.&nbsp; What has a dimension is a vector space, which is an algebraic structure whose elements are vectors with the additional notions of vector addition and subtraction and scalar multiplication.&nbsp; That additinal structure is what is responsible for the notion of dimension (see a text on linear algebra for the details).Once you have a notion for the dimensin of a vector space, that notion provides the notion of dimension for Euclidean spaces, which are just vector spaces with additional structure that allows the measurement of distance (also angles), which in turn provides a topology, and multi-variable calculus.&nbsp; That in turn permits the development of the theory of topological and differentiable manifolds which is the theory required for general relativity (Riemannian geometry).There would be a problem with a multi-dimensional notion of time, the inability to provide, even locally, the ordering necessary to what we experience as the "flow" of time.&nbsp; There have been some speculations to the contrary -- I seem to recall Hawking contemplating complex time, but I don't think it ever went anywhere.&nbsp; You can already think about curves related to time and that is&nbsp;pretty much what happens in general relativity with curved space-time, and in fact time gets a bit mixed up with space over large expanses.The bottom line is that time is not all that well understood, and in particular the reason that time seems to flow in one direction is not explainable from more fundamental principles.&nbsp; It is a topic of research.&nbsp; Einstein was of the opinion that time is basically statistical in nature, but no one has ever provided a rigorous and useful formulation of that notion as a predictive theory.&nbsp; <br />Posted by DrRocket</DIV></p><p>&nbsp;</p><p><font size="2">Thanks.&nbsp; With the exception of differential manifolds, I understand what you're saying.&nbsp; Please excuse my magazine approach on this forum and misuse of terminology.&nbsp; I'm just trying to challenge assumptions and inspire thought by presenting alterrnative concepts.&nbsp; I'm seeing Occam's razor used to&nbsp;chop off heads&nbsp;as if ideas&nbsp;were a challenge to religion.&nbsp; Granted, some ideas should be chopped, shredded and used to&nbsp;line hamster cages.</font><br /></p>

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#### UncertainH

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<p>Replying to:</p><p>All of the above</p><p>&nbsp;We already ready know from relativity that time is not linear I wonder why it is so hard to accept time as a non-linear dimension. I would suggest that energy cannot exist without time. Pure energy (perhaps the photon), we say, sees no length in its frame of reference but it must exist for a certain amount of time until it gets absorbed. Pretty simple laws of physics that must always hold true. It has no need for anything like euclidian space to describe itself. I suppose you could say that it is not even 'flatland' it is 'pointland'. So time is the first dimension, energy is the second. Mass energy has a different relationship to time and is second order to pure energy we know this from e=mc2. Perhaps mass energy is the third dimension. It may be possible to construct a vector space of forms of energy separated by time but that time would be non-linear. Spacetime as we know it would be&nbsp;an emergent phenomenon and applied on top of the vector space and the time that we use in that space would have a fixed relationship to the time used in&nbsp;the vector space. In the vector space when two mass energies move towards each other they are lessening the amount of second order time between them this would be viewed as acceleration in the emergent spacetime. How the energy translates to volume and time separation to distance would likely be governed by things like planks constant and other constants.</p>

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#### SHU

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Replying to:All of the above&nbsp;We already ready know from relativity that time is not linear I wonder why it is so hard to accept time as a non-linear dimension. I would suggest that energy cannot exist without time. Pure energy (perhaps the photon), we say, sees no length in its frame of reference but it must exist for a certain amount of time until it gets absorbed. Pretty simple laws of physics that must always hold true. It has no need for anything like euclidian space to describe itself. I suppose you could say that it is not even 'flatland' it is 'pointland'. So time is the first dimension, energy is the second. Mass energy has a different relationship to time and is second order to pure energy we know this from e=mc2. Perhaps mass energy is the third dimension. It may be possible to construct a vector space of forms of energy separated by time but that time would be non-linear. Spacetime as we know it would be&nbsp;an emergent phenomenon and applied on top of the vector space and the time that we use in that space would have a fixed relationship to the time used in&nbsp;the vector space. In the vector space when two mass energies move towards each other they are lessening the amount of second order time between them this would be viewed as acceleration in the emergent spacetime. How the energy translates to volume and time separation to distance would likely be governed by things like planks constant and other constants. <br />Posted by UncertainH</DIV></p><p>&nbsp;</p><p><font size="2">Glad and astonished to see someone following along and taking the lead.&nbsp; You must be reading between the lines.&nbsp; The multidimensional time was a bit of a sidetrack but hopefully illustrated the previous point of abstraction as spacetime transforms from "3+1" to 1.&nbsp; Don't know of anyone actually travelling through time, much less laterally.&nbsp; We're just along for the ride.&nbsp; Sometimes, I feel it rides me.&nbsp; Did I mention that I took a trip around the Sun&nbsp;last year?&nbsp; Thinking of doing it again this year.</font></p><p><font size="2">You might have gone a bit off path there but that's fine.&nbsp; Frankly, I'm not sure.&nbsp; Never pretended to have answers, just questions.&nbsp; It's a bizarre concept with no established rules or proof.&nbsp; I suppose that it might be just common sense for Bhuddists.&nbsp; The hierarchy you mentioned, or something like that,&nbsp;might be a foundation of causality, forming an alignment.&nbsp;</font></p><p>&nbsp;</p>

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#### UncertainH

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>&nbsp;Glad and astonished to see someone following along and taking the lead.&nbsp; You must be reading between the lines.&nbsp; The multidimensional time was a bit of a sidetrack but hopefully illustrated the previous point of abstraction as spacetime transforms from "3+1" to 1.&nbsp; Don't know of anyone actually travelling through time, much less laterally.&nbsp; We're just along for the ride.&nbsp; Sometimes, I feel it rides me.&nbsp; Did I mention that I took a trip around the Sun&nbsp;last year?&nbsp; Thinking of doing it again this year.You might have gone a bit off path there but that's fine.&nbsp; Frankly, I'm not sure.&nbsp; Never pretended to have answers, just questions.&nbsp; It's a bizarre concept with no established rules or proof.&nbsp; I suppose that it might be just common sense for Bhuddists.&nbsp; The hierarchy you mentioned, or something like that,&nbsp;might be a foundation of causality, forming an alignment.&nbsp;&nbsp; <br />Posted by SHU</DIV></p><p>Yes I did get a little carried asway there. It has always struck me as odd that people insist that time is linear yet in the next breath can talk about clocks ticking slower near massive bodies, clocks ticking slower at speed approaching c and even time stopping inside a black hole or in the frame of reference of a photon. The slowing of clocks is a well measured phenomenon. Energy however cannot be created or destroyed, this seems to me to be the most fundamental quantity with which to form a dimensional model.&nbsp;But it&nbsp;also seems that time&nbsp;should&nbsp;be&nbsp;a fundamental dimension that energy would require in order to exist however that might not be the same as the clocks that we use to measure events in our frame of reference. The passage of time would logically seem to be related to entropy and inertia so mass energy will continue to follow its time line until some work is done. In this model, work would be calculated as energy moving through time. This doesn't have to be translated as some kind of sci-fi time travel but when you transform that motion into a more familiar spacetime, the energy becomes mass and the time separation is distance with a clock to join the two together. One problem here though is that the transformation is likely arbitrary, you could probably choose a variety of values for G, c and h&nbsp;to make it fit our current observations</p>

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#### nova_explored

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>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)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&nbsp;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&nbsp;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 &nbsp;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 <br /> Posted by vandivx</DIV></p><p>&nbsp;</p><p>Dimensions, regardless of their number, come from the fact that no two points can occupy the same space.&nbsp; More specifically, no two states of energy (on subatomic level) can hold the same space.&nbsp; The disentegration of one state into another is a measure of time, a change of state becomes another dimension because the physical space that a particle held has changed.&nbsp; That describes the four dimensions subatomically.&nbsp; As for additional states, yes, they are part theory and part experimental. &nbsp;</p><p>You have the photon which acts in two different states at the same time as both a wave and a particle, an eye into the quantum world not describable by the physics of particles that build a 3D world. &nbsp;&nbsp; Oh and someone said that ALL dimensions are something of a subjective experience not really quantifiable (I am paraphrasing), this is not so.&nbsp; If our 3D world were subjective and indeterminable through experiment, it wouldn't behave according to fundamental laws.&nbsp; Remember 3D with time as a 4th dimension is simply the law that two particles cannot occupy the same space, giving them a certain quantifiable dimension, and a change in state determines time- the change from one state to another. &nbsp;</p><p>&nbsp;(edited) okay I opened up a can of worms with photons so I should lay the groundwork a little.&nbsp; This quantum field study is theory, BUT it is experimental.&nbsp; We can quantify it.&nbsp; Photons, and similar particle behavior is unique in that it exists as both a wave function and a particle.&nbsp; It can interact with other particles without any decay (again, simply without any change in energy) but it's physical properties can change. &nbsp;</p><p>&nbsp;<br />Now every other particle builds up to form a 3D world, has a decay rate, and so on.&nbsp; But not these particles on the quantum field, but they interact with other particles, so we have a conundrum.&nbsp; How does one describe such interactions?&nbsp; Additional dimensions is one route.&nbsp; It doesn't mean a hidden universe is one of the dimensions, it means a bridge between the two totally opposite forms found in the physical universe. </p><p>&nbsp;</p> <div class="Discussion_UserSignature"> </div>

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#### bruceralph

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With regard to the universe, it does appear to be flat (ie parallel lines don't cross). So the reality of 4 dimensions does have some basis in describing the universe. Extra dimensions could be curled up small or they could be very large. This does open the possibility of large extra dimensions&nbsp;existing in&nbsp;the larger cosmos of which our universe is a smaller part. <div class="Discussion_UserSignature"> </div>

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#### xmo1

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<p>You've managed to postulate on what is, or what seems to be (existence). What dimension do you give things of the imagination? I know, they live in ... "The Twilight Zone." No, seriously. What of dreams?</p> <div class="Discussion_UserSignature"> <p>DenniSys.com</p> </div>

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#### DrRocket

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Replying to:All of the above&nbsp;We already ready know from relativity that time is not linear I wonder why it is so hard to accept time as a non-linear dimension.[/quote]</p><p>It makes no sense to state that time or any other parameter is or is not linear.&nbsp; Linearity is a quality of one function with respect to another.&nbsp; Unless you relate time to something else a statementof linearlity is utterly meaningless.</p><p>Dimensions are dimensions.&nbsp;They are neither linear nor non-linear.&nbsp; </p><p>
&nbsp;I would suggest that energy cannot exist without time. Pure energy (perhaps the photon), we say, sees no length in its frame of reference but it must exist for a certain amount of time until it gets absorbed. Pretty simple laws of physics that must always hold true. It has no need for anything like euclidian space to describe itself.
</p><p>You can suggest anything you like.&nbsp; But unless you make that suggestion sensible and substantitate it somehow, it is just words.&nbsp;&nbsp;Gibberish.&nbsp; &nbsp;</p><p>&nbsp;</p><p>
I suppose you could say that it is not even 'flatland' it is 'pointland'. So time is the first dimension, energy is the second.
</p><p>Please review the definiton of the term "dimension".&nbsp; This statement makes no sense.</p><p>
&nbsp;Mass energy has a different relationship to time and is second order to pure energy we know this from e=mc2. Perhaps mass energy is the third dimension. It may be possible to construct a vector space of forms of energy separated by time but that time would be non-linear.
</p><p>While reviewing the definition of "dimension" also review the definiton of "vector space".&nbsp; This is more gibberish.</p><p>
&nbsp;Spacetime as we know it would be&nbsp;an emergent phenomenon and applied on top of the vector space and the time that we use in that space would have a fixed relationship to the time used in&nbsp;the vector space. In the vector space when two mass energies move towards each other they are lessening the amount of second order time between them this would be viewed as acceleration in the emergent spacetime. How the energy translates to volume and time separation to distance would likely be governed by things like planks constant and other constants. <br />Posted by UncertainH</DIV></p><p>Gibberish squared.&nbsp; <br /></p> <div class="Discussion_UserSignature"> </div>

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#### UncertainH

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Gibberish squared.&nbsp; <br />Posted by DrRocket</DIV></p><p>Thanks for you useful suggestions As I said I did get a bit carried away. So I understand what you are saying that to say something is or is not linear has to be said with respect to something else. So time is not linear in relation to relative velocity. Time is not linear in relation to relative position in a gravitational fields. Would these statements be correct ?</p><p>What I was trying to get at with the whole 'emergence' thing was that maybe there is a way to describe the universe using dimensions other than length width and height and as you mentioned even something like ticket price can be considered a dimension. Obviously one cannot determine one's position in the theatre by using the ticket price but perhaps there is a way to describe the energy and time of a objects in a system independently of a euclidean space and then apply the space afterwards. By way of analogy of a person at a theatre perhaps we know his row and seat number and the date of his ticket purchase, if we had a blueprint of the building we could translate his position into how many feet he is from the door</p>

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#### emperor_of_localgroup

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>&nbsp; Posted by nova_explored</DIV><br /><br /><font size="2"><font color="#ff0000">Dimensions, regardless of their number, come from the fact that no two points can occupy the same space.</font>&nbsp;</font></p><p><font size="2">One exception is projections. A projection of&nbsp;a 3D object on 1D world will appear as a line where several points of the cube(3D) will be mapped onto one point in 1D. </font></p><p><font size="2">With this thinking, how can we say for sure that what we are seeing are not projections on 3D world from a higher (3+)dimensional world?</font></p><p><font size="2">Some food for thought</font>.</p> <div class="Discussion_UserSignature"> <font size="2" color="#ff0000"><strong>Earth is Boring</strong></font> </div>

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#### DrRocket

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Thanks for you useful suggestions As I said I did get a bit carried away. So I understand what you are saying that to say something is or is not linear has to be said with respect to something else. So time is not linear in relation to relative velocity. Time is not linear in relation to relative position in a gravitational fields. Would these statements be correct ?What I was trying to get at with the whole 'emergence' thing was that maybe there is a way to describe the universe using dimensions other than length width and height and as you mentioned even something like ticket price can be considered a dimension. Obviously one cannot determine one's position in the theatre by using the ticket price but perhaps there is a way to describe the energy and time of a objects in a system independently of a euclidean space and then apply the space afterwards. By way of analogy of a person at a theatre perhaps we know his row and seat number and the date of his ticket purchase, if we had a blueprint of the building we could translate his position into how many feet he is from the door <br />Posted by UncertainH</DIV></p><p>Dimension is related to the notion of a basis for a vector space, but is an invariant whereas a vector space can have many bases.&nbsp; The dimension of a vector space is the maximum number of linearly independent vectors, call that number N.&nbsp; Any set of linearly independent vectors with N elements is a basis, and any point in the vector space can be uniquely written as a liinear combination of the vectors in that basis.&nbsp; So while, time, and 3 spatial vectors may be a "natural" basis or set of coordinates with which to work, there may be other coordinate systems that could be used.&nbsp;</p><p>So one answer to your question is yes, there may be another set of coordinates in which time and/or space are not explicit that might shed more light on the problem.&nbsp; But I have no idea what those coordinates might be.</p><p>A set of corrdinates may or may not represent a Euclidean space.&nbsp; The notion of bases and of dimension is independent of the notion required to make a space Euclidean.&nbsp; A Euclidean space occurs when there is, in addition to a coordinate system, a notion of distance that makes the Pythagorean theorem true.&nbsp; The most common way in which this occurs is for the space to come equipped with an inner product or dot product which is used to define the notion of length of vector and of angles between vectors.&nbsp; Other geometries can arise with different notions of distance or with notions of curvature.&nbsp; In classical general relativity one deals with a non-Euclidean geometry and with curvature, also the notion of a metric arises which is a sort of inner product, a quadratic form, but is not positive-definite (a positive-definite inner product is required for a Euclidean geometry).&nbsp; So with some, as yet unidentified, new set of coordinates the natural ideas for a metric may indeed result ina non-Euclidean geometry.&nbsp; <br /></p> <div class="Discussion_UserSignature"> </div>

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#### DrRocket

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Here are some links to articles that discuss the notion of spacetime as an emergent phenomenon:http://arxiv.org/abs/hep-th/0601234http://www.citeulike.org/user/spinwave/article/3356454There are lots more too. It seems that this is a common notion in a lot of string theories. <br />Posted by UncertainH</DIV></p><p>Those appear to be interesting papers.&nbsp; They are fairly sophisticated, so read then with care and recognize that the mathematics involved is pretty advanced.&nbsp; Seidman in particular is pretty high-powered theoretician, primarily a string theorist.&nbsp; Also recognize that the physics under discussion is speculative.<br /></p> <div class="Discussion_UserSignature"> </div>

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#### nova_explored

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>Here are some links to articles that discuss the notion of spacetime as an emergent phenomenon:http://arxiv.org/abs/hep-th/0601234http://www.citeulike.org/user/spinwave/article/3356454There are lots more too. It seems that this is a common notion in a lot of string theories. <br /> Posted by UncertainH</DIV></p><p>essentially the first paper is an example of string theory's interaction into our macro world.&nbsp; It shows through mathematics, toplogically, how a smooth surface can degenerate to the isosemetry of a string, and back: energy wise. &nbsp;</p><p>&nbsp;The whole field of string theory relies in connecting it to our macro world through function and proving a dual 1D can energize into a 3D.&nbsp; But it is theory only.</p><p>&nbsp;As for dimension itself- that is a classical definition of a coordinate system only.&nbsp; For as many dimensions as you have you have additional members to explain an energy state existence.&nbsp; in 2D we use vectors, in 3D its vectors with space, with string theory (1D) its a caluab/yau space, etc..&nbsp; but all essentially mean the same thing- a description of the state of energy at some specific period.</p> <div class="Discussion_UserSignature"> </div>

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#### DrRocket

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>essentially the first paper is an example of string theory's interaction into our macro world.&nbsp; It shows through mathematics, toplogically, how a smooth surface can degenerate to the isosemetry of a string, and back: energy wise. &nbsp;&nbsp;The whole field of string theory relies in connecting it to our macro world through function and proving a dual 1D can energize into a 3D.&nbsp; But it is theory only.&nbsp;As for dimension itself- that is a classical definition of a coordinate system only.&nbsp; For as many dimensions as you have you have additional members to explain an energy state existence.&nbsp; in 2D we use vectors, in 3D its vectors with space, with string theory (1D) its a caluab/yau space, etc..&nbsp; but all essentially mean the same thing- a description of the state of energy at some specific period. <br />Posted by nova_explored</DIV></p><p>You seem to be a bit confused.</p><p>http://mathworld.wolfram.com/Calabi-YauSpace.html</p> <div class="Discussion_UserSignature"> </div>

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#### nova_explored

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>You seem to be a bit confused.http://mathworld.wolfram.com/Calabi-YauSpace.html <br /> Posted by DrRocket</DIV></p><p>No, really not.&nbsp; Look up harmonic functions.&nbsp; specifically scalar.&nbsp; electromagnetism.&nbsp; String theory is the attribute of scaling a mulit dimensional model into a 1D function.&nbsp; It exists 'underneath' our real experience or natural universe on plane, not 2D or 3D or 10 or 11, but 1D where harmonics are the sequences by which more dimensions are added. &nbsp;</p><p>&nbsp;I am not using the calaub yau structure used in defining natural world, but the model as according to string theory. </p> <div class="Discussion_UserSignature"> </div>

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#### DrRocket

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<p><BR/>Replying to:<BR/><DIV CLASS='Discussion_PostQuote'>No, really not.&nbsp; Look up harmonic functions.&nbsp; specifically scalar.&nbsp; electromagnetism.&nbsp; String theory is the attribute of scaling a mulit dimensional model into a 1D function.&nbsp; It exists 'underneath' our real experience or natural universe on plane, not 2D or 3D or 10 or 11, but 1D where harmonics are the sequences by which more dimensions are added. &nbsp;&nbsp;I am not using the calaub yau structure used in defining natural world, but the model as according to string theory. <br />Posted by nova_explored</DIV></p><p>http://en.wikipedia.org/wiki/Harmonic_function</p><p>I think your problem may go a bit beyond simple confusion.&nbsp; Please explain what the theory of harmonic functions has to do with any of your assertions.</p><p>For that matter pleas just explain your last two sentences in any language that is relatable to standard mathematics.&nbsp; Your assertion uses neither Calabi-Yau spaces not any intelligible version of string theory.&nbsp; <br /></p> <div class="Discussion_UserSignature"> </div>

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