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Proof for the Impossibility of Higher and Lower Dimensions

  1. Aug 8, 2011 #1
    Introduction

    The premise that our universe might have more spatial dimensions than the three that are immediately apparent is so widespread and popular that it is nearly accepted already as a fact despite the absence of evidence. This is primarily the result of two separate influences: the stronger being that most of the world’s professional physicists and mathematicians have embraced the concept of higher dimensions within their respective fields for technical reasons, and the weaker being that the concept has a romantic appeal which has also captured the imagination of the public at large. Given such synchronicity across the intellectual spectrum it would seem to be utter folly to try and restore the old three dimensional limits previously ascribed to space.

    Nevertheless it is precisely the intention of this essay to demonstrate through elementary logical argumentation that not only are fields and objects of greater than three dimensions impossible but also that fields and objects having less than three dimensions are equally impossible. The reasoning justifying these conclusions is held by myself to completely adhere to strict necessity and therefore to be definitive – I can no more imagine these arguments being invalid than that ten multiplied by ten could equal something other than a hundred – but this fact of course does not make the arguments beyond reproach and if an error in reasoning were discovered then I would be foremost among those desiring to know that: rather my certainty is merely mentioned to account for the hubris of contradicting so many people.

    Only certainty and madness can inspire one to defy the whole world but in virtue of the simplicity of the arguments I employ I am unable to convince myself that I could possibly be deluded in this context. Of course my reasoning must be convincing in its own right and so I will not waste any more words on self justification. The arguments will be presented as follows: first in one section it will be established that lower dimensions cannot exist within higher dimensions and then in a second section it will be established that higher dimensions cannot exist within lower dimensions. This should suffice to satisfy the claim in the title of this essay. Lastly a short conclusion will be provided to summarize what has been argued and to comment on it. The significance of the claims presented here if true must require their own greatly detailed discussion and accordingly will only be given the briefest treatment within the present essay.


    Section One

    So why can’t lower dimensional spaces exist in higher dimensional ones? Well to begin with one should clarify what one means here. A dimensional space for the present purposes is any spatial object or spatial field that has a specific number of dimensions. A three dimensional space of course has exactly three dimensions providing separate perpendicular axes and these are commonly denoted by the letters x, y, and z. This leaves only two possible lower forms of dimensional space: two dimensional space and one dimensional space. Since it does not matter which of the dimensions from three dimensional spaces the lower forms of dimension share, we will assign to the former x and y while assigning to the latter x only.

    Now consider a piece of paper within a three dimensional space. It can be regarded as a kind of pseudo two dimensional object since it is very flat in appearance but is still obviously three dimensional. What we want to know however is whether a truly two or one dimensional object can exist within a three dimensional space so let’s use our imaginations and try and transform the paper into the former. Picture the paper resting on a table while it becomes thinner and thinner. When we are looking at it from a top down perspective nothing much is noticeable. In fact after a certain point we can’t really tell from any angle whether the paper is truly getting thinner but returning to the top down perspective the white plane of the paper remains visible on the desk so we are reassured that it still has some thickness. This is true right up until the paper is a mere quantum width, until it is as thin as the elementary constituents of physical reality will allow. Then suddenly though it just disappears.

    Why has it disappeared? Because it no longer exists: without the thickness provided for it by its third dimension – its z dimension – it has ceased to exist in the other two dimensions as well. Think about it this way: if it no longer has any extension along the z axis then it no longer as any substance within the x and y axes. The paper – having no thickness – obviously cannot project itself from the surface of the desk: every point where it existed in the x and y dimensions was contingent on their having some existence in z. Without any extension in z, the x and y axes cannot have any thickness and so cannot have any extension. The cause of this is that we have added a zero to our multiplication. Let’s assume that the piece of paper was the regular 8 x 11 inches: if you multiply 8 inches by 11 inches by 0 inches what do you get? The answer is obvious: you get nothing. Any number multiplied by zero equals zero.

    So a two dimensional object or field obviously cannot exist embedded in a three dimensional object or field: it could not project itself into the requisite third dimension. But then no lower dimensional space can exist in any higher dimensional space because all of them would suffer from the same inadequacy. As such it is established that the number of dimensions which a dimensional space possesses provides its lower boundary and that nothing could possibly exist within it of a lower dimensional order. The boundary here is as conspicuous as it is immutable.


    Section Two

    The question may seem to remain as to whether higher dimensional spaces can exist within lower dimensional ones but in fact the inferences within the previous section have substantially dealt with it. For the sake of exposition though I have separated the two arguments and will strive furthermore to clarify why the two claims may seem at first independent of one another. In so doing I hope to show why this distinction is fallacious and that the statement that higher dimensional spaces cannot exist within lower dimensional spaces is just as certain as its counterpart.

    To be honest, when I was first considering the questions which I propose to answer here I was initially unsure of the equivalence between the two forms of embedding. It was immediately apparent that lower dimensional spaces couldn’t satisfy the extension demands required by higher dimensional spaces but conversely higher dimensional spaces didn’t seem to suffer from the same deficiency when the reverse was being considered. It was easy to imagine that dimensionality was hierarchical and concentric: that within lower dimensional spaces there were higher dimensional centers, perhaps in the form of ultramicroscopic realms and so, and therefore that lower dimensions could provide a surrounding medium for higher dimensions with the very lowest – zero dimensionality – providing the ultimate all encompassing foundation. In hindsight I see how absurd it was that I could even for a moment think that lower dimensions could – in any sense of the term – encompass higher dimensions.

    After all it would actually seem intuitively more plausible that higher dimensional spaces would encompass lower dimensional ones: this is in fact how people have traditionally imagined two and one dimensional objects existing within three dimensional spaces. Accepting the arguments in the previous section though eliminate that theory and provide the basis for eliminating its counterpart.

    The idea that lower dimensional elements could exist within higher dimensional spaces was found to be false because lower dimensional elements are incapable of projecting themselves into higher dimensions: as a result of this incapacity they would be utterly negated upon insertion. When one considers the opposite conjecture though it is not a negation that prohibits such a possibility but rather the meaninglessness of the conjecture itself: a lower dimensional space after all cannot accommodate higher dimensional elements, a dimensional element could only exist within a lower dimensional space in as much as it adhered to the limits of said lower dimension. For example: an object could only project a surface within a plane, within the limits of a two dimensional space any cube for example can only intrude as a two dimensional face.

    One might be inclined to suggest that the cube would still exist separately as a three dimensional object in such a context and that its two dimensional appearance would merely be a contingent aspect but this more sophisticated suggestion still fails. A two dimensional space after all can only accommodate the existence of two dimensional objects: in order for an object to exist meaningfully within a two dimensional space then it would have to be wholly flattened. From these insights one can discern with confidence that the upper boundaries of dimension are just as strict as the lower boundaries and that together they fully provide for the general framework required by a true concept of dimension. Existing as we do within a three dimensional space all forms of higher and lower dimension are by definition inaccessible and as such not open to a speculative existence for us in any meaningful sense.


    Conclusion

    The surprising corollary that one must make if subscribing to the inferences so far presented is that the three spatial dimensions of our physical universe provide the only conceivable arrangement of space since both higher and lower dimensional arrangements are impossible given the three dimensional state existent. Even the idea that the initial conditions at the origin of the universe might have possibly generated different dimensional states is rendered untenable since the only elements one can properly consider now as logically accessible must themselves be three dimensional: elements can exist for us meaningfully only as far as they could be manifest within the x, y, and z axes. Our universe then cannot contain elements of unequal dimension just as surely as it cannot itself be the element of any kind of hyperspace.

    Pure mathematics however might seem to provide for research into other forms of dimension on merely intellectual grounds but such a belief would result from confusion. For instance: when one uses a pencil to draw, say, a circle on a piece of paper, one is in fact creating a three dimensional construct. One may try to convince oneself that the circle represents an ideal two dimensional object but the fact is it can only exist at all on paper because the graphite of the pencil leaves a three dimensional residue that in virtue of its shape and color is capable of projecting into our minds a circular form. This of course applies to all forms of representation. A computer model for instance is only possible because we are looking through our screen at a three dimensional physical state. The two dimensional aspects of such object conditions is a mere intellectual simplification and provides no justifiable basis whatsoever for making the mental leap that because we can invent the concept of two dimensional reality syntactically that this provides us with any real semantic content.

    The concept of higher and lower dimensions as such results from false analogies generated by grammatical nonsense. One may think that because our reality has three spatial dimensions that it is therefore logical to consider other realities with different numbers of spatial dimension but this is as false as thinking that one can meaningfully consider triangles with other than three angles. What I mean to say here is that space is – by definition – three dimensional. We can speak of other forms of space and generate names for them syntactically in the manner that one conjures up the phrase “Non Three Angled Triangle” but such contrivance cannot produce meaningful objects with real semantic content. All we have done is smash words together.

    The reasoning I have presented here is of the simplest form I can imagine and it is in accordance with this simplicity that I will not go on to venture much with regards to the significance of these conclusions. It would seem to me though that even if none of the problems or the stagnation which currently plagues contemporary physics and mathematics is resolved by these insights still said insights would be necessary eventually to resolve some impasse that minds today have yet to dream: regarding this matter however I must demur and leave it to the experts to decide for themselves.
     
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  3. Aug 8, 2011 #2
    A couple thoughts:

    1) To a photon moving at the speed of light, the world is two dimensional.

    2) It's possible that we live in an 11 dimensional world (or whatever), but that, due to the limits on our senses, we can only detect phenomena that are transferred in three dimensions. I would think that it would be completely possible to imagine an entity that only moved in a single direction, or along a plane. Following that line of thought, maybe the waves which constitute the reality which we see can only be propagated in the three dimensions that we see.
     
  4. Aug 8, 2011 #3

    bapowell

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    This is all rather exaggerated. Most physicists do not take extra-dimensional space seriously. Even string theorists, who might strongly advocate this possible view, do not believe it to be fact.

    You must be more exact with your terminology. How are you defining "substance" and why must it be 3-dimensional?

    Why? You have not shown this to be true 1) logically, 2) mathematically, or 3) physically. You just wrote it in a sentence.


    Yeah. But its area is nonzero.
     
  5. Aug 8, 2011 #4

    WannabeNewton

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    How did you come to that conclusion?
     
  6. Aug 8, 2011 #5
    bapowell,

    Okay, I suppose I can try and reformulate the argument in even more concise terms.


    1) The known universe is, at its macroscopic limits, at its overall boundary, three dimensional.

    2)Therefore everything that exists within it exists within those three dimensional confines.

    3) In so far as an object exists within our known universe then it must have three spatial coordinates at any given time OR - for quanta - a three dimensional probability area.

    4) Therefore every possible object, even a point particle, would have to have three dimensions of spatial extension: this would be denoted by x=n1, y=n2, and z=n3 where n cannot be less than 1. A point particle is extended 1x1x1, a hypothetical string would be say 8x1x1, a flat rectangular object say 2300x17000x1, and so on. Obects cannot have a zero in their volume equation: therefore they cannot be two or one dimensional within a three dimensional space.

    5) Logically then our universe cannot exist within any kind of hyperspace. In general lower dimensional objects (or constructs) cannot exist within higher dimensional spaces.

    6) Now considering higher dimensional objects existing within lower dimensional spaces: what would this look like?

    7) What exists within our universe would have to have exactly three spatial coordinates: a fourth coordinate - and logically then a four dimensional object - could not be introduced in any meaningful way into a three dimensional space. And this also applies for all attempts to embed higher dimensional objects into lower dimensional spaces.

    Lastly, by substance I simply mean, in the most general neutral terms, what has a independent form that persists in space. Of course at the quantum level substance does not exist in absolute Newtonian terms but still quanta can only exist in as much as they have substantial effect within some specific spatial area.
     
  7. Aug 8, 2011 #6
    I don't believe this is true and I don't see how anyone can. The path of a bee is a line. The edge of a black hole is a surface. Why can't we put 1,2,...N dimensional objects in N+1 dimensional space?

    This isn't a proof, it's a collection of unsupported statements.
     
  8. Aug 8, 2011 #7

    bapowell

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    I disagree. You have not proven this to be the case. You're just saying it. What is your operational definition of "exist"?

    Except for objects that don't have volumes....
    You are ostensibly making a mathematical argument here...not a physical one. But mathematical objects of dimension D exist perfectly happy in spaces of dimension D+p. For example, the Standard Model of particle physics is built out of zero-dimensional point particles.

    You've not really constructed a logical argument in the form of premise ---> conclusion. You are simply stating a conclusion: objects with other than 3 dimensions can't exist. You've still not proven why.

    I agree that this is a related argument that simply extends from the one for lower dimensional one. In fact, it's the same argument.

    I see nothing in this definition that forbids the existence of a 2D object in 3D space.
     
  9. Aug 8, 2011 #8
    Can't reply to any further questions for today. Will post replies to any that are left here some time in the near future though.
     
  10. Aug 8, 2011 #9
    I mean, due to length contraction, if all distances in the light's direction of travel are infinitesimal, then there doesn't appear to be a third dimension according to the photon. Then again, from our reference point photons progress through time at an infinitely slow rate, so I guess its pretty pointless to think about what a photon might see.
     
  11. Aug 9, 2011 #10
    "You can't fit a three dimensional object in a two dimensional box" : obvious fact.
    "A three dimensional object can't exhibit itself within a two dimensional area" : obviously wrong.

    The essay that opened this thread seems like more a play on words than a logical argument or proof of anything. It may be founded on confused or ill defined terms.

    Think of the cave analogy, or generally think about shadows. A three dimensional object, moving and rotating, can cast a shadow on a two dimensional plane. The object is not CONTAINED within the plane, but the changes to the shadowy form ultimately can only be understood by reference to the third dimension. An observer limited to viewing the plane only can draw meaningful conclusions by reference to the third dimension.
     
  12. Aug 9, 2011 #11

    bapowell

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    But these observations are gotten by making the unphysical equality [itex]v=c[/itex].
     
  13. Aug 9, 2011 #12

    bcrowell

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  14. Aug 9, 2011 #13
    Sorry if I'm being pedantic here, but if photons are massless, why wouldn't v = c?
     
  15. Aug 9, 2011 #14

    bcrowell

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