Do Partial Spatial Dimensions Exist?

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So I'm sure everyone here knows of the basic spacial dimensions. 1D is a line, 2D a plane and 3D a cube. There is even a 4th dimension (theoretical), the tesseract. And an infinite number of dimensions beyond, represented by various hypercubes. Finding the space taken up by one of these objects (length, area, volume) is easy. The length of a line is x^1, the area of a plane is x^2, and the volume of a cube is x^3 continuing ad infinitum.

But what about partial dimensions, do they exists (theoretically of course)? For example, a 2.5 dimension. It might make sense to think of dimensions as discrete, but then how would you account for the ability to find the space taken up by these objects (length, area, volume)? For example, you can find the "area/volume" of a 2.5 dimension "square/cube" by x^2.5. If the length of a side of this "square/cube" was 2 than 2^2.5 = 5.65685425 is the "area/volume" of this mythical partial dimensional object. If partial dimensions don't exists, what am I measuring?

I have searched the internet far and wide and have found no mention of partial dimensions. Can someone please elaborate on them or if they are even possible? And if they are, could someone explain how I could go about drawing one?
 

Answers and Replies

  • #2
DaveC426913
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I have searched the internet far and wide and have found no mention of partial dimensions.
Read up on fractional dimensions, better known as fractals.

The essence behind a fractional or fractal dimension is thus: on a straight line you have one degree of freedom - back and forth on the line. But kink that line up into a squiggly line, like a very craggy coastline, and you effectively have more than a single degree of freedom. The craggier the coastline gets, the closer you get to having a full two degrees of freedom. But you won't really reach two, it'll fall short, some fraction between 1 and 2 dimensions. This fraction is quantifiable and there is quite a bit of science developed around fractals.
 
  • #3
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Wow that's complicated.

So it seems that the dimension of fractals can be calculated, and they end up somewhere in between two dimensions. Can the reverse be done? Can I take a partial dimension and conceive a fractal? Can I conceive the fractals of any recognizable objects?
 
  • #4
I've been thinking about this some. Mathematically, time can be treated as a dimension. We also know that time is relative and approaches a stand still on the event horizon of a black hole. In this sense, as one approaches the event horizon of a black hole, four dimensional space-time would approach a three dimensional space. I would be interested if anyone here can comment on this.
 
  • #5
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I've been thinking about this some. Mathematically, time can be treated as a dimension.
yes you are right time is not a geometrical dimension. It is only considered a dimension mathematically.

as one approaches the event horizon of a black hole, four dimensional space-time would approach a three dimensional space
I too think the same. But we should wait for an experts comment before reaching to conclusion.
 
  • #6
DaveC426913
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yes you are right time is not a geometrical dimension. It is only considered a dimension mathematically.
Not true. It is very much a dimension. It just happens to be a timelike dimension, as opposed to a spacelike dimension.

I've been thinking about this some. Mathematically, time can be treated as a dimension. We also know that time is relative and approaches a stand still on the event horizon of a black hole. In this sense, as one approaches the event horizon of a black hole, four dimensional space-time would approach a three dimensional space. I would be interested if anyone here can comment on this.
1] It only approaches a stand still from the point of view of a distant observer. For the observer at the horizon, time marches on. This is an issue of differing frames of reference, not of time "actually" stopping.

2] To suggest that, because time appears to stop means that dimension disappears, is to likewise suggest that, if a ball's radius were kept uniform and unchanging along its x-axis (i.e. a cylinder), it has somehow become two dimensional.
 
  • #7
BWV
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are the dimensions of Hilbert space infinitely countable or uncountable? i.e. is it an infinite number of discrete integer dimensions or a continuum across all the real numbers?
 
  • #8
dimensionless said:
I've been thinking about this some. Mathematically, time can be treated as a dimension. We also know that time is relative and approaches a stand still on the event horizon of a black hole. In this sense, as one approaches the event horizon of a black hole, four dimensional space-time would approach a three dimensional space. I would be interested if anyone here can comment on this.
1] It only approaches a stand still from the point of view of a distant observer. For the observer at the horizon, time marches on. This is an issue of differing frames of reference, not of time "actually" stopping.

2] To suggest that, because time appears to stop means that dimension disappears, is to likewise suggest that, if a ball's radius were kept uniform and unchanging along its x-axis (i.e. a cylinder), it has somehow become two dimensional.
I see your point on number 2. However, there is still the point of view of the distant observer. To the distant observer, the flow of time near an event horizon would be much less when compared to the local flow of time. If the local region of space-time had four dimensions, I could see an argument that the region of space-time near an event horizon has between three and four dimensions(comparatively speaking). I'm just brainstorming here. I'm not quite sure what a partial dimension would look like if it does indeed exist.
 
  • #9
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You're getting very off topic. Can we please keep this to the spacial dimensions? Time is irrelevant here as I'm using dimensions as a mathematical concept, not a physical one.
 
  • #10
DaveC426913
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You're getting very off topic. Can we please keep this to the spacial dimensions? Time is irrelevant here as I'm using dimensions as a mathematical concept, not a physical one.
Typically, once a question has been answered, a post is free to wander where its posters take it, at least until it gets locked for being way off.

Has this not been answered? Do you have more questions?
 
  • #11
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Read up on fractional dimensions, better known as fractals.

The essence behind a fractional or fractal dimension is thus: on a straight line you have one degree of freedom - back and forth on the line. But kink that line up into a squiggly line, like a very craggy coastline, and you effectively have more than a single degree of freedom. The craggier the coastline gets, the closer you get to having a full two degrees of freedom. But you won't really reach two, it'll fall short, some fraction between 1 and 2 dimensions. This fraction is quantifiable and there is quite a bit of science developed around fractals.
I don't quite get this concept. Whatever the shape of a line, it still only has one degree of freedom. Like, I could walk along the "squiggliest" coastline in the world and still parameterise it as the number of steps I've taken from my starting point.
 
  • #12
SpectraCat
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I don't quite get this concept. Whatever the shape of a line, it still only has one degree of freedom. Like, I could walk along the "squiggliest" coastline in the world and still parameterise it as the number of steps I've taken from my starting point.
Right, and the number of steps you take would also depend on the "squiggly-ness" of the coastline, so that is the second partial dimension. That is a simplification of the real mathematical answer, but I believe it captures the essence of the fractal dimension. Note also that such a fractal coastline must exist in a space of (at least) two dimensions. So, you can think of a fractal as a restricted exploration of the full two-dimensional space ... it's been a while since I read about this, but I believe that this is one way to calculate the dimension of a fractal. In other words, you calculate the fraction of the two dimensional space that can be accessed by moving along the fractal edge.
 
  • #13
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I think I might have an idea what's going on. Essentially as the line becomes more squiggly it covers more two dimensional space while still only existing in one dimension. So in some limit of infinite-squigglyness, it should in theory cover every point in the plane?
 
  • #14
Khashishi
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Typically, people use the Hausdorff dimension and not the number of degrees of freedom when talking about non-integer values of dimension. In this case, infinite squigglyness can have non-integer dimension.
 
  • #15
SpectraCat
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Take a look at the book "Flatterland" (the sequel to the classic, "Flatland"), for an excellent and accessible introduction to fractal dimension, as well as many other mathematical topics related to topology and geometry.
 
  • #16
DaveC426913
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I think I might have an idea what's going on. Essentially as the line becomes more squiggly it covers more two dimensional space while still only existing in one dimension. So in some limit of infinite-squigglyness, it should in theory cover every point in the plane?
Correct.

Say I have one of those path-following robots (Black tape is placed on the white ground.The robot reads the tape with a sensor and is free to move along the line but nowhere else). It lives in a 20ft square room.

The robot has one degree of freedom - back and forth along the line.

But what if that line were very squiggly? The robot, despite the fact that it is constrained to one degree of freedom, can do pretty good job of getting most places in the room. (Practically speaking, the robot has a non-zero size, so can cover any spot in the room, however to be perfectly rigid, the robot has a centre point, which is zero size, that we are concerned with.)

You can see that there is virtually no upper limit on how squiggly the line can be; the robot's freedom of movement approaches 2, though it never reaches it. To have exactly 2 dimensions of movement, the line of tape would have to be infinitely long.

The squiggliness of the line can be associated with a value of its fractality, which will be a number somewhere between 1 and 2.
 
  • #17
So extrapolating from your example, can we assume we have full freedom of motion in 3 dimensions, and only partial freedom in the fourth? Does this suggest that in reality we live in the 3.5 dimension, or something of this nature due to relativistic effects that make parts of the fourth dimension inaccessible to us (as far as we know, at least)?
 
  • #18
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Time can be considered a dimension but it really shouldn't be. It's kind of like calling tomatoes fruit. Especially since the fourth dimension has so many interesting properties, calling it time is a bit of a cop out. It's better just to call our perceivable universe spacetime and call the fourth dimension the fourth spatial dimension.

You can think of time as a limited spacial dimension but it's going to reveal the exact reason why it doesn't make sense to call it a dimension.
 
  • #19
DaveC426913
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You can think of time as a limited spacial dimension but it's going to reveal the exact reason why it doesn't make sense to call it a dimension.
Reveal for us.
 
  • #20
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Just remember I'm not formally educated in any of this, I've just read books.

Like I said before, the length of a 1D line is x^1, the area of a 2D plane is x^2, and the volume of a 3D cube is x^3. If we were to treat time as a whole dimension then the "hyper volume" of a 3D cube moving through time (4D space) would be determined by x^3. Which makes sense.

But when you take into account that time only moves in one direction (or at least appears to for us) those equations don't really work. Can you imagine a volume moving in solely one direction? What would the "hyper volume" of that object be? To say that time is half a dimension gives a nonsensical answer. It's possible you can represent the single directionness of time some other way, but I'm pretty sure the volume should remain the same as a regular 4D object. As far as I can tell it really only makes sense to call time a whole dimension or none at all.
 
  • #21
DaveC426913
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Just remember I'm not formally educated in any of this, I've just read books.

Like I said before, the length of a 1D line is x^1, the area of a 2D plane is x^2, and the volume of a 3D cube is x^3. If we were to treat time as a whole dimension then the "hyper volume" of a 3D cube moving through time (4D space) would be determined by x^3. Which makes sense.
I think you meant x^4.

Yes, a unique point in space time requires 4 coordinates: x, y, z and t.

But when you take into account that time only moves in one direction (or at least appears to for us) those equations don't really work.
They do.

There are 3 space-like dimensions and 1 time-like dimension. Timelike dimensions are characterized by the fact that we move through them in 1 direction only.
 
  • #22
I'm not a physicist, and I've only taken the first year of college Physics (along with lower division math, including differential eq. and intro linear algebra), so I certainly do not have the mathematical basis to describe what I'm pondering. I find it fascinating that time (for us) seems to move in one direction, and I wonder about the "why". I guess I'm pondering whether or not its possible that the reason why we only move in one direction in time is because we actually do not live in a "4D" universe, but in fact only inhabit a partial dimension somewhere betwen 3 and 4 (and thus are only able to traverse "part" of the 4th dimension). Is it conceivable that a GUT could be postulated where instead of us living in a whole-numbered dimensional universe (4,5,6,9,10,11,etc.) we actually inhabit something in between. As a thought experiment, what would be the consequences of living in a world of, say, 3.5 dimensions? I'm not sure this is the right forum for this discussion, so please let me know if I should take my rambelings elsewhere...:wink:
 
  • #23
Also, I'd love to get some advice on what branches of mathematics/physics to explore that might allow me to explore fractional dimensions as a potentinal unification theory...any advice would be greatly appreciated...
 
  • #24
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Another interesting concept is that of 'many worlds'. That might mean an infinite number of parallel universes, maybe spawned from each other every time a quantum choice occurs... so every time an 'either- or choice' occurs here, trillions upon trillions every second, both choices are carried forward but in 'different' places and times....if true, EVERY imaginable spatial dimension has already been created...partial ones included. Those which are unable to evolve die a rapid death,,,maybe immediately, for example, if no time dimension occurs, maybe remain 'small' if one fractal dimension is born.

I would also mention not to avoid 'time'....relativity suggests an intimate link between space and time and as noted above sometimes time slows' and when it does what always happens,,,space, distance, changes....that's call 'time dilation' and 'length contraction' in relativity....

Finally consider space and dimensions 'geometry'....and follow that term if of interest....

Here are a few links to 'sink your teeth' into:

(I suggest if you follow any of these, skim past the parts you don't understand and focus on
those that are of interest or seem to make sense...Nobody understands all of this, as is evident in the opening post of #1. ]

[1]Spinfoam discussion [posted by Marcus]

A view of reality emerging from current LQG research is somewhat analogous to the "seething vacuum" of quantum field theory. But it is a "seething geometry" in which bits of area and volume constantly come into existence and go out of existence......

The formalism uses a more complicated version of Feynman diagrams. A spin network is an instantaneous state of geometry: the nodes represent bits of volume and the links represent bits of area......
https://www.physicsforums.com/showthread.php?p=2775507&posted=1#post2775507


[2] Check this discussion for a lot of links to interesting discussions.....
Group Field Theory:

https://www.physicsforums.com/showthread.php?t=415915&highlight=causal+dynamic+triangulation

[3] Need more?... Search within these forums for 'causal dynamic triangulation'....
that brings up a LOT of discussions with links within.
 
  • #25
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"Partial" spacetime has another aspect to it nobody mentioned and I forgot on the previous post: Is space time continuous or discrete... analog or digital??

You'll have to decide what constitutes a 'partial' spacial dimension [to you] in a continuous versus discrete formulation.


Here is an interesting perspective from another discussion in these forums:

http://pirsa.org/09090005/

Spacetime can be simultaneously discrete and continuous, in the same way that information can.
Maybe "continuous" and "discrete" are two sides of the same coin, analogous to wave-particle duality.

ok, I also found this in my notes:

Ben Crowell posted this question...

https://www.physicsforums.com/showthread.php?t=391989

"argument for the discreteness of spacetime",

The following is a paraphrase of an argument for the discreteness of spacetime, made by Smolin in his popular-level book Three Roads to Quantum Gravity. The Bekenstein bound says there's a limit on how much information can be stored within a given region of space. If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete.



and one more:
http://arxiv.org/abs/1010.4354


Spacetime could be simultaneously continuous and discrete in the same way that information can

Achim Kempf

(Submitted on 21 Oct 2010)
“The equivalence of continuous and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any bandlimited signal it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the bandlimit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possess an ultraviolet cutoff.”
 

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