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posted by Fnord666 on Sunday November 29 2020, @08:51AM   Printer-friendly
from the The-B-stands-for-Benoit-B-Mandelbrot dept.

Noninteger dimensions are fairly well known in mathematics, but they are also used in various branches of physics and engineering to explain the emergence of scale invariant phenomena, such as atmospheric turbulence, or for measuring coastlines. Subhash Kak of Oklahoma State University in the United States published a paper that shows that if you look at physical space through the lens of information theory, the optimal number of dimensions turns out to be not an integer.

Kak shows the optimal dimension associated with the representation of information is e = 2.71828... and he argues that physical space is e-dimensional instead of 3-dimensional if one accepts optimality as a fundamental physical principle. He argues the discrepancy between 3 and e can be seen in existing data, and the example used has to do with the large scale structure of the universe.

One of the "crises" in physics is reconciling the two different values of the Hubble constant, H0. The two values are 67 km s-1 Mpc-1 if you use early universe data, and 74 km s-1 Mpc-1 if you use late universe data. If physical reality is e-dimensional and we insist on viewing it as being 3-dimensional then there is a discrepancy equal to e/3=0.9060. This number is very close to the divergence of 67/74=0.9054 from the experimental data.

Journal Reference:
Subhash Kak. Information theory and dimensionality of space [open], Scientific Reports (DOI: 10.1038/s41598-020-77855-9)


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  • (Score: 1) by shrewdsheep on Sunday November 29 2020, @11:48AM

    by shrewdsheep (5215) on Sunday November 29 2020, @11:48AM (#1082010)

    A *--------------------------------* B
    Two points separated by distance r

    Figure 1

    Well, that's already the information content of e, therefore information content of the rest is e - e = 0.

  • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @11:54AM (2 children)

    by Anonymous Coward on Sunday November 29 2020, @11:54AM (#1082011)

    The optimal number of dimensions is, in actual fact, zero. Just because a zero-dimensional space does not appear to exist - it wouldn't be detectable anyhow - surely all points to it indeed being the most optimal. In zero dimensional space there is no light, no speed of light, no measurements, or other thorny issues. Travel anywhere is instantaneous and unbounded. Totally optimal.

    • (Score: 3, Touché) by unauthorized on Sunday November 29 2020, @12:15PM (1 child)

      by unauthorized (3776) on Sunday November 29 2020, @12:15PM (#1082014)

      The paper asserts that e-dimensional space is optimal for the representation of information. A 0-dimensional space cannot contain any information and is therefore actually the least optimal according to the criteria set by the authors.

      • (Score: 1) by shrewdsheep on Monday November 30 2020, @02:08PM

        by shrewdsheep (5215) on Monday November 30 2020, @02:08PM (#1082359)

        Well, a 0-dimensional space contains a point, or a single number. You can encode any amount of information in a single real number. What you describe is a -1 dimensional space, which is the empty set, which just is. Just being is optimal in many senses, however, think Buddha.

  • (Score: 2) by legont on Sunday November 29 2020, @01:15PM (12 children)

    by legont (4179) on Sunday November 29 2020, @01:15PM (#1082016)

    So, flat earthers are right after all? We live in two dimensional world just folded here and there a bit.

    --
    "Wealth is the relentless enemy of understanding" - John Kenneth Galbraith.
    • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @03:08PM (11 children)

      by Anonymous Coward on Sunday November 29 2020, @03:08PM (#1082034)

      Well they're partly right according to the article?

      We live in a 2.7... dimensional world according to the article so the world isn't quite 3D but it's not 2D either. It's somewhere in-between?

      As they say, the truth lies somewhere in between.

      Uhm ... does time count as a dimension?

      • (Score: 1) by khallow on Sunday November 29 2020, @04:40PM (10 children)

        by khallow (3766) Subscriber Badge on Sunday November 29 2020, @04:40PM (#1082054) Journal

        Uhm ... does time count as a dimension?

        Yes. So we're at e+1 dimensions by this argument.

        • (Score: 1, Interesting) by Anonymous Coward on Sunday November 29 2020, @05:26PM (9 children)

          by Anonymous Coward on Sunday November 29 2020, @05:26PM (#1082065)

          Are you sure the time dimension is an integer number of dimensions? Maybe some of those weird quantum quirks can be explained with time not being a full integer? This might also help explain time dilation????

          • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @05:28PM (1 child)

            by Anonymous Coward on Sunday November 29 2020, @05:28PM (#1082067)

            Perhaps if you include space + time it would equal e + (pi - e) = pi and that explains why physics is so confusing it makes my head spin.

            • (Score: 0) by Anonymous Coward on Tuesday December 01 2020, @02:25PM

              by Anonymous Coward on Tuesday December 01 2020, @02:25PM (#1082814)

              Also the pi number of dimensions makes sense with respect to information presentation because pi relates to a circle and if you are at the center of the circle you can have information presented to you in all directions uniformly and hence more efficiently.

              So space is not a cube ... it's a sphere?

              Now my head is spinning again .... starting to space out (is that the loss of information?). At least I'm not blocking out?

          • (Score: 1) by khallow on Sunday November 29 2020, @05:47PM (6 children)

            by khallow (3766) Subscriber Badge on Sunday November 29 2020, @05:47PM (#1082070) Journal
            Guess I'm not sure. But I figure the actual dimension still is between e and e+1, including e+1.
            • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @08:16PM (5 children)

              by Anonymous Coward on Sunday November 29 2020, @08:16PM (#1082103)

              A dimension is two directional but time can only go in one direction .... kinda ... some weird quantum oddities might suggest otherwise .... so it's more than just half a dimension but less than an entire dimension ... ?

              • (Score: 1) by khallow on Monday November 30 2020, @06:39AM (4 children)

                by khallow (3766) Subscriber Badge on Monday November 30 2020, @06:39AM (#1082294) Journal
                "Going" is not part of the definition of dimension.
                • (Score: 0) by Anonymous Coward on Tuesday December 01 2020, @03:12AM (3 children)

                  by Anonymous Coward on Tuesday December 01 2020, @03:12AM (#1082670)

                  What about "stopping"?

                  • (Score: 1) by khallow on Tuesday December 01 2020, @04:13AM (2 children)

                    by khallow (3766) Subscriber Badge on Tuesday December 01 2020, @04:13AM (#1082684) Journal
                    No on that one too.
                    • (Score: 0) by Anonymous Coward on Tuesday December 01 2020, @02:00PM

                      by Anonymous Coward on Tuesday December 01 2020, @02:00PM (#1082808)

                      Because time slows down as you speed up. Does time "stop" when you reach the speed of light?

                    • (Score: 0) by Anonymous Coward on Tuesday December 01 2020, @02:09PM

                      by Anonymous Coward on Tuesday December 01 2020, @02:09PM (#1082811)

                      Also does time go backwards when you go faster than light? Can you then go back in time like the flash? If so, does this affect our understanding of how many dimensions time has?

                      Since going back to the past could change the past this could create a branched timeline like you see in the flash, you can't really go back to the original past as it was. So time is not exactly one dimension but it's more than half a dimension.

  • (Score: 2) by Runaway1956 on Sunday November 29 2020, @01:54PM

    by Runaway1956 (2926) Subscriber Badge on Sunday November 29 2020, @01:54PM (#1082024) Journal

    Bending and stretching space has been a staple of science fiction for a long, long time. Even little kids know about it.

    https://m.youtube.com/watch?v=esBGhMwJX4E [youtube.com]

    Oh yeah, motorcyclists too. Can't go fast enough to ditch the cops, just create a shortcut wherever you need it! Youtube has loads of videos if you care to watch.

  • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @02:41PM (6 children)

    by Anonymous Coward on Sunday November 29 2020, @02:41PM (#1082028)

    Noninteger dimensions only make mathematical sense, not physical sense.

    There is a problem with the Hubble constant, but it's almost certainly either an observation problem or that it isn't really constant, either of which is much more likely than a strange mathematical coincidence turning out to be the key to the nature of reality.

    • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @03:14PM (1 child)

      by Anonymous Coward on Sunday November 29 2020, @03:14PM (#1082035)

      uhm ... I wonder if relativistic distance skewing could be explained by the universe not quite being 3D? I wonder if time dilation may also have some non-nonintegral / non-integer dimensional time explanation?

      • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @03:17PM

        by Anonymous Coward on Sunday November 29 2020, @03:17PM (#1082036)

        err ... non-integral

    • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @03:36PM (2 children)

      by Anonymous Coward on Sunday November 29 2020, @03:36PM (#1082040)

      Noninteger dimensions only make mathematical sense, not physical sense.

      What is the argument that it does not make physical sense? In the physical sciences, space is assumed to be 3-dimensional--it doesn't naturally fall out of the equations.

      • (Score: 3, Touché) by mhajicek on Sunday November 29 2020, @06:06PM

        by mhajicek (51) on Sunday November 29 2020, @06:06PM (#1082078)

        Any CNC mill is proof enough.

        --
        The spacelike surfaces of time foliations can have a cusp at the surface of discontinuity. - P. Hajicek
      • (Score: 1, Interesting) by Anonymous Coward on Tuesday December 01 2020, @06:45AM

        by Anonymous Coward on Tuesday December 01 2020, @06:45AM (#1082728)

        Sorry for not responding to this sooner, I had a migraine. If you read this, perhaps you will soon too ;) I certainly don't specialize in topology, which is a difficult field, so I can only give you an overview.

        Our measurements of real spacetime indicate that it is either flat, or very close to flat. On large scales of the universe, Euclidean geometry describes reality. But not all mathematical structures work this way. The formal definition is actually based on set theory [wikipedia.org], but you don't have to think of it that way. Colloquially, you can think of a non-integer dimension as one which is a subset of a higher dimension, in which not every state that can be expressed in the higher dimension can be expressed in the partial lower dimension, but in which there is too much information to express in the next lower integer dimension.

        Let's start with the example of a circle. We think of a line as one-dimensional, and a circle as a two-dimensional object. But if you are working in a space where you *only* have circles and only care about the edge of the circle, they are really one-dimensional, because you can define a circle with only one number, its radius, and you can identify any point on that circle with only one number as well. (Assume by convention that "0" is straight up, and then any point on the circle can be defined by a number from 0 to 2pi representing the angle from that starting point). This is effectively still a line, it's just a line that's bent. It takes the same amount of information as it would take to describe a line: one number for the size, and one number can specify any point on it. (If we wanted to instead think about the area of the circle rather than its perimeter, it would be two-dimensional).

        But now consider that the circle might have irregularities on the surface, defined by some function which describes the variation from an ideal circle at any point. Assume the irregularities are confined within some epsilon due to the range of the function, and that the function is expressable in some compact way. Now we have this "roughness" on the surface of the circle, which increases the dimension past one. But it's not really two-dimensional either, because it's still more or less circular. The dimension of the surface is now non-integer. The exact "size" of that fractional dimension depends on the information content of the function used to describe the variation from an ideal shape!

        The common example of this is the Koch snowflake [wikipedia.org], which is a simple fractal but like all fractals it has a finite area but an infinite perimeter. This infinity indicates that there's too much information for one dimension to contain. But it's not a true two-dimensional object either, because it doesn't have a full degree of freedom in its second dimension. The dimensionality of the Koch snowflake is about 1.262.

        For a fractal-like object that is fully two-dimensional, you can take a look at the Peano space-filling curve (https://en.wikipedia.org/wiki/Space-filling_curve). This is an expanding blob that, despite being a line, fills *all* space within its area, and therefore, requires two full dimensions to describe.

        The physical world... just isn't like this. We have equal freedom in all three dimensions, and there does not appear to be any restriction on position or velocity. Position and velocity can be any real-valued vectors. There is, of course, some fuzziness at the small scales around the Planck length, and due to the Heisenberg uncertainty principle, but this doesn't affect the dimensionality of the universe because there are no excluded states.

    • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @06:06PM

      by Anonymous Coward on Sunday November 29 2020, @06:06PM (#1082079)

      Stuff like this makes the Electric Universe guys seem sane.

      I wonder how many physicists have suddenly thrown up their hands, ran away screaming "None of it makes any sense!", and are now murmuring to themselves in their new cave homes.

  • (Score: 1) by khallow on Sunday November 29 2020, @02:45PM (6 children)

    by khallow (3766) Subscriber Badge on Sunday November 29 2020, @02:45PM (#1082029) Journal
    First, the article ignores the time dimension. Second, from our point of view, N(\epsilon) increases as \epsilon^{-4}*e^(\lambda/\epsilon), where \lambda is the scaled cosmological constant. The end result is that we get, crudely for our alleged dimension, 4+\lambda/(\epsilon*ln(\epsilon)) due to the "negative energy" expansion we appear to observe in the universe.
    • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @03:45PM (1 child)

      by Anonymous Coward on Sunday November 29 2020, @03:45PM (#1082041)

      Your equation assumes a posteriori space is integer dimensional.

      • (Score: 1) by khallow on Sunday November 29 2020, @04:26PM

        by khallow (3766) Subscriber Badge on Sunday November 29 2020, @04:26PM (#1082049) Journal

        Your equation assumes a posteriori space is integer dimensional.

        Hence, the "+\lambda/(\epsilon*ln(\epsilon))" term? I assure you, that's nowhere near an integer.

    • (Score: 0) by Anonymous Coward on Monday November 30 2020, @12:24AM (3 children)

      by Anonymous Coward on Monday November 30 2020, @12:24AM (#1082164)

      First, the article ignores the time dimension.

      You are free to ignore it too. Growing body of evidence all points to it being an emergent phenomenon; an "illusion" instead of a "dimension".
      https://en.wikipedia.org/wiki/Problem_of_time#Proposed_solutions_to_the_problem_of_time [wikipedia.org]

      The article's ignoring the question of how *gravity* should be working in 2.71828... dimensions and how to reconcile that with observation, on the other hand, is a total showstopper.

      • (Score: 1) by khallow on Monday November 30 2020, @02:42AM (2 children)

        by khallow (3766) Subscriber Badge on Monday November 30 2020, @02:42AM (#1082202) Journal

        Growing body of evidence all points to it being an emergent phenomenon; an "illusion" instead of a "dimension".

        It's going to have to grow a lot more before it can overcome the fully grown body of evidence supporting time as a dimension.

        • (Score: 0) by Anonymous Coward on Monday November 30 2020, @04:12AM (1 child)

          by Anonymous Coward on Monday November 30 2020, @04:12AM (#1082239)

          Calling a presupposition "evidence" does not make it such.

          You learning to move through your imagined "dimension" as freely as we do through the real ones, that would be evidence. A mere linguistic exercise like calling eggs "hen fruit"... isn't.

          • (Score: 1) by khallow on Monday November 30 2020, @05:05AM

            by khallow (3766) Subscriber Badge on Monday November 30 2020, @05:05AM (#1082253) Journal

            Calling a presupposition "evidence" does not make it such.

            That did not happen with my post. For a glaring counterexample, we have clocks which can parameterize time to incredible precision.

  • (Score: 3, Interesting) by The Mighty Buzzard on Sunday November 29 2020, @05:27PM

    by The Mighty Buzzard (18) Subscriber Badge <themightybuzzard@proton.me> on Sunday November 29 2020, @05:27PM (#1082066) Homepage Journal

    I wonder how many instances of "3d" will have to be replaced with "ed" if this turns out to be the case. I also wonder how much confusion this will cause to guys named ed and to our eds here at SN.

    --
    My rights don't end where your fear begins.
  • (Score: 3, Interesting) by istartedi on Sunday November 29 2020, @06:41PM (2 children)

    by istartedi (123) on Sunday November 29 2020, @06:41PM (#1082084) Journal

    I had a hard time wrapping my head around this, so I slept on it. It was late anyway.

    This morning it occurred to me that if you had a cubic crystal with N spacing between atoms, you can easily compute the number of atoms in a specific crystal at a given size. If the dimension of space is e instead of 3, there should be fewer atoms in the crystal and it should have less mass than we expect.

    Obviously this isn't the case, because a discrepancy that large would have left every lab in the world scratching its head and we'd have come up with this a long time ago.

    Given that, the only other alternative I could come up with is that the atoms would be hanging out int "gap-space" for some period of time, and that gap-space would have properties very similar or even the same as ordinary space, such that the atoms would continue to interact with their neighbors and provide the crystal with mass, structure, etc.

    So. What makes these vast cosmological distances different from ordinary crystals?

    --
    Appended to the end of comments you post. Max: 120 chars.
    • (Score: 0) by Anonymous Coward on Monday November 30 2020, @04:39AM

      by Anonymous Coward on Monday November 30 2020, @04:39AM (#1082246)

      I don't see the problem. It's just like a polygon with a non-integer number of sides.

    • (Score: 2) by Common Joe on Tuesday December 01 2020, @07:16PM

      by Common Joe (33) <{common.joe.0101} {at} {gmail.com}> on Tuesday December 01 2020, @07:16PM (#1082919) Journal

      This morning it occurred to me that if you had a cubic crystal with N spacing between atoms, you can easily compute the number of atoms in a specific crystal at a given size. If the dimension of space is e instead of 3, there should be fewer atoms in the crystal and it should have less mass than we expect.

      Not necessarily. If I understood you correctly, you're assuming mass and space are linearly related. A certain amount of mass fits into e dimensional space. According to this idea, what we perceive as 3 dimensional isn't really 3 dimensions.

      Disclaimer: I suck at physics. I didn't read the article, just the summary.

  • (Score: 0) by Anonymous Coward on Sunday November 29 2020, @07:43PM

    by Anonymous Coward on Sunday November 29 2020, @07:43PM (#1082094)

    So we live in a big 2.5D sidecroller arcade platformer brawler beat'em up

  • (Score: 2) by hendrikboom on Sunday November 29 2020, @10:42PM (5 children)

    by hendrikboom (1125) Subscriber Badge on Sunday November 29 2020, @10:42PM (#1082138) Homepage Journal

    So I tried to read the paper to find out just what space he's talking about that's e-dimensional.
    I could follow along easily for a while, but I got off at the point he sucks a potential function UAB out of his thumb. Why that one? Why is there even a potential function and what's it used for? It's not as if potential functions are unique.
    So then I started wondering what the dimension of an information space could possibly mean. I mean, what are the unit cubes (or spheres, I'd be happy with spheres) in information space? For that matter, what is the distance that determines the size of cubes or spheres?
    I could imagine that he could be talking about ordinary distance as measured by a ruler, and then look at the fractal density of information in space. It's extremely plausible that information isn't uniformly distributed in space, and so a fractal dimension might even be plausible. Now there's a movement to formulate quantum mechanics primarily in terms of information instead of uncertainty, but I think that's far beyond the content of the article. Besides, quantum mechanics makes it doubtful that information is even localized in space.
    But I can see nothing in the article that says what kind of space he's talking about.
    Maybe I'm just missing the point. It's a respected journal, so I suspect there may be a point. But I don't even see in what space of ideas the point might lie.

    • (Score: 3, Interesting) by maxwell demon on Monday November 30 2020, @12:56AM (1 child)

      by maxwell demon (1608) on Monday November 30 2020, @12:56AM (#1082169) Journal

      I'm not getting it either.

      First, if asked for a potential that corresponds to d dimensions, the one the author invents (without giving either a justification or a reference) is not the one I'd have chosen (and I'm pretty sure I'm not alone with that). Rather, the potential I would have given (for d≠2) is K/r(d-2). Because that gives a field that is proportional to 1/rd-1 and therefore fulfils Gauß's law in d dimensions.

      Second, WTF is “the use of one of the dimensions”? It's not as if we put one object in the first dimension and another in the second dimension. All objects “use” all of the dimensions at the same time!

      And the rest doesn't really make sense to me either.

      --
      The Tao of math: The numbers you can count are not the real numbers.
      • (Score: 0) by Anonymous Coward on Monday November 30 2020, @04:50AM

        by Anonymous Coward on Monday November 30 2020, @04:50AM (#1082250)

        It's basic math - you have one dimension (x) and another dimension (y) leaving 0.7 of a dimension for z. You can think of it as a diagonal line.

    • (Score: 1) by khallow on Monday November 30 2020, @05:32AM (2 children)

      by khallow (3766) Subscriber Badge on Monday November 30 2020, @05:32AM (#1082263) Journal

      I could follow along easily for a while, but I got off at the point he sucks a potential function UAB out of his thumb. Why that one? Why is there even a potential function and what's it used for? It's not as if potential functions are unique.

      That's a good point. I suppose it could come from the observation that gravity and electric field potentials are of that form, proportional to 1/r where r is the radius.

      • (Score: 1) by khallow on Monday November 30 2020, @01:50PM (1 child)

        by khallow (3766) Subscriber Badge on Monday November 30 2020, @01:50PM (#1082351) Journal
        Classic model not relativistic.
        • (Score: 0) by Anonymous Coward on Tuesday December 01 2020, @02:19PM

          by Anonymous Coward on Tuesday December 01 2020, @02:19PM (#1082812)

          I would like to see what the fake guy has to say about all this. He confused quantum physics with relativity the other day and insisted he was such a genius (before I schooled him on both and then he eventually admit that he doesn't know anything about this stuff, lol). Maybe he was such a genius he was unifying the two theories together and I was too dumb to realize it ;)

          Where did he go by the way? I hope I didn't scare him off ... I did like reading his comments, many of them were insightful when it came to stuff he was actually familiar with. I know I can be a jerk sometimes, lol, but it's not like he wasn't a jerk either. I'm just the bigger, more persistent, jerk I suppose ;)

          and most of my comments above about dimensions that you responded to were made jokingly, the discussion wasn't meant to be a serious discussion.

  • (Score: 2) by bzipitidoo on Tuesday December 01 2020, @10:08PM

    by bzipitidoo (4388) on Tuesday December 01 2020, @10:08PM (#1082993) Journal

    Something that's been known for quite a while is that the most efficient numeric base is e, not binary, and definitely not decimal or hexadecimal. The closest integer is, of course, 3, and, yes, takes slightly less info to represent numbers in base 3. 3 is the most efficient integer base.

    We should've made our computers trinary.

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