Of Particular Significance

What is an “Extra” Dimension? Some Examples

Matt Strassler [10 Jan 2012]

It’s time now to learn something new from a two-dimensional space that we couldn’t learn from a one-dimensional space. In particular, I want to focus your attention on the strip — a space that looks like a ribbon, with one very long (perhaps infinite) dimension and one very short one. (Most of what I’ll say for the strip is also true for the tube, but I’ll stick with the strip because it is easier to draw.)

First, let me remind you of a concept I introduced at the end of Worlds of 1 Spatial Dimension. That was the idea that even when a physical world is three-dimensional, such as the one we are used to, it is possible for certain aspects of the world to behave as though they are one-dimensional. For example, a person walking on a high-wire truly exists in three dimensions, but her motion is going to be essentially one-dimensional. In this case, it is the constraint of her own safety that makes her world one-dimensional. Well, on the strip we have another reason why the world could be effectively one-dimensional. It’s because there is a large dimension and a small one, and whether you can move around in the small dimension depends upon what your own shape is… and how big you are relative to the distance across the small dimension.

Fig. 1: A small boat moving on a ship canal can move both along and across the canal — two-dimensional motion. If it were a conscious creature, it would likely have developed senses that would allow it to produce a two-dimensional picture of the world.

Here’s an example of a strip: a ship canal. A long strip of water allows a small boat to move around in a two-dimensional way: it can move up and down the canal, or back and forth across it. But a giant freighter that barely fits inside the canal can only move forward or backward. For the freighter, the canal is effectively one-dimensional. If you watched it move, you would see no sign that the canal is truly two-dimensional. The motion of a large object — large compared to the width of the strip — reveals the long dimension but does not show signs of the second, short dimension; the second dimension is only revealed by the motion of a small object, like the small boat. This is an important lesson!

Fig. 2: A giant freighter on the same ship canal can move only forwards or backwards, along the strip’s long dimension. Its motion shows no sign of the second dimension across the strip. If it were a conscious creature, its senses might well not provide it with any knowledge whatsoever of the existence of the second dimension. (But if the freighter were very smart, it might occur to it to speculate as to whether the ship canal might have a second dimension, very small and not easily detected by the freighter’s motion or senses!)

What’s another example of the same thing? For you and me, the surface of a human hair is essentially a one-dimensional object, as far as our eyes can see. We could cut a hair into two strands of half the length, but we would have a tough time slicing a strand of hair the long way: indeed, in English we refer colloquially to “splitting hairs” as something not worth the time and trouble. A tiny dust mite, however, can crawl around on the surface of a strand of hair as though it were a giant tube. For the mite, the surface of the hair seems two-dimensional, just as the surface of a flag pole seems two-dimensional to us.

Let’s get back to the freighter. Imagine for a moment that the freighter were a conscious creature, with a brain that obtained information from sense organs, just as do all animals, but whose only sense organ was one that allowed it to detect its own motion. [Why am I introducing consciousness here?  It has nothing to do with the dimensions of space that we’re talking about, which are present whether the freighter is conscious or not.  What I am doing here is trying to explain to you how a conscious creature — such as yourself — could be misled by its senses and brain as to the true features of nature, and could consequently be completely unaware of a dimension of space.]  The only information coming in from that sense to the freighter’s brain would be something to the effect of “forward at five kilometers per hour” or “backward at two kilometers per hour” or “stationary”. What picture of the world would the brain of the freighter build? With no information coming in from outside about the presence of the second dimension across the strip, it would naturally build a one-dimensional image of the world, as though the canal had but one dimension.  The freighter would have no knowledge of the second dimension, even though a much smaller boat could easily detect it.

This is one example of how “extra” dimensions can work (though not the only one, as we’ll see later.) What we mean by “extra” is this: it is a dimension that the freighter didn’t (and couldn’t, given its senses) know was there.

Imagine now that something similar applies to us. Imagine we live in a sort of “strip” that, in addition to three large spatial dimensions that we normally move around in, detect with our senses, and conceive of in our minds, there is also a fourth spatial dimension, so incredibly small that we fill it up just like the freighter, so that we can’t move across it. Moreover, neither our sight, hearing, touch, smell or taste faculties can detect the presence of this extra dimension. As far as our brains know, it isn’t there. But maybe it really is there, and sufficiently tiny creatures might be able to move across it and sense it.

To suggest that one or more extra dimensions exist is just speculation — pure imagination .  We certainly do not know that this is true. But what I am trying to convey to you is that we do not know it is not true.

So then the next question is this: how could human scientists, whose senses and brains cannot detect this fourth dimension of space, discover it is there? That is where a mixture of human intelligence and advanced scientific instruments comes into play.

[New additions as of 10 January begin here] 

And as we’ll see later, scientific study reveals that although one or more extra dimensions of this type may exist in nature, if they do they must be extraordinarily small.  The distance across any such dimensions can be no more than 1/10,000 the distance across a proton, which is about 1/1,000,000,000 times smaller than the distance across an atom, or if you prefer, about 0.000,000,000,000,000,000,1 meters (and a meter is just over 3 feet.)  Research at the Large Hadron Collider [LHC] is gradually pushing out this frontier.

Another Example

The type of extra dimension that I’ve just described here is just one of several general possibilities.  Here’s another example.  Instead of a ship of huge width that spans the entire canal, let’s consider a much narrower ship that is tethered tightly to the side of the canal, so that although in principle it could move across the canal it is unable to do so in practice. The relation of the boat to the extra dimension is different, but the effect — that the short dimension of the two-dimensional ship canal is undetectable in the boat’s motion — is the same.

Fig. 3: Bottom: Another boat, possibly longer than the canal is wide, but much narrower than the freighter of Figure 2, is tightly tethered to one wall of the canal and towed forward or backward along the canal.  Like the freighter, this boat is similarly unable to move across the canal, though for a different reason.    Top: This is to be contrasted with the small boat that can move freely across the canal.

Specifically, as shown in Figure 3, imagine a boat that moves forward or backward along the canal, using its own engine, but also with the assistance of vehicles that run on rails alongside the ship canal — a modern form of a tow-path. In contrast to the freighter, this boat does not does not fill the canal from side to side. In principle it could move across the canal from one side to the other. But in practice, there are forces — exerted by the ropes that attach it to the vehicles on rails — that hold it tight against the left wall of the canal. So strong are these forces that the boat never strays from the left wall. And so, as it moves forward and backward along the canal, its motion is, like that of the freighter, entirely one-dimensional.  And as we did for the freighter, we may imagine that if this boat were conscious, and its senses provided information only about its motion, its brain would likely construct a one-dimensional image of the canal, despite the fact that, if it were possible for the ropes to be cut, the boat would then be able to turn and move into the center or across to the right-hand wall of the canal.

The same could be true for us. We — more precisely, every particle from which our bodies are made, and from which everything that we can see and feel or even detect by modern scientific instruments is made — may be attached, by forces of extraordinary strength that we cannot counter, and of which we are not aware, to the three-dimensional wall of a space with four spatial dimensions. And because our motions are constrained to the wall, and our senses detect only objects that are similarly stuck to the wall, our brains build for us a three-dimensional image of the world, despite the fact that in truth the world has an extra dimension.

Again, there’s nothing that assures us that this is true. It’s pure speculation. But neither are we sure that it is not true. The only way to find out is to design and carry out scientific experiments, whose nature I will explain soon.  It turns out (as was pointed out, somewhat to the shock and incredulity of the community [including yours truly], in a famous 1999 paper by Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali) that this second type of extra dimension can be much larger than the type I described earlier — maybe almost as large across as the width of a human hair!  [In fact when the paper was first written, experiments still allowed for extra dimensions to be as large as a millimeter; but new experiments have been done, as I’ll describe later, that have put better constraints on this possibility.] This is rather amazing.  Despite all the experiments that we perform using microscopes, and the tiny electronic equipment that we build into our computers, we would still not know about an extra dimension of small but still macroscopic size.

More Examples

Once we start going down these roads we find many more possibilities than just these two.

For instance, there’s no reason there couldn’t be more than one extra dimension. We can get an example using our ship canal — which, if it is deep enough, should really be thought of as three-dimensional, very long in one dimension but with a finite width and a finite depth. Then we can have all sorts of objects for which the canal has zero, one or two extra short dimensions, as shown in Figure 4.  A submarine can move around three-dimensionally within the canal. A small boat is stuck to the two-dimensional surface of the canal, and moves unaware that the canal has depth. Its two-dimensional view of the world would miss one extra dimension. And three different types of boats would be unaware of two of the dimensions. The towed boat, stuck at the wall and at the surface, is one example. A wide barge that fills the canal from side to side but is stuck at the surface is a second. And finally, the giant freighter, whose keel reaches to the bottom of the canal, essentially filling it completely, gives a third example. For three very different reasons, these three boats would have motions that show no sign of the depth and width of the canal, and were they conscious creatures they might well be surprised to learn that there are two extra dimensions to their world.

Fig. 4: When we remember its finite depth, a ship canal really forms a finite three-dimensional world. Top to bottom, from left to right: Although a submarine can move throughout this three-dimensional world, a small boat can move across the canal but is “tethered” to its surface, so for it the depth of the canal is an extra dimension.  The last three boats can move only along the canal: the boat tethered to the side of the canal, the barge, which spreads across the canal but is trapped at its surface, and the freighter, which completely fills the canal.  These three boats all move one-dimensionally and their world naively appears one-dimensional; for them, the two finite dimensions of the canal are unexpected extras.

More examples will follow over time.  But at this point it is important to learn more about how the scientific experiments to search for these and other types of extra dimensions are actually carried out.

48 Responses

  1. Firstly I’d like to thank you for all the work you’ve done here. I’ve only recently discovered this site and I think it ranks right up there with the invention of sliced bread. I’ve read a lot of the popular (and not so popular) physics literature and I have to say that your ability to clearly illustrate and explain ideas reminds me of Richard Feynman’s style.

    Secondly, is it possible that we cannot detect any extra spatial dimensions because the fundamental particles (quanta) have a structure that extends into and requires those spatial dimensions for their very existence? Much like my body–due to its complexity and structure–requires three dimensions to exist, perhaps the fundamental quanta require tiny spatial dimensions to give them complexity and structure. That may explain why they are so difficult to detect: the only particles we have at our disposal to detect these extra dimensions may be incapable of turning inward and looking at themselves.

  2. We humans live constantly in multidimensional states. We use them to create our world in endless ways. Dreaming is a gateway to another dimension we can access. Once science grasps this, it will solve the theory of extra dimensions.

    1. Why would dreams have to have anything to do with extra dimensions? Especially extra spatial dimensions. Have you ever had a 5-dimensional dream?

  3. I feel the example of ship-canal is misleading. The ship is indeed in the 2nd dimension, but cannot feel it because itself fills the canal side by side. But we humam beings cannot feel any extra dimension, if it exists, because we are much much larger than it. We (and even a proton) are not in any extra dimension at any time. An analog is a ship even larger than the canal, it can never manage to go down into the canal. (Or there are very small holes in the ground, a walking person cannot feel them, but an ant can.)
    If this is true, could it mean extra dimension is not everywhere, but only SOMEWHERE?

    1. If you think it is misleading for the reasons that you stated, then you are the one who is misled; you haven’t understood extra dimensions properly. My analogy is, in fact, correct, in that it is consistent with the mathematics that we use to describe extra dimensions in scientific papers; your statements critiquing it, and your analogies, are inconsistent with the mathematics and with the physics.

  4. Prof Strassler, once again thank you for your patient, thoughtful, logically-sequenced tutorials. I, for one, enjoy the “virtual constructionism” style (to coin an awkward and probably inapt phrase) that you employ here.

    You manage to remain scientifically rigorous (judging from comments by your professional peers) and widely accessible at the same time, which is no mean feat, particularly in explaining today’s cutting edge physics explorations (I’ve greatly enjoyed Sean Carroll’s theoretical physics writing for precisely this too-rare talent).

    As a layman self-educating relatively late in life, I anticipate each installment like an unfolding episodic mystery, ending with a “tune in next week” cliff-hanger. Knowing that many chapters are being written as we speak, and many are yet to be written, just makes following the journey even more exciting.

    As someone seemingly right in the cross-hairs of your intended audience, I offer my anecdotal support of the success of your entertaining and enlightening teaching methodology, to counterbalance some of its critics.

    Short version: Works for me!

    Thanks again!

  5. Matt: “In my view we should never get into the habit of simply discarding theories because we find them aesthetically displeasing, or because they state certain things instead of predicting them. Nor should we get in the habit of accepting theories because they are aesthetically pleasing or because they make predictions as yet untested. We should apply criteria of this type with more care and nuance, remembering that our goal is not to find the theory that looks the prettiest and/or is the most predictive, but rather to find a theory that is consistent with nature, which is for the experimentalists to decide.

    That said, internal theoretical consistency, and consistency with previous experiments, are important criteria for a theory. …”

    This is a more advanced view on physics epistemology, a much more open one than the traditional. Thanks again for this.

  6. I understand the point, and don’t want to take up too much space here, but my own view is that if we taught foundations in high school we would not have to tear down Freshman “intuitions” and there would be a better general appreciation of both physics and mathematics. Unfortunately, we educate for perceived needs in the market place, and not learning for it’s own sake – that is, I believe, an error.

  7. I have to say that this argument worries me. I do not see how any logical or scientific argument can be supported on the basis that “we do not know it is not true.” Unless, of course, you are making the falsification argument.

    Falsification relies upon verification and observation. Now, you have developed the conventional way of speaking about dimensionality but you have not identified any observation that justifies this way of speaking in the first place.

    I will argue that dimensionality is an epistemic convenience but that there is no existential foundation for the conception. You speak here as though these dimensions have some existential basis.

    So, what is your observation, aside from mathematical convention, that establishes that dimensionality is anything other than a convenient way of speaking about the world? (My challenge is consistent with Mach and Einstein’s epistemic view).

    1. Your comment would potentially be well-justified except for the fact that I have promised you that the next article in this series will be on the observational consequences of extra dimensions… which are very substantial.

      I agree that the fact that something might be true — that it is not known to be false — does not make it science. It becomes science under the following condition: that if it is true, it has well-defined and precise measurable effects. This is what makes it falsifiable, and at that point a scientific research effort may ensue.

      Without observational consequences, all of this discussion of extra dimensions would be metaphysics. What makes it physics is that it is possible to discover extra dimensions — and from the fact that they have not yet been discovered in various experiments, it is possible to state (as is done briefly in the text) that if extra dimensions of various types exist, they must be smaller than such and such, etc.

      As for your last point: it is worse even than you think. But this subject is very advanced and I am not ready to take my readers into the subtleties of what is and is not a matter of convenience. Let us walk before we try to fly.

      1. Understood. ‘Though you do not answer the main question, I hope you will do so in following articles. Before dealing with “extra” dimensions it seems to me important to justify what a dimension is in the first place and why one would or would not be considered independent of any other. That we fail to make these matters of convenience and existential construction clear is very misleading pedagogically in my view.

        1. Well, I tend to disagree with you about the pedagogical issues. The problem with doing what you suggest is that for a person who is not already educated in physics, such a conceptual step is even more difficult than the one I am trying to map out. In talking to non-experts who do not have mathematics to serve as their foundation, it seems to me one has to start with the intuition that people bring to the table, and use this as the foundation, only gradually pointing out its problems and replacing it with a better foundation. What you suggest — tearing layperson’s intuition apart right at the beginning, on the grounds that it is misleading — is what is done in freshman physics, where everything a nonexpert intuitively knows about the world is torn apart in the first week, through Newton’s laws. The dismal pedagogical success record of freshman physics courses should make you wonder about the wisdom of such an approach.

  8. While in a sense the story about dimensions is a historical one used in relation to Flatland, by E. A. Abbott , is projective geometry then a leading perspective about projections? Expressions, finally leading to a relation with topology? What does this mean of Genus development in the valley?

    http://www.nature.com/nature/journal/v423/n6941/images/423695a-f1.0.jpg – “Juan Maldacena: The strings move in a five-dimensional curved space-time with a boundary. The boundary corresponds to the usual four dimensions, and the fifth dimension describes the motion away from this boundary into the interior of the curved space-time. ” High-energy physics: Into the fifth dimension- http://www.nature.com/nature/journal/v423/n6941/full/423695a.html;jsessionid=AA09912C40810B705B2CE65ABC4DE672

    How then in any form can one describe the unification of electromagnetism and gravity with that geometrical sense other then what has been describe by Maldacena?


    240 Eโ‚ˆ polytope vertices using 5D orthographic_projection to 2D using 5-cube (Penteract) Petrie_polygon basis_vectors overlaid on electron diffraction pattern of an Icosahedron Zn-Mg-Ho Quasicrystal.

    E8_(mathematics) and Quasicrystals

    Is there not a relation?


    1. What Maldacena is doing (building on work of many authors, especially of ‘t Hooft and of Polyakov) is rather more subtle than what we are dealing with so far. Patience.

  9. Thanks, Professor, for your blog. While reading the update, I was struck by how much the tethered freighter sounds like the surface of the earth for animals. The discovery of the bounded sphere that defines the earth’s surface illustrates your point. There were observations of the motions of heavenly bodies, and even the Greek measurement of the earth’s size using geometry and the sun. They told us that our flat landscape is embedded in a higher-dimensional space.

    1. Well… my analogies have their limits, and you have to be careful not to push them too far. There are two senses in which my analogy fails for animals on the surface of the earth. First, many animals walk by lifting their feet off the surface of the earth, and many can actually jump, under their own power; not only are these three-dimensional forms of motion, but most animals are surely intuitively aware of this third dimension, as we are. And most animals can see upward as well as straight ahead, and can see trees and the sky and other things well off the surface; their picture of the world is not merely that of the earth’s surface itself. Thus it is not appropriate to view the third dimension perpendicular to the surface as an “extra” dimension, since we and other animals are quite clear about of the presence of this third dimension.

      The analogy might be better for snails traveling on flat ground. I am not sure how aware they are of the third dimension. Of course I am not sure how aware they are of anything.

      This has to be contrasted with the possibility of extra dimensions of which we have absolutely no sensory perception and in which direction we cannot move at all (at least not under our own power.) Those dimensions are truly unknown to us and in this sense would be “extra” — certainly extra-surprising!

  10. Matt,
    If you propose an extra spacial dimension, can’t you create a science to go out and look for it? We as humans can dream. Every dream has a possibility(probability) of existance.

    1. Yes, there are indeed ways to look for these things (if there were not, I would be talking metaphysics, not physics) and I will provide you soon with an article or two which explains how it is done.

  11. May I suppose that if we look closer at a smaller scale (the small boat) to our world of particles we could imagine that point like particles posess more dimensions in the form of complex forms or even compound knots?
    Then quarks could be all compound knots with different geometry. A Quantum Form theory could be the result.

  12. Since the ‘extra dimension’ is ‘incredibly small’ and not like the other spatial dimensions, it seems to me to be misleading to refer to it as a spatial dimension. What is the motivation for referring to it as spatial?

    One of the nice things about the three spatial dimensions is that they are the all equivalent in the sense that space is uniform. There is plenty of direct evidence for the uniformity of space at all scales, so why doesn’t this evidence preclude an extra spatial dimension that is different from the others?

    Or is the idea that you need to add multiples of 3 more dimensions, so that XY and Z each get a new companion dimension?

    1. I wrote about this in an earlier article, http://profmattstrassler.com/articles-and-posts/some-speculative-theoretical-ideas-for-the-lhc/extra-dimensions/extra-dimensions-how-to-think-about-them/dimensions-of-physical-space/ . This article explicitly explains the justification for calling them spatial dimensions.

      You ask: “There is plenty of direct evidence for the uniformity of space at all scales, so why doesnโ€™t this evidence preclude an extra spatial dimension that is different from the others?”

      The uniformity of the three spatial dimensions we know about has no bearing on the properties of others that we do not know about. In my example of a strip or ribbon of infinite length and finite width, one dimension is uniform and the other isn’t. No problem.

      1. It sounds to me like the idea is that two objects could be in the same location at the same time, but at a different place along the smaller dimension. There is more ‘room’ created by the smaller dimensions and the objects, if they were small enough, would not necessarily touch.

        If the two objects at the same XYZ but displaced along the smaller dimension were charged particles, would the force k*q1*q2/r^2 get a little extra r due to the smaller dimension? Is the distance function for r something obvious like the square root of the sum of the squares along all the dimensions?

  13. “But what I am trying to convey to you is that we do not know it is not true.”
    Technically true, but in practice any theory that depends on such an unnatural construct should be dismissed out of hand *unless* it can be shown that the theory itself *predicts* that some dimensions should be vastly smaller than others. Are there any such theories?

    1. You make a remarkable statement here. “…in practice any theory that depends on such an unnatural construct should be dismissed out of hand *unless* it can be shown that the theory itself *predicts* that some dimensions should be vastly smaller than others…”

      You make this statement as though it is a fact, or as though it is accepted doctrine. But I don’t agree with it in the slightest. On what grounds do you make it?

      1. I fully agree with Prof. Strassler and I think I can guess from which direction in the blogosphere Willy Soon comes from ๐Ÿ™‚ …

        1. Well — if what you mean is that perhaps Mr. Soon is politically opposed to string theory, all I can say is that this should be irrelevant to all of us, because (a) extra dimensions are common in string theory, but not necessary, and (b) as far as we currently know, it may be that extra dimensions can exist without string theory, so we need not discuss string theory here at all. Moreover I have been discussing the experimental situation — what it is we know from measurements — not the theoretical one.

          Generally, my view on Mr. Soon’s comment is that what is at stake is not extra dimensions, which we can take or leave, but rather the very methodology of science, in particular theoretical endeavor. Imagine we accept his type of argument. The same argument could have been used to discard Einstein’s general relativity (which does not *predict* that spacetime is curved, it simply states it — and for a Newtonian physicist the notion would seem highly unnatural). It could be used to dismiss the Standard Model, which demands we believe something highly unnatural (a Higgs particle whose mass is much smaller than the gravitational mass scale, something which the Standard Model does not predict but merely assumes.) It could be used to dismiss the possibility of dark energy, whose small but non-zero value is highly unnatural (and which no theory currently predicts). It could even be used to dismiss the Standard Model on the grounds that it does not predict the large ratio of the top quark mass to the electron mass.

          In my view we should never get into the habit of simply discarding theories because we find them aesthetically displeasing, or because they state certain things instead of predicting them. Nor should we get in the habit of accepting theories because they are aesthetically pleasing or because they make predictions as yet untested. We should apply criteria of this type with more care and nuance, remembering that our goal is not to find the theory that looks the prettiest and/or is the most predictive, but rather to find a theory that is consistent with nature, which is for the experimentalists to decide.

          That said, internal theoretical consistency, and consistency with previous experiments, are important criteria for a theory. Extra dimensions of certain types are certainly consistent with previous experiments, and have enough internal theoretical consistency to be worth pursuing.

          In my view a theoretical idea is useful if it predicts phenomena that have not yet been searched for in experiments. The notion of extra dimensions, as it was developed in the late 1990s, was very useful, because it made particle physicists recognize gaps in their reasoning, opened their minds to new ideas, and propelled a new experimental research program that turned out to be valuable in searching not only for extra dimensions but for a number of other speculative effects.

  14. Great analogy for spatial dimensions, but what about time? Our human minds do not seem to have direct means of experiencing time the same way we experience space. It sounds counterintuitive, yet to have concepts of ‘past’ and ‘future’ we have to rely on memory and deduction.

    1. Time is much, much more subtle than space, for many reasons, some of them not obvious. For instance, why not have two dimensions of time instead of one? Answer: there are no sensible dynamical equations with two time dimensions — things do not “happen” properly in such a world, and predictions don’t seem to be possible. (And of course, nothing “happens” in a world with zero time dimensions, whereas a world with no space dimensions and one time dimension can still be quite interesting.) A single time dimension seems to be very special, whereas any number of space dimensions seems to give an interesting world. There is plenty of research trying to get a better handle on this, but I for one have nothing more intelligent to say. Of course you should read Sean Carroll’s book about time for a nice summary and many interesting insights, but you won’t get an answer for this question.

      1. By the doctrine “physical effects occur locally in space” of spacial dimensions, wouldn’t time be a spacial dimension? After all, things could easily exist in the same location in 3-space without affecting each other, if it is at different times.

        1. You’re right to ask the question, but the reason it seems puzzling is that you have the doctrine wrong. The doctrine (or more precisely, what we infer from many decades of experiments) is that physical effects occur locally in space and time. Yet time is still different from space. Time differs from space in various ways. For instance, cause and effect are oriented with respect to time. Crudely, what is to my right can affect what is to my left, and what is to my left can affect what is to my right, but (as far as we can tell in daily life and controlled experiments) the future cannot affect the past.

    1. That’s a much more complicated story — and has to do with the details of fluids and “viscosity” and how fluids flow. So no, it’s not related. My story here is really just geometry; the water really plays no role.

  15. I’ve used a similar example when talking about the phenomenon of emergence. For most purposes a thread is one dimensional. Weave them together, and you get a 2-dimensional piece of cloth.

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