# Other, Completely Different Infinities

January 26, 2012 8:05 AM Subscribe

Japanese scientists think they may have an explanation for how a three-dimensional universe emerged from the original nine dimensions (plus time) of space: the universe had nine spatial dimensions at its birth, but only three of them experienced expansion. A Hollywood sound designer tries to explain ten dimensions. Efforts to portray an eighth dimension have already been explored.

I watched Primer and had a headache. Now my nose is bleeding.

posted by the_very_hungry_caterpillar at 8:23 AM on January 26, 2012 [4 favorites]

posted by the_very_hungry_caterpillar at 8:23 AM on January 26, 2012 [4 favorites]

Right there with you, caterpillar. Just reading the post caused some sort of non-Euclidean event inside my skull.

posted by magstheaxe at 8:42 AM on January 26, 2012

posted by magstheaxe at 8:42 AM on January 26, 2012

It's actually pretty easy to visualize (well, for certain values of the word "visualize") more than 3 dimensions. Here's a simple example: Imagine two people moving around in the world. Each person's location, of course, is described by 3 coordinates, so the location of two people is described by 6 coordinates. Voila, 6-dimensional space: a point in 6-dimensional space gives the location of two people (or two objects, in general).

posted by Frobenius Twist at 8:43 AM on January 26, 2012 [3 favorites]

posted by Frobenius Twist at 8:43 AM on January 26, 2012 [3 favorites]

That ten dimensions video is one of those things that drives me properly crazy whenever I see it linked (xkcd-386 times a thousand).

posted by edd at 8:47 AM on January 26, 2012 [1 favorite]

posted by edd at 8:47 AM on January 26, 2012 [1 favorite]

*"In particular, why do we only experience three spatial dimensions in our universe, when superstring theory, for instance, claims that there are ten dimensions -- nine spatial dimensions and a tenth dimension of time?"*

Why do I only have ten dollars in my pocket, when the crazed raccoon on the street corner is telling me I have fifty?

posted by curious nu at 8:55 AM on January 26, 2012 [5 favorites]

Okay, then why did only three of them experience expansion?

posted by benito.strauss at 9:00 AM on January 26, 2012

posted by benito.strauss at 9:00 AM on January 26, 2012

Apparently, "the sheet shrunk in the wash" is all they've been able to figure out so far.

posted by BYiro at 9:01 AM on January 26, 2012 [1 favorite]

posted by BYiro at 9:01 AM on January 26, 2012 [1 favorite]

Here, I will show you what ten dimensions looks like:

(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)

posted by Wolfdog at 9:09 AM on January 26, 2012 [5 favorites]

(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)

posted by Wolfdog at 9:09 AM on January 26, 2012 [5 favorites]

*Okay, then why did only three of them experience expansion*

Obviously the most stable configuration. If space time fractured due to the incredibly high tension required to maintain symmetry one could suppose that upon breakup it would assume the most stable configuration. Alternatively, it was the first stable configuration. Your guess is as good as mine.

I'm pretty sure that the nature of the other dimensions is beyond our ability to sense and grasp given that fact that we evolved in this 3 dimensional universe. It would be a waste genetic resources to develop the ability to understand and detect dimensions that cannot be manipulated.

posted by mygoditsbob at 9:14 AM on January 26, 2012 [1 favorite]

The "of" that I put in the above response was cleverly hidden in the 8th dimension.

posted by mygoditsbob at 9:15 AM on January 26, 2012 [1 favorite]

posted by mygoditsbob at 9:15 AM on January 26, 2012 [1 favorite]

*Here, I will show you what ten dimensions looks like:*

(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)

(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)

Looks too much like math. Could you rephrase that as an interpretive dance cycle?

posted by qxntpqbbbqxl at 9:15 AM on January 26, 2012 [2 favorites]

So the other six dimensions are Plancking? I thought it was just a fad.

posted by OHenryPacey at 9:20 AM on January 26, 2012 [7 favorites]

posted by OHenryPacey at 9:20 AM on January 26, 2012 [7 favorites]

*The only difficulty imagining it is that we have no physical conception of what that last extrusion means on a gut level. However, it shouldn't cause us to discard the concept.*

Actually (as per my example above), I think we do ourselves a disservice to say that we don't understand, say, 4-dimensional space on a gut level. We really do; it's just that the insistence on seeing 4-dimensional space as representing 4

*spatial*dimensions is limiting, and in fact doesn't represent how we usually live in the world.

Here's another simple example: when driving a car, one is acutely aware of their location (which is a point in 3-dimensional space) and their speed (a point in 1-dimensional space). Thus a point in 4-dimensional space represents the location and speed of our car, and that is definitely something we all have an intuitive notion of.

posted by Frobenius Twist at 9:22 AM on January 26, 2012

But our awareness is not very mathematical.

posted by benito.strauss at 9:35 AM on January 26, 2012 [1 favorite]

location (which is a point in 3-dimensional space)Of course.

speed (a point in 1-dimensional space)Nope. We can practice thinking about it that way, but 'speed' does not feel like a point in 1-space to most people.

posted by benito.strauss at 9:35 AM on January 26, 2012 [1 favorite]

Okay, I've seen the imagining the tenth dimension video before.

It's mostly bunk. It's correct for the first half, but his imaginings of dimensions 5-10 are not those that physicists are talking about. He's mashing together different ideas in physics (Everett Interpretation, bubble universes, the Ultimate Ensemble etc.) and incorrectly labeling them dimensions 1-10.

posted by justkevin at 9:37 AM on January 26, 2012 [2 favorites]

It's mostly bunk. It's correct for the first half, but his imaginings of dimensions 5-10 are not those that physicists are talking about. He's mashing together different ideas in physics (Everett Interpretation, bubble universes, the Ultimate Ensemble etc.) and incorrectly labeling them dimensions 1-10.

posted by justkevin at 9:37 AM on January 26, 2012 [2 favorites]

I think if I was clever enough to understand this it might blow my mind.

posted by biffa at 9:39 AM on January 26, 2012

posted by biffa at 9:39 AM on January 26, 2012

Obviously the most stable configuration. ..... Your guess is as good as mine.

My guess is that it is obviously because three is a magic number, yes it is.

My broader point is that so many of these explanations seem to just kick the can down the road. "Why are there three (large) dimensions?" "Because all but three collapsed" isn't really an explanation.

The cool fact from the article is that they constructed a simulation, ran it, and found all but three collapsed.

posted by benito.strauss at 9:46 AM on January 26, 2012

My guess is that it is obviously because three is a magic number, yes it is.

My broader point is that so many of these explanations seem to just kick the can down the road. "Why are there three (large) dimensions?" "Because all but three collapsed" isn't really an explanation.

The cool fact from the article is that they constructed a simulation, ran it, and found all but three collapsed.

The new simulations may help shed some light on why this symmetry breaking might have unfolded the way it did.So we've got a tool for exploring, but I don't think it yet qualifies as an explanation.

posted by benito.strauss at 9:46 AM on January 26, 2012

I thought string theory had fallen out of favor a while ago when it became clear that it was a bunch of untestable faddish handwavy hokum? Has it become more respectable in the last few years or something?

my favorite interpretation of string theory is still Jamie Braddock

posted by FatherDagon at 10:02 AM on January 26, 2012

my favorite interpretation of string theory is still Jamie Braddock

posted by FatherDagon at 10:02 AM on January 26, 2012

Yeah that 10 dimensions video is really frustrating. People see a well-produced sciencey sort of thing that has the same style as stuff they've seen before, and it takes a shortcut to credibility. Is there a term for that? Bunk in legitimate clothing?

There's nothing mysterious or hard to understand about extra dimensions if they exist as physicists think they do: so completely tiny that they are not perceivable in any real sense. They only matter as math. All it means is that there's a lot more space than we can really tell by looking. Simple.

posted by danny the boy at 10:12 AM on January 26, 2012

There's nothing mysterious or hard to understand about extra dimensions if they exist as physicists think they do: so completely tiny that they are not perceivable in any real sense. They only matter as math. All it means is that there's a lot more space than we can really tell by looking. Simple.

posted by danny the boy at 10:12 AM on January 26, 2012

FatherDagon: "

Whoever told you that was being pretty dishonest?

posted by danny the boy at 10:16 AM on January 26, 2012

*I thought string theory had fallen out of favor a while ago when it became clear that it was a bunch of untestable faddish handwavy hokum? Has it become more respectable in the last few years or something?*"Whoever told you that was being pretty dishonest?

posted by danny the boy at 10:16 AM on January 26, 2012

*I thought string theory had fallen out of favor a while ago when it became clear that it was a bunch of untestable faddish handwavy hokum? Has it become more respectable in the last few years or something?*

My understanding (as a mathematician and [very] lay physics enthusiast) is that there's a disjunction between the popular view of string theory (not so good) and the view of mathematicians and physicists. In the mathematical community there is still a reasonable amount of interest in string theory because it's given rise to some interesting mathematics. (I personally view this with some ambivalence). Physically speaking, there's no theory that

*everybody*is stoked about for reconciling quantum mechanics and relativity, so perhaps for many physicists there's a feeling of "eh, this is the best we've got right now."

posted by Frobenius Twist at 10:18 AM on January 26, 2012 [1 favorite]

Are there any physicists here that might improve on the journalistic platitude that dimensions beyond the 4 familiar ones are "very small". I can't make sense of that. Perhaps a flatland approach might help. What would it mean for, say one of our three spatial dimensions to be very small? Small requires a comparator. Dimensions are not objects.

posted by stonepharisee at 10:22 AM on January 26, 2012 [3 favorites]

posted by stonepharisee at 10:22 AM on January 26, 2012 [3 favorites]

*People see a well-produced sciencey sort of thing that has the same style as stuff they've seen before, and it takes a shortcut to credibility. Is there a term for that? Bunk in legitimate clothing?*

Cargo Cult Science.Cargo Cult Science.

posted by Schmucko at 10:28 AM on January 26, 2012

For a dimension to be very small means that if you move along that direction for a small distance, you get back where you started. For example, the 2 dimensions of the Earth's surface are not infinite. A torus (doughnut shape) is another way to have 2 dimensions that are finite. The ways to have 9 dimensions shrivel down to 3 are too many, leading some to think string theory has lost predictive power.

posted by Schmucko at 10:30 AM on January 26, 2012 [1 favorite]

posted by Schmucko at 10:30 AM on January 26, 2012 [1 favorite]

FatherDagon: "I thought string theory had fallen out of favor a while ago when it became clear that it was a bunch of untestable faddish handwavy hokum? Has it become more respectable in the last few years or something?"

There is a small possibility that results from the LHC could verify string theory. String theory is still very difficult to

posted by jiawen at 10:31 AM on January 26, 2012

There is a small possibility that results from the LHC could verify string theory. String theory is still very difficult to

*disprove*, though.posted by jiawen at 10:31 AM on January 26, 2012

*I thought string theory had fallen out of favor a while ago when it became clear that it was a bunch of untestable faddish handwavy hokum? Has it become more respectable in the last few years or something?*

String theory has dominated and continues to dominate theoretical physics departments because for all its faults it still makes lots of interesting predictions. There was a backlash against it starting a few years ago -- Lee Smolin's The Trouble with Physics summed up the arguments against string theory nicely. But the backlash wasn't so much that string theory is

*wrong*(though obviously it may be) so much as it's not healthy for an unverified theory to have such a pervasive lock on theoretical research. Much of Smolin's book discussed how if you aren't working on string theory it's pretty much impossible to get a job in a top physics department. So don't knock theorists for pursuing string theory, knock them for neglecting the alternatives.

posted by no regrets, coyote at 10:52 AM on January 26, 2012

Here's Peter Woit's take on the Japanese paper- This Week’s Hype

Basically it's the problem of string theory only predicting (or 'postdicting') things that we already know exist- like gravity and the 3 dimensions of space.

posted by bhnyc at 10:53 AM on January 26, 2012

Basically it's the problem of string theory only predicting (or 'postdicting') things that we already know exist- like gravity and the 3 dimensions of space.

posted by bhnyc at 10:53 AM on January 26, 2012

benito.strauss: "

Ask Edward Witten. I have a feeling even he doesn't know.

(Actually - a really great book is "Elegant Universe" by Brian Greene. It discusses all of this stuff with the latest (as of the time it was written) info. I highly recommend. I may have to go back and re-read it.)

posted by symbioid at 10:58 AM on January 26, 2012

*Okay, then why did only three of them experience expansion?*"Ask Edward Witten. I have a feeling even he doesn't know.

(Actually - a really great book is "Elegant Universe" by Brian Greene. It discusses all of this stuff with the latest (as of the time it was written) info. I highly recommend. I may have to go back and re-read it.)

posted by symbioid at 10:58 AM on January 26, 2012

*Could you rephrase that as an interpretive dance cycle?*

Bees? In my Metafilter?

posted by Malice at 11:00 AM on January 26, 2012

Yeah - I think one of the problems when people hear "dimensions" is that they're thinking a vast amount of "space". Perhaps infinite, with no geometry, but, Schmucko's reply is really a good point.

Dimensions can have geometry. We talk about the 3 spatial dimensions of our universe and can ask about the topology (toroidal, spherical, saddle -- non-euclidean geometry is the type of geometry you do on these warped/curved dimensions, if my understanding is correct).

You can do the same for any dimension... I think Brian Green uses an analogy of an ant walking along a telephone wire (I can't recall exactly how he uses the analogy) but it's basically him saying how you can curl up a sheet into a tube, and that alters its properties and the ants relationship to it (and how it can move along that space depending on how it's folded). That picture at the top of the article is a Calabi-Yau manifold which, if my understanding is correct, is one form of folding of space into certain configurations.

posted by symbioid at 11:08 AM on January 26, 2012

Dimensions can have geometry. We talk about the 3 spatial dimensions of our universe and can ask about the topology (toroidal, spherical, saddle -- non-euclidean geometry is the type of geometry you do on these warped/curved dimensions, if my understanding is correct).

You can do the same for any dimension... I think Brian Green uses an analogy of an ant walking along a telephone wire (I can't recall exactly how he uses the analogy) but it's basically him saying how you can curl up a sheet into a tube, and that alters its properties and the ants relationship to it (and how it can move along that space depending on how it's folded). That picture at the top of the article is a Calabi-Yau manifold which, if my understanding is correct, is one form of folding of space into certain configurations.

posted by symbioid at 11:08 AM on January 26, 2012

Speaking as a theoretical physicist who isn't a string theorist, I should clarify no regrets, coyote's post. String theory had a hayday of hiring a decade or so ago, meaning that today a lot of departments are overstocked on string theorists and understocked on other specialties. However, in reaction to that, and also the fact that a lot of the early promise of string theory hasn't panned out, the hiring situation now is very bad for new string theorists, especially since all those string theory professors are pumping out new string theory grad students in excess of real demand.

I think the turning point was the discovery of the string landscape, when it became apparently that there were a lot of possible solutions that string theory provided to the laws of nature, and it would be very hard to pick out the "right one" that our Universe selected. The kind of questions string theorists are trying to answer have changed as a result, and could still be very important, but that took a lot of wind out of people's sails.

(these days a lot of people, not just string theorists, are looking at something called the AdS/CFT correspondence - anti-de-Sitter space/conformal field theory - which relates intractable problems in field theory to tractable ones in in a different theory. This has to the potential to answer some previously impossible questions in strong nuclear forces, as well as in condensed matter physics).

Since I was a bit too young to really live through the "string collapse," I might be simplifying too much, but my view is that a lot of string theorists got hired, possibly dumping too many resources into a field that had a lot of promise but didn't quite live up to all the hype, and right now we're in a corrective phase.

As for whether string theory will be vindicated by the LHC, my opinion is that no matter what is found, it could be claimed as a success. If they find supersymmetry, well that's necessary for string theory. If they don't find supersymmetry, well string theory never said it needed to be at a TeV energy scale.

stonepharisee, I'll try to come back with a good explanation in a bit, I don't want to dash something off quickly and be even more confusing.

posted by physicsmatt at 11:15 AM on January 26, 2012 [4 favorites]

I think the turning point was the discovery of the string landscape, when it became apparently that there were a lot of possible solutions that string theory provided to the laws of nature, and it would be very hard to pick out the "right one" that our Universe selected. The kind of questions string theorists are trying to answer have changed as a result, and could still be very important, but that took a lot of wind out of people's sails.

(these days a lot of people, not just string theorists, are looking at something called the AdS/CFT correspondence - anti-de-Sitter space/conformal field theory - which relates intractable problems in field theory to tractable ones in in a different theory. This has to the potential to answer some previously impossible questions in strong nuclear forces, as well as in condensed matter physics).

Since I was a bit too young to really live through the "string collapse," I might be simplifying too much, but my view is that a lot of string theorists got hired, possibly dumping too many resources into a field that had a lot of promise but didn't quite live up to all the hype, and right now we're in a corrective phase.

As for whether string theory will be vindicated by the LHC, my opinion is that no matter what is found, it could be claimed as a success. If they find supersymmetry, well that's necessary for string theory. If they don't find supersymmetry, well string theory never said it needed to be at a TeV energy scale.

stonepharisee, I'll try to come back with a good explanation in a bit, I don't want to dash something off quickly and be even more confusing.

posted by physicsmatt at 11:15 AM on January 26, 2012 [4 favorites]

I eagerly await physicsmatt's take on this, but in the meantime it might be helpful to think about what life would be like if we had three spatial dimensions... but one of them was really small. Like really really small. Like you wouldn't notice your movement in that dimension, even though it existed.

It'd be a lot like living in flatland, wouldn't it?

posted by danny the boy at 11:23 AM on January 26, 2012

It'd be a lot like living in flatland, wouldn't it?

posted by danny the boy at 11:23 AM on January 26, 2012

That's just the smell of money!

posted by mygoditsbob at 11:47 AM on January 26, 2012

posted by mygoditsbob at 11:47 AM on January 26, 2012

Fuck, the cookies are burning. I forgot to grease the manifold.

posted by smidgen at 11:49 AM on January 26, 2012 [1 favorite]

posted by smidgen at 11:49 AM on January 26, 2012 [1 favorite]

Metafilter: beyond our ability to sense and grasp.

posted by herbplarfegan at 12:51 PM on January 26, 2012 [1 favorite]

posted by herbplarfegan at 12:51 PM on January 26, 2012 [1 favorite]

Thanks for all this. I hope we can get a bit more on this. I come from cognitive science, and indulge in some speculation about consciousness, or as I look at it, the phenomenal. When we say "big", "small", "brief", or "long", a core sense of these must refer to the reference scale of the human being. We inhabit, and only have experience of, human sized worlds. Even microscopes, telescopes, etc serve to make patterns present in a phenomenal sense. So if we (the experiencers) think of ourselves as co-extensive with spatially 3-D bodies, then these comparative adjectives can only refer to something expressable in those three spatial dimensions. The hidden 6 can not be "small" in that sense. You have to give meaning to that tricky notion of a "small distance".

More!

posted by stonepharisee at 2:12 PM on January 26, 2012

More!

posted by stonepharisee at 2:12 PM on January 26, 2012

Physicsmatt, sorry to be dumb, but could you (or someone similarly well-informed) give an layperson's description of what these other six dimensions of space are and how we might have experienced them?

posted by foxy_hedgehog at 2:15 PM on January 26, 2012

posted by foxy_hedgehog at 2:15 PM on January 26, 2012

*> I'm pretty sure that the nature of the other dimensions is beyond our ability to sense and grasp given that fact that we evolved in this 3 dimensional universe. It would be a waste genetic resources to develop the ability to understand and detect dimensions that cannot be manipulated.*

Absolutely. We sense and grasp with the action of 3-D bodies. But to what extent should that rule out whole other dimensions. We could not speak of "things" in those other dimensions. Nothing there would make any phenomenal sense any more, because it is a Newtonian physics that describes this familiar world of experience.

We could never experience structure in those other 6 dimensions. But nothing in quantum physics makes that kind of experiential sense either - things at that scale don't appear even rational to us.

But we could sure as shootin' do some science on them.

posted by stonepharisee at 2:20 PM on January 26, 2012

I wouldn't necessarily argue that one can't grasp 6 dimensions on an intuitive level. Well 6 is a bit high, but I would argue that after consuming, let's say, 5 grams of P. Cubensis mushrooms, you may have an experience that renders intuition capable of experiencing reality from a 4th dimensional framework such that you KNOW what it is to step outside the characteristic bounds of 3 dimensional reality into a "higher" frame of reference. You are not physically capable of literally doing this, but you can feel it from within, and your brain will understand.

That uh - is merely a personal ... experience... which, of course, isn't actual science, but we're asking about experiences of higher dimensionality, and all experience is subjective, so...

But the mathematics of higher dimensions, AFAIK, aren't that terribly odd, and are relatively mundane, no?

posted by symbioid at 2:25 PM on January 26, 2012

That uh - is merely a personal ... experience... which, of course, isn't actual science, but we're asking about experiences of higher dimensionality, and all experience is subjective, so...

But the mathematics of higher dimensions, AFAIK, aren't that terribly odd, and are relatively mundane, no?

posted by symbioid at 2:25 PM on January 26, 2012

It is not clear to me, and some others, that the body is the correct and sole frame of reference with which to understand phenomenal experience. A view that identifies both with the here and now, and with a body, is a 3-D view, and it vastly underdetermines the experience of the world.

posted by stonepharisee at 2:30 PM on January 26, 2012

posted by stonepharisee at 2:30 PM on January 26, 2012

*But the mathematics of higher dimensions, AFAIK, aren't that terribly odd, and are relatively mundane, no?*

Absolutely. It's 3 dimensions that are weird - only in 3 dimensions can you tie a knot, and only in 3 dimensions or lower do you have stable enough orbits in gravity, I believe.

posted by edd at 2:44 PM on January 26, 2012 [3 favorites]

Seconding edd. Three dimensions is a funny place, mathematically speaking. Two dimensions is so small that you can just enumerate everything that can happen, while in four or more dimensions everything and anything can happen. But at n = 3 it gets tricky.

If you asked many mathematicians why it seems like the world is three dimensional they'd say it's because that's where the math is most interesting.

posted by benito.strauss at 3:24 PM on January 26, 2012 [2 favorites]

If you asked many mathematicians why it seems like the world is three dimensional they'd say it's because that's where the math is most interesting.

posted by benito.strauss at 3:24 PM on January 26, 2012 [2 favorites]

OK: dimensions. Picture the old computer game Asteroids. It's a two-dimensional game: left-right and up-down. However, those dimensions are not infinite: going one screen-length left brings you around to the right, and the same for up and down. So, we can identify those edges. What is the shape of this? Well, you can't display it in 2-D (though I could mathematically, easily). But in three-dimensions we can imagine bending the screen around, gluing the top and bottom, and then left and right. The result is a donut: a torus (the screen, in additional to being flexible, will have to be a bit bendy).

Some additional things we would care about in Asteroids-space: the 2D space is flat, which is a statement about non-local properties. A triangle dropped anywhere in that space has angles adding to 180 degrees, for example. THat would not be true for a triangle painted onto a real donut. So while the topology is similar to a torus in 3D, it's not quite the same. So the Asteroids space is two dimensions with particular curvature and particular boundary conditions. The fact that the edge of the space "wraps around" makes it compact.

Now, how BIG are those dimensions, what makes it compact in a real sense? Well, we could define them to a be a unit length. This gets to stonepharisee's question: dimension isn't a unit! Well, it is, actually, or it is in physics. In pure mathematics, I agree you could just call the width of the Asteroids-space equal to 1 by fiat, and then rescale the vertical distance to be 1 as well, and no one would notice. However, this is the real world, or at least the black-and-white, multiplying-asteroids version of the real world. There are THINGS in this world, and those things have a dimension. For example, the width of the little space-ship is a scale, and the fact that the space-ship is somewhat small compared to the screen means that these dimensions aren't TOO compact. Or, we could take the fundamental size to be the pixel on your screen, in which case the 2D space is a few thousand pixels in each direction. Not exactly the biggest universe, but it's a start.

Ah, you say, that's fine for Asteroids, but in OUR Universe, what's the scale? How can you say anything is big or small in our Universe? It's all

The answer that it isn't. There are plenty of fundamental scales against which to measure. They're just not the scales you usually use (or, they are, its just you don't think of them that way typically).

Start off by asking why we think of the 3 non-compact spatial dimensions as "big." We think of them that way because we can fit a lot of things into it, planets, galaxies, New York, and so on. What sets the size of those things? Well, they're set by the size of atoms, which is set by the strength of electromagnetism and the strong nuclear force (though the latter results in a far smaller scale than the EM one). It's also big compared to most of the things gravity likes to build: planets, stars, galaxies, even clusters of galaxies. So we think of these dimensions as non-compact because their fundamental scale is so big (perhaps infinite) compared to the things in it. Back to the Asteroids example, if the screen was as big as the Solar System, that would be hard for the little spaceship to ever notice it was on a bounded surface with odd edge conditions.

So, can we give numbers to these different scales? And why aren't they the same? Well, to start, you need to know that to me as a physicist, there is no functional difference between distance and energy. Energy has the unit of inverse length, or, as I usually use it, length is an inverse energy. So SMALL distances correspond to BIG energies, and vice versa. For example, a high energy photon has very small wavelength, and you can think of the Large Hadron Collider at CERN as a way of measuring very small distances. For comparison, a GeV (1 billion electron volts, or about the rest energy of a proton) is about 5 femtometers inverse. These days, I think better in energy, so I'll give the various scales in terms of energy, and remember, big energies are small scales.

As I said, a proton is 1 GeV, so that's your fundamental scale of the strong interaction. An atom is about 10 eV, so it's about 10^8 times BIGGER. The scale at which gravity would be important (important here is "builds black holes") is the Planck mass, 1.2 10^19 GeV. Planets, stars, mountains, tables and people are all performing a balancing act between these scales, which is why they are the size they are.

These are all possible "pixels" against which to measure the size of a dimension. The non-compact dimensions are all about 10 billion light years across, at minimum, which corresponds to a mass scale of 10^-41 GeV. So, the Universe seems big just because everything we're built out of has energy scales way too large to notice any "edges" to our dimensions (and again, there's no evidence that these scales have edges). So, the reason we can talk about "big" and "small" is that we have many things to compare against, and those things are fundamental scales that everyone in the Universe would agree on. The technical term is that these energy scales of physic break conformal invariance, which is just a fancy way of saying that the Universe has a yardstick that actually matters (or several, really).

So, what constitutes a small dimension? Well, something that has edges or boundaries that are close together compared to all the scales above. The compactified dimensions of string theory are going to have a length of the Planck mass (again, mixing energy and distance, so 1.2 10^19 GeV becomes 1.6 10-35 m). We, built of atoms with electromagnetic interactions, have much lower energy scales, so our characteristic length (really I'm talking about the quantum wavelength of our atoms) is just so large compared to the dimension that we just don't notice. Back to the asteroids example, if you made the spaceship 10^36 m long, and painted it on the 2D toroidal universe only a third of a meter across, it wouldn't have much room to maneuver and the game would be very boring.

In our Universe, you would only notice the dimension if you could dump enough energy (10^19 GeV worth) into a small area. In that case, the length-scale of particles excited by that energy would be smaller than the dimensional scale, and you could see a difference in how they moved, due to the fact that they have "more options" in which to travel. However, this is hard to do, as a) we can only get to 10000 GeV in our best colliders with no way to scale up 10^15 orders of magnitude, and b) if we did, we'd create a black hole first.

(One last technical note, the conversion between energy and mass I used above comes from the Planck constant. What it really is doing is measuring the pixel size in the SIX-dimension phase-space of position (x,y,z) and velocity (speed in x, y, and z). The fact that it's such a huge scale compared to the energies we live at is why you don't notice that you're living in a pixelated Universe.)

OK, I probably didn't answer the question fully, and there was a lot of other conversations starting up in this thread that I could comment on, but I've gotta run and so I'll leave it at this for now. Hopefully I'll stop back in when I have a minute later tonight, as I'd love to talk about the WHY of some of these scales (though not the compact dimensions, no idea there), but I've already written an essay and I'd like to leave work now.

posted by physicsmatt at 3:40 PM on January 26, 2012 [23 favorites]

Some additional things we would care about in Asteroids-space: the 2D space is flat, which is a statement about non-local properties. A triangle dropped anywhere in that space has angles adding to 180 degrees, for example. THat would not be true for a triangle painted onto a real donut. So while the topology is similar to a torus in 3D, it's not quite the same. So the Asteroids space is two dimensions with particular curvature and particular boundary conditions. The fact that the edge of the space "wraps around" makes it compact.

Now, how BIG are those dimensions, what makes it compact in a real sense? Well, we could define them to a be a unit length. This gets to stonepharisee's question: dimension isn't a unit! Well, it is, actually, or it is in physics. In pure mathematics, I agree you could just call the width of the Asteroids-space equal to 1 by fiat, and then rescale the vertical distance to be 1 as well, and no one would notice. However, this is the real world, or at least the black-and-white, multiplying-asteroids version of the real world. There are THINGS in this world, and those things have a dimension. For example, the width of the little space-ship is a scale, and the fact that the space-ship is somewhat small compared to the screen means that these dimensions aren't TOO compact. Or, we could take the fundamental size to be the pixel on your screen, in which case the 2D space is a few thousand pixels in each direction. Not exactly the biggest universe, but it's a start.

Ah, you say, that's fine for Asteroids, but in OUR Universe, what's the scale? How can you say anything is big or small in our Universe? It's all

*relative*, man.The answer that it isn't. There are plenty of fundamental scales against which to measure. They're just not the scales you usually use (or, they are, its just you don't think of them that way typically).

Start off by asking why we think of the 3 non-compact spatial dimensions as "big." We think of them that way because we can fit a lot of things into it, planets, galaxies, New York, and so on. What sets the size of those things? Well, they're set by the size of atoms, which is set by the strength of electromagnetism and the strong nuclear force (though the latter results in a far smaller scale than the EM one). It's also big compared to most of the things gravity likes to build: planets, stars, galaxies, even clusters of galaxies. So we think of these dimensions as non-compact because their fundamental scale is so big (perhaps infinite) compared to the things in it. Back to the Asteroids example, if the screen was as big as the Solar System, that would be hard for the little spaceship to ever notice it was on a bounded surface with odd edge conditions.

So, can we give numbers to these different scales? And why aren't they the same? Well, to start, you need to know that to me as a physicist, there is no functional difference between distance and energy. Energy has the unit of inverse length, or, as I usually use it, length is an inverse energy. So SMALL distances correspond to BIG energies, and vice versa. For example, a high energy photon has very small wavelength, and you can think of the Large Hadron Collider at CERN as a way of measuring very small distances. For comparison, a GeV (1 billion electron volts, or about the rest energy of a proton) is about 5 femtometers inverse. These days, I think better in energy, so I'll give the various scales in terms of energy, and remember, big energies are small scales.

As I said, a proton is 1 GeV, so that's your fundamental scale of the strong interaction. An atom is about 10 eV, so it's about 10^8 times BIGGER. The scale at which gravity would be important (important here is "builds black holes") is the Planck mass, 1.2 10^19 GeV. Planets, stars, mountains, tables and people are all performing a balancing act between these scales, which is why they are the size they are.

These are all possible "pixels" against which to measure the size of a dimension. The non-compact dimensions are all about 10 billion light years across, at minimum, which corresponds to a mass scale of 10^-41 GeV. So, the Universe seems big just because everything we're built out of has energy scales way too large to notice any "edges" to our dimensions (and again, there's no evidence that these scales have edges). So, the reason we can talk about "big" and "small" is that we have many things to compare against, and those things are fundamental scales that everyone in the Universe would agree on. The technical term is that these energy scales of physic break conformal invariance, which is just a fancy way of saying that the Universe has a yardstick that actually matters (or several, really).

So, what constitutes a small dimension? Well, something that has edges or boundaries that are close together compared to all the scales above. The compactified dimensions of string theory are going to have a length of the Planck mass (again, mixing energy and distance, so 1.2 10^19 GeV becomes 1.6 10-35 m). We, built of atoms with electromagnetic interactions, have much lower energy scales, so our characteristic length (really I'm talking about the quantum wavelength of our atoms) is just so large compared to the dimension that we just don't notice. Back to the asteroids example, if you made the spaceship 10^36 m long, and painted it on the 2D toroidal universe only a third of a meter across, it wouldn't have much room to maneuver and the game would be very boring.

In our Universe, you would only notice the dimension if you could dump enough energy (10^19 GeV worth) into a small area. In that case, the length-scale of particles excited by that energy would be smaller than the dimensional scale, and you could see a difference in how they moved, due to the fact that they have "more options" in which to travel. However, this is hard to do, as a) we can only get to 10000 GeV in our best colliders with no way to scale up 10^15 orders of magnitude, and b) if we did, we'd create a black hole first.

(One last technical note, the conversion between energy and mass I used above comes from the Planck constant. What it really is doing is measuring the pixel size in the SIX-dimension phase-space of position (x,y,z) and velocity (speed in x, y, and z). The fact that it's such a huge scale compared to the energies we live at is why you don't notice that you're living in a pixelated Universe.)

OK, I probably didn't answer the question fully, and there was a lot of other conversations starting up in this thread that I could comment on, but I've gotta run and so I'll leave it at this for now. Hopefully I'll stop back in when I have a minute later tonight, as I'd love to talk about the WHY of some of these scales (though not the compact dimensions, no idea there), but I've already written an essay and I'd like to leave work now.

posted by physicsmatt at 3:40 PM on January 26, 2012 [23 favorites]

I tried to wrap my brain around the idea of 10 dimensions, but only got to six before my head exploded.

It expanded 4 dimensionally, however, thanks to Frobenius Twist's exploitation of my intuitive grasp of the speed with which it did so.

posted by BlueHorse at 5:20 PM on January 26, 2012

It expanded 4 dimensionally, however, thanks to Frobenius Twist's exploitation of my intuitive grasp of the speed with which it did so.

posted by BlueHorse at 5:20 PM on January 26, 2012

So I haven't seen the PRL that the first link refers to. This means I can't really comment on the particular simulation these particular physicists are working on.

But I can make a few comments, being as I work in weird numbers of dimensions a bunch-- being a string theorist and all.

1) Getting a job as a string theorist right now kinda sucks, yep. But honestly getting a job as *any* kind of academic is pretty challenging these days; we stringy folk don't have it significantly harder than anyone else.

2) "String theory" as a coherent thing that we study, as one theory that holds together and explains everything and that we all study, doesn't really exist. When I was an undergrad, I read Brian Greene's Elegant Universe and I caught the feeling in high energy physics departments that string theorists were on the verge of finding a "theory of everything".

That isn't really what string theorists are doing, these days. (Well, some of them are, but as a whole). People who call themselves string theorists study everything from mathematical abstractions like mock modular forms, all the way through to the AdS/CFT duality physicsmatt mentioned above and its applications to, e.g., superconductors, or quark gluon plasmas produced in gold ion collisions. String theory is much more a set of tools, a method of inquiry, maybe a perspective on thinking about physical problems, than it is a coherent "theory of everything".

3) I'm ok with (2). Well, I am now; I was perhaps a little crushed at some point, but honestly I don't think there's anyone involved any line of serious academic inquiry that doesn't find the nature of their work somewhat different than they originally expected.

4) Flatland is a fabulous book that everyone should read. Especially everyone who's interested in trying to visualize extra dimensions. Thinking about it in the context of the time it was written is also fun.

5) Working in too many dimensions still messes with me. It's always a relief when I get to do a project that's only in 4 (or better yet, 2!) I can draw pictures of 2 dimensions!

posted by nat at 8:10 PM on January 26, 2012

But I can make a few comments, being as I work in weird numbers of dimensions a bunch-- being a string theorist and all.

1) Getting a job as a string theorist right now kinda sucks, yep. But honestly getting a job as *any* kind of academic is pretty challenging these days; we stringy folk don't have it significantly harder than anyone else.

2) "String theory" as a coherent thing that we study, as one theory that holds together and explains everything and that we all study, doesn't really exist. When I was an undergrad, I read Brian Greene's Elegant Universe and I caught the feeling in high energy physics departments that string theorists were on the verge of finding a "theory of everything".

That isn't really what string theorists are doing, these days. (Well, some of them are, but as a whole). People who call themselves string theorists study everything from mathematical abstractions like mock modular forms, all the way through to the AdS/CFT duality physicsmatt mentioned above and its applications to, e.g., superconductors, or quark gluon plasmas produced in gold ion collisions. String theory is much more a set of tools, a method of inquiry, maybe a perspective on thinking about physical problems, than it is a coherent "theory of everything".

3) I'm ok with (2). Well, I am now; I was perhaps a little crushed at some point, but honestly I don't think there's anyone involved any line of serious academic inquiry that doesn't find the nature of their work somewhat different than they originally expected.

4) Flatland is a fabulous book that everyone should read. Especially everyone who's interested in trying to visualize extra dimensions. Thinking about it in the context of the time it was written is also fun.

5) Working in too many dimensions still messes with me. It's always a relief when I get to do a project that's only in 4 (or better yet, 2!) I can draw pictures of 2 dimensions!

posted by nat at 8:10 PM on January 26, 2012

Oh hey nat!

foxy_hedgehog, so I'm not sure if my previous comment helped or just hurt so very much, but the idea is that we can have an arbitrary number of extra spatial dimensions along side the 3 non-compact ones we are familiar with (and 1 in time). What this means is that, if you want to describe your location and momentum in the Universe, you need not just 8 numbers (time, energy, x coordinate, x momentum, y coordinate and momentum, z coordinate and momentum), but and additional 2n numbers (x_4, p_4, x_5, p_5... where x_n is the nth space location and p_n is the nth momentum component). Practically though, we know these dimensions are all tiny. That is, their characteristic length is incredibly small compared to the distances over which our constituent particles spread out. Using the energy-length relation, that means we are at too low of an energy to prove these other directions. So every particle in the Universe has the same x_4, x_5... p_4, p_5... coordinates. So these aren't active degrees of freedom for us (degree of freedom is just the technical term for "how many numbers do I need to specify to describe something). The number of degrees of freedom has measurable consequences, from basic things like "how does the strength of gravity and electromagnetism act at a distance?" to more subtle effects, like the thermal history of the early Universe would be different if there were more, so this isn't just a question of our perception, but a real statement about how objects from electrons and quarks on up work.

Basically, these other dimensions that string theory postulates haven't affected the Universe we live in directly since the average temperature was high enough to be approximately the Planck scale. For comparison, the background temperature today is 2.7 K, or 2 10^-4 eV, 32 orders of magnitude lower than the Planck scale. The Universe hasn't been Planck-hot since basically instant 0 after the Big Bang; the only things we have any real direct knowledge about happened when the Universe was 1 GeV; 18 orders of magnitude later, so this is completely speculative stuff. When things were that hot though, the wavefunctions of particles were smaller than the scale of the extra dimension, and so they could move freely in those directions, whether that caused anything interesting is another question.

So why does string theory want extra dimensions? Basically because string theory doesn't work without them. The theory isn't well defined mathematically without 10 dimensions (though nat would have to say more on why, I'm not sure if there's a good layman's answer), and since we obviously only have 3+1, we need to hide six by making them small. This paper claims to have figured out why, but I'm not very convinced. There are so many string vacua, I just am suspicious that all of them find it energetically favorable to have exactly 3 big space dimensions.

As people pointed out above, there are anthropic arguments to make about needing 3 big space dimensions (1 and 2 doesn't allow interesting physics, but 4+ might not allow necessary things like knots, which probably necessary for things like protein folding), but don't I really believe these arguments at all. Mostly because I hate anthropics.

Also, in addition to the stringy Planck-scale dimensions, you could imagine dimensions small, but not too small. If they were say a TeV in energy (a few tens of femtometers across), then we could conceivably probe them at the LHC, and then there could have played a critical role in the history of the Universe, in breaking electroweak symmetry.

I still can't picture more than 3 space dimensions, and to be honest I've never met anyone who's convinced me they are really are, and not just using very clever mathematical tricks and symmetries to get a handle on this subject (which is not to say that sort of thing isn't very impressive).

posted by physicsmatt at 8:40 PM on January 26, 2012 [5 favorites]

foxy_hedgehog, so I'm not sure if my previous comment helped or just hurt so very much, but the idea is that we can have an arbitrary number of extra spatial dimensions along side the 3 non-compact ones we are familiar with (and 1 in time). What this means is that, if you want to describe your location and momentum in the Universe, you need not just 8 numbers (time, energy, x coordinate, x momentum, y coordinate and momentum, z coordinate and momentum), but and additional 2n numbers (x_4, p_4, x_5, p_5... where x_n is the nth space location and p_n is the nth momentum component). Practically though, we know these dimensions are all tiny. That is, their characteristic length is incredibly small compared to the distances over which our constituent particles spread out. Using the energy-length relation, that means we are at too low of an energy to prove these other directions. So every particle in the Universe has the same x_4, x_5... p_4, p_5... coordinates. So these aren't active degrees of freedom for us (degree of freedom is just the technical term for "how many numbers do I need to specify to describe something). The number of degrees of freedom has measurable consequences, from basic things like "how does the strength of gravity and electromagnetism act at a distance?" to more subtle effects, like the thermal history of the early Universe would be different if there were more, so this isn't just a question of our perception, but a real statement about how objects from electrons and quarks on up work.

Basically, these other dimensions that string theory postulates haven't affected the Universe we live in directly since the average temperature was high enough to be approximately the Planck scale. For comparison, the background temperature today is 2.7 K, or 2 10^-4 eV, 32 orders of magnitude lower than the Planck scale. The Universe hasn't been Planck-hot since basically instant 0 after the Big Bang; the only things we have any real direct knowledge about happened when the Universe was 1 GeV; 18 orders of magnitude later, so this is completely speculative stuff. When things were that hot though, the wavefunctions of particles were smaller than the scale of the extra dimension, and so they could move freely in those directions, whether that caused anything interesting is another question.

So why does string theory want extra dimensions? Basically because string theory doesn't work without them. The theory isn't well defined mathematically without 10 dimensions (though nat would have to say more on why, I'm not sure if there's a good layman's answer), and since we obviously only have 3+1, we need to hide six by making them small. This paper claims to have figured out why, but I'm not very convinced. There are so many string vacua, I just am suspicious that all of them find it energetically favorable to have exactly 3 big space dimensions.

As people pointed out above, there are anthropic arguments to make about needing 3 big space dimensions (1 and 2 doesn't allow interesting physics, but 4+ might not allow necessary things like knots, which probably necessary for things like protein folding), but don't I really believe these arguments at all. Mostly because I hate anthropics.

Also, in addition to the stringy Planck-scale dimensions, you could imagine dimensions small, but not too small. If they were say a TeV in energy (a few tens of femtometers across), then we could conceivably probe them at the LHC, and then there could have played a critical role in the history of the Universe, in breaking electroweak symmetry.

I still can't picture more than 3 space dimensions, and to be honest I've never met anyone who's convinced me they are really are, and not just using very clever mathematical tricks and symmetries to get a handle on this subject (which is not to say that sort of thing isn't very impressive).

posted by physicsmatt at 8:40 PM on January 26, 2012 [5 favorites]

Thanks Physicsmatt. That goes much further than the standard "they're small" description. The janus-like relation of distance and energy is food to chew on.

posted by stonepharisee at 11:50 PM on January 26, 2012

posted by stonepharisee at 11:50 PM on January 26, 2012

*degree of freedom is just the technical term for "how many numbers do I need to specify to describe something"*

This is one part I don't understand. Wouldn't the question more accurately be not "how many numbers" but how many bits - i.e. if you said I need the numbers 3, 4 and 5 I could just use some encoding/decoding function to create a single number and then get 3, 4 and 5 back out

It seems to me actually like various recent physics ideas/work are actually pointing to the idea that just about everything can be understood as emergent from lower-order rules (e.g. "it from bit", the holographic principle, AdS/CFT unless I'm misunderstanding)

posted by crayz at 12:18 AM on January 27, 2012

Thanks for the really enlightening comments Matt. One very numpty question: how can more than 3 dimensions not allow knotting.

I can understand that there are things you can do and "directions" you can move in an n-dimensional universe that you can't travel along in an n-1 dimensional universe (so, you can travel along the z axis a 3d universe, but not a 2d universe) but don't see why you can't tie a knot in a 4d universe. Surely a 4d universe allows you a whole extra plane in which to knot?

(I ask as a literature and drama graduate who weeps at the mere smell of a simultaneous equation)

posted by Pericles at 12:25 AM on January 27, 2012

I can understand that there are things you can do and "directions" you can move in an n-dimensional universe that you can't travel along in an n-1 dimensional universe (so, you can travel along the z axis a 3d universe, but not a 2d universe) but don't see why you can't tie a knot in a 4d universe. Surely a 4d universe allows you a whole extra plane in which to knot?

(I ask as a literature and drama graduate who weeps at the mere smell of a simultaneous equation)

posted by Pericles at 12:25 AM on January 27, 2012

A knot boils down to one line bumping in to another line (or itself). If you add an extra dimension then at the point the lines meet you can just nudge one across in the extra dimension and let it slip by.

Naturally shoelaces and proteins are a lot bigger than any compactified dimensions in reality so this isn't an option in our universe (if those extra dimensions are real) but if the line has space to move past in the extra dimension then it's easy to unravel any tangle of them.

posted by edd at 1:40 AM on January 27, 2012

Naturally shoelaces and proteins are a lot bigger than any compactified dimensions in reality so this isn't an option in our universe (if those extra dimensions are real) but if the line has space to move past in the extra dimension then it's easy to unravel any tangle of them.

posted by edd at 1:40 AM on January 27, 2012

I suppose in four dimensions you could knot a sheet rather than a line - but there's still an argument that it's most interesting with lines as they're the simplest 'object' with any extension at all.

posted by edd at 1:53 AM on January 27, 2012

posted by edd at 1:53 AM on January 27, 2012

**physicsmatt**: Write a book. I'll buy it.

posted by metaBugs at 3:26 AM on January 27, 2012 [2 favorites]

metaBugs, thanks, that was very kind. Right now I'm finding that it's a lot easier for me to come up with interesting physics explanations in response to specific questions than if I just sit around and free associate. Which is why I always enjoy metafilter physics FPPs. Also, I need to get a faculty job and tenure at some point, so a book might take a while to get to. Though I guess we could get a collaborative effort going: someone asks me a question, then stands back while I head off into the sunset, babbling incoherently about extra dimensions.

Pericles, one thing to take away from edd's answer on knots is that math in more dimensions is, while not easy by any normal sense of the word, not impossibly hard. We may not be able to visualize 4+ space dimensions, but we can figure out some complicated things in it if we just trust the math.

crayz, it's not that I could encode all the information in 2n bits. A counterexample would be that it will take way more than three bits of information to encode your position in space relative to the entire Universe.

(I had something really long here about the holographic principle about half written, but it ended up drifting WAY off topic from what this thread is ostensibly about, and I started to feel like Grandpa Simpson going on about onions on his belt. and I'm too young to be Grandpa Simpson.)

Let me end with a plug for Lisa Randall's book "Warped Passages." Lisa Randall made a major contribution to physics working on "large" extra dimensions, where large here is TeV scale, rather than Planck scale (so large in distance, small in energy). These extra dimensions are the type of thing that can looked for at the LHC, and Warped Passages is a good popular science book on the topic, and one of the better ones in general. She also has a more recent book out, "Knocking on Heaven's Door," but I haven't had a chance to read it yet.

posted by physicsmatt at 6:39 AM on January 27, 2012

Pericles, one thing to take away from edd's answer on knots is that math in more dimensions is, while not easy by any normal sense of the word, not impossibly hard. We may not be able to visualize 4+ space dimensions, but we can figure out some complicated things in it if we just trust the math.

crayz, it's not that I could encode all the information in 2n bits. A counterexample would be that it will take way more than three bits of information to encode your position in space relative to the entire Universe.

(I had something really long here about the holographic principle about half written, but it ended up drifting WAY off topic from what this thread is ostensibly about, and I started to feel like Grandpa Simpson going on about onions on his belt. and I'm too young to be Grandpa Simpson.)

Let me end with a plug for Lisa Randall's book "Warped Passages." Lisa Randall made a major contribution to physics working on "large" extra dimensions, where large here is TeV scale, rather than Planck scale (so large in distance, small in energy). These extra dimensions are the type of thing that can looked for at the LHC, and Warped Passages is a good popular science book on the topic, and one of the better ones in general. She also has a more recent book out, "Knocking on Heaven's Door," but I haven't had a chance to read it yet.

posted by physicsmatt at 6:39 AM on January 27, 2012

Physicsmatt, i loved your explanation above. Like many I have been trying to wrap my head around these complicated topics for years, and having easy(er) to grasp analogies and metaphors really helps.

In this case I was also aided by a link by another Mefite several weeks ago in another thread. Bread-eater linked to a lecture by Nima Arkani-Hamed which goes into depth on the correspondence between energy and scale (or distance).

I would recommend it to all.

posted by OHenryPacey at 9:42 AM on January 27, 2012

In this case I was also aided by a link by another Mefite several weeks ago in another thread. Bread-eater linked to a lecture by Nima Arkani-Hamed which goes into depth on the correspondence between energy and scale (or distance).

I would recommend it to all.

posted by OHenryPacey at 9:42 AM on January 27, 2012

how can more than 3 dimensions not allow knotting?

pericles, as others have said above, you should read Flatland. It's a romance, and a critique of the Victorian class system as well. And I'm going to shamelessly rip it off to talk about knots.

Imaging points living on a line - or ants who are restricted to living at the bottom of a steep groove, so steep that they can't climb the walls to get past each other. Now suppose we look at four ants in the groove, who are colored Red, Blue, Red, and then Blue (RBRB). No matter how they move back and forth, they can't re-order themselves to Red, Red, Blue, then Blue (RRBB). The Red and the Blue points (0-dimensional objects) are knotted together along the line (1-dimensional space). But if you pick them up and put them in a 2-D space, they can easily re-order themselves to RRBB.

Now let's move to a table top, 2-D space. Our things-that-will-be-knotted are a rubber band (1-D object) and an M&M (0-D object). Put the rubber band down as a nice cirlce, and put the M&M in the middle of the circle. No matter how you push them around (staying on the table), you can't turn them into the configuration where the M&M is outside the rubber band. So there are two distinct configurations, one knotted and the other un-knotted. And you can't turn one into the other without breaking one or pushing one through the other.

Notice, though, that if instead of staying on the table (2-D) you let them move around in 3-D. Then you can easily turn one configuration into the other by just picking up the M&M and putting it outside the rubber band. You can't say a 0-D object and a 1-D object are 'knotted' in 3-D space — any configuration of the two objects can be turned into any other configuration by pushing them around.

Now we get to two 1-D objects in 3-D space. Make an "okay" sign with your right hand; your thumb and index finger make a circle, the first 1-D object. Make a similar circle with your left thumb and forefinger; that's the other 1-D object. If you link the two circles, you can't unlink them without (temporarily) breaking one of them. So in 3-dimensional space there are two configurations, one knotted and one un-knotted. But in more than 3-dimensions there's no difference between these configurations — If you let your hands move around in 4-dimensional space you can change one configuration into the other without breaking either of the circles.

That last sentence makes no sense to us 3-D creatures. The sweet thing about

[To talk about knotting in higher dimensions, I'm pretty sure it's generally true that to talk about knotted objects you need

[After writing all this up I realized that I've been talking about two different objects being tangled up with each other, which is more properly called 'linking', whereas 'knotting' is one object being tangled up with itself. And everywhere I said 'knotted' I should have said 'linked'. Rats. But I'll post it anyways because the dimensional stuff is still valid, and it might actually be easier to understand links than knots.]

posted by benito.strauss at 12:45 PM on January 27, 2012 [1 favorite]

pericles, as others have said above, you should read Flatland. It's a romance, and a critique of the Victorian class system as well. And I'm going to shamelessly rip it off to talk about knots.

Imaging points living on a line - or ants who are restricted to living at the bottom of a steep groove, so steep that they can't climb the walls to get past each other. Now suppose we look at four ants in the groove, who are colored Red, Blue, Red, and then Blue (RBRB). No matter how they move back and forth, they can't re-order themselves to Red, Red, Blue, then Blue (RRBB). The Red and the Blue points (0-dimensional objects) are knotted together along the line (1-dimensional space). But if you pick them up and put them in a 2-D space, they can easily re-order themselves to RRBB.

Now let's move to a table top, 2-D space. Our things-that-will-be-knotted are a rubber band (1-D object) and an M&M (0-D object). Put the rubber band down as a nice cirlce, and put the M&M in the middle of the circle. No matter how you push them around (staying on the table), you can't turn them into the configuration where the M&M is outside the rubber band. So there are two distinct configurations, one knotted and the other un-knotted. And you can't turn one into the other without breaking one or pushing one through the other.

Notice, though, that if instead of staying on the table (2-D) you let them move around in 3-D. Then you can easily turn one configuration into the other by just picking up the M&M and putting it outside the rubber band. You can't say a 0-D object and a 1-D object are 'knotted' in 3-D space — any configuration of the two objects can be turned into any other configuration by pushing them around.

Now we get to two 1-D objects in 3-D space. Make an "okay" sign with your right hand; your thumb and index finger make a circle, the first 1-D object. Make a similar circle with your left thumb and forefinger; that's the other 1-D object. If you link the two circles, you can't unlink them without (temporarily) breaking one of them. So in 3-dimensional space there are two configurations, one knotted and one un-knotted. But in more than 3-dimensions there's no difference between these configurations — If you let your hands move around in 4-dimensional space you can change one configuration into the other without breaking either of the circles.

That last sentence makes no sense to us 3-D creatures. The sweet thing about

*Flatland*is how Abbot shows that "moving the M&M out of the rubber band" seems equally nonsensical to a 2-D creature.[To talk about knotting in higher dimensions, I'm pretty sure it's generally true that to talk about knotted objects you need

dim of Objso in 4-D space you could have a spherical shell (2-D) be knotted with a circle (1-D).]_{1}+ dim of Obj_{2}= dim of Space - 1

[After writing all this up I realized that I've been talking about two different objects being tangled up with each other, which is more properly called 'linking', whereas 'knotting' is one object being tangled up with itself. And everywhere I said 'knotted' I should have said 'linked'. Rats. But I'll post it anyways because the dimensional stuff is still valid, and it might actually be easier to understand links than knots.]

posted by benito.strauss at 12:45 PM on January 27, 2012 [1 favorite]

Benito Strauss

thanks - makes sense. But I could presumably have 4D knots that can't be broken (except in 5D space), so 4D protein chains etc are possible?

(I don't see that you can only have knotting in 1,2 or 3D space)

posted by Pericles at 11:28 AM on January 29, 2012

thanks - makes sense. But I could presumably have 4D knots that can't be broken (except in 5D space), so 4D protein chains etc are possible?

(I don't see that you can only have knotting in 1,2 or 3D space)

posted by Pericles at 11:28 AM on January 29, 2012

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posted by resurrexit at 8:13 AM on January 26, 2012 [6 favorites]