Even Dimensional Waves

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Even Dimensional Waves

Postby anderscolingustafson » Fri Mar 21, 2014 9:55 pm

I read in an article about the fourth dimension that waves in an even number of dimensions double back on themselves.

http://www.askamathematician.com/2012/0 ... imensions/

I was wondering why do waves double back on themselves in an even number of dimensions but not in an odd number of dimensions?
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Re: Even Dimensional Waves

Postby quickfur » Sat Mar 22, 2014 1:42 am

This is the first time I've heard of this effect. I searched online but could find no other reference to it. Are you sure this information is reliable?

EDIT: Actually nevermind, I just found an explanation that talks about this. Apparently it's a very real effect of waves in even dimensions indeed! :o Wow. This was unexpected. :P It will definitely have profound consequences for 4D physics!
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Re: Even Dimensional Waves

Postby quickfur » Sat Mar 22, 2014 2:05 am

Here's another explanation of this effect. Apparently it was known as early as 1954(!).

And another paper that points out something even more striking: 3D is the only dimension in which a spherical wave propagates past an observer undistorted. In even dimensions, the waves will double back and "echo" repeatedly, but in odd dimensions, the wave front will propagate only outwards and pass the observer exactly once. However, in all odd dimensions higher than 3D (i.e., 5D, 7D, ...), what the observer sees of the wave will not be the same as it was at its source; 3D is the only space where the original signal is transmitted in an undistorted way. Edit: more precisely, the solution to the wave function in n dimensions is shown to be the derivative of order (n-3)/2 of the original signal. Precisely when n=3, the wave function transmits the original signal as-is; when n=5,7,9,..., the receiver doesn't get the original signal, but a derivative of it. (Although, now that I think of it, if the original signal is a harmonic function like sin or cos, then the derivative is merely a phase shift, so the signal isn't fundamentally distorted; the original signal can still be recovered.) When n is even, however, it's a so-called fractional derivative, which is a non-local function corresponding with a signal that repeatedly echoes after the original wavefront passes the observer.
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Re: Even Dimensional Waves

Postby ICN5D » Sat Mar 22, 2014 2:43 am

Wow, that's really weird. Never heard of it either. That also determines how high-D beings can hear, or the fact that it won't be as reliable. Or, maybe needs multiple ears for 4-D triangulation. It seems like 3D is the only " normal " dimension, where strange things aren't allowed to happen. Be it self-induced echos or distorted perceptions, 3D is stable. This also makes me think about the concepts with string theory. It's all about undulating waves in high-D space. Not to forget the whole extended/compactified dimension idea. We are huge, so we only perceive the extended, opened up directions of freedom. And right under our noses, the realm of the tiny works in the way of waves sloshing around in extra dimensional spaces. This is what makes planets and solar systems work in a simpler way than atoms and molecules. I forget what the "official " number of D's is, but it might be an even number. Like 26 or something. Revolving around Ramanujan functions, which I have no knowledge about. Sounds amazing, though.
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Re: Even Dimensional Waves

Postby quickfur » Sat Mar 22, 2014 3:44 am

ICN5D wrote:Wow, that's really weird. Never heard of it either. That also determines how high-D beings can hear, or the fact that it won't be as reliable. Or, maybe needs multiple ears for 4-D triangulation. It seems like 3D is the only " normal " dimension, where strange things aren't allowed to happen. Be it self-induced echos or distorted perceptions, 3D is stable. This also makes me think about the concepts with string theory. It's all about undulating waves in high-D space. Not to forget the whole extended/compactified dimension idea. We are huge, so we only perceive the extended, opened up directions of freedom. And right under our noses, the realm of the tiny works in the way of waves sloshing around in extra dimensional spaces. This is what makes planets and solar systems work in a simpler way than atoms and molecules. I forget what the "official " number of D's is, but it might be an even number. Like 26 or something. Revolving around Ramanujan functions, which I have no knowledge about. Sounds amazing, though.

The same references also describe how in 1D, the signal will never attenuate, so it becomes a persistent wave that fills all space. So noise would be a big problem in 1D, since it never goes away! :P
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Re: Even Dimensional Waves

Postby anderscolingustafson » Sat Mar 22, 2014 4:44 am

I wonder how a 4d life form would sense the world around it considering how waves work differently in an even number of dimensions and in dimensions above 3d.
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Re: Even Dimensional Waves

Postby quickfur » Sat Mar 22, 2014 4:56 am

Some further thoughts about this. One of the sources I linked above explain this strange effect approximately as the result of back-propagation from the expanding wavefront.

Huygen's principle is one of the early theories that explained the cause of light diffraction: it proposed that the way light propagates can be predicted by assuming every point of its wavefront is a point source of a spherical light wave, and that the mutual interference of these secondary waves produce the new wavefront. However, one lingering problem always remained: why does light propagate only forwards, since in theory the wavelets should also propagate backwards and produce a back-propagating wavefront (or waveback :P)?

Apparently, the answer comes from the way the differential wave equations work out in n dimensions: when n is odd, these back propagations (crudely speaking) destructively interfere, so that only forward propagation is observed. This is indicated by the wave functions have a pole in the complex plane at the origin of the signal. When n is even, however, there is no pole, but a branch point, meaning that, crudely speaking, the back propagations do not destructively interfere, so the wave will echo back and forth with diminishing amplitude. This can be observed from the way the ripples in a pond propagate when you drop a stone into it: the initial circular wave moves outward, but there's a repeated back propagation that causes "echoes" that converge back to the original point of impact and the propagate outwards again, so instead of a single circular ripple, you have a complex series of concentric ripples that spread out from the source with gradually diminishing amplitude (rather than a sharp signal that you get with, say, a 3D sound wave like a gunshot, or a flash of light).

This has far reaching consequences for 4D: assuming that 4D light still takes place via some kind of electromagnetism analogue that produces 4D EM waves, this means that light in 4D does not travel in a single direction, but will have back propagations that causes "echoes"! A 4D flash of light will not appear as a single flash, but as multiple flashes with decreasing amplitude, fading out gradually. This also means that light reflected off objects would behave the same way, thus objects will not have a sharp, focused image, but a blurry "echo-ey" image caused by the repeated back-propagations!

The same thing will happen with 4D sound waves: the spreading wavefront will back-propagate, causing "echoes" that muffle the sound. One nice illustration of this effect that occurred to me, is the kind of sound different musical instruments make. String instruments, for example, have what's effectively a 1D wave on the string, which produces a clear, crisp note. Drums, however, have 2D waves on their surfaces, and guess what kind of sound they produce? That's right, a muffled "thud" sound with uncertain pitch. There are drums that can produce specific pitches, of course (e.g. timpani), but they still have that "dull" quality of a 2D wave rather than the crisp tone of a 1D string. Of course, we only know this because it just so happens that in 3D, the waves on the string or drum can be transmitted in an undistorted, non-echoing form to our ears. :) In 4D, a string instrument would produce a crisp tone in the string itself, but as soon as that sound propagates through the air, it will start to echo and become muffled. So a 4Der may never learn the difference between a violin and a drum, because both sounds become muffled as soon as they travel through the air. :\

OTOH, if you drop a stone into a pond in 4D, you'll see a single, perfect spherical ripple travel outwards, because the surface of the pond will be 3D. :P There will be no concentric ripples, and only a single echoing ripple when it hits the edge of the pond. This has the very interesting consequence that ponds and lakes in 4D are likely to be very calm, with only the occasional isolated ripple passing through, quite unlike our 3D ponds where the surface of the water is almost always filled with little ripples. The same thing may happen in 4D oceans, I believe. The water will be almost always calm, unless there's a persistent source of disturbance like an underwater volcano or something. So 4D beaches are probably very unlikely to have the persistent rolling waves that 3D beaches do, but instead will be as calm as a lake most of the time! The sound of lapping water and the sound of rolling waves will be mostly absent in 4D... except that a single ripple, making a single sound when it hits the edge of the water, produces a sound wave that echoes through the air, so it will sound like lapping water when it's actually just a single ripple! :P :lol: And a single ocean wave striking the shore will sound like echoing thunder and persist for some time after the actual impact.

I knew before that 4D was pretty weird compared to what we've come to expect in 3D, but this one really takes the cake. :roll: Every little sound will echo like shattering glass or a trickling stream, and every loud sound will reverberate like booming thunder. Light travels bidirectionally, causing everything to appear blurry and out-of-focus, planetary orbits don't exist, and electrons in atoms have no non-trivial minima -- chemistry is impossible. Yet the lakes and oceans are in an almost perpetual calm. What a strange, strange world! :sweatdrop:
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Re: Even Dimensional Waves

Postby quickfur » Sat Mar 22, 2014 5:06 am

anderscolingustafson wrote:I wonder how a 4d life form would sense the world around it considering how waves work differently in an even number of dimensions and in dimensions above 3d.

Well, I think it's pretty clear by this point that if 4D life forms do exist, they would have to be so fundamentally different from us that we probably won't be able to recognize it as a life form, or even if we did, we wouldn't understand what the world is like from its point of view, since its perceptions will be radically different from ours. Not to mention that since atoms don't exist in 4D, if there's such a thing as matter at all, it must be of a fundamentally different nature that we have no idea about. Things like light, sound, touch, etc., would have to be so fundamentally different, if they even exist in 4D.

Unless, of course, we give up trying to extend 3D physics to 4D, since it obviously isn't working out so well. :P As far as 4D dimensional analogy is concerned, I'm of the opinion that we should just adopt Aristotelian physics and be done with it. :P It may not be how 3D works, but who knows, maybe it's just the thing we need in 4D? :lol:
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Re: Even Dimensional Waves

Postby anderscolingustafson » Sat Mar 22, 2014 6:28 pm

If our physical laws work very well in 3d but doesn't work very well when applied to 4d I wonder if that means that a set of physical laws that would work really well in 4d would not work very well when applied to 3d. I mean if our physical laws only work well in 3d then could we come up with a set of physical laws that only works well in 4d but which has disastrous implications when applied outside 4d?
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Re: Even Dimensional Waves

Postby quickfur » Sat Mar 22, 2014 7:26 pm

anderscolingustafson wrote:If our physical laws work very well in 3d but doesn't work very well when applied to 4d I wonder if that means that a set of physical laws that would work really well in 4d would not work very well when applied to 3d. I mean if our physical laws only work well in 3d then could we come up with a set of physical laws that only works well in 4d but which has disastrous implications when applied outside 4d?

Sure! If we're allowed to write the rules, then we can make anything. Just don't expect it to look anything like the physics we're familiar with, though! :)
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Re: Even Dimensional Waves

Postby Hugh » Sat Mar 22, 2014 7:30 pm

ICN5D wrote:This also makes me think about the concepts with string theory. It's all about undulating waves in high-D space. Not to forget the whole extended/compactified dimension idea. We are huge, so we only perceive the extended, opened up directions of freedom. And right under our noses, the realm of the tiny works in the way of waves sloshing around in extra dimensional spaces.


It's interesting to think of the consequences of string theory. If it is correct, you'd have all these extra degrees of freedom for things to move in. Also, the issue of certain things not being possible if you had an even number of dimensions, for example if there was an even 4 spatial dimensions rather than the odd 3 that we currently think there are, kind of becomes less of an issue when you think that there might be 11 rather than 12, or 25 rather than 26. Things are more easily worked out and understood if you have the extra dimensions to explain them.

Michio Kaku explained it in this quote:

"To see how higher dimensions helps to unify the laws of nature, physicists use the mathematical device called “field theory.” For example, the magnetic field of a bar magnet resembles a spider's web which fills up all of space. To describe the magnetic field, we introduce the field, a series of numbers defined at each point in space which describes the intensity and direction of the force at that point. James Clerk Maxwell, in the last century, proved that the electro-magnetic force can be described by four numbers at each point in four dimensional space-time. These four numbers, in turn, obey a set of equations (called Maxwell's field equations).

For the gravitational force, Einstein showed that the field requires a total of 10 numbers at each point in four dimensions. The gravitational field, in turn, obey Einstein's field equations. The key idea of Theodore Kaluza in the 1920s was to write down a five dimensional theory of gravity. In five dimensions, the gravitational field has 15 independent numbers, which can be arranged in a five dimensional array. Kaluza then re-defined the 5th column and row of the gravitational field to be the electromagnetic field of Maxwell. The truly miraculous feature of this construction is that the five dimensional theory of gravity reduces down precisely to Einstein's original theory of gravity plus Maxwell's theory of light. In other words, by adding the fifth dimension, we have trivially unified light with gravity. In other words, light is now viewed as vibrations in the fifth dimension. In five dimensions, there is “enough room” to unify both gravity and light.

This trick is easily extended. For example, if we generalize the theory to N dimensions, then the N dimensional gravitational field can be split-up into the following pieces. Now, out pops a generalization of the electromagnetic field, called the “Yang-Mills field,” which is known to describe the nuclear forces. The nuclear forces, therefore, may be viewed as vibrations of higher dimensional space. Simply put, by adding more dimensions, we are able to describe more forces. Similarly, by adding higher dimensions and further embellishing this approach (with something called “supersymmetry), we can explain the entire particle “zoo” that has been discovered over the past thirty years, with bizarre names like quarks, neutrinos, muons, gluons, etc. Although the mathematics required to extend the idea of Kaluza has reached truly breathtaking heights, startling even professional mathematicians, the basic idea behind unification remains surprisingly simple: the forces of nature can be viewed as vibrations in higher dimensional space."
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Re: Even Dimensional Waves

Postby ICN5D » Sat Mar 22, 2014 7:31 pm

quickfur wrote:This has far reaching consequences for 4D: assuming that 4D light still takes place via some kind of electromagnetism analogue that produces 4D EM waves, this means that light in 4D does not travel in a single direction, but will have back propagations that causes "echoes"! A 4D flash of light will not appear as a single flash, but as multiple flashes with decreasing amplitude, fading out gradually. This also means that light reflected off objects would behave the same way, thus objects will not have a sharp, focused image, but a blurry "echo-ey" image caused by the repeated back-propagations!



Sounds like the double-slit experiment! A single photon travels through both slits and interferes with itself. This is also thought to occur through the past, a photon mingles with itself in its own past.



OTOH, if you drop a stone into a pond in 4D, you'll see a single, perfect spherical ripple travel outwards, because the surface of the pond will be 3D. :P There will be no concentric ripples, and only a single echoing ripple when it hits the edge of the pond. This has the very interesting consequence that ponds and lakes in 4D are likely to be very calm, with only the occasional isolated ripple passing through, quite unlike our 3D ponds where the surface of the water is almost always filled with little ripples.



This seems like the way gravity waves propagate. Very interesting stuff!
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Re: Even Dimensional Waves

Postby quickfur » Sat Mar 22, 2014 10:34 pm

ICN5D wrote:
quickfur wrote:This has far reaching consequences for 4D: assuming that 4D light still takes place via some kind of electromagnetism analogue that produces 4D EM waves, this means that light in 4D does not travel in a single direction, but will have back propagations that causes "echoes"! A 4D flash of light will not appear as a single flash, but as multiple flashes with decreasing amplitude, fading out gradually. This also means that light reflected off objects would behave the same way, thus objects will not have a sharp, focused image, but a blurry "echo-ey" image caused by the repeated back-propagations!


Sounds like the double-slit experiment! A single photon travels through both slits and interferes with itself. This is also thought to occur through the past, a photon mingles with itself in its own past.

It's not quite the same, though, since these "light echoes" persist across time, so if you switch a light bulb on and off very quickly, it will appear to continue flickering on and off over and over again like a dying lighthouse until it becomes too faint to see. A very counterintuitive behaviour for light, which we've come to expect to be near-instantaneous! This makes it virtually impossible to transmit signals in a recognizable way to a receiver: if you try to send Morse code via a flashing light bulb, for example, the "light echoes" will completely mangle the signal from the recipient's POV, and it will be almost impossible to tell which pulses are the original signal and which are the after-echoes.

I've yet to think through what this means to 4D vision in general, but the initial impressions I got are that this effect is going to drastically affect how vision works in 4D (assuming it happens via the 4D analogue of EM radiation). A stationary object may be relatively easy to see, but a moving object will leave a complex trail of echoing light ripples that will in all likelihood completely distort its image. It will probably appear as a smeared-out image that looks like motion blur, except more persistent and quite disruptive because it will also distort background objects as well.

OTOH, if you drop a stone into a pond in 4D, you'll see a single, perfect spherical ripple travel outwards, because the surface of the pond will be 3D. :P There will be no concentric ripples, and only a single echoing ripple when it hits the edge of the pond. This has the very interesting consequence that ponds and lakes in 4D are likely to be very calm, with only the occasional isolated ripple passing through, quite unlike our 3D ponds where the surface of the water is almost always filled with little ripples.

This seems like the way gravity waves propagate. Very interesting stuff!

I have this hypothesis that the 3D universe we see around us is actually the surface of an expanding 4D sphere...
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Re: Even Dimensional Waves

Postby Hugh » Sun Mar 23, 2014 1:50 am

quickfur wrote:I have this hypothesis that the 3D universe we see around us is actually the surface of an expanding 4D sphere...

It would be interesting to explore this idea further.

I've read about branes, do you mean like them? Like we'd be confined to the brane but actually be in higher dimensional space?
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Re: Even Dimensional Waves

Postby anderscolingustafson » Sun Mar 23, 2014 2:03 am

I have this hypothesis that the 3D universe we see around us is actually the surface of an expanding 4D sphere...


Are you saying you think the Universe is a 4d wave?
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Re: Even Dimensional Waves

Postby ICN5D » Sun Mar 23, 2014 2:25 am

A stationary object may be relatively easy to see, but a moving object will leave a complex trail of echoing light ripples that will in all likelihood completely distort its image. It will probably appear as a smeared-out image that looks like motion blur, except more persistent and quite disruptive because it will also distort background objects as well.



Holy crap, dude. That's amazing, the light is stirred up like waves on the surface of a pond. But, in a 3D photon way. Wow, that's quite a mind blowing analysis of the end result. When you look at something, and an object passes in front, the light waves start rippling like water. Just like how a boat passes by and leaves multiple waves behind, the water echo of the boat works like 4D photons and sound.



I have this hypothesis that the 3D universe we see around us is actually the surface of an expanding 4D sphere...



Me too, for some time. I have recently added to this idea. Our now moment is the surface of the 3-sphere, and it originated towards the center of a glome when it was a point. Now, I've had the interesting idea that all atoms and particles are actually interception points in our expanding 3-sphere, from a single entangled mass of energy strands. All particles came from the same instant, the primordial energy-matter blob, when all was once united. As the 3-sphere expanded, an increase in volume and decrease in pressure lead to cooling, and separation of the strands. We see trillions upon trillions upon trillions of individual particles, that are really joined like some cosmic toratope. I found that a certain 15 dimensional toratope (((((II)I)I)I)((((II)I)I)I)((((II)I)I)I)), the triotritorus tiger, has a 3D midsection cut of 4,096 spheres in a 16x16x16 cubic lattice. These spheres seem like they could be analogous to atoms, and when moving the cut planes through the 15D shape, we see spheres merge and split, perhaps imitating some level of chemistry, in an oversimplified far reaching context. Now, consider a much higher dimensional toratope, with nested symmetry upon nested symmetry. The mid-cut lattice would contain many subgroupings of lattices themselves, corresponding to entirely different behavior and symmetry. This effect kind of feels like atoms and molecules on the small scale, clustered into life forms and planets on the large scale. All atoms of which are joined by one single continuous mass, connected by the past, dangling towards the future. We're moving along the 3D cross cut of it by passing through time. How many dimensions? Well, a good place to start would be to ask, " How many particles are in the universe? ". That is, how many intercept points into our 3-plane. From there, we can then ask, how many dimensions are required to simulate atom symmetry. Just pure speculation.....
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Re: Even Dimensional Waves

Postby Keiji » Sun Mar 23, 2014 7:33 pm

quickfur wrote:Unless, of course, we give up trying to extend 3D physics to 4D, since it obviously isn't working out so well. :P As far as 4D dimensional analogy is concerned, I'm of the opinion that we should just adopt Aristotelian physics and be done with it. :P It may not be how 3D works, but who knows, maybe it's just the thing we need in 4D? :lol:


Hence why I came up with my particle system :)
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Re: Even Dimensional Waves

Postby quickfur » Mon Mar 24, 2014 5:02 pm

Keiji wrote:
quickfur wrote:Unless, of course, we give up trying to extend 3D physics to 4D, since it obviously isn't working out so well. :P As far as 4D dimensional analogy is concerned, I'm of the opinion that we should just adopt Aristotelian physics and be done with it. :P It may not be how 3D works, but who knows, maybe it's just the thing we need in 4D? :lol:


Hence why I came up with my particle system :)

Hmm. I went back to re-read that proposal, and it's certainly interesting. I'm not sure what the macroscopic consequences of it would look like, though. I was thinking more in terms of taking certain macroscopic properties as axioms, and then working backwards to see what kind of physical laws might have produced that result. Generally, I think this is far easier than trying to design something from ground-up with the hope that the emergent properties will resemble what we want. Emergent properties depend in a very complicated way on the fine details of the underlying system, and it's almost impossible to predict what kind of properties will emerge when you change things around in the underlying system. It's far easier to just shed all pretense and postulate straight-up the desired macroscopic effects as a given, and then trying to rationalize what physical laws may have led to those effects. :)
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Re: Even Dimensional Waves

Postby anderscolingustafson » Mon Mar 24, 2014 7:55 pm

quickfur wrote:
Keiji wrote:
quickfur wrote:Unless, of course, we give up trying to extend 3D physics to 4D, since it obviously isn't working out so well. :P As far as 4D dimensional analogy is concerned, I'm of the opinion that we should just adopt Aristotelian physics and be done with it. :P It may not be how 3D works, but who knows, maybe it's just the thing we need in 4D? :lol:
<br abp="662"><br abp="663">Hence why I came up with my particle system :)
<br abp="664">Hmm. I went back to re-read that proposal, and it's certainly interesting. I'm not sure what the macroscopic consequences of it would look like, though. I was thinking more in terms of taking certain macroscopic properties as axioms, and then working backwards to see what kind of physical laws might have produced that result. Generally, I think this is far easier than trying to design something from ground-up with the hope that the emergent properties will resemble what we want. Emergent properties depend in a very complicated way on the fine details of the underlying system, and it's almost impossible to predict what kind of properties will emerge when you change things around in the underlying system. It's far easier to just shed all pretense and postulate straight-up the desired macroscopic effects as a given, and then trying to rationalize what physical laws may have led to those effects. :)


I was just thinking the same thing. I know that we figured out the effects of our own physical laws before we figured out our physical laws whether than the other way around. Whether than deriving the effects of Gravity from the equations for Gravity for instance we looked at the effects Gravity had and then worked back to figure out what equations were producing those effects. It seems that when trying to understand 4d we do the exact opposite of what we did when we worked out the physical laws that govern our universe as we tend to apply the physical laws of our universe to see how they effect 4d whether than coming up with certain effects of the physical laws in 4d and then working out what physical laws would produce those effects.

According to the anthropic principle we can only obverse the physical laws to have certain effects as the laws of physics must have effects that make it possible for us to exist. For 4d we could consider what kind of effects of physical laws would be necessary for someone to exist to observe those effects and then work out the equations that would produce those effects in order to see what kinds of physical laws an observer living in 4d would find.
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Re: Even Dimensional Waves

Postby quickfur » Mon Mar 24, 2014 8:33 pm

Personally, I find the anthropic principle to be nonsensical tautological circular reasoning that doesn't answer any questions, only shovels them under the carpet. :roll: But since we're dealing with pure speculation here, we can let that slide. :P

It's clear, in any case, that if an actual 4D universe exists somewhere out there (and that's a very big "if"), it has to be so completely alien that we would probably be unable to comprehend it. The whole point of the exercise behind this forum in the first place, is, according to Garrett Jones, the guy who first started this forum, to speculate on what consequences a 4th spatial dimension would have if our world (presumably the world "as we know it", that is, with matter, energy, living beings, planets, comparable physical laws, etc.) were 4D instead of 3D.

Of course, we've learned a lot since those early days of cute analogies about 2D beings living in a wall painting of a 3D being's room, who in turn lives in a wall painting of a 4D being's room. Patrick Stein ("pat" on this forum) discovered that if we assume Newtonian physics, which implies a 1/r^3 potential for gravity, then stable 4D planets cannot exist. This means analogies like weather patterns and day/night cycles don't work in a "real", "native" 4D universe; it can only work in a contrived universe where we tweak the laws of physics so that these things "magically" work out. Then someone else (I forgot who) discovered that the Schrödinger wave equation for the 4D hydrogen atom has no non-zero minima: that is, a 4D hydrogen atom would simply collapse, it has no discrete electron orbitals, so chemistry as we know it cannot exist, and indeed, matter as we know it cannot exist! And now we learn that Huygen's principle in 4D leads to light (i.e. EM radiation) being unsuitable as a signal carrier due to the "back-echoing" effect of 4D waves.

So to me, there are two different lines of research going on here: (1) research into "native" 4D phenomena, to explore what a true 4D universe might look like, and (2) a contrived 4D world built via dimensional analogy, where we apply "naïve" generalizations of 3D phenomena as a way of understanding the geometric consequences of 4 dimensions of space.

So far, most of our discussions have been centered around (2), which is itself very interesting, because the additional degree of freedom conferred by the 4th dimension lets us do many things that are impossible in 3D. And by translating familiar 3D-centric phenomena into 4D, it allows us to tackle the geometry of 4D space using familiar objects and concepts, which is very helpful in learning to visualize 4D.

But I'd say that research into (1) can also be very interesting in itself; for example, after reading about the "echoing" waves in even dimensions, now I'm all curious about what implications it may have on a "native" 4D analogue of electromagnetism, and whether this can be turned into something useful. Who knows, maybe an interesting system could arise from this effect? It's also additional motivation for me to dust off my rusty knowledge of differential equations to work through more 4D effects that might turn out to have surprising answers. :) I'm pretty sure the consequences would be rather hard to understand unless you've already learned to visualize 4D, though, so (2) still serves a foundational purpose to that end.
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Re: Even Dimensional Waves

Postby ICN5D » Mon Mar 24, 2014 11:43 pm

Yeah, that's kind of the way I did it. I started with the immediate physical structures and properties, then developed on an atomic scale universe that could make it happen. But, I also tied in some of the atomic stuff along with the larger, in that radical universe I described.
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Re: Even Dimensional Waves

Postby anderscolingustafson » Tue Mar 25, 2014 5:19 am

I was thinking would the fact that waves would double back on themselves in 4d have any effects on gravity in 4d? Changes in gravitational fields are carried through gravity waves so would the behavior of gravity waves effect the behavior of gravity in 4d?
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Re: Even Dimensional Waves

Postby quickfur » Tue Mar 25, 2014 5:37 am

anderscolingustafson wrote:I was thinking would the fact that waves would double back on themselves in 4d have any effects on gravity in 4d? Changes in gravitational fields are carried through gravity waves so would the behavior of gravity waves effect the behavior of gravity in 4d?

Now that's an interesting thought. I wonder what are the consequences of the force carriers of a field, be it EM or gravity, being "echo-ey" waves. Will that fundamentally change the way the field behaves? If so, what does that mean for planetary orbits? Note that Patrick Stein's proof of the instability of 4D orbits was based on the assumption that 4D gravity behaves just like 3D gravity, except that it follows an inverse cube law by the principle of flux density (strength of force is proportional to the surface area of an n-sphere centered on the source of that force). But if the gravitons no longer behave like finite particles due to their "echo-ey" nature in 4D, would it change the way gravity behaves? If so, what consequences will it have for orbital motion?

Note that this would have consequences not just for gravity, but for basically 99% of physics, because ultimately, the interaction between macroscopic objects can be boiled down to EM interactions between atoms -- e.g., you can't walk through a wall because the EM repulsion between the electrons in your body and the electrons in the wall prevents you from doing so. But since this repulsion happens via the EM force carrier, which is just the photon, this means that the properties of the photon will basically determine the properties of the EM force, and if the 4D photon has "echo-ey" properties, this may cause fundamental changes in the way EM works, which in turn may lead to different macroscopic laws.

(I fear, though, that the answer might be that the results are extremely bizarre and far too strange for us to comprehend. :sweatdrop: )
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Re: Even Dimensional Waves

Postby ICN5D » Wed Mar 26, 2014 12:04 am

Sounds like a good approach for discovering 4D macroscopic effects. Especially in how the planetary scale would work. The idea of rippling gravity waves makes me think of Lagrange points. Perhaps there is some equivalent to them, induced by interference. Be it constructively or destructively, there may be some neutral zones, or perhaps entirely different structures altogether. I remember reading way back about this neat circular wave pool that could make alphabet letters in standing water waves. Maybe, if gravity is rippling more water-like, there could be linear neutral zones of aggregation, giving rise to toroidal, linear, or tiger-like planets that emerge from undiscovered stabilities.

If sphere and torus shaped planets are possible in 3D, then it would stand to reason that all of the 4D toratopes are possible physical structures, on the macro scale. Given that these 5 toratopes are by default a reasonable suggestion for 4D, it may be just as reasonable to suggest explaining how 4D gravity could create these structures. A tiger-shaped ring world could have a double rotation, at different rates even. Or, a ditorus-shaped world, with an internal cavern system that's a separate inside out world, in its own ecosystem. Hmmm...... Or, a dyson sphere as a torispheric world!
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Re: Even Dimensional Waves

Postby ICN5D » Wed Mar 26, 2014 6:46 pm

I also thought about the relationship with the duocylinder and a tiger world. If a duocylinder is an artifact of the Hopf fibration, and the tiger is the inflated margin of said duocylinder, then the hopf fibration could be used to help explain how 4D gravity could manifest a tiger-shaped 4D ringworld. Also noting how the 3D magnetic field is a potential Hopf artifact as well, and we explored the idea of a pseudo-EM flowing gravity field in 4D. Maybe the tiger is some sort of harmonic 4D gravity structure, made by standing 4D gravity waves like the circular wave pool. And, perhaps a good direction for exploration of such possibilities :)
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Re: Even Dimensional Waves

Postby CollIB » Thu Mar 27, 2014 1:01 am

If these light echoes are real, then how would look like the 3D shadow of a tesseract? If there are too many light sources (like a sunny day with many reflections and other echoes), probably it will be so much distorted, it would look beyond recognition for us. But if there is only one point-like source of light, maybe it would just "periodically distorted".
But look at the bright side, you can really see 4D, you need just 100 stroboscope with different colors and with different speed in a dark room, turn them on, and there you go 8)
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Re: Even Dimensional Waves

Postby ICN5D » Thu Mar 27, 2014 2:41 am

Hmm, yeah, maybe you'd have to wear 3D vision goggles that would cancel out the echo effect. Like Bose noise cancellation headphones, but for use with 4D photons.
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Re: Even Dimensional Waves

Postby quickfur » Thu Mar 27, 2014 4:20 am

I've been thinking a little more about this. Consider your typical 3D photon. It moves at the speed of light (obviously), and while the human eye is sensitive enough to actually detect a single photon, generally speaking it takes a lot more than just one photon to convey an image. Moreover, suppose you have a brief flash of light, say about 0.01 seconds or so. To us macroscopic beings, that's an extremely short time. But relative to the speed and frequency of light, that's a huge amount of time, during which billions (probably more) of photons would have been emitted and absorbed.

Now consider the situation in 4D. Due to the echoing effect in even dimensions, 4D light propagates not as a single wavefront, but a complex superimposition of "echoes" of the primary wavefront, each of which diminishes in amplitude over time. Now from the POV of a single 4D photon (or whatever the echo-ey equivalent might be), it takes many such secondary wavefronts to pass before the amplitude has fallen low enough to no longer be detected by the eye. However, from a macroscopic POV, this could easily happen within 0.01 seconds or so, considering the speed of light and its frequency relative to macroscopic time. As we all know from 3D games, a video rendering of about 0.05 seconds per frame will already appear "smooth" to our eye, and by about 0.025 seconds per frame, even the keenest eye won't detect the jerkiness anymore.

This means that perhaps this "echo-ey" effect of light in 4D is really only noticeable when you start getting to microscopic scales, but in everyday life, it will hardly be noticeable, except perhaps in fast-moving objects where they might appear elongated because light reflected off their previous position will persist for about 0.01 seconds before fading from view. In other words, you get a sort of "motion blur" effect.

Moreover, consider how the cells in our retina actually work: while they are sensitive enough to detect a single photon, the image that forms in the retina also persists for about 0.05 seconds (or thereabouts) before fading. This means that if the 4D "echo-ey" effect lasts only for about 0.01 seconds or so, nobody will notice any difference at all!

Furthermore, consider a static (or mostly static) scene. The papers that I found above were all analysing the behaviour of wave propagation in a single pulse -- i.e., the equivalent of emitting a single photon. But in real-life, in a mostly-static scene, light is continuously being reflected off the surfaces of the objects in view. This means we're not dealing with a single pulse, but with a continuous signal. Now AFAIK, if you set up a vibrator with constant frequency on the surface of a pond, eventually the water waves will settle into a simple harmonic pattern -- i.e., a simple sine wave -- any complexity that arises from the echo-ey effect of waves in the 2D surface of the pond gets overwhelmed by the continuous signal, so that a receiver at the far end of the pond will eventually get a constant signal virtually free of the echo-ey effects. So I'd expect the same thing to happen in 4D: in a (mostly) static scene, the light reflecting off the objects will settle into a constant, simple sine wave, so there's really no distortion going on. The only time the echo-ey effects make a difference is when something is moving or changing, which I've already said that if it's only noticeable for about 0.01 seconds or so, doesn't really make too much of a difference except add motion blur to the scene.

Of course, the 0.01 seconds analysis breaks down when the amplitude of the waves is too big -- because the bigger the initial amplitude, the longer it takes for the secondary echoes to fade away. So this means that if the ambient lighting is too bright, then the echo-ey effects become very prominent, and will cause significant visual distortions. So it seems likely that sunbathing will not be a very desirable thing to do in 4D, because it will practically blind you until you get out of the light. :P At night or in a dimly-lit room, however, vision will probably not be as big a problem as we initially thought.

(Caveat: all of the above is just speculation. I haven't actually worked out the wave equations to prove what actually happens.)
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Re: Even Dimensional Waves

Postby anderscolingustafson » Thu Mar 27, 2014 4:37 am

Removed full last post quote. ~Keiji

What about the effect on sound? Sound travels much more slowly than light and so there are fewer sound waves that pass by within a given amount of time. So even if the effects of echoing of light weren't noticeable in daily life would the effects of echoing of sound be noticed?
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Re: Even Dimensional Waves

Postby quickfur » Thu Mar 27, 2014 5:27 am

Removed full last post quote. ~Keiji

I'm guessing that the effect on sound will be very noticeable. Which kinda puts a damper on verbal communication. :\

OTOH, this kinda reminds me of how the Bible describes God's voice as being "like the sound of many waters". Could it be alluding to this effect? :)
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