Teragon wrote:I know it doesn't really belong here, but just a few words on your comment, quickfur.
quickfur wrote:Being an even-numbered dimensionality, 4D behaves similarly to 2D, in that if you have a point source of a wave (the signal), as the wave propagates outwards from the source it will actually send back echoes of itself back to the source, which then interferes with subsequent parts of the signal and combines with it as it bounces back again, causing subsequent parts of the signal to be distorted in complex ways. You can see this in 2D by dropping a pebble into a still pond and seeing how it's not just a single wavefront, but multiple ripples moving back and forth in a complex way behind the initial main ripple.
Having glanced at the corresponding thread this seems well-founded to me. However I can't understand this analogy argument. Throwing a stone into a pond leads to a vertical elongation of the surface of the water at this location. With gravity as a restoring force it is inevitable that the surface oscillates back and forth, producing a train of circular waves rather than a single pulse. Because of this also in 3D waves are not single peaks, but longer wave trains encompassing several oscillations.
It's not the fact that it oscillates that's the issue. It's the fact that the relationship between the final waveform and the original signal (presumably some kind of oscillator producing a sine wave, say, at the point of origin) is non-linear. IIRC the paper describes the effect as the net effect of the summation of wavelets that make up the front of the expanding waveform. As the waveform expands, the peaks at its frontier produces wavelets that travel both forwards
and backwards; in 3D, however, the backward-travelling wavelets cancel out and so you have a clear sine wave, for example, throughout the entire spherical region from the origin to the frontier. However, in other dimensions, the backward-travelling wavelets do
not cancel; thereby producing a residual effect that changes the shape of the tail of the waveform via interference. In even dimensions such as 2D and 4D, this causes every peak of the original sine wave signal to produce backward travelling peaks that significantly alters the overall shape of the waveform, so that the overall shape can no longer be described by a simple sine wave but a complex combination of forward and backward travelling sine waves. In odd dimensions l ike 5D, the backward-travelling wavelets "mostly" cancel out, but not quite, thus leaving their trace in altering the final waveform so that instead of being a faithful representation of the original sine wave signal, it ends up being an n'th derivative of the original signal. Only in 3D is the original signal propagated without being mutilated in some way.
quickfur wrote:The net effect of this, is that a single peak, say a blip of light from some point source, will be perceived by the receiving end as one big blip followed by "echoes". So in both 2D and 4D, a single flash of light gets perceived as multiple afterflashes.
This may be true for sound, but not for light in everyday life. For the same reason that if you stand in a room with mirrored walls and switch off the light it will still get dark immediatly. An impulse of light travels so fast that in the time span of 0.01 s it gets reflected 600,000 times on the walls of a walls of a 5 m diameter room. Even if the mirrors reflect 99.999% of the light and nothing else is in the room, only 0.25% of the initial intensity would be left after 10 ms. In reality light is always emitted from something, so each time the reflected wave hits the source a large fraction of the light will be absorbed.
I think you're misunderstanding what I wrote. :-) The point isn't that the light gets reflected, the point is that the waveform of the light pulse (or steady wavetrain, whichever) takes on a fundamentally different form from the energy source. Thus communicating the original signal becomes difficult or perhaps even impossible, because by the time it gets to the receiving end it has mutated into a very different waveform.
Or perhaps you're taking issue with my interpretation of the equations, which is perfectly understandable. I think at a fundamental level we actually agree. We could save ourselves the trouble if you read the paper yourself and looked at the differential equations and their solutions as presented there; it makes it quite clear that there is something unique about 3D that causes wave propagation to retain the pristine form of the original signal, whereas in all other dimensions some kind of fundamental alteration occurs by the time the signal reaches the receiver, in odd dimensions as a derivative of the original waveform, and in even dimensions a far more complex alteration that likely makes recovering the original signal very difficult, if not impossible.
quickfur wrote:And that's not to mention that there are no stable orbits in 4D (except for the perfect circle, which is so unlikely it probably will never happen -- the slightest deviation causes the planet to either crash into the star or fly off into cold, dark space), which means things like galaxies and solar systems are unlikely to exist in 4D.
Well, that's only true for a 4D universe with otherwise identical physical laws to our own universe. We shouldn't talk about it, as if it was a general property of 4D. A lot of fine-tunig is necessary in 3D to produce a universe where solar systems and stable matter can exist. Adding a forth spacial dimension reshuffles the pack, a 4D universe needs different physics in order to form complex matter.
Of course the context is that we're talking about identical (or at least analogous) physical laws to our own universe here! Otherwise what do planetary orbits even mean?? In a 4D universe with fundamentally different physical laws to our own, it seems meaningless, or at least presumptuous, to even speak of planets, as if they necessarily exist under arbitrary physical laws! For that matter, what does "matter" even mean? How are we so sure that such a thing would even exist in a 4D universe that operates according to completely alien physical laws?