Scaling laws in (very) high dimensions

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.

Scaling laws in (very) high dimensions

Postby Teragon » Wed Sep 23, 2015 11:59 am

Have you ever asked yourself why a light source looks fainter and fainter, the farther away you go, but an extended object like a house or a mountain does not?

A light source that's small enough (or distant enough) may be considered a point light source. From there light will spread equally in any direction and as the surface of a sphere grows with r², the light intensity will decrease with 1/r². Extended objects may be considered as a composition of many point light sources. The light emitted by each one decreases with 1/r², but the individual points also go closer together in the same manner, so the light intensity coming from a fixed angle smaller than the extent of the object stays the same.

The consequence of this scaling law is that in higher and higher dimensions lights gets fainter faster and faster with the distance to the light source. As long as we're close enough to see an object extended, the fraction the object takes up in our field of vision decreases with 1/r^n. Doing a few steps in very high dimensions means that object shrinks very fast in our field of vision (while lengths still shrink as they do in 3D). In a 10.000 dimensional space for example, going 1 cm further apart will cause an object to shrink by a cosmological factor of 10^43. Point light sources vanish very rapidly in high dimensions. If two light sources have the same intensity at a distance of 1 m, 100 km off in 3D gives the same intensity as 10 m off in 10D. The same holds for sounds. Things are getting more and more local in high dimensions.

The other way round, going to higher and higher dimensions, space grows faster and faster with the distance. As a result "the world gets smaller", houses and other structures with big lengths won't make much sense anymore. If we assume an n-sphere-like beeing in a spherical world with three times the perimeter of the beeing (and leave hight apart), the world is 9 times the volume of the being in 3D, 27 times its hypervolume 4D, 243 times in 6D, 19.683 times in 10D, 10.460.353.203 times in 22D. An 21-surface in a 22D world with a perimeter of 3 m corresponds to a 2-surface in a 3D world with a perimeter of 100 km. Still all the beeings in this world are just one or two steps away from each other!

For a being in very high dimensions it might become less practical to move, because everything it needs is directly around it and orientation would be virtually impossible. It seems resonable that beeings in very high dimensions would perceive the world very locally as there is way too much information for them to process on an imaginairy hypersphere only slightly apart. This raises the idea that beeings in very high dimensions might not be able to evolve and develop a consciousness about space, because the environment might be to chaotic and unpredictable to learn, so in their experience they might live in a 0D world.

For a beeing in intermediate dimensions, things would look quite different from low dimensions too. As the available space of a house and the amounts of material needed to build it would grow very fast with its side length, the individual dimensions of the houses might become smaller and smaller, only slightly larger that the length of the beiing itself. Movement might either be confined to a few dimensions, where the house is spread, or distributed equally to all degrees of freedom (or a combination of both). Every place in one room might be just one step away from each other.

It's already been discussed in the forum that rectangular shape get more "starlike" in higher and higher dimensions (http://hi.gher.space/forum/viewtopic.php?f=25&t=1891&hilit=volume+sphere+cube). In 10.000 dimensions a hypercube with a sidelength of 1 m has 2^9999 diagonals with 100 m in length (and many shorter ones). It would feel very unnatural for a rather round beeing to live in such a space. In just 20 dimensions a cubic room with the same side length as the perimeter of a roughly spherical being would provide 200.000.000 times as much space as the being itself occupies, but it could not even move! So it seems more natural that rooms would be spherical, maby extented only into a few directions, in high dimensions.
What is deep in our world is superficial in higher dimensions.
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Re: Scaling laws in (very) high dimensions

Postby Teragon » Wed Sep 23, 2015 1:20 pm

In any case these huge corners where no one can go would be inevitable for whole cities. Building would become very inefficient, a city would take up far more space than is directly useable by its inhabitants, but in return cities could remain reservoirs for nature, leaving the habitat of smaller creatures practically untouched (except for the shading by the walls) and large amounts of ressources could be stored in corners without any need of additional space. Paradoxically free land could become rare on a densly populated high dimensional planet. With the same population density the average distance between two creatures becomes smaller and smaller as the dimensionality grows. So it could be necessairy to build high buildings or underground cities in high dimensions. This is quite contrairy to the prevalent assumption that flat buildings might be common in higher dimensions. Of course this is only the case under the premise the reachable population density is independent of the dimensionality.
What is deep in our world is superficial in higher dimensions.
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Re: Scaling laws in (very) high dimensions

Postby fallfromgrace » Thu Sep 24, 2015 1:29 pm

Teragon wrote:Things are getting more and more local in high dimensions.


You missed to take into account other facts. In higher dimensions, chemical and biological possibilities would increase as well, not only physical. Visual and auditive receptors could have their functions increased allowing animals to detect light and sound of lesser intensity than they can in 3D.

Teragon wrote:As a result "the world gets smaller", houses and other structures with big lengths won't make much sense anymore.


As if higher dimensional weather conditions wouldn't be scary? And casual civil measures of security.

Teragon wrote:For a being in very high dimensions it might become less practical to move, because everything it needs is directly around it and orientation would be virtually impossible.


Orientation of animals is down to basic instinct level. Pseudoscientific properties would increase along with dimension number. Egocentric self spreads radially and they would get to desire to reach all possible directions to be happy.
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Re: Scaling laws in (very) high dimensions

Postby PatrickPowers » Fri Dec 04, 2015 5:32 pm

Teragon wrote:In any case these huge corners where no one can go would be inevitable for whole cities. Building would become very inefficient, a city would take up far more space than is directly useable by its inhabitants, but in return cities could remain reservoirs for nature, leaving the habitat of smaller creatures practically untouched (except for the shading by the walls) and large amounts of ressources could be stored in corners without any need of additional space. Paradoxically free land could become rare on a densly populated high dimensional planet. With the same population density the average distance between two creatures becomes smaller and smaller as the dimensionality grows. So it could be necessairy to build high buildings or underground cities in high dimensions. This is quite contrairy to the prevalent assumption that flat buildings might be common in higher dimensions. Of course this is only the case under the premise the reachable population density is independent of the dimensionality.



I think round/spherical buildings like yurts would be much more practical in such a space.
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Re: Scaling laws in (very) high dimensions

Postby Teragon » Sun Dec 06, 2015 9:42 pm

I think so too, but on the other hand space between buildings would be huge.
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