See Through Forests

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.

See Through Forests

Postby anderscolingustafson » Sun Mar 09, 2014 10:23 pm

The higher the number of dimensions the more see through forests would become and the further it would be possible to see through a dense forest. This is because the higher the number of dimensions the greater the amount of empty space between trees and tree branches would be in comparison to the amount of space occupied by trees and tree branches. In any number of dimensions the amount of space not occupied by a tree or tree limb is given by the equation dn-1-1 where d is the distance between any given tree or tree branch measured using the diameter of the trees and tree branches and the closest tree or tree branch to it and n is the number of dimensions. So the greater the number of dimensions the more likely that any randomly chosen space in a forest will be just empty space. So the higher the number of dimensions the further on average your line of sight in a dense forest will extent before ending on a tree.
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Re: See Through Forests

Postby wendy » Mon Mar 10, 2014 6:56 am

Regardless of the dimension, the forrests come ever-thick. Þat is the nature of the beast.
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Re: See Through Forests

Postby Keiji » Mon Mar 10, 2014 8:30 am

Trees take positions in the laterals, because frontal is taken up by line of sight, and vertical is taken up by the heightspan of the tree.

Clearly in 2D, you would not be able to see past the first tree - there are no laterals.

In 3D, trees arrange themselves as points on a line, giving distance to the first tree as a function of lateral coordinate. Notice that the further away the trees are from an observer, the closer they appear to each other; but also, the thinner they appear. The distance-from-coordinate could well be represented as a fractal.

In 4D, trees arrange themselves as points in a plane. Now you have a function for distance from two coordinates. Those trees further away will again appear closer to each other, though now that everything is squared, it's hard to say which wins out between empty space, and space blocked by trees.

I'm on the fence on this one until someone does some actual statistics to figure out the expected value of distance for a given tree thickness and separation over a varying number of dimensions.
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Re: See Through Forests

Postby quickfur » Mon Mar 10, 2014 6:47 pm

Keiji wrote:[...]
In 4D, trees arrange themselves as points in a plane. Now you have a function for distance from two coordinates. Those trees further away will again appear closer to each other, though now that everything is squared, it's hard to say which wins out between empty space, and space blocked by trees.

I'm on the fence on this one until someone does some actual statistics to figure out the expected value of distance for a given tree thickness and separation over a varying number of dimensions.

It shouldn't be that hard to estimate. As a crude approximation, assume each tree is shaped like a lollipop: a tall thin cylinder with a sphere on top. In projection, then, each tree is represented as two concentric (n-1)-spheres. Assume a random sphere packing of the outer spheres on the (n-1)-dimensional groundspace. This packing should be relatively compact. It seems reasonable to assuming minimal overlap between the outer spheres (i.e., the shadow of the leaves and branches), since otherwise there would not be enough space for the leaves and branches to spread out and capture enough sunlight, and it would be less favorable for trees to grow in such a condition.

The total volume of the inner spheres relative to the total groundspace, then, is the density of trees per unit groundspace, and should be roughly proportional to how deep your line-of-sight can penetrate into the forest before your view is blocked. Assuming a fixed ratio of outer radius to inner radius, the total volume of the inner spheres would be the volume of an n-sphere minus the volume of a smaller n-sphere. Now, from Wikipedia's volume of an n-sphere page, the volume of an n-sphere of radius R is:

V(R) = pin/2 / Gamma(n/2 + 1) * Rn

Let K = pin/2 / Gamma(n/2 + 1). Then we have V(R) = K*Rn. Let R2 be the outer radius, and R1 the inner radius. Then the difference between these two volumes is:

D = K*R2n - K*R1n = K*(R2n - R1n)

Since we're assuming a fixed ratio between R2 and R1, say this ratio is A (i.e., R2 = A*R1), then we have:

D = K*((A*R1)n - R1n) = K*(An-1)*R1n = (An-1)*(K*R1n) = (An-1)*V(R1)

Now, we know that V(R1) approaches zero as n increases without bound, but the fate of (An-1) depends on the choice of A. If A<2, that is, if the average radius of the branches and leaves are less than twice the radius of the tree trunk, then (An-1) will shrink to zero as n increases, meaning that the forest will become almost completely opaque as n increases (the volume of unoccupied groundspace shrinks to zero).

The case where A≥1 is a bit more complicated, because now you have two quantities, one is shrinking and the other is growing. I'm not sure how to analyze this case, since the limit of exponential functions is non-trivial to compute. My gut feeling is that the two quantities should flatten out asymptotically to a constant factor, meaning that the volume of unoccupied groundspace will approach some fixed value dependent on A, so how deeply your line-of-sight can penetrate the forest will depend on the value of A. Either way, it seems to be showing that the answer to this question isn't as straightforward as it might appear at first glance. Are your trees tall and thin (small value of A), or squat and bushy (large value of A)? That could potentially make a big difference. The magic value of A=2 seems to be some kind of pivotal point, which is unexpected, intuitively speaking.

Now, all of the above analysis assumes that A remains constant as n increases. This may not be the case, due to various other complicating factors.

One is that if we assume circular planetary orbits (BIG assumption here! :P But hey, if we're assuming forests exist, then this may not be that unreasonable of a stretch), then the amount of sunlight that the forest will get, will depend on where it's located on the planet. For very high dimensions, we have the counterintuitive fact that almost all of the planet's surface is in a twilight zone (i.e., the sun never rises above the horizon by more than a small angle). Only a narrow equatorial band will actually receive overhead sunlight during daytime. This would mean that forests can only exist near this narrow band, because elsewhere, the trees on the boundary of the forest will block almost all sunlight from the interior of the forest, so trees probably won't grow very well inside the forest -- so there will be no forest at all! Furthermore, any trees that would grow outside of the equatorial band will only get sunlight horizontally, so its leaves and branches would be best arranged laterally rather than vertically! Moreover, it would be advantageous for the tree to be as tall as possible, in order to maximize exposure to the sun that never rises far above the horizon (and to rise above any obstacles in front that will block sunlight). So trees will likely be T-shaped, with a tall trunk, and long horizontal branches at the top, with leaves restricted to the periphery of the branches. A rather different shape from what we'd expect. :)

Of course, in the narrow equatorial band, trees will have more "normal" shapes because the sun will actually rise overhead at noon. But for very large n, this equatorial band will become extremely narrow, so past a certain point, it would seem that the only "forest" in the normal sense of the word that can exist would be limited to a single great circle of trees lined up around the equator, because everywhere else is a twilight zone that doesn't have enough light for trees to grow (or can only support T-shaped trees)! So the very concept of forests become questionable as n increases without bound.
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Re: See Through Forests

Postby ICN5D » Mon Mar 10, 2014 7:45 pm

T-shaped trees. Hmm, reminds me of some ancient forest artwork of dinosaur-era forests

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Re: See Through Forests

Postby anderscolingustafson » Mon Mar 10, 2014 10:24 pm

As the number of dimensions increases the surface area of a leaf would also increase as there would be more available space for a leaf to expand into. In large numbers of dimensions trees might be able to survive in much dimmer light than trees on Earth because their leaves would have more area to capture the available light. In 3d a leaf has a 2d surface area but in 4d a leaf would have a 3d surface area and with every number of dimensions leaves have a surface with one fewer dimensions than the number of dimensions.
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Re: See Through Forests

Postby quickfur » Mon Mar 10, 2014 11:24 pm

anderscolingustafson wrote:As the number of dimensions increases the surface area of a leaf would also increase as there would be more available space for a leaf to expand into. In large numbers of dimensions trees might be able to survive in much dimmer light than trees on Earth because their leaves would have more area to capture the available light. In 3d a leaf has a 2d surface area but in 4d a leaf would have a 3d surface area and with every number of dimensions leaves have a surface with one fewer dimensions than the number of dimensions.

But an n-dimensional tree also has to collect enough energy to fuel all the cells in its n-dimensional bulk, which grows geometrically as n increases. So the question is, does the leaf surface area increase fast enough to balance out the dimming sunlight and the increasing body mass?
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Re: See Through Forests

Postby ICN5D » Tue Mar 11, 2014 2:29 am

Hmm, good question. The light gathering field is a 3D array of cells. This would be a more efficient use of space, in the collected energy density. Maybe it's offset perfectly, by the increase in tree matter vs the increase in light gathering matter. Then, of course, we haven't begun to question the root system, and how it may be spherinder-shaped tendrils.

Perhaps a torisphere-shaped tree is best at this level, where there is an encasing spherical boundary zone of leaf material, feeding a spherinder-shaped trunk, connected by spherinder-shaped roots. These roots, of course, would be in a 4D array, with 3D arrays on the surface to absorb nutrients. I apologize if this made it more complex to reason the possibility. I don't think anyone has ever explored 4D flora biology, yet. History in the making....
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Re: See Through Forests

Postby wendy » Tue Mar 11, 2014 7:37 am

Consider trees as spheres (eg circles) on the ground.

The densest packing of spheres in eight dimensions, the E<sub>8</sub>, covers just over 1/4 of space. But the hamilton-tiling (that is, of omnitruncated simplexes), is supposed to be the one providing the lightest cover (ie least overlap of spheres). It's not all that much denser. Increasing the folage from 4 to 5, would completely cover this factor of 4 in eight dimensions.

I suppose in very large dimensions the light might get through (don't forget the inverse-nth power law), but i suspect for the low dimensions (ie n<24), it would not take too much forrest to black out the sky.
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Re: See Through Forests

Postby anderscolingustafson » Tue Mar 11, 2014 5:40 pm

As the number of dimensions increases surface area tends to increase faster than volume meaning that as the number of dimensions increases the amount of surface area relative to unit volume also increases. So as the number of dimensions increases a higher and higher percentage of a tree would be concentrated in the leaves. So in a very large number of dimensions most of a tree would be concentrated in the leaves.
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Re: See Through Forests

Postby ICN5D » Wed Mar 12, 2014 1:56 am

I guess that depends on which kind of tree structure you're referring to. In 3D, a pine tree has long, thin needles for leaves, which has a dramatic increase in surface area and light gathering power, than an oak or maple. But, when considering a Baobab tree, it's almost entirely a trunk, with a small array of leaves on top. Same with 4D, we just add an extra direction to build things with. I'm pretty sure that there is some equivalent to 4D baobab trees or 4D pine-needle trees, that stray from generalized high-D effects.
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