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) = pi
n/2 / Gamma(n/2 + 1) * R
nLet K = pi
n/2 / Gamma(n/2 + 1). Then we have V(R) = K*R
n. Let R2 be the outer radius, and R1 the inner radius. Then the difference between these two volumes is:
D = K*R2
n - K*R1
n = K*(R2
n - R1
n)
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 - R1
n) = K*(A
n-1)*R1
n = (A
n-1)*(K*R1
n) = (A
n-1)*V(R1)
Now, we know that V(R1) approaches zero as n increases without bound, but the fate of (A
n-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 (A
n-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!
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.