Atoms, Flight, and Muscles in 4D

[PatrickPowers]

The smart money says atoms can't exist in 4D. However if we go with that then we have an empty Universe. No fun in that. So let's assume that atoms or some equivalent exist. (I'd put my money on knotted electromagnetic vortices but nothing much is known about them. Better to pretend our familiar atoms exist. )

The one dimensional quantity of distance is the same no matter how many dimensions you may have. A diameter is a distance. So let's assume our 4D atoms have the same diameter as 3D ones. We are already in fantasy land so we may as well have simplicity. The same mass too. Then for the sake of definiteness let's say that 4D things have the same number of atoms as the corresponding 3D things. Surface area is measured in the number of atoms exposed on that surface. Now we can compare 3D and 4D objects in a systematic way.

An extreme example is a string of single atoms. Its length and surface area is the same in 3D and 4D. At the other extreme is a sphere, but let's make it easy and go with a cube. A 4D cube has more surface area than the corresponding 3D one. Consider a cube with a trillion atoms. The 3D cube has ten thousand atoms per edge, the 4D cube has only a thousand. The 3D cube has 6*10^8 atoms on its surface, the 4D cube has 8*10^9, that is 13 1/3 more surface area. Go to a trillion trillion atoms and the ratio becomes 130 1/3. The ratio increases with the twelfth root of the number of atoms. By the time one gets to a human body the increase in surface area and decrease in proportions is about a thousand, while with the Earth it is ten thousand.

The huge increase in surface area means that objects have much more drag. Jump off of a cliff and with a thousand times more drag your terminal velocity is something like 10 mph. And with a thousand times more lift one might have difficulty approaching that terminal velocity. Not only that, muscles though tiny are a thousand times stronger. This is because the strength of a muscle depends on its cross section. A cross section is an virtual area, so it is a thousand times greater. So a muscle though only 500 micrometers long would be a thousand times stronger than ours. With all that lift and strength one can imagine flying elephants. Dumbo would be only a centimeter tall but equally massive as a 3D elephant. Not only that, bones would be a thousand times stronger as would be the connection between bone and muscle. The wings can be fairly small as with insects, not dominating as in birds. Indeed the bird design would have too much surface area, too much drag and difficulty with control in winds. Dumbo rules. Birds lose.

The same sort of thing plays out in our world. The scarab can lift over a thousand times its own weight for these very same reason. It's surface to volume ratio is about thousand times greater than ours so its muscles have a proportionally greater cross section. A scarab can achieve equivalent to a human lifting a hundred tons in a single go.

I thought there would be an increase in friction. Could 4D people climb up blank walls? I'm not sure yet but I think no. Insects owe such abilities to specially adapted feet. Friction is supposed to be independent of surface area. How about walking on water, taking advantage of surface tension? Dunno, but today I'm guessing not. Insects do it with special hydrophobic feet.

[PatrickPowers]

Some more consequences. With all that extra drag flight speeds would not be that high. Let's guess ten kilometers an hour. That might not seem like much, but relative to you body size that's like being able to fly 10,000 kilometers an hour. Recall that the world is ten times smaller relative to you than in 3D. You would be able to fly around the entire world in about twenty minutes. Since the world is round, this means that assuming good weather you can get anywhere you like in ten minutes. With that sort of travel ability a passenger airplane would be severe overkill. It would be going to a good deal of trouble to solve a problem you do not have. Indeed any sort of personal transportation machine seem excessive. Automobiles and roads and their support systems are difficult to make. It seems not at all worthwhile. You wouldn't even need a bicycle. The only purpose for machines would seem to be to move freight. But you are so strong you can carry ten tons without difficulty. Golly, that's a truly different world. All the machines we have to increase our strength and speed begin to seem superfluous. There's got to be a catch, yes?

The first catch I can think of is that the energy expended during such tasks increases. For such feats food intake has to increase drastically. Eating literally a ton of food for a meal is a bit too mind boggling. I'm going to stop here, today at least.

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The speed of sound would be about the same.. You can measure density in average distance between atoms times average atom mass. We're assuming that atoms behave the same in 4D so densities would be the same in 4D. That means the speed of sound doesn't change. Then could you fire a bullet or shell around the world? Maybe, but there might be too much drag. Not an avenue I want to go down today.

[PatrickPowers]

Now turn attention to 1D things like pipes, rods, and spaghetti. The upshot is that rods win big, pipes maybe win too, while spaghetti is unchanged.

Let's say you got a rod that gets the 4D atoms treatment. You have three basic choices. You get a rod that is proportionally either X times stronger, X times longer, or X times lighter. Compromises are possible. X is maybe 50 for the smallest rods to 500 for massive construction beams.

Same thing with pipes but here the main concern is capacity to move liquids or gasses. Either the capacity is X times higher, the pipe is X times longer, or the diameter is X times less.

Spaghetti is food so you want the same length and mass. That makes it X times stronger, but who cares about that.

[quickfur]

Be careful with generalizations to higher dimensions. The n-cube increases in volume much faster than the n-sphere as n increases, so using an n-cube as a simplication may lead to completely different results than using an n-sphere! The volume ratio of sphere to cube approaches zero as dimension increases without bound. So how much your surface-to-volume ratio changes depends a lot on what kind of shape you take on as you transition to a higher dimension.

[AsgharH247]

I found this discussion very interesting. I found it very awesome how people can apparently fly around the entire 4D world in just 20 minutes!

I have also been wondering what atomic orbitals would look like in 4 dimensions. I'm pretty sure there is one s-orbital and four p-orbitals instead of three, but I have no idea about d, f, and g+ orbitals. I think this would make Na a halogen, Mg a noble gas, and Al an alkali metal. Other than that, I'm not really sure.

When it comes to orbital hybridization, the 4D analog of carbon actually has 7 protons/electrons, and it can do sp, sp2, sp3, or sp4 hybridization. It can form 5 bonds and quadruple bonds.

(I'm a newcomer, so I'm sorry if I went off topic)

[quickfur]

I found this discussion very interesting. I found it very awesome how people can apparently fly around the entire 4D world in just 20 minutes!

I have also been wondering what atomic orbitals would look like in 4 dimensions. I'm pretty sure there is one s-orbital and four p-orbitals instead of three, but I have no idea about d, f, and g+ orbitals. I think this would make Na a halogen, Mg a noble gas, and Al an alkali metal. Other than that, I'm not really sure.

When it comes to orbital hybridization, the 4D analog of carbon actually has 7 protons/electrons, and it can do sp, sp2, sp3, or sp4 hybridization. It can form 5 bonds and quadruple bonds.

(I'm a newcomer, so I'm sorry if I went off topic)

4D chemistry is pretty much up to your imagination.

Because, unfortunately, Schroedinger's equation for the 4D hydrogen atom has no local minima; the only solution is at r=0. I.e., the direct analogues of 3D atoms in 4D are inherently unstable and will instantly collapse. There are no orbitals (no local minima), so none of the familiar 3D chemistry would work in 4D; if any atoms were to exist in 4D at all, they must necessarily be of a completely foreign structure that's different from the way 3D chemistry works. Which means that anything we try to generalize from 3D chemistry is essentially just based on fantasy and speculation, so it can literally be whatever you want it to be.

[AsgharH247]

Oh, sorry. Thanks for the clarification.

[PatrickPowers]

Be careful with generalizations to higher dimensions. The n-cube increases in volume much faster than the n-sphere as n increases, so using an n-cube as a simplication may lead to completely different results than using an n-sphere! The volume ratio of sphere to cube approaches zero as dimension increases without bound. So how much your surface-to-volume ratio changes depends a lot on what kind of shape you take on as you transition to a higher dimension.

A thought provoking post.

My view is that once the number of dimensions becomes high the cube becomes impractical. A 10D cube has 1024 corners and even a square has 512. It is just too hard to make such things and the surface-to-volume ratio is too high requiring too much in the way of materials. A cylinder is much simpler. Top, bottom, sides, that's it. I dare say that the cylinder dominates. The cylinder is quite common even here in 3D, and it's advantages increase exponentially with the dimension.

The N-cube has the advantage of packing without any wasted space. It maximizes contact between boxes when tightly packed, increasing strength. But it seems to me that in higher dimensions it isn't worth it. The corners get sharp, pointy, and very numerous,
hence either weak or dangerous.

So how about as the unit of volume. If you are in N-D then you have to multiply N numbers to get your volume. With a cylinder, three numbers, height and diameter and pi.

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So I calculated some things that would happen if you had a cube and cylinder of equal volume. As N increases the cube's diameter (maximum distance from surface to surface) and surface area go to infinity while the length of an edge goes to zero. These seem to me like problems.

As to how much space is wasted by packed cylinders I haven't worked it out. It seems to me that it's going to be 1-pi/4 in any number of dimensions, but I'm not sure about that.

In sum I get the feeling that in dimensions higher than 5 the cube will get used about as often as the octahedron is in ours.

[quickfur]

Why not spheres? Spheres have the advantage of minimal surface area to volume ratio. As the number of dimensions go up, the number of directions a pointy shape needs to extend increases. An n-sphere minimizes how many vertices you need to enclose an n-dimensional volume. Heat dissipates very fast in higher dimensions; a unit of heat decreases in proportion to 1/r^d as the distance from the source increases. To maintain vital temperatures, an n-sphere would seem to be the ideal shape for minimizing heat loss. An n-cube shape would have far too much surface area relative to the enclosed volume, meaning that past a certain number of dimensions you'd be losing heat much faster than your cells can generate it, which does not bode well. You want to maximize the clustering of cells in order to maintain heat, and minimize surface area to reduce heat loss.

Of course, for practical purposes you probably wouldn't end up with a perfect n-sphere; I'd speculate that you'd probably do OK if you extended yourself in one or more dimensions to get higher-dimensional equivalents of cylinders or cubinders. Though you'd want to keep yourself rounded in the majority of your n dimensions, to prevent catastrophic heat loss. For the higher-dimensional equivalent of a food pipe, I'd surmise an n-dimensional cylinder with extension in 1 dimension, with the other (n-1) dimensions rounded. So basically an extruded (n-1)-dimensional sphere as the basic body shape. Limbs would likely be short, and only as numerous as necessary for locomotion / support. Too many limbs extending in too many directions will also suffer from the surface area to volume ratio problem.

[PatrickPowers]

Why not spheres? Spheres have the advantage of minimal surface area to volume ratio. As the number of dimensions go up, the number of directions a pointy shape needs to extend increases. An n-sphere minimizes how many vertices you need to enclose an n-dimensional volume. Heat dissipates very fast in higher dimensions; a unit of heat decreases in proportion to 1/r^d as the distance from the source increases. To maintain vital temperatures, an n-sphere would seem to be the ideal shape for minimizing heat loss. An n-cube shape would have far too much surface area relative to the enclosed volume, meaning that past a certain number of dimensions you'd be losing heat much faster than your cells can generate it, which does not bode well. You want to maximize the clustering of cells in order to maintain heat, and minimize surface area to reduce heat loss.

Of course, for practical purposes you probably wouldn't end up with a perfect n-sphere; I'd speculate that you'd probably do OK if you extended yourself in one or more dimensions to get higher-dimensional equivalents of cylinders or cubinders. Though you'd want to keep yourself rounded in the majority of your n dimensions, to prevent catastrophic heat loss. For the higher-dimensional equivalent of a food pipe, I'd surmise an n-dimensional cylinder with extension in 1 dimension, with the other (n-1) dimensions rounded. So basically an extruded (n-1)-dimensional sphere as the basic body shape. Limbs would likely be short, and only as numerous as necessary for locomotion / support. Too many limbs extending in too many directions will also suffer from the surface area to volume ratio problem.

Packing of cylinders is a lot simpler than spheres, which is well know as a difficult problem. In most dimensions it remains unsolved. Even in 3D it was 1953 before it could be proved the familiar arrangement was optimal. I have learned to stay well away from problems like these,

Cylinders are a linear unit of volume if the only thing that gets varied is the "height," so I figure that's what they would do.

As far as heat and moisture loss what would be done? Even in 4D it makes a big difference. It's beyond me. Maybe the denizens resemble the Michelin Man.

[quickfur]

Just because sphere packing is unsolved mathematically doesn't preclude it from occurring in nature. Or in this case, speculated n-dimensional nature.