How Does Drag Affect Airplane Dynamics in a Four-Dimensional Universe?

[Hornbein]

Suppose ia Universe with four equal Euclidean space dimensions. Suppose further the unlikely situation that atoms and molecules like ours exist, there can be airplanes and ships and so forth. Suppose further that atoms have the same diameter as ours. Then an airplane of the same mass will have proportions about one thousandth of ours. Instead of one hundred meters long it is one hundred millimeters long. That's fine, since people will be the same, standing two millimeters tall. The question is, what about drag? That airplane has a thousand times the surface area. We measure this by calculating the number of atoms exposed on the surface. Redesign should be able to reduce that to maybe four hundred times as much as what we have here on 3D Earth but that is still quite a bit. A jet engine or propeller should be able to move just as much mass of air as they do in our world, so the thrust should be no less.

So I went to Wikipedia. It says drag is proportional to (the density of the fluid) * (the relative speed of the object)^2 * (the cross sectional area) * (the drag coefficient). While the cross sectional area is four hundred times higher, we can reduce the speed by a factor of twenty to compensate. That leaves the density of the "fluid." We can calculate the number of air molecules close to the surface of the plane. I don't know what that distance should be, but since the number of molecules is much smaller than with the airplane itself it might be an increase of only a factor of ten. To compensate we reduce the speed by another factor of three for a total of sixty. Our top speed is about 12km/hour. That's very good, because the Earth itself is even more compact than the plane, its circumference being about two and a half kilometers. The drag situation for ships is similar, so a two week voyage is reduced to a three hour tour. Heck, with ocean travel that fast what would be the point of enduring the discomforts of air travel?

So...how am I doing so far?

 

[hutchphd]

Suppose further the unlikely situation that atoms and molecules like ours exist,

What does this mean?

 

[Algr]

Suppose further that atoms have the same diameter as ours. Then an airplane of the same mass will have proportions about one thousandth of ours.

There is an inherent assumption that the 4d airplane would have the same number of atoms as a 3d one. This is valid, but not the most obvious extrapolation from our 3d world.

Picture a 2d square that can fit four circles inside. If you make a 3d cube with the same edge length as the square, and spheres with the same diameter as the circles, you would fit eight, not four of the spheres. Do that into the 4th dimension, and you get sixteen hyperspheres in a tesseract.

Suppose further the unlikely situation that atoms and molecules like ours exist

Again valid, but with complex consequences. You would not have analogs of our familiar elements because the number of electrons fitting in a given orbit would likely be higher, and there would be more room for different combinations of atoms to attach to each other in different ways. in 3D, helium is a noble gas because only two electrons can fit in the inner orbit. In 4d, I have no idea what would happen.

It occurs to me that the inverse square law for light would become distance cubed in four dimensions. You should fear the four dimensional realm. It is a dark, heavy, and complex place that would reduce our 3d mass to tiny helpless specks in its vast horror.

 

[jack action]

Suppose further that atoms have the same diameter as ours.

What is an equivalent 4D diameter to our 3D diameter?

Then an airplane of the same mass will have proportions about one thousandth of ours.

How do you come up with that conclusion?

If I go backward and compare the "2D world" to our "3D world", I can say "atoms have the same diameter as ours", no problem. But how can I compare the masses between the two worlds? Why would the masses be different if they have the same diameter? Maybe, comparing the area with the volume? If so, the "mass" of a circle compared with the mass of a sphere would be $\frac{m_{3d}}{m_{2d}} = \frac{\frac{4}{3}\pi r^3}{\pi r^2} = \frac{4}{3}r$, so the mass ratio depends on the object size?

 

[Algr]

What is an equivalent 4D diameter to our 3D diameter?

Diameter has the same meaning in 2d and 3d, so why not 4d?

But how can I compare the masses between the two worlds?

A two dimensional object has zero mass in our three dimensional world because one of its dimensions is zero, and hence it has no volume. The three dimensional equivalent of that object is a different story - Since we are into imaginary physics at this point, I think that you could pick any arbitrary number, say that that was the mass of a 4d proton, and then build the universe out from there. This is Sci-Fi Writing and World Building after all.

 

[jack action]

I think that you could pick any arbitrary number, say that that was the mass of a 4d proton, and then build the universe out from there.

Exactly. You could say the same about a 2D space as well: The mass of a 2D object in a 2D space can be found with $m = \frac{F}{a}$. What would that mean though? Somehow, someone would have to imagine what is the mass of a 2D proton, to begin with. There is no logic behind it when it comes to relating this to our 3D space. But the OP states:

Suppose further that atoms have the same diameter as ours. Then an airplane of the same mass will have proportions about one thousandth of ours.

What does it mean "same mass for the same proportions" in 3D versus 4D spaces? For example, if I have a 2D object represented as a circle with a given 2D mass, does the "equivalent" object in 3D a cylinder or a sphere? What does a 4D airplane look like? If the 3D airplane is made of aluminum molecules, what does a 4D aluminum molecule look like? What will be the properties of that material? It is impossible to look for equivalency.

Trying to define logic when comparing the two worlds is pure speculation. Anything can go.

 

[Algr]

Trying to define logic when comparing the two worlds is pure speculation. Anything can go.

On the one hand, that means Hornbein's ideas can't be disproved. But we can throw him off of the ship like Hippasus.

 

[Hornbein]

Again valid, but with complex consequences. You would not have analogs of our familiar elements because the number of electrons fitting in a given orbit would likely be higher, and there would be more room for different combinations of atoms to attach to each other in different ways. in 3D, helium is a noble gas because only two electrons can fit in the inner orbit. In 4d, I have no idea what would happen.

I've been told that the Schroedinger equation has no stable solutions in 4D, so atoms cannot exist. So already we are in science fiction land. Assuming that quantum mechanics somehow remains similar,there would be more orbitals to fill up a shell so the periodic table would be wider. But there would still be shells full or not, so noble gasses, halides, and metals would still exist. It seems chemistry would be essentially the same with just the atomic weights higher.

It occurs to me that the inverse square law for light would become distance cubed in four dimensions. You should fear the four dimensional realm. It is a dark, heavy, and complex place that would reduce our 3d mass to tiny helpless specks in its vast horror.

While there would be an inverse cubed law, distant things would tend to be closer so it would at least partially cancel out in a non-linear way. Besides, aren't we already tiny specks in vastness? Look at the advantages. It keeps the extraterrestrial hordes away.

 

[Hornbein]

Exactly. You could say the same about a 2D space as well: The mass of a 2D object in a 2D space can be found with $m = \frac{F}{a}$. What would that mean though? Somehow, someone would have to imagine what is the mass of a 2D proton, to begin with. There is no logic behind it when it comes to relating this to our 3D space. But the OP states:

What does it mean "same mass for the same proportions" in 3D versus 4D spaces? For example, if I have a 2D object represented as a circle with a given 2D mass, does the "equivalent" object in 3D a cylinder or a sphere? What does a 4D airplane look like? If the 3D airplane is made of aluminum molecules, what does a 4D aluminum molecule look like? What will be the properties of that material? It is impossible to look for equivalency.

Trying to define logic when comparing the two worlds is pure speculation. Anything can go.

That's the whole point of these dubious assumptions. They make comparisons possible. Diameters are one dimensional no matter how many dimensions there may be in the space. It's also nice that there are a number of unique and curious consequences. Proportions shrink while surface areas grow. The change is non-linear. For a solid sphere these differences increase in magnitude with the twelfth root of the number of atoms. The shape of a object is important too. Hollow objects follow a different rule, cylinders are different from spheres, thread is a yet different case, and so on. But I didn't bother to figure all this out. If drag makes an airplane impractical, then why bother?

I also implicitly assumed that elementary particles existed and had the same mass. There is no reason to believe that, but there you go. It's also worth noting that planetary systems are expected to be unstable and apparently impossible. But this example is so interesting and informative that I nevertheless use it heavily.

If one makes no assumptions then you're stuck with 4D Platonic solids and spheres. There is a Platonic solid that is possible only in 4D, that's nice. But there's only so much one can say if limited to such things.

 

[Hornbein]

If you find the idea of 4D atoms unacceptable then call it sphere packing in 4D. But then it isn't science fiction.

 

[jack action]

In the OP, you made another assumption that the thrust would be the same because "a jet engine or propeller should be able to move just as much mass of air as they do in our world". This implies that the air has the "same" density in both worlds even though the density would be now in kg/m⁴ in 4D. What does it mean to have $1\ kg/m^3 \equiv 1\ kg/m^4$?

About the drag equation, what is the equivalent "frontal area" in 4D? Does it still have two dimensions or does it have three now?

You are still stuck in making more assumptions which can lead to any conclusion you like. None of them more logical than the previous one.

 

[Hornbein]

In the OP, you made another assumption that the thrust would be the same because "a jet engine or propeller should be able to move just as much mass of air as they do in our world". This implies that the air has the "same" density in both worlds even though the density would be now in kg/m⁴ in 4D. What does it mean to have $1\ kg/m^3 \equiv 1\ kg/m^4$?

About the drag equation, what is the equivalent "frontal area" in 4D? Does it still have two dimensions or does it have three now?

You are still stuck in making more assumptions which can lead to any conclusion you like. None of them more logical than the previous one.

I don't reply to insults. If you are unwilling to discuss in a civil manner then please stay out of my thread. You can have your fun somewhere else.

 

[jack action]

They are not insults. Everywhere you see an interrogation point in my posts, it is a genuine question to which I cannot imagine an answer. I wouldn't even know where to begin to find one. (Full disclosure: I'm not an expert in these physics/mathematics fields.)

But, without any answers to these questions, I cannot see how anyone will answer yours:

So...how am I doing so far?

Let me go back to the OP such that you can help me answer your question:

Then an airplane of the same mass will have proportions about one thousandth of ours. Instead of one hundred meters long it is one hundred millimeters long. [...] That airplane has a thousand times the surface area. We measure this by calculating the number of atoms exposed on the surface.

I don't understand this. Where did that 1000 comes from? The way I understand it is that the airplane is smaller but has a greater surface area which seems weird. I would like to see those calculations you talk about.

Redesign should be able to reduce that to maybe four hundred times as much as what we have here on 3D Earth

How do you arrive at that conclusion?

It says drag is proportional to (the density of the fluid) * (the relative speed of the object)^2 * (the cross sectional area) * (the drag coefficient).

Is this equation relevant to 4D? I have a hard time believing it will be proportional to a 2D cross-sectional area. If so, the drag coefficient would be greatly affected for "comparable" airplanes. How? I cannot imagine. Even in our 3D world drag coefficients are difficult to evaluate theoretically.

That leaves the density of the "fluid." We can calculate the number of air molecules close to the surface of the plane. I don't know what that distance should be, but since the number of molecules is much smaller than with the airplane itself it might be an increase of only a factor of ten.

I really have a hard time following this. I would really appreciate seeing how you came up with the number "10".

because the Earth itself is even more compact than the plane, its circumference being about two and a half kilometers.

Why is the Earth more compact than the plane? What is the circumference of a 4D Earth? I would like to see how you arrived at the distance "two and a half kilometers" as well.

 

[Algr]

They are not insults.

Those last two sentences could have been better.

 

[hutchphd]

I really have a hard time following this. I would really appreciate seeing how you came up with the number "10".

There are many places one can hide numbers.
This is the time to walk away slowly. Do not touch the tar baby.

 

[Algr]

Surface area becomes volumetric area. Drag coefficient could be significantly reduced because in 4d the air molecules have many more ways to get out of the way.

On the other hand, the density of the air could be absolutely anything from zero to infinity. So ultimately there is no way to answer the question.

 

[Hornbein]

Assume a 4+1 Euclidian space. Then make the unrealistic assumption that life is more or less the same there, with real estate agents and pizza delivery men and so forth. If you don’t, then there can be no science fiction. This implies that the elementary particles and the basic fields exist, and the fundamental theories are somehow more or less the same. If you go with inverse cube laws then everything falls apart and again there can be no science fiction.

In four spatial dimensions solids are 4D and areas are 3D. Some things remain the same. Distances and diameters are 1D no matter how many dimensions you may have, so these are already comparable. A meter here is a meter there. To compare objects between 3D and 4D we assume that atoms have the same diameter in both. One can do better than that but it won’t make that much difference so I went for simplicity. Besides, they say quantum orbitals aren’t possible in 4D so why mess with it at all.

In summary
1. 4+1D Universe
2. Elementary particles, the fundamental fields, and QED and QCD are magically the same. Then atoms can exist and things have mass. We have gasses, liquids, metals, solar systems, and so forth.

We make two more assumptions to simplify calculations and standardize things so that everyone gets the same results of any calculation.
3. Atoms of the same element have the same diameter in 4D
4. Familiar objects have the same number of atoms

If you feel like doing more complicated things than these, go ahead.

With these four assumptions it is possible to systematically compare 3D and 4D objects with the same materials and proportions. The result depends on their shape. Large symmetrical solid things have smaller proportions and larger surface areas in 4D. If you somehow had a string of single atoms its proportions and surface area wouldn’t change at all. Other cases vary between these extremes. It’s possible to derive equations about common simple special cases that then can be used to make rough estimates for complex things like airplanes.

Surface areas of things can be compared by counting the number of exposed atoms in the 3D and 4D cases. Division gives you the ratio. If a 4D thing has a thousand times more exposed atoms of the same type than does the comparable 3D object then it’s surface area is a thousand times greater.

I expect that the importance of the cross sectional surface of the drag equation is the number of molecules of the dragging medium that are displaced. We can calculate this.

Velocity is one dimensional so it doesn’t change.

The drag coefficient requires some fiddling around with the units but what really matters is the effect the value has on drag. Would it be different in 4D? I don’t know but can’t think of any reason for it to differ. Maybe an expert would know more.

That’s everything you need for the drag equation. The incommensurate units are avoided by counting atoms instead.

In summary, I have attempted to come up with two minimal assumptions that make a science fiction world possible, and two more inessential assumptions to simplify and standardize calculations.

 

[Algr]

In four spatial dimensions solids are 4D and areas are 3D. Some things remain the same.

My opinion is that it is better to keep the names the same and invent new ones for 4d. So a square has 4 edges enclosing an area, a cube has 6 areas enclosing a volume, and a tesseract has 8 volumes enclosing a hypervolume.

Surface area becomes volumetric area.

In light of above, I made a mistake here. (It is too late for me to edit it.) I should have said surface area becomes surface volume.

 

[Hornbein]

"A 1974 NASA study found that for subsonic aircraft, skin friction drag is the largest component of drag, causing about 45% of the total drag." -- Wikipedia

Aha. In 4D skin friction increases drastically, so much so the lift-induced drag shouldn't matter at all. That means instead of using the drag equation I should use the skin friction equation, which depends on the surface area, not the cross sectional area.

This means that 4D transportation moves at lower speeds. For an airplane we likely will be moving at a speed low enough to allow laminar flow. This has about half as much drag as turbulent flow. The transition is gradual and around 20 knots. Worth dealing with but not a big difference.

In sum, the estimates I made in ignorance turn out to be pretty much correct. Or so I believe right now.