Hyperdimensional Earth

[Hornbein]

Let’s dubiously assume that these 4D atoms would have the same radius as 3D atoms. Let’s further dubiously assume that a 4D HyperEarth has the same number of atoms as 3D Earth. Then the diameter of HyperEarth would be less than a kilometer. Four dimensional Earth is much more compact.

Wouldn’t that be too small? Not really. We don’t care about the diameter, we live on the surface. What matters is the surface area. There would be a thousand times more atoms on the surface of 4D Earth than there are on the surface of 3D Earth. Could it be that there is much more space for real estate on the surface of HyperEarth? That HyperEarth provides a much bigger living space than does Earth? It seems that way.

 

[andrewkirk]

Very interesting.
But it's not comparing like with like. We measure real estate in our 3D world in squared length units. On the 3D hypersurface of 4D earth they'd measure it in cubed length units. The two are not comparable.
When we use incommensurable units, comparisons and orderings will change as we change the scale. In the OP's scale the breadth of the object is more than one unit. If we instead chose a unit in which the breadth was less than one unit, we'd get a different result: the longer the unit, the smaller the 4D 'space' appears compared to the 3D one.
How do they compare if we use units of nautical miles, instead of km?
What about light years?

 

[Hornbein]

Very interesting.
But it's not comparing like with like. We measure real estate in our 3D world in squared length units. On the 3D hypersurface of 4D earth they'd measure it in cubed length units. The two are not comparable.
When we use incommensurable units, comparisons and orderings will change as we change the scale. In the OP's scale the breadth of the object is more than one unit. If we instead chose a unit in which the breadth was less than one unit, we'd get a different result: the longer the unit, the smaller the 4D 'space' appears compared to the 3D one.
How do they compare if we use units of nautical miles, instead of km?
What about light years?

While it does appear that way I didn't actually pick a unit. The assumption is that the diameter and mass of atoms remains the same. (There's no reason to think atoms or even elementary particles could exist at all in 4D, but if we exclude them then there is little fun to be had.) So different atoms work out differently. The Earth is mostly iron so it shrinks significantly less than does a human being. In short, this is a property of the materials, not of the space.

Using this scheme a 4D sumo wrestler with the same mass of atoms as ours would be 3 millimeters tall, about the length of a small 3D ant, but still weighing 150kg. I say he is more "compact" than are we. To be quantitative an atom has a compactness of one. As the number of atoms in an aggregate grows its compactness also grows. This however happens remarkably slowly, with the twelfth root (!) of the number of atoms. Consider a 3D volume that holds a trillion atoms. In 4D such a volume has a tenth the diameter. I say it has a compactness of ten.

Getting back to 4D Earth, it is only ten times as compact as Mr. Sumo. (It would be about 80 times if the Earth were made of water instead of iron.) So a 4D Akebono would see ten times more curvature of the horizon than do we. Not much.

So if Akebono hops on an airplane does he get across the Pacific ten times faster? Or is it that California is only 250 meters from Nihon so it's almost no time at all? Hmm, gotta work on that one. 4D human-scale things have a lot more surface area than ours, so there is much more friction and drag. Cross-Pacific travel could take either more time or less time.

 

[Hornbein]

I'm thinking like this. Let's say our 4D airplane with the same mass as a 3D one has a hundred times as much surface area. That means it has about fifty times as much drag. Fortunately the area of its lifting surfaces also increases a hundred times, so it would have one hundred times as much lift. That seems like enough lift to take off at quite a low speed.

Now there is a big advantage that the 4D Earth has only about 1/10,000 the circumference of 3D Earth. So even if your airplane can fly at a maximum of one knot, you are ahead of the game. Akebono can fly to LAX in half an hour.

 

[Algr]

I'm not clear on why a larger object would see a different percentage of linear reduction than a smaller one. It does seem to me that your 4D airplane would have an additional challenge - far more ways to get lost!

 

[snorkack]

I'm not clear on why a larger object would see a different percentage of linear reduction than a smaller one.

On the initial and expressly dubious assumption that the number of atoms is the same.
Clearly if 1,7 moles of water - 1*10²⁴ molecules - are the same number in 3D and 4D then in 3D 30 ml of water has about 3√(1*10²⁴)=10⁸ molecules on each edge, while in 4D, the same volume of water has just 4√(1*10²⁴)=10⁶ molecules per edge. Pack 2¹²=4096 cubes together - in 3D, you need 16 times bigger cube, in 4D just 8 times bigger cube.

 

[Hornbein]

I'm not clear on why a larger object would see a different percentage of linear reduction than a smaller one. It does seem to me that your 4D airplane would have an additional challenge - far more ways to get lost!

The diameter of a 3D sphere grows with the cube root of the number of identical atoms. The diameter of a 4D sphere grows with the fourth root of the number of identical atoms. The ratio of the cube root and fourth root is the twelfth root.

n^1/3 / n^1/4 = n^1/12

 

[Hornbein]

Surface area of 4D relative to 3D grows with the twelfth root.

n^3/4 / n^2/3 = n^1/12

 

[Hornbein]

Surface area of 4D relative to 3D grows with the twelfth root.

n^3/4 / n^2/3 = n^1/12

This means that a 4D human body with the same number of atoms as a 3D one has two hundred times more atoms in its surface area. The increase in drag would be such that it might be possible to skydive without a parachute.

 

[snorkack]

This means that a 4D human body with the same number of atoms as a 3D one has two hundred times more atoms in its surface area. The increase in drag would be such that it might be possible to skydive without a parachute.

Surface volume, that is. Or is it even "surface"?

 

[Hornbein]

Surface volume, that is. Or is it even "surface"?

I'm counting the number of atoms of a solid that are in contact with to air, water, or whatever surrounds the object. I say that in 4D surfaces are 3D. Paint comes in quartic centimeters.