How would your body change if you were in a 4D space?

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Hornbein
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I don't believe this question will interest anyone else, but remains worthy of notice because the solution is a formula that depends on the 36th root of a number.

You unexpectedly find yourself in a space with four Euclidean dimensions. Fortunately your molecules have been rearrange into a four dimensional configuration. The dimensions of your body thusly have been reduced by a factor of the twelfth root of the number of your molecules. Calculations at endnote [0]. [We're dubiously assuming that atoms exist in 4D and that their diameter is the same as in 3D.]

That comes out to be a factor of 200 reduction in the proportions of a human body. Rearranged people would be about a centimeter tall.

An ant has a million times fewer atoms than a human. So the reduction is less by a factor of the twelfth root of a million, which is about 3. This means that in your new 4D world rearranged ants look three times longer, wider, and taller, because they ARE bigger relative to rearranged you. Their mass hasn't changed though.

An elephant goes the other way. An African elephant is reduced to the relative proportions of the Indian variety.

Now for the 36th root formula. Here in 3D you have a water pipe with diameter of half a meter. In 4D you want a pipe of the same length and the same capacity to carry water. What will be the diameter of that transformed pipeline relative to 4D you? Bigger or smaller? It turns out the answer is the diameter will be relatively smaller by a factor of the 36th root of the number of molecules in your body. Relative to your new 4D form it will be 6 times smaller than it was in 3D. To the new 4D you the diameter of the pipe would appear to be about 8 centimeters. Calculations at endnote [1].

Let's ditch that convenient but strange thing about molecules in the body and go direct to cross sections. Take the cross section to be half a millimeter, like a hypodermic needle. The 4D rearranged needle with the same flow will be 2 times larger in diameter relative to rearranged you, making for a one millimeter opening. Not bad. Better than ants. Calculations at endnote [2].------

ENDNOTES

[0] Take a cube full of n molecules. Each edge will contain n^(1/3) molecules. In a 4D space you can rearrange these molecules into a 4D cube with n^(1/4) molecules on each edge. The length of each edge decreased by a factor of n^(1/3) / n^(1/4) = n^(1/3-1/4) = n^(1/12).

The number of atoms in a 100kg human body is about N= 3*10^27.

N^(1/12) = 3*10^(27/12) = 3000^(1/12) * 10^2 = 2*100[1]The capacity to carry water depends on the number of molecules in a cross section of the pipeline. The one meter pipe for water was chosen because a sphere of water one half meter in diameter has about the same number of molecules as does a 100kg human body. (That's where that weird dependency comes from.) Make that number N. It is about 3*10^27. The number of molecules in a cross section of the sphere can be found by taking N to the 2/3 power. The diameter of the cross section in 4D with the same number of atoms is found by taking the sixth root of this number and dividing the diameter by this. (You can show this with plain ordinary 3D geometry/algebra with a disc full of molecules reformed to a sphere shape.)

N= 3*10^27
N^(2/3) = 2*10^18 molecules in the cross section
N^(2/3 * 1/6) = N^(1/9) = 1.2*10^3 = 1200

1200 is the factor by which the diameter is reduced. Now we see how the diameter is reduced relative to the brave new 4D man.
1200/200 = 6

More abstractly N^(1/9)/N^(1/12) = N^(1/9-1/12) = N^(1/36)[2] Calculation for hypodermic diameter in 4D.
2*10^18 water atoms in cross section of big pipe from endnote [1]
0.5m/0.5mm= m/mm = 10^3
(m/mm)^2 = 10^6 so the number of atoms in the cross section of the hypo is a million times less than that of the big pipe.

n = 2*10^18 / 10^(3*2) = 2*10^12 is the molecules in the hypo's cross section. Take the sixth root to get 100.

That needle will be 200/100 = 2 times larger in diameter in the 4D world. Not bad.
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No doubt reader is concerned. What if I should find myself cast into a 12 dimensional Euclidian space? To which proportions should I have my molecules rearranged? 1/3 - 1/12 = 1/4. You proportions would be reduced by the fourth root of the number of molecules in your body. That's a factor of about

3*10^(27/4) = 3000^(1/4) * 10^(24/4) = 3000^(1/4) * 10^6

which is about seven million. That gives a height of about 300 nanometers.

The number atoms in the Earth are about 10^50. So its diameter would be reduced by a factor of three trillion from 13 million meters to four micrometers. Its circumference would be 25 micrometers, so our human could walk around it in 150 steps. That seems about the same size as the Little Prince's planet.

The surface of said planet would be three trillions times greater so population would be limited not by crops but by the amount of breathable air.
 
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