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Premise

On viewing this question, I wondered what the small-scale behavior of drops of liquid in that world would be. What would happen to small volumes of liquids in an atmosphere?

Calculations

For the purposes of this calculation, I believe gravity can be ignored, and to prevent immediate evaporation in a vacuum, an appropriate atmosphere is available.

The equations and derivations remain the same as in the linked question and the resulting answers.

Question

What shape would small volumes of liquids take in this world, if any?

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    $\begingroup$ Can you put the relevant part of that question in this post? $\endgroup$
    – L.Dutch
    Oct 12, 2021 at 8:37
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    $\begingroup$ Since the droplets will take the shape of any equipotential surface that encloses the volume using the cohesion forces - surface tension (small range, stronger than gravity, but no different in action from a spherical symmetry, like gravity is) why do you feel the answer in the question you cite is not appropriate here too? $\endgroup$ Oct 12, 2021 at 8:45
  • $\begingroup$ I don't know if it applies. However, if that's true, then the question can probably be closed. Thanks for the help, Adrian. $\endgroup$
    – Mous
    Oct 12, 2021 at 9:48
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    $\begingroup$ The equipotential surface must stand, otherwise any "particle" with higher energy will move to a position with a lower one. I'm not quite sure about the "spherical symmetry" in spaces with weird metrics, it's more of an intuition. Howevs, I know for sure that the surface will need to be minimum that is still able to enclose the droplet volume (that's what superficial tension forces guarantees - the cohesion forces are way higher than any "attraction" from the air molecules can be) $\endgroup$ Oct 12, 2021 at 12:13
  • $\begingroup$ I'm fairly certain that the minimum surface is a cube. $\endgroup$
    – Mous
    Oct 12, 2021 at 20:54

1 Answer 1

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Unlike gravity, which incidentally happens to produce surface-area-minimizing shapes in 3D under the L2 norm, surface tension actually does produce surface-area-minimizing shapes when minimizing energy, and that holds regardless of the space in which it is operating.

So, "what is the equilibrium, minimum-energy shape of a small liquid drop in the cubiverse?" turns out to be equivalent to "what shape has the minimum surface area for a given volume?"

Well, in our universe, the answer is "a sphere", and in the absence of any obviously better starting point, we might as well go ahead and ask what the geometric equivalent of a sphere is in the cubiverse, and see if we can do any better by perturbing that solution.

The cubiverse-geometry equivalent of a circle is a square, and the equivalent of a sphere is a(n axis-aligned) cube. Now, what happens if you try to shave off the corners of a square or a cube to make it look more like a Euclidean circle or sphere? Well, the internal area or volume will decrease... but the perimeter and area won't. Which means that altering a cube will result in a larger surface area to volume ratio--or, larger surface area for a fixed volume.

So, there you go. The equilibrium, energy-minimizing and surface-area-minimizing shape of a liquid droplet held together by surface tension in a universe operating under the infinity norm must be a cube--and more specifically, an axis-aligned cube.

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  • $\begingroup$ Thank you. I'll wait a while before accepting this, but this seems to be the best answer that one could give. $\endgroup$
    – Mous
    Oct 13, 2021 at 4:46

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