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In a lab at Caltech Ken Libbrecht has been simulating snowflake formation and under absurdly exact laboratory conditions, they have been able to create almost identical snowflakes.

Clearly, in the real world, we do not observe identical snowflakes because each snow flake has different conditions as it falls from the sky to hit the Earth. Humidity, wind, temperature are not uniform enough to produce "identical" snowflakes. The process is too stochastic. Here is a cool infographic portraying all the different possibilities:

enter image description here

Note: To prevent a straw-man stalemate, I do not propose the snowflakes have to be 100% identical down to the very quark or subatomic level -- I'm not taking an absolutist position. I'm just using "identical" in the reasonable sense. Suppose even after studying two snowflakes carefully for a few minutes, you still can't find any differences.

Question

How can I construct a world such that the humidity, temperature (and other relevant factors that the answerer wants to address) are as uniform as possible?

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    $\begingroup$ Are you talking about a proper realistic world? I would image that planets which are too hot or too cold, it would be very easy to achieve uniform snowflakes simply because the snow flakes would not exist. I also doubt you would be able to have an entire world with uniform humidity and temperature because the equator will always end up a tiny bit closer to the sun than the poles and hence be warmer. $\endgroup$ – Shadowzee Jul 29 at 5:36
  • $\begingroup$ I disagree with both the OP and @L.Dutch's answer. The whole point of a stochastic process is that each outcome is not dependent on another or earlier outcome. There is no reason why two snowflakes in a million should not be found to be identical. If you don't get a match the first time, repeat with another million snowflakes, and sooner or later you will end up with a snowflake identical to another one. It's the same as rolling a die and working out the probability of getting 2 of the same number; except in this case, the die has a few billion permutations $\endgroup$ – nzaman Jul 29 at 12:51
  • $\begingroup$ I just like how P8 refutes Neil DeGrasse Tyson's critique of the Frozen2 Poster -- no 8-sided flakes. $\endgroup$ – April --Un-Slander Monica-- Jul 29 at 15:56
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You cannot: the size of a world is so large that, even with an extremely narrow statistical distribution of physical values, you will end up with noticeable differences, which would not emerge on a lab size environment.

At most you can scale up the lab environment to the size of a small building, but as you get larger than that, statistics will beat you.

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    $\begingroup$ And "small building" is the important point. A large building has its own weather, with air currents and temperature variations and humidity etc. $\endgroup$ – AlexP Jul 29 at 7:31
  • $\begingroup$ Really small. And really carefully controlled. My basement is cooler than the rest of my house during the summer, and the other way around in the winter. Sigh. I should get more insulation. $\endgroup$ – puppetsock Jul 29 at 13:27
  • $\begingroup$ What if we somehow segment the atmosphere into small boxes? I’m thinking like a world with a giant crystal lattice structure that hosts the atmosphere and maintains homogenous conditions in each of its cells. $\endgroup$ – SRM - Reinstate Monica Jul 29 at 14:12
  • $\begingroup$ Not to mention, humidity and temperature aren't even the biggest factors for determining snowflake shape, the size, shape, and composition of the dust particle around which the snowflake forms is. $\endgroup$ – Shufflepants Jul 29 at 17:54
  • $\begingroup$ @L.Dutch You make a good point. Maybe by compartmentalizing the whole planet into small building-sized labs? Now I'm just getting silly... $\endgroup$ – Arash Howaida Jul 31 at 9:39
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I'd like to offer a frame challenge, based in part on your note on straw-man stalemates:

Snowflakes are already all but identical

Snowflake classification does not have a particularly long history; while the ancients remarked that there were columns and planes, that is about as close a study as anyone was able to make until the 1880s.

Now, as it happens, I don't recall ever really seeing columnar snow crystals; I suppose I must have at some point, but all the ones I actually remember are planar. Similarly, I happen to recall that the Ancient Greeks described columnar snow crystals. All of this leads me to believe that for a given climate, you should expect to mostly see one type or the other, or combination ones in transition climates (or at any rate, than any given cloud should give one kind of snowflake or the other).

In any case; to the typical naked-eye observer, snow should be expected to consist of "reasonably" identical particles.

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If the temperature is uniform, snow won't form.

The only way I can imagine there being n ear-identical snowflakes is for there to be some kind of curse on the world, it's a simulation, or there's some funny-business going on with nano-machines.

Maybe with some future-technology it's possible that all of the snowflakes are "made uniform" after their formation. (this sort of reminds me of "Tree diagram" from raildex.)

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Frame-challenge

You are assuming that the relevant variables controlling ice crystallization are thermodynamic variables. While admittedly not an expert in fractals, the relevant variables to control crystal formation are microscopic variables such as the initial rotation, velocity, etc. of the water particles.

For a simple example, look at the Ising model. It has far less variables than a Ice Process, but already produces fractal formation highly dependent on environmental noise and initial conditions.

It's pretty messy math that I only have minimal experience with, but fractals are very difficult and certainly not something that can be identically reproduced.

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