Taking into account the estimated age of the Earth, the estimated number of snowfalls which occurred over its history, and the estimated number of snowflakes which formed in each snowfall, what is the probability that an exact snowflake configuration has occurred at least twice? At what point does the old saying that "no two snowflakes are alike" become false due to the sheer quantity of snowflakes which have formed in the history of time?
The most likely answer is that there are no two equal snowflakes, as the formation of snowflakes is extraordinarily complex.
It is estimated that there are 10 to 18 molecules of water in a snowflake and that these can be ordered in an almost infinite number of forms.
No one can say for sure that there are not a couple of identical snowflakes, but with a rare possibility, the bet seems pretty safe.
Snowflakes' shapes might seem random at first, but they form according to a series of simple physical interactions. It all starts when six water molecules slow down and settle into order with each other, freezing into a crystal. That crystal starts as a tiny speck of dust or pollen which catches water vapor out of the air and eventually forms the simplest of snowflake shapes: tiny hexagons called diamond dust. Then, randomness takes over. The snowflake’s arms extend outward from the hexagon as water molecules land on it.
Differences in temperature and humidity can give rise to an infinite number of snowflake shapes. In the rare case when two flakes look identical at the microscopic level, they still differ from each other at the atomic level.
That’s because about one in 3,000 water molecules in a snowflake contain a variant of the ordinary hydrogen molecule known as deuterium. These deuterium atoms are scattered across the millions of molecules in the flakes and it’s very unlikely that any could possibly fall in the exact same pattern. So the chance of two snowflakes being alike on the atomic level, even factoring in deuterium, is infinitesimally small. The number of possible arrangements of the 10^18 water molecules (in a snowflake) is a large number.
I second Daniel in the number of possible arrangements.
The issue is more complex. If you have only 2 snowflakes, the chance being identical would be 1/(10^18/3000)! assuming that the shapes are entirely random. This assumption is clearly not true. Also, the level of arrangement can be important, so you can nest the same shapes on a level of magnification, although the arrangement of the atoms are different. So, even 2 randomly selected snowflakes have the chance being identical (not too high, but have).
It is a false statement that all snowflakes must be different.
In spite of that, it is highly possible that the Earth has never had two identical snowflakes, since 70! (10^100) is a larger number than the number of atoms in the universe (10^82), so obviously (10^18/3000)! (it is approximately (3.33*10^15)!) is way larger number than the number of all the snowflakes all the time since even 70! is a way larger number.
You could try modeling snowflakes. This might be instructive if one condition were to apply: that snowflake formation is not random in time or space.
One way to approach parameterization in your model might be to use a genetic algorithm model based on coding a few real snowflakes. Alternatively, you could adapt a biological neural network.
The advantage of the evolutionary approach is that you could determine if the development path of real snowflakes is a random walk or alternatively if there is memory in the process, at least to the extent that each step in the development process is constrained by the previous step. Do time-series rules apply either in time or in space?
You may have to do a literature search to discover if anyone has done time-lapse photography of snowflake formation. Alternatively, crystallographers may have done comparative evolution of crystallization for various molecules including water. I would be surprised if nobody has done time-lapse photography of chrystal formation. And if not, that would be an opportunity to develop a new branch of the science.
For an applied mathematician this would be more fun than watching television.
Two Snowflakes Ever Alike?
1. "No Two Snowflakes the Same" Likely True, Research Reveals
https://news.nationalgeographic.com/news/2007/02/070213-snowflake...
Researchers do know enough to confirm that the "no two snowflakes are alike" adage is likely true for ... a microscope and claimed they found two that look alike. .
2. Who Ever Said No Two Snowflakes Were Alike? - The New York ...
https://www.nytimes.com/2016/01/23/science/who-ever-said-no-two...
Jan 23, 2016 · As the East Coast braces for the first ferocious snowstorm of the season, we wanted to revisit a frosty adage: Is it true that no two snowflakes are the ... .
3. Why no two snowflakes are the same - The Washington Post
https://www.washingtonpost.com/national/health-science/why-no-two...
The early snow was a perfect excuse to ponder an age-old question: Is it true that no two snowflakes are alike? ... it’s possible for two snowflakes to be identical. .
4. Is it true that no two snowflakes are alike? | Mental Floss mentalfloss.com/article/20296/it-true-no-two-snowflakes-are-alike Is it true that no two snowflakes are alike? ... Bentley sometimes told people that he had never seen two snowflakes that looked alike and published several magazine ...
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