This is more of a meme than a scientific fact, but it’s not _just_ an extension of the idea that no two objects are identical because there will always be atoms in different places, etc. There’s a reason we go around saying that “no two _snowflakes_ are identical” and not “no two pebbles or twigs or fish or whatever are identical”.
This reason is that classic big, flat, 6 sided snowflakes come in such a diversity of shapes that if you pop a bunch under a microscope (or even just look at them closely) you will be impressed at a huge diversity of variations on the basic six-sided pattern.
See [this picture](https://i2.wp.com/www.brainpickings.org/wp-content/uploads/2020/01/wilsonbentley_snowflakes22.jpg?resize=680%2C820&ssl=1) or [this one](https://s3.amazonaws.com/media.harvardartmuseums.org/production/file_uploads/Events/images/000/002/188/hero/Homer%20lecture%20with%20Nemerov%20image.jpg) for example. Unlike rocks or fish or twigs, which often look pretty similar even if they aren’t actually identical, snowflakes all _look_ distinctly different from each other.
The reason for this is that the ice crystal growth of a snowflake is very sensitive to things like the original particle that seeded the growth, the temperature, the humidity, wind currents, etc, that change from moment to moment as the crystal grows. And tiny changes can get magnified as the crystal grows. This produces the stunning variety of snowflake details which gave rise to the saying.
Basically, the saying “no two snowflakes are alike” really means “if you look at a collection of snowflakes, you will be impressed by how different they all look from each other in a way that is not true for most other things”.
But this also really only holds for “classic” snowflakes. Sometimes snow comes down as a bunch of tiny hexagons or as long thin columns, which pretty much all look alike.
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