The Fascinating Mathematics Behind Snowflakes
It’s so mystical when you leave home in the morning in the snow. Snowflakes fly around in the measureless universe, falling to the ground and blanketing the ground. Or a snowflake landing on your nose. You don’t see any flowers because flowers cannot flourish in the cold. It’s so noteworthy that you come to realize that no two snowflakes are alike. It is like the uniqueness of a snowflake is so perfectly controlled by a divine force. The individuality of a snowflake’s structure is parallel to human life. Like snowflakes, everyone has a different story to tell.
I am not the only one who thinks about snowflakes. Many mathematicians also think about snowflakes, and they think about patterns because snowflakes are particularly important for three basic mathematical ideas; pattern, symmetry, and symmetry breaking.
For instance, an unfamous scientist, “the Snowflake Man,” Wilson Bentley, took pictures of snowflakes almost every day and observed them until he died. He shared his thoughts in his notes.
“Under the microscope, I found that snowflakes were miracles of beauty; and it seemed a shame that this beauty should not be seen and appreciated by others. Every crystal was a masterpiece of design, and no one design was ever repeated. When a snowflake melted, that design was forever lost. Just that much beauty was gone, without leaving any record behind.”
- Wilson Bentley
When I checked the Oxford dictionary, there are three definitions of the word “pattern.” Two of the descriptions below are very important for this post.
Pattern: A repeated decorative design or an example for others to follow.Here are some pictures of astonishing structures of snowflakes. When we check the images and delve deeper into each snowflake one by one, we will see that although the detailed structures of the snowflakes are utterly different. However, they have something in common, which is symmetry and a hexagonal structure.
To see more mesmerizing snowflakes pictures, you should check Wilson Bentley’s mesmerizing snowflakes collection or Vivian Wu's snowflake generator below:When I take a close look at a snowflake, the beauty of the combination of ice molecules fascinates me every time because they are unique. However, uniqueness is not the point here. The things that make snowflakes important objects for mathematicians are symmetry and their hexagonal structure. Math people have a lot of interest in transformations, and they like moving objects a lot. And surprisingly, if a thing is symmetric, transformations would not even be noticed by someone.
To be more precise, when you have a hexagonal symmetric snowflake or a different object, and when you rotate it any direction 60°, 120°, 180°, 240°, 300°, and 360°, people are watching, you wouldn’t realize anything. If you check the images below, you will see rotated shapes but no difference: the same shape and place.
Snowflakes also have reflectional symmetry. If we stand in front of a mirror, the reflection of ourselves looks the same. Hence, there will be a reflection if we put a mirror in the middle of a snowflake. For a snowflake, we can arrange a mirror in six different ways. We can now say that a snowflake has 12 symmetries. 6 from reflections and six from rotations.
I think now we can define symmetry as a transformation that leaves things unchanged. And also, we can claim that a combination of any of the transformations will give us precisely the same shape. For instance, we can rotate our snowflake 60° two or three times in a row and flip it over, and then it will be unchanged.
At this point, you can ask this question:
“You have all these fancy symmetries for this particular snowflake. But, can you have the same group of symmetry for all other snowflakes?”
Snow is a molecular structure of an ice crystal. And ice is a structured substance. It is a different form of water. When the water cools down, and the molecules move more slowly, they begin to affect how the molecules line up — hydrogen atoms of one water molecule bond with two oxygen atoms. And, as water freezes, the molecules arrange into hexagons. They prefer to stay as far away from each other as possible, making them take up more space. And that large space affects density. The density of ice becomes less dense than water, and it means that ice floats. Almost all other liquids have a higher density when they freeze.When we examine an ice crystal profoundly under normal conditions, we always see a combination of molecules with six-fold symmetry. Snowflakes molecules make a honeycomb structure, and that causes a limitless number of hexagonal symmetries. By the way, molecules are three-dimensional structures.
Okay, we saw the structure of a snowflake under normal conditions. But, what if we change the conditions?
After his experiments, Johannes Kepler answered this question and wrote a book about snowflakes, The Six-Cornered Snowflake.
Two key elements affect the structure of a snowflake; temperature and moisture. Whenever temperature and the amount of moisture change, the structure of a snowflake changes. If you check the snow crystal morphology diagram below, you will see that when the temperature closes to 0° and humidity is high, the structure of a snowflake will be flowery. Flowery structures are called dendrites. When you make it a little colder, the structure will be fancy hexagonal plates. We can apply so many combinations and get different structures.
Professor of physics Kenneth G. Libbrecht is the owner of the diagram below. And he said, “It’s a mystery as to why snowflake shapes go from plates to columns to plates to columns as the temperature lowers. That’s one of the things I’ve been trying to understand. It has been a mystery for about 75 years, and it’s still unsolved.” in a PBS interview. He also has a pretty interesting book about snowflakes.
In the end, although almost all the structures of snowflakes are hexagonal, some of them are not mainly hexagonal. For instance, some snowflakes have tree structures. The snowflake you can see below has branches, and each branch has tiny branches.
But why the structure of this snowflake is not hexagonal?
So far, we have talked about a picture taken at a particular instant. I mean, we have seen the picture of the snowflakes’ motion for the smallest amount of time that can be measured. (This is about calculus.) However, a snowflake never stops going around in the air. They tend to oscillate. That means the shape of the snowflake is changing all the time. But how? When you see a snowflake in the air, it changes its place after a second because it would be whirled around and under different conditions at that time. And this will happen until the snowflake fall to the ground. We know from the diagram, temperature and the amount of moisture always affect the shape of a snowflake. Yes, on a small scale, conditions are almost the same, but conditions are different on a larger time scale. And this change will change every corner of a hexagonal snowflake every time, and we will get a different structure. That is the main reason behind the variety of snowflakes structures and uniqueness.
Finally, we can say that a snowflake can preserve its six-fold symmetry all the time. I think we have another reason to love mathematics. I want to finish my writing with Hermann Hankel’s words:
“In most sciences, one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.”
- Hermann Hankel** Note: I get commissions for purchases made through links in this post.
