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News from Mathnasium of South Jordan

Symmetry and Snowflakes

Jan 30, 2020

The year 2020 is upon us, bringing in a new decade. Who knows what that will bring? New technologies, new scientific developments, new experiences... The possibilities are endless. 

 
Today we are going to look at the cold, crystalline shape that is associated with January- a snowflake. Snowflakes are beautiful, intricate, and as we all know, unique. They are also closely studied by scientists, mathematicians, and enthusiasts alike. The mathematic appeal comes in the form of three basic mathematical ideas; pattern, symmetry, and symmetry breaking. 
 
So, what is a snowflake? A single snowflake is a molecular structure of an ice crystal. If we closely examined an ice crystal under normal conditions, we will always see a combination of molecules that have a six-fold symmetry. Snowflake molecules then make a honeycomb structure which causes an incalculable number of hexagonal axes of symmetry. 
 
But that's under normal conditions. If we change the conditions, that will change. Two key elements will change the structure of a snowflake, and they are temperature and moisture. When the temperature closes to 0° and humidity is high, the structure of a snowflake will be flowery, which is called a dendrite. If the temperature drops more, the snowflake will have a hexagonal plate structure. We don't exactly know why the structures change like this.
 
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The most important things that snowflakes have that mathematicians look for is symmetry and their hexagonal structure. These are important factors because they allow for transformations that are essentially unnoticeable. 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°, it does not appear that anything changes. Check out the images below for an example. 
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1- Counter-clockwise rotation by 120° | 2- Reflection through a vertical axis | 3- Reflection axes of a snowflake | Source

Snowflakes have also reflectional symmetry. If we put a mirror in the middle of a snowflake, there will be a reflection. For a snowflake, we can put a mirror along six different axes. With this, we can say that a snowflake has 12 symmetries- six from reflections, six from rotations. We can also claim that a combination of any number of transformations will give us the same shape. For instance, we can rotate our snowflake 60° two or three times in a row and flip it over, then it will be unchanged.
 
Snowflakes are a marvel of nature, and they captivate the mind of scientists and mathematicians alike. So next time you're caught in a snowstorm and enjoying a nice cup of hot chocolate, think about how such a small crystal can be so, well, perfect. 
- by Nicole Hamaker