The Math Mom: Snowstorm Warning. No Shoveling is Required.

In the America's Northeast, we take pride in surviving and enjoying the thick snowy winters. Many of you who live much closer to the belly of the globe do not get to experience the joy and frustration of real snowstorms. What if math could help me sprinkle snowflakes, to share with you online? Here is a fascinating Snowflake Maker application created by a friend and colleague, Henry Kaufman:

This beautiful holiday card from a few years ago looks like a snowman's needlework. Try pressing “Stop,” and “Start “ to observe the creation process at work. Press “Restart” and then “Start” to create your own snowflakes and notice how each is very different from another.

In nature, each snowflake is unique, but they are almost all symmetric and have six equal sides (aka regular hexagons). Snowflakes are ice crystals, and the water molecules in an ice crystal form a hexagonal symmetrical shape:

Wikipedia references studies that explain why no two snowflakes are alike. Each snowflake is composed of roughly 10¹⁸ water molecules that stick together, at different rates and in different patterns, depending on the changing temperature and humidity within the atmosphere when the snowflake falls through it on its way to the ground. A bit similar to us navigating through life and accumulating experiences, friends and objects that shape our views, behaviors and appearances.

The virtual snowflakes that you can interactively create above are designed by a process imitating a real evolution of a snowflake and are similarly unique in their patterns. These snowflakes grow gradually and symmetrically in each of its six directions. On each step, a circle representing a water molecule is dropped at some random distance from the existing snowflake cluster. This circle is a starting point from which a thin thread is elegantly woven until its end is within some small distance from the existing water droplets. At this point, a new droplet of water is added to the snowflake and the thread disappears. The same process is repeated over and over with new water droplets. The thin thread’s beautiful path is created by simulating a process called Brownian motion. The name of which comes from the Scottish botanist, Robert Brown, who first observed and recorded the interesting motion pattern of pollen particles floating in water. This motion is observed and simulated by letting one big particle be pushed from the sides by many smaller, chaotically moving particles. Imagine yourself at the center of a four-year-olds’ soccer practice. From the sky, your bumpy motion path will look similar to the thread weaves that appear and disappear on the card above. A more scientific demonstration can be found here.

Enjoy observing how snowflake is knitted from thousands of particles and use the panning and zooming controls to wrap yourself in this crafty blanket. Imagine the gentle tickle of snow on your face. Share the Snowflake Maker and fascinating snowflake geometry with your kids and reveal a secret: when in doubt whether the snow you are skiing on is artificial or real, look closer at the pattern. Only nature or a computer simulation like above can create elaborate crystal arrangements: