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# Mathematics of Snowflakes

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## Gabrielle Day

on 22 November 2013

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#### Transcript of Mathematics of Snowflakes

Mathematics of Snowflakes
What is a snowflake?
A snowflake is either a single ice crystal or a series of ice crystals which falls through the Earth's atmosphere. They begin as snow crystals which develop when microscopic supercooled cloud droplets freeze. Snowflakes come in a variety of sizes and shapes. Complex shapes emerge as the flake moves through differing temperature and humidity regimes, such that individual snowflakes are nearly unique in structure.
Snowflakes & Math
Many times we overlook the beauty and mathematical property's in nature, such as the Snowflake. In 1611 Johannes Kepler published a short report on the Six-Cornered Snowflake, which was the first scientific reference to snow crystals or snowflakes. Philosopher and mathematician René Descartes was the first to find a reasonably accurate description of snow crystals. His careful notes included observations of capped columns and 12-sided snowflakes this is a very rare forms.

Koch Snowflake
The Koch Snowflake (also known as the Koch Star) demonstrates infinity, one of the most intriguing concepts of math. Created by Swedish mathematician Helge Von Koch, it all begins as a line segment and is divided into three equal parts. An equilateral triangle is then created, using the middle section of the line as its base, and the middle section is removed. After this process, the result is a shape similar to the Star of David. The process is then repeated indefinitely.

Fractal Properties
The Koch Snowflake displays a property known as self-similarity. This means that as we continue to magnify the Koch Snowflake, each magnified section continues to look similar to the larger.
Symmetry
Uniqueness
A snowflake often has six-fold symmetry. The symmetry occurs because the crystalline structure of ice is six-fold. The six "arms" of the snowflake grow independently, and each side of each arm grows independently. Most snowflakes are not completely symmetric. The micro-environment in which the snowflake grows changes dynamically as the snowflake falls through the cloud, and tiny changes in temperature and humidity affect the way in which water molecules attach to the snowflake. Since the micro-environment are nearly identical around the snowflake, each arm can grow in nearly the same way. However, being in the same micro-environment does not guarantee that each arm grows the same.

Snowflakes form in a wide variety of intricate shapes,so no two are alike. Although possible, it is very unlikely for any two randomly selected snowflakes to appear exactly alike due to the many changes in temperature and humidity the crystal experiences during its fall to earth. Attempts to find identical snowflakes by photographing thousands of them with a microscope from 1885 by Wilson Alwyn Bentley found the wide variety of snowflakes we know about today.
Project Objective
Math and nature are complex structures. Using simple geometrical shapes and measurements to replicate a snowflake on paper or actually growing an ice crystal with the proper materials. Projecting snowflake structures is used as a way to simulate these geometrically simple structures.
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