{"id":446,"date":"2019-06-27T17:54:47","date_gmt":"2019-06-27T17:54:47","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=446"},"modified":"2019-12-03T17:50:45","modified_gmt":"2019-12-03T17:50:45","slug":"koch-snowflake","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/koch-snowflake\/","title":{"rendered":"Koch Snowflake"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Snowflakes are amazing creations of nature. They seem to have intricate detail no matter how closely you look at them. One way to model a snowflake is to use a\u00a0fractal<\/em>\u00a0which is any mathematical object showing “self-similarity” at all levels. <\/p>\n\n\n\n

The Koch snowflake is constructed as follows. Start with a line segment. Divide it into 3 equal parts. Erase the middle part and substitute it by the top part of an equilateral triangle. Now, repeat this procedure for each of the 4 segments of this second stage. See Figure 1. If you continue repeating this procedure, the curve will never self-intersect, and in the limit you get a shape known as the Koch snowflake<\/em>.<\/p>\n\n\n\n

Amazingly, the Koch snowflake is a curve of infinite length!<\/p>\n\n\n\n

And, if you start with an equilateral triangle and do this procedure to each side, you will get a snowflake, which has finite area, though infinite boundary!<\/p>\n\n\n\n

Presentation\u00a0Suggestions:<\/strong>
Draw pictures. If they like this Fun Fact, ask them: can you figure out how to construct a 3-dimensional example? [Hint: start with a regular tetrahedron. See\u00a0Koch Tetrahedron\u00a0for what happens.]<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
You can see that the boundary of the snowflake has infinite length by looking at the lengths at each stage of the process, which grows by 4\/3 each time the process is repeated. On the other hand, the area inside the snowflake grows like an\u00a0infinite series, which is geometric and converges to a finite area! You can learn about fractals in a course on\u00a0dynamical systems.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Koch Snowflake.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
K. Falconer, Fractal Geometry.<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
Jorge Aarao <\/p>\n","protected":false},"excerpt":{"rendered":"

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