{"id":57,"date":"2014-06-19T11:37:04","date_gmt":"2014-06-19T18:37:04","guid":{"rendered":"http:\/\/www.bellevuecollege.edu\/math\/?page_id=57"},"modified":"2022-02-07T15:42:38","modified_gmt":"2022-02-07T23:42:38","slug":"snowflake","status":"publish","type":"page","link":"https:\/\/www.bellevuecollege.edu\/math\/links\/mathsnips\/snowflake\/","title":{"rendered":"The Snowflake Curve"},"content":{"rendered":"\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><a href=\"https:\/\/www.bellevuecollege.edu\/wp-content\/uploads\/sites\/137\/2014\/06\/Snowflake2.gif\"><img loading=\"lazy\" decoding=\"async\" width=\"493\" height=\"144\" src=\"https:\/\/www.bellevuecollege.edu\/wp-content\/uploads\/sites\/137\/2014\/06\/Snowflake2.gif\" alt=\"Snowflake Curve Diagram\" class=\"wp-image-296 img-fluid \"\/><\/a><\/figure><\/div>\n\n\n\n<ol class=\"wp-block-list\"><li>Start with an equilateral triangle whose sides have length 1.<\/li><li>On the middle third of each of the three sides, build an equilateral triangle with sides of length 1\/3. Erase the base of each of the three new triangles.<\/li><li>On the middle third of each of the twelve sides, build an equilateral triangle with sides of length 1\/9. Erase the base of each of the twelve new triangles.<\/li><li>Repeat the process with this 48-sided figure. See the likeness to a crystal of snow emerge?<\/li><\/ol>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img loading=\"lazy\" decoding=\"async\" width=\"225\" height=\"257\" src=\"https:\/\/www.bellevuecollege.edu\/wp-content\/uploads\/sites\/137\/2014\/06\/Snowflake3.gif\" alt=\"Snowflake Curve Diagram\" class=\"wp-image-297 img-fluid \"\/><\/figure><\/div>\n\n\n\n<p>At the right, figure 4 is magnified by a power of two.<\/p>\n\n\n\n<p>The &#8220;limit curve&#8221; defined by repeating this process an infinite number of times, adding more and more, smaller and smaller triangles at each stage, is called the <strong>Koch&#8217;s SNOWFLAKE CURVE<\/strong>, named after Niels Fabian Helge von Koch (Sweden, 1870-1924).<\/p>\n\n\n\n<p>The snowflake curve has some interesting properties that may seem paradoxical.<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The snowflake curve is connected in the sense that it does not have any breaks or gaps in it. But it&#8217;s not smooth (jagged, even), because it has an infinite number of sharp corners in it that are packed together more closely than pebbles on a beach.<\/li><li>The snowflake never escapes the dashed square you see in figures 1-4, so it encloses a finite amount of area no larger than a credit card. On the other hand, at each step building the new little triangles adds more than one unit of length to the curve. To be precise, [4 \u00f7 3]n &#8211; 1 units are added at the nth step, so the length of the snowflake is larger than 3 + 1 + 1 + 1 + 1 + 1 + &#8230;&#8230;. = infinity. <strong>The snowflake curve is infinitely long, yet it would fit in your wallet!<\/strong><\/li><li>Under a magnifying glass, a little piece of the snowflake looks identical to a larger, unmagnified chunk. Objects that exhibit this kind of self-similarity are called <strong>FRACTALS<\/strong> and are of great research and applied interest in modern science and mathematics.<\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Start with an equilateral triangle whose sides have length 1.On the middle third of each of the three sides, build an equilateral triangle with sides of length 1\/3. Erase the base of each of the three new triangles.On the middle third of each of the twelve sides, build an equilateral triangle with sides of length <a class=\"read-more\" href=\"https:\/\/www.bellevuecollege.edu\/math\/links\/mathsnips\/snowflake\/\">...more about The Snowflake Curve<\/a><\/p>\n","protected":false},"author":115,"featured_media":0,"parent":61,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-57","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages\/57","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/users\/115"}],"replies":[{"embeddable":true,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/comments?post=57"}],"version-history":[{"count":3,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages\/57\/revisions"}],"predecessor-version":[{"id":2119,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages\/57\/revisions\/2119"}],"up":[{"embeddable":true,"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/pages\/61"}],"wp:attachment":[{"href":"https:\/\/www.bellevuecollege.edu\/math\/wp-json\/wp\/v2\/media?parent=57"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}