{"id":535,"date":"2019-06-29T22:19:35","date_gmt":"2019-06-29T22:19:35","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=535"},"modified":"2019-12-03T17:52:07","modified_gmt":"2019-12-03T17:52:07","slug":"koch-tetrahedron","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/koch-tetrahedron\/","title":{"rendered":"Koch Tetrahedron"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

In\u00a0Koch Snowflake\u00a0we saw an interesting fractal snowflake-like object that is obtained when gluing smaller triangles iteratively to the sides of a big triangle.<\/p>\n\n\n\n

So, what happens if you do something similar to a tetrahedron? That is, suppose you take a regular tetrahedron (all side lengths the same), and glue to each of its triangular faces some smaller regular tetrahedra, as in Figure 1? (Each smaller tetrahedron is scaled down by a factor of 1\/2 from the larger one, and placed on each face in an inverted fashion, so that it divides the face into 4 equilateral triangles and covers the center one.)<\/p>\n\n\n\n

Then iterate this process: at each stage, take the new object, and glue still smaller regular tetrahedra (scaled by 1\/2 in the length of each side) on each of its triangular faces.<\/p>\n\n\n\n

You might think that you get a very jagged object after all the stages are completed, but surprisingly, in the limit, you get a perfect\u00a0cube!<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Draw a picture to help students to see what is going on. Challenge them to think about the object they get before telling them.<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
The cube you obtain has side length T\/Sqrt[2], where T is the length of one of the edges of the regular\u00a0tetrahedron\u00a0you started with.<\/p>\n\n\n\n

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

References:<\/strong>
R. Vakil, A Mathematical Mosaic<\/em><\/a>, 1996.<\/p>\n\n\n\n

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

In\u00a0Koch Snowflake\u00a0we saw an interesting fractal snowflake-like object that is obtained when gluing smaller triangles iteratively to the sides of...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[57,8,163],"class_list":["post-535","page","type-page","status-publish","hentry","tag-advanced","tag-geometry","tag-triangles"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/535","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=535"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/535\/revisions"}],"predecessor-version":[{"id":1506,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/535\/revisions\/1506"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=535"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}