A point has dimension 0, a line has dimension 1, and a plane has dimension 2. But did you know that some objects can be regarded to have “fractional” dimension?<\/p>\n\n\n\n
You can think of dimension of an object X as the amount of information necessary to specify the position of a point in X. For instance, a block of wood is 3-dimensional because you need three coordinates to specify any point inside.<\/p>\n\n\n\n
The standard Cantor set has fractional dimension! Why? Well it is at most 1-dimensional, because one coordinate would certainly specify where a point is. However, you can get away with “less”, because the object is self-similar. At each stage, you only need to specify which 2 out of 3 segments a point is in. Mathematicians have developed a notion of “dimension” which for the standard Cantor set works out to be: <\/p>\n\n\n\n
ln(2)\/ln(3) = 0.6309… <\/p>\n\n\n\n
Most other “fractals” have fractional dimension; for instance a curve whose boundary is very, very intricate can be expected to have dimension between 1 and 2 but closer to 2. This concept has been applied in other sciences to describe structures that appear to have some self-similarity, such as the coast of England or gaseous nebulae in interstellar space.<\/p>\n\n\n\n
Presentation Suggestions:<\/strong> The Math Behind the Fact:<\/strong> How to Cite this Page:<\/strong> References:<\/strong> Fun Fact suggested by: <\/strong> A point has dimension 0, a line has dimension 1, and a plane has dimension 2. But did you know...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,8,12,113],"class_list":["post-303","page","type-page","status-publish","hentry","tag-easy","tag-geometry","tag-other","tag-standard-cantor-set"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/303","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=303"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/303\/revisions"}],"predecessor-version":[{"id":1450,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/303\/revisions\/1450"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=303"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=303"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
It makes sense to do this Fun Fact after doing the one on the standard Cantor set.<\/p>\n\n\n\n
Actually, the notion of “dimension” can be extended to crazy sets in many different ways. One notion isbox dimension<\/em>, and another is Hausdorff dimension<\/em>. These notions agree for the standard Cantor set and many other sets.<\/p>\n\n\n\n
Su, Francis E., et al. “Fractional Dimensions.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
K. Falconer, Fractal Geometry<\/em>.<\/p>\n\n\n\n
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"