![\"gasket.jpg\"](\"https:\/\/i0.wp.com\/scientopia.org\/img-archive\/goodmath\/img_183.jpg?resize=99%2C115\")
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I thought in addition to the graph theory (which I’m enjoying writing, but doesn’t seem to be all that popular), I’d also try doing some writing about fractals. I know pretty much nothing<\/em> about fractals, but I’ve wanted to learn about them for a while, and one of the advantages of having this blog is that it gives me an excuse to learn about things that that interest me so that I can write about them.<\/p>\n Fractals are amazing things. They can be beautiful: everyone has seen beautiful fractal images – like the ones posted by my fellow SBer Karmen. And they’re also useful: there are a lot of phenomena in nature that seem to involve fractal structures.<\/p>\n But what is a fractal?<\/p>\n The word is a contraction of fractional dimension<\/em>. The idea of that is that there are several different ways of measuring the dimensionality of a structure using topology. The structures that we call fractals are things that have a kind of fine structure that gives them a strange kind of dimensionality; their conventional topological dimension is smaller than their Hausdorff<\/em> dimension. (You can look up details of what topological dimension and Hausdorff dimension mean in one of my topology articles.) The details aren’t all that important here: the key thing to understand is that there’s a fractal is a structure that breaks the usual concept of dimension: it’s shape has aspects that suggest higher dimensions. The Sierpinski carpet, for example, is topologically one-dimensional. But if you look at it, you have a clear sense of a two-dimensional figure.<\/p>\n That’s all frightfully abstract. Let’s take a look at one of the simplest fractals. This is called Sierpinski’s carpet<\/em>. There’s a picture of a finite approximation of it over to the right. The way that you generate this fractal is to take a square. Divide the square into 9 sub-squares, and remove the center one. Then take each of the 8 squares around the edges, and do the same thing to them: break them into 9, remove the center, then repeat on the even smaller squares. Do that an infinite number of times.<\/p>\n When you look at the carpet, you probably think it looks two dimensional. But topologically, it is a one-dimensional space. The “edges” of the resulting figure are infinitely narrow – they have no width that needs a second dimension to describe. The whole thing is an infinitely complicated structure of lines: the total area covered by the carpet is 0! Since it’s just lines, topologically, it’s one-dimensional.<\/p>\n In fact, it is more than just a one dimensional shape; what it is is a kind of canonical<\/em> one dimensional shape: any<\/em> one-dimensional space is topologically equivalent (homeomorphic) to a subset of the carpet.<\/p>\n But when we look at it, we can see it has a clear structure in two dimensions. In fact, it’s a structure which really can’t be described as one-dimensional – we defined by cutting finite sized pieces from a square, which is a 2-dimensional figure. It isn’t really two dimensional; it isn’t really one dimensional. The best way of describing it is by its Hausdorff dimension, which is 1.89. So it’s almost<\/em>, but not quite, two dimensional.<\/p>\n Sierpinski’s carpet is a very typical fractal; it’s got the traits that we use to identify fractals, which are the following:<\/p>\n I thought in addition to the graph theory (which I’m enjoying writing, but doesn’t seem to be all that popular), I’d also try doing some writing about fractals. I know pretty much nothing about fractals, but I’ve wanted to learn about them for a while, and one of the advantages of having this blog is […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[86],"tags":[],"class_list":["post-463","post","type-post","status-publish","format-standard","hentry","category-fractals"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-7t","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/463","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=463"}],"version-history":[{"count":2,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/463\/revisions"}],"predecessor-version":[{"id":3662,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/463\/revisions\/3662"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=463"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=463"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=463"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"carpet.jpg\"](\"https:\/\/i0.wp.com\/scientopia.org\/img-archive\/goodmath\/img_184.jpg?resize=239%2C240\")
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