{"id":5991,"date":"2020-07-02T16:27:49","date_gmt":"2020-07-02T16:27:49","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=5991"},"modified":"2020-07-02T19:41:40","modified_gmt":"2020-07-02T19:41:40","slug":"estimating-the-fractal-dimension-of-the-spiders-of-mars","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2020\/07\/02\/estimating-the-fractal-dimension-of-the-spiders-of-mars\/","title":{"rendered":"Estimating The Fractal Dimension of the Spiders of Mars"},"content":{"rendered":"\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"731\" src=\"http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/Spiders.jpg\" alt=\"\" class=\"wp-image-5992\" srcset=\"http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/Spiders.jpg 974w, http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/Spiders-300x225.jpg 300w, http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/Spiders-768x576.jpg 768w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><figcaption> Spider Terrain   (NASA\/JPL\/UArizona )<\/figcaption><\/figure>\n\n\n\n<p>Above is an image of &#8220;dry ice spiders&#8221; on Mars.  Every spring the Sun warms up the  Martian south polar icecap and causes   jets of carbon-dioxide  gas  to erupt through the icecap.  These jets carrying dark sand into the air and spraying it for hundreds of  feet around each jet forming these wonderful spider-like structures. Notice how they look very much like fractals.<\/p>\n\n\n\n<p>Just for fun, and because I had a week off,  I wondered if I could calculate, or really estimate the fractal dimension of these structures.  To do this, I decided to use the box-counting dimension, which gives a bound on the fractal dimension.  Rather than give a careful description here I will point to the following <a href=\"https:\/\/en.wikipedia.org\/wiki\/Minkowski%E2%80%93Bouligand_dimension\"> link<\/a>.<\/p>\n\n\n\n<p>I did this in a few stages using Mathematica. First I needed to get a form of the image that can be used.  <\/p>\n\n\n\n<figure class=\"wp-block-gallery columns-2 is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\"><ul class=\"blocks-gallery-grid\"><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"731\" src=\"http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/data.jpg\" alt=\"\" data-id=\"5995\" data-full-url=\"http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/data.jpg\" data-link=\"http:\/\/blogs.scienceforums.net\/ajb\/?attachment_id=5995\" class=\"wp-image-5995\" srcset=\"http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/data.jpg 974w, http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/data-300x225.jpg 300w, http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/data-768x576.jpg 768w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" decoding=\"async\" width=\"974\" height=\"731\" src=\"http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/overlay.jpg\" alt=\"\" data-id=\"5996\" data-full-url=\"http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/overlay.jpg\" data-link=\"http:\/\/blogs.scienceforums.net\/ajb\/?attachment_id=5996\" class=\"wp-image-5996\" srcset=\"http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/overlay.jpg 974w, http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/overlay-300x225.jpg 300w, http:\/\/blogs.scienceforums.net\/ajb\/wp-content\/uploads\/sites\/10\/2020\/07\/overlay-768x576.jpg 768w\" sizes=\"auto, (max-width: 974px) 100vw, 974px\" \/><\/figure><\/li><\/ul><\/figure>\n\n\n\n<p>The image of the left is what I then used as the fractal that I wanted to estimate the box-counting dimension of.  The image of the right is the  overlay of the original image and the &#8220;extracted&#8221; image. This shows that it is not perfect, but it will do for now.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>Then I needed to write a Mathematica notebook that will give an estimate of the box-counting dimension.   Of course, before I applied it to the spider question, I tested it on fractals with know fractal dimensions. It worked very well in general.  Thus, I am confident that the estimate for the spider landscape is reasonable (modulo details of how I created the simpler black and white image). <\/p>\n\n\n\n<p>So, what do I get for the box-counting dimension?  <\/p>\n\n\n\n<p>dim<sub>box<\/sub>= 1.758  <code>\u00b1<\/code>  0.013<\/p>\n\n\n\n<p>Before anyone takes that figure too seriously, one should study many more of these spiders and see what range of values ones gets.  Also the sensitivity to how I have &#8220;extracted&#8221; the fractal from the original should be tested.    I have no idea if this has been done before or if it is interesting to anyone, like i said just for fun.  <\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Above is an image of &#8220;dry ice spiders&#8221; on Mars. Every spring the Sun warms up the Martian south polar icecap and causes jets of carbon-dioxide gas to erupt through the icecap. These jets carrying dark sand into the air and spraying it for hundreds of feet around each jet forming these wonderful spider-like structures. &hellip; <a href=\"http:\/\/blogs.scienceforums.net\/ajb\/2020\/07\/02\/estimating-the-fractal-dimension-of-the-spiders-of-mars\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Estimating The Fractal Dimension of the Spiders of Mars<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2,9],"tags":[],"class_list":["post-5991","post","type-post","status-publish","format-standard","hentry","category-astronomy","category-mathematical-art"],"_links":{"self":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5991","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/comments?post=5991"}],"version-history":[{"count":10,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5991\/revisions"}],"predecessor-version":[{"id":6004,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5991\/revisions\/6004"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=5991"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=5991"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=5991"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}