{"id":1178,"date":"2015-03-16T12:00:14","date_gmt":"2015-03-16T17:00:14","guid":{"rendered":"https:\/\/blogs.mathworks.com\/cleve\/?p=1178"},"modified":"2015-03-12T16:15:10","modified_gmt":"2015-03-12T21:15:10","slug":"an-ornamental-geometric-inequality","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/cleve\/2015\/03\/16\/an-ornamental-geometric-inequality\/","title":{"rendered":"An Ornamental Geometric Inequality"},"content":{"rendered":"\r\n<div class=\"content\"><!--introduction--><p>I came across this \"ornamental geometric inequality\" in a tribute to Lothar Collatz.<\/p><!--\/introduction--><h3>Contents<\/h3><div><ul><li><a href=\"#514d1937-8143-425c-b2da-35ff5e9d7f96\">An Inequality<\/a><\/li><li><a href=\"#f27ad13b-3f00-4adb-b396-1d0f1b40f817\">References<\/a><\/li><\/ul><\/div><h4>An Inequality<a name=\"514d1937-8143-425c-b2da-35ff5e9d7f96\"><\/a><\/h4><p>I mentioned the German mathematician Lothar Collatz in my post in January on his <a href=\"https:\/\/blogs.mathworks.com\/cleve\/2015\/01\/19\/the-three-n-plus-one-conjecture\/\">3n+1 Conjecture<\/a>. Here is a beautiful inequality from his 1934 paper titled \"Ornamental Geometric Inequalities\". Consider the subregion of $-4 \\le (x,y) \\le 4$ for which<\/p><p>$$ ||||x| - 1| - 1| - |||y| - 1| - 1|| \\ge \\frac{1}{3} $$<\/p><p>I've never seen four absolute value signs in a row before.<\/p><p>Let's visualize it with MATLAB.<\/p><pre class=\"codeinput\">   [x,y] = meshgrid(-4:1\/256:4);\r\n   z = abs(abs(abs(abs(x)-1)-1) - abs(abs(abs(y)-1)-1)) &gt;= 1\/3;\r\n   imagesc([-4,4],[-4,4],z);\r\n   axis <span class=\"string\">image<\/span>\r\n   colormap(copper)\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/cleve\/collatz_inequality_01.png\" alt=\"\"> <h4>References<a name=\"f27ad13b-3f00-4adb-b396-1d0f1b40f817\"><\/a><\/h4><p>Lothar Collatz, 1910-1990, <a href=\"http:\/\/www.math.uni-hamburg.de\/home\/collatz\/auflage3-beta2.pdf\">&lt;http:\/\/www.math.uni-hamburg.de\/home\/collatz\/auflage3-beta2.pdf<\/a>&gt;<\/p><p>Lothar Collatz, Gleichungen geometrischer Ornamente, (Kurvengleichungen mit Absolutstrichen). Z. mathem. u. naturw. Unterr. 64 (1934), 165-169.<\/p><script language=\"JavaScript\"> <!-- \r\n    function grabCode_cb48a6bf188a443fb4e238efc301463d() {\r\n        \/\/ Remember the title so we can use it in the new page\r\n        title = document.title;\r\n\r\n        \/\/ Break up these strings so that their presence\r\n        \/\/ in the Javascript doesn't mess up the search for\r\n        \/\/ the MATLAB code.\r\n        t1='cb48a6bf188a443fb4e238efc301463d ' + '##### ' + 'SOURCE BEGIN' + ' #####';\r\n        t2='##### ' + 'SOURCE END' + ' #####' + ' cb48a6bf188a443fb4e238efc301463d';\r\n    \r\n        b=document.getElementsByTagName('body')[0];\r\n        i1=b.innerHTML.indexOf(t1)+t1.length;\r\n        i2=b.innerHTML.indexOf(t2);\r\n \r\n        code_string = b.innerHTML.substring(i1, i2);\r\n        code_string = code_string.replace(\/REPLACE_WITH_DASH_DASH\/g,'--');\r\n\r\n        \/\/ Use \/x3C\/g instead of the less-than character to avoid errors \r\n        \/\/ in the XML parser.\r\n        \/\/ Use '\\x26#60;' instead of '<' so that the XML parser\r\n        \/\/ doesn't go ahead and substitute the less-than character. \r\n        code_string = code_string.replace(\/\\x3C\/g, '\\x26#60;');\r\n\r\n        copyright = 'Copyright 2015 The MathWorks, Inc.';\r\n\r\n        w = window.open();\r\n        d = w.document;\r\n        d.write('<pre>\\n');\r\n        d.write(code_string);\r\n\r\n        \/\/ Add copyright line at the bottom if specified.\r\n        if (copyright.length > 0) {\r\n            d.writeln('');\r\n            d.writeln('%%');\r\n            if (copyright.length > 0) {\r\n                d.writeln('% _' + copyright + '_');\r\n            }\r\n        }\r\n\r\n        d.write('<\/pre>\\n');\r\n\r\n        d.title = title + ' (MATLAB code)';\r\n        d.close();\r\n    }   \r\n     --> <\/script><p style=\"text-align: right; font-size: xx-small; font-weight:lighter;   font-style: italic; color: gray\"><br><a href=\"javascript:grabCode_cb48a6bf188a443fb4e238efc301463d()\"><span style=\"font-size: x-small;        font-style: italic;\">Get \r\n      the MATLAB code <noscript>(requires JavaScript)<\/noscript><\/span><\/a><br><br>\r\n      Published with MATLAB&reg; R2014b<br><\/p><\/div><!--\r\ncb48a6bf188a443fb4e238efc301463d ##### SOURCE BEGIN #####\r\n%% An Ornamental Geometric Inequality\r\n% I came across this \"ornamental geometric inequality\" in a tribute to\r\n% Lothar Collatz.\r\n\r\n%% An Inequality\r\n% I mentioned the German mathematician Lothar Collatz in my post in January\r\n% on his\r\n% <https:\/\/blogs.mathworks.com\/cleve\/2015\/01\/19\/the-three-n-plus-one-conjecture\/\r\n% 3n+1 Conjecture>. Here is a beautiful inequality from his 1934 paper titled\r\n% \"Ornamental Geometric Inequalities\".\r\n% Consider the subregion of $-4 \\le (x,y) \\le 4$ for which\r\n%\r\n% $$ ||||x| - 1| - 1| - |||y| - 1| - 1|| \\ge \\frac{1}{3} $$\r\n%\r\n% I've never seen four absolute value signs in a row before.\r\n\r\n%%\r\n% Let's visualize it with MATLAB.\r\n\r\n   [x,y] = meshgrid(-4:1\/256:4);\r\n   z = abs(abs(abs(abs(x)-1)-1) - abs(abs(abs(y)-1)-1)) >= 1\/3;\r\n   imagesc([-4,4],[-4,4],z);\r\n   axis image\r\n   colormap(copper)\r\n\r\n%% References\r\n% Lothar Collatz, 1910-1990, \r\n% <http:\/\/www.math.uni-hamburg.de\/home\/collatz\/auflage3-beta2.pdf\r\n% http:\/\/www.math.uni-hamburg.de\/home\/collatz\/auflage3-beta2.pdf>\r\n\r\n%%\r\n% Lothar Collatz, Gleichungen geometrischer Ornamente,\r\n% (Kurvengleichungen mit Absolutstrichen).\r\n% Z. mathem. u. naturw. Unterr. 64 (1934), 165-169.\r\n\r\n##### SOURCE END ##### cb48a6bf188a443fb4e238efc301463d\r\n-->","protected":false},"excerpt":{"rendered":"<div class=\"overview-image\"><img decoding=\"async\"  class=\"img-responsive\" src=\"https:\/\/blogs.mathworks.com\/images\/cleve\/collatz_inequality_01.png\" onError=\"this.style.display ='none';\" \/><\/div><!--introduction--><p>I came across this \"ornamental geometric inequality\" in a tribute to Lothar Collatz.... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/cleve\/2015\/03\/16\/an-ornamental-geometric-inequality\/\">read more >><\/a><\/p>","protected":false},"author":78,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[5,23,4,8],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/1178"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/users\/78"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/comments?post=1178"}],"version-history":[{"count":1,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/1178\/revisions"}],"predecessor-version":[{"id":1179,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/1178\/revisions\/1179"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media?parent=1178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/categories?post=1178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/tags?post=1178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}