{"id":2608,"date":"2018-05-02T13:04:02","date_gmt":"2018-05-02T12:04:02","guid":{"rendered":"http:\/\/blogs.reading.ac.uk\/the-forum\/?p=2608"},"modified":"2021-11-16T16:30:46","modified_gmt":"2021-11-16T16:30:46","slug":"broccoli-19th-century-maths-and-polymers","status":"publish","type":"post","link":"https:\/\/research.reading.ac.uk\/research-blog\/2018\/05\/02\/broccoli-19th-century-maths-and-polymers\/","title":{"rendered":"Broccoli, 19th century maths and polymers"},"content":{"rendered":"<p><strong>Professor Howard Colquhoun tells the story of how an infinitely repeating pattern discovered in a plastic molecule relates to obscure 19<sup>th<\/sup> century maths \u2013 and Romanesco broccoli.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2612\" src=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/chemical-science-journal-art-1024x757.jpg\" alt=\"\" width=\"540\" height=\"399\" srcset=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/chemical-science-journal-art-1024x757.jpg 1024w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/chemical-science-journal-art-300x222.jpg 300w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/chemical-science-journal-art-768x568.jpg 768w\" sizes=\"auto, (max-width: 540px) 100vw, 540px\" \/><\/p>\n<p>In 1874, Henry J. S. Smith, Professor of Geometry in the University of Oxford, had an interesting thought. &#8220;What would happen,&#8221; he wondered, &#8220;if you took a line, divided it into four, and then threw away the end quarter?&#8221;<\/p>\n<p>Well, obviously, you would be left with a line three-quarters as long as the original.<\/p>\n<p>But Smith&#8217;s next question was more subtle. &#8220;What would happen if you repeated this operation on the <em>line that was left<\/em>, and then continued to do this indefinitely?&#8221;<\/p>\n<p>When Smith worked through the problem, it turned out that he had discovered the first ever example of a <em>fractal<\/em>, a mathematical structure that is made up of an infinite number of progressively smaller copies of itself.<\/p>\n<p><!--more--><\/p>\n<p>A few years later, the German mathematician Georg Cantor discovered a number of closely-related examples and, despite Smith&#8217;s prior discovery which was published in 1875 in the <em>Proceedings of the London Mathematical Society<\/em>, such fractals are now universally known as Cantor Sets.<\/p>\n<p>The best-known &#8216;real&#8217; object that shows fractal geometry is probably Romanesco Broccoli (below), a vegetable that has a spiral structure made up of many smaller spirals, which are themselves made up of smaller spirals still, and so on.<\/p>\n<p><a href=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/romanesco_broccoli.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2611\" src=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/romanesco_broccoli.jpg\" alt=\"\" width=\"540\" height=\"404\" srcset=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/romanesco_broccoli.jpg 550w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/romanesco_broccoli-300x225.jpg 300w\" sizes=\"auto, (max-width: 540px) 100vw, 540px\" \/><\/a><\/p>\n<p>And the best known &#8216;mathematical&#8217; fractal is certainly the Mandelbrot set, which, on magnification reveals a vast wealth of remarkably beautiful detail.<\/p>\n<p><a href=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandlebrot.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2616\" src=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandlebrot-1024x768.jpg\" alt=\"\" width=\"540\" height=\"405\" srcset=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandlebrot-1024x768.jpg 1024w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandlebrot-300x225.jpg 300w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandlebrot-768x576.jpg 768w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandlebrot.jpg 1280w\" sizes=\"auto, (max-width: 540px) 100vw, 540px\" \/><\/a><\/p>\n<p><a href=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandelbrot-set.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2615\" src=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandelbrot-set.jpg\" alt=\"\" width=\"541\" height=\"405\" srcset=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandelbrot-set.jpg 988w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandelbrot-set-300x224.jpg 300w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Mandelbrot-set-768x574.jpg 768w\" sizes=\"auto, (max-width: 541px) 100vw, 541px\" \/><\/a><\/p>\n<p><em>The Mandelbrot set (top image), and the detail within it (lower image).<\/em><\/p>\n<p>The fractal that Smith had discovered remained just a rather obscure mathematical construct for well over a hundred years. Then in 2010, a PhD student, John Shaw, working with\u00a0me at the University of Reading, discovered a very unusual pattern in the nuclear magnetic resonance (NMR) spectrum of a polymer he was analysing.<\/p>\n<p>The pattern seemed, astonishingly, to be made up of several smaller copies of itself (does this start to sound familiar&#8230;.?). Subsequent work by other researchers in the Department of Chemistry confirmed John&#8217;s result and enabled the construction of a detailed atomic model for the polymer system that had produced the fractal-like spectrum.<\/p>\n<p>An analysis of the data by Dr Ricardo Grau-Crespo, Lecturer in Chemistry at Reading, then revealed that the model leads directly to an equation that is the <em>definition<\/em> of Smith&#8217;s original fractal, now called the \u2018Fourth-Quarter Cantor Set\u2019.<\/p>\n<p>Even more intriguingly, this equation accurately <em>predicts<\/em> the spectrum discovered by John Shaw. So that an apparently obscure result in 19th Century mathematics has now emerged as a complete description of some new and fascinating real-world polymer chemistry.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2613\" src=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Fourth-quarter-Cantor-set-and-prediction-of-NMR-spectrum-of-polymer-complex-1024x739.jpg\" alt=\"\" width=\"547\" height=\"394\" srcset=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Fourth-quarter-Cantor-set-and-prediction-of-NMR-spectrum-of-polymer-complex-1024x739.jpg 1024w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Fourth-quarter-Cantor-set-and-prediction-of-NMR-spectrum-of-polymer-complex-300x216.jpg 300w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/Fourth-quarter-Cantor-set-and-prediction-of-NMR-spectrum-of-polymer-complex-768x554.jpg 768w\" sizes=\"auto, (max-width: 547px) 100vw, 547px\" \/><\/p>\n<p><em>Construction of the fourth-quarter Cantor set and its prediction of the NMR spectrum of the polymer-complex shown at right.<\/em><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/NMR-spectrum-of-polymer.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2617\" src=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/NMR-spectrum-of-polymer-892x1024.jpg\" alt=\"\" width=\"541\" height=\"621\" srcset=\"https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/NMR-spectrum-of-polymer-892x1024.jpg 892w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/NMR-spectrum-of-polymer-261x300.jpg 261w, https:\/\/research.reading.ac.uk\/research-blog\/wp-content\/uploads\/sites\/72\/2018\/05\/NMR-spectrum-of-polymer-768x882.jpg 768w\" sizes=\"auto, (max-width: 541px) 100vw, 541px\" \/><\/a><\/p>\n<p><em>Experimental NMR spectrum of the polymer-complex showing how it consists of several smaller copies of itself.<\/em><\/p>\n<p>Going back to Henry Smith&#8217;s original idea, the images shown above illustrate the construction of his fractal and show how the construction predicts the NMR spectrum observed at Reading. The work has just been accepted for publication by the Royal Society of Chemistry, <a href=\"http:\/\/pubs.rsc.org\/en\/content\/articlelanding\/2018\/sc\/c8sc00830b#!divAbstract\">in their flagship journal <em>Chemical Science<\/em><\/a>.<\/p>\n<p><em>Read today&#8217;s\u00a0<a href=\"http:\/\/www.reading.ac.uk\/news-and-events\/releases\/PR764333.aspx\">press release about this paper<\/a>\u00a0and the future potential of fractal patterns in molecules as a way of efficiently storing information. Howard Colquhoun is Professor of Materials Chemistry at the University of Reading.<\/em><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Professor Howard Colquhoun tells the story of how an infinitely repeating pattern discovered in a plastic molecule relates to obscure 19th century maths \u2013 and Romanesco broccoli. In 1874, Henry&#8230;<a class=\"read-more\" href=\"&#104;&#116;&#116;&#112;&#115;&#58;&#47;&#47;&#114;&#101;&#115;&#101;&#97;&#114;&#99;&#104;&#46;&#114;&#101;&#97;&#100;&#105;&#110;&#103;&#46;&#97;&#99;&#46;&#117;&#107;&#47;&#114;&#101;&#115;&#101;&#97;&#114;&#99;&#104;&#45;&#98;&#108;&#111;&#103;&#47;&#50;&#48;&#49;&#56;&#47;&#48;&#53;&#47;&#48;&#50;&#47;&#98;&#114;&#111;&#99;&#99;&#111;&#108;&#105;&#45;&#49;&#57;&#116;&#104;&#45;&#99;&#101;&#110;&#116;&#117;&#114;&#121;&#45;&#109;&#97;&#116;&#104;&#115;&#45;&#97;&#110;&#100;&#45;&#112;&#111;&#108;&#121;&#109;&#101;&#114;&#115;&#47;\">Read More ><\/a><\/p>\n","protected":false},"author":127,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"__cvm_playback_settings":[],"__cvm_video_id":"","footnotes":""},"categories":[13],"tags":[385,554,558,700],"class_list":["post-2608","post","type-post","status-publish","format-standard","hentry","category-environment","tag-fractal-pattern","tag-mandelbrot-set","tag-materials-chemistry","tag-polymers"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.8.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Broccoli, 19th century maths and polymers - Connecting Research<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/research.reading.ac.uk\/research-blog\/2018\/05\/02\/broccoli-19th-century-maths-and-polymers\/\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Broccoli, 19th century maths and polymers - Connecting Research\" \/>\n<meta property=\"og:description\" content=\"Professor Howard Colquhoun tells the story of how an infinitely repeating pattern discovered in a plastic molecule relates to obscure 19th century maths \u2013 and Romanesco broccoli. 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