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{"id":35669,"date":"2023-06-22T21:58:07","date_gmt":"2023-06-22T21:58:07","guid":{"rendered":"https:\/\/scienceandnerds.com\/2023\/06\/22\/the-simple-geometry-that-predicts-molecular-mosaics\/"},"modified":"2023-06-22T21:58:08","modified_gmt":"2023-06-22T21:58:08","slug":"the-simple-geometry-that-predicts-molecular-mosaics","status":"publish","type":"post","link":"https:\/\/scienceandnerds.com\/2023\/06\/22\/the-simple-geometry-that-predicts-molecular-mosaics\/","title":{"rendered":"The Simple Geometry That Predicts Molecular Mosaics"},"content":{"rendered":"

Source:https:\/\/www.quantamagazine.org\/the-simple-geometry-that-predicts-molecular-mosaics-20230621\/#comments<\/a><\/br>
\nThe Simple Geometry That Predicts Molecular Mosaics<\/br>
\n2023-06-22 21:58:07<\/br><\/p>\n

\n

Over the next year, Domokos and his colleagues used geometric thinking to unpack the rules of molecular self-assembly \u2014 devising a new way<\/a> to constrain the mosaics that molecules can form, using only the simple geometry of tessellation.<\/p>\n

\u201cAt first, they did not believe that you can do it,\u201d Domokos said. \u201cThey were doing artificial intelligence, supercomputing and all this kind of jazz. And now they are just looking at formulas. And this is very relaxing.\u201d<\/p>\n

From Planets to Atoms<\/strong><\/h2>\n

\u00a0<\/strong>After Decurtins got in touch, Domokos tried to sell the idea to Krisztina Reg\u0151s<\/a>, his graduate student. Decurtins had sent a handful of images depicting patterns at an atomic scale \u2014 tilings of a molecule that had been designed and synthesized by his colleague\u00a0Shi-Xia Liu<\/a> \u2014 viewed through the eye of a powerful microscope. Domokos wanted to see if Reg\u0151s could use the geometry that he had originally developed to describe geological fractures to characterize the patterns in Decurtins\u2019 images.<\/p>\n

To get started, Reg\u0151s treated the 2D materials as simple polygonal tessellations \u2014 patterns that fit together with no gaps and infinitely repeat. Then, following Domokos\u2019 approach, she calculated two numbers for each pattern. The first was the average number of vertices, or corners, per polygon. The second was the average number of polygons surrounding each vertex.<\/p>\n

Together, those two average values are like a pattern\u2019s GPS coordinates. They give its location within a landscape of all possible tessellations.<\/p>\n

This landscape is called the symbolic plane. It is a simple 2D grid with the average number of shapes per vertex on the x<\/em>-axis and the average number of vertices per shape on the y<\/em>-axis. Each tessellation should plot to exactly one point within the plane. A perfect honeycomb pattern, for instance, is a tessellation of six-pointed hexagons that meet in trios at each vertex \u2014 a point at (3, 6) in the symbolic plane.<\/p>\n

But most natural mosaics, from rock cracks to molecular monolayers, are not perfectly periodic tessellations.<\/p>\n

For example, the cells of a real wax honeycomb are not all perfect hexagons. Bees make mistakes. But messy as it may be, a honeycomb is still, on average, a honeycomb. And on average, it still plots to a point at (3, 6) in the symbolic plane. Rather than being an oversimplification, Domokos\u2019 method of calculating averages is insightful, said the mathematician Marjorie Senechal<\/a> of Smith College, who reviewed the new study. By throwing out the mistakes and treating patterns as averages, it reveals a sort of ideal reality that\u2019s normally buried beneath heaps of happenstance.<\/p>\n

But when Reg\u0151s tried to apply this method to Decurtins\u2019 molecular pictures, she quickly ran into trouble. \u201cI started to put them on the symbolic plane,\u201d she said, \u201cand then I realized that I can\u2019t.\u201d<\/p>\n

The problem was scale. Unlike the geological patterns Domokos had worked with before, the molecular mosaics are really patterns within patterns. Viewed at different magnifications, they have different geometries. Reg\u0151s couldn\u2019t describe the molecular mosaics with a single pair of values because the patterns plotted different points on the symbolic plane, depending on an image\u2019s magnification. It was a bit like zooming in on a hexagonal tiling and finding that its basic building blocks are really triangles.<\/p>\n<\/div>\n

<\/br><\/br><\/br><\/p>\n

Uncategorized<\/br>
\n<\/br>
\nSource:https:\/\/www.quantamagazine.org\/the-simple-geometry-that-predicts-molecular-mosaics-20230621\/#comments<\/a><\/br><\/br><\/p>\n","protected":false},"excerpt":{"rendered":"

Source:https:\/\/www.quantamagazine.org\/the-simple-geometry-that-predicts-molecular-mosaics-20230621\/#comments The Simple Geometry That Predicts Molecular Mosaics 2023-06-22 21:58:07 Over the next year, Domokos and his colleagues used geometric thinking to unpack the rules of molecular self-assembly \u2014 devising a new way to constrain the mosaics that molecules can form, using only the simple geometry of tessellation. \u201cAt first, they did not believe that […]<\/p>\n","protected":false},"author":1,"featured_media":35670,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","om_disable_all_campaigns":false,"pagelayer_contact_templates":[],"_pagelayer_content":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-35669","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"yoast_head":"\nThe Simple Geometry That Predicts Molecular Mosaics - Science and Nerds<\/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:\/\/scienceandnerds.com\/2023\/06\/22\/the-simple-geometry-that-predicts-molecular-mosaics\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Simple Geometry That Predicts Molecular Mosaics - Science and Nerds\" \/>\n<meta property=\"og:description\" content=\"Source:https:\/\/www.quantamagazine.org\/the-simple-geometry-that-predicts-molecular-mosaics-20230621\/#comments The Simple Geometry That Predicts Molecular Mosaics 2023-06-22 21:58:07 Over the next year, Domokos and his colleagues used geometric thinking to unpack the rules of molecular self-assembly \u2014 devising a new way to constrain the mosaics that molecules can form, using only the simple geometry of tessellation. \u201cAt first, they did not believe that […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/scienceandnerds.com\/2023\/06\/22\/the-simple-geometry-that-predicts-molecular-mosaics\/\" \/>\n<meta property=\"og:site_name\" content=\"Science and Nerds\" \/>\n<meta property=\"article:published_time\" content=\"2023-06-22T21:58:07+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-06-22T21:58:08+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/scienceandnerds.com\/wp-content\/uploads\/2023\/06\/the-simple-geometry-that-predicts-molecular-mosaics_6494c3f012cdd.webp\" \/>\n\t<meta property=\"og:image:width\" content=\"2560\" \/>\n\t<meta property=\"og:image:height\" content=\"1440\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/webp\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/scienceandnerds.com\/2023\/06\/22\/the-simple-geometry-that-predicts-molecular-mosaics\/\",\"url\":\"https:\/\/scienceandnerds.com\/2023\/06\/22\/the-simple-geometry-that-predicts-molecular-mosaics\/\",\"name\":\"The Simple Geometry That Predicts Molecular Mosaics - 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