topology
https://plus.maths.org/content/taxonomy/term/613
enMaths in a minute: Local connectivity
https://plus.maths.org/content/maths-minute-local-connectivity
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Marianne Freiberger </div>
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<p>How crinkly is crinkly?</p>
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<p>The Mandelbrot set. Image <a href="http://en.wikipedia.org/wiki/File:Mandel_zoom_00_mandelbrot_set.jpg">Wolfgang Beyer</a>. </p>
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<p>We know that there are shapes in the plane whose outline is incredibly crinkly. Examples are fractals, like the famous <em>Mandelbrot set</em>. But just how complex can a shape be?</p><p><a href="https://plus.maths.org/content/maths-minute-local-connectivity" target="_blank">read more</a></p>https://plus.maths.org/content/maths-minute-local-connectivity#commentsconnectednessFP-belowfractallocal connectivitytopologyMon, 20 Apr 2015 09:42:02 +0000mf3446350 at https://plus.maths.org/contentIntroducing the Klein bottle
https://plus.maths.org/content/introducing-klein-bottle
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Marianne Freiberger </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/5_jan_2015_-_1744/kleinbottle_icon.jpg?1420479898" /> </div>
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<p>A Klein bottle can't hold any liquid because it doesn't have an inside. How do you construct this strange thing and why would you want to?</p>
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<p>If you like a drink, then a Klein bottle is <em>not</em> a recommended receptacle. It may look vaguely like a bottle, but it doesn't enclose any volume, which means that it can't actually hold any liquid. Whatever you pour "in" will just come back out again.</p><p><a href="https://plus.maths.org/content/introducing-klein-bottle" target="_blank">read more</a></p>https://plus.maths.org/content/introducing-klein-bottle#commentsklein bottlemobius striporient abilitysurfacetopologyTue, 06 Jan 2015 10:37:52 +0000mf3446292 at https://plus.maths.org/contentJohn Milnor: A conversation with a mathematical legend
https://plus.maths.org/content/mathematical-legend-our-interview-john-milnor
<div class="rightimage" style="width: 250px;"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/news/2011/abel/milnor.jpg" alt="" width="250" height="264" /><p>John Milnor</p><p><a href='http://plus.maths.org/content/sites/plus.maths.org/files/podcast/milnor_edit.mp3'>Listen to our conversation with John Milnor</a></p><p><a href="https://plus.maths.org/content/mathematical-legend-our-interview-john-milnor" target="_blank">read more</a></p>https://plus.maths.org/content/mathematical-legend-our-interview-john-milnor#commentsdynamical systemICM 2014topologyMon, 18 Aug 2014 04:25:24 +0000mf3446167 at https://plus.maths.org/contentMaryam Mirzakhani: counting curves
https://plus.maths.org/content/mm
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/12_aug_2014_-_0143/f1_icon.jpg?1407804201" /> </div>
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<p>Maryam Mirzakhani is being honoured for her "rare combination of superb technical ability, bold ambition, far-reaching vision, and deep curiosity".</p>
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<p><em> Maryam Mirzakhani has been awarded the <a href="http://www.mathunion.org/general/prizes/fields/details/">Fields Medal</a>, the most prestigious prize in maths, at this year's <a href="http://www.icm2014.org">International Congress of Mathematicians</a> in Seoul.</em></p>
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<p></p><p><a href="https://plus.maths.org/content/mm" target="_blank">read more</a></p>https://plus.maths.org/content/mm#commentsfields medalFields Medal 2014geometryICM 2014topologyWed, 13 Aug 2014 00:24:46 +0000mf3446156 at https://plus.maths.org/contentPlaying billiards on doughnuts
https://plus.maths.org/content/billiards-donuts
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Marianne Freiberger </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/27_may_2014_-_1025/icon.jpg?1401182706" /> </div>
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<p>The paths of billiard balls on a table can be long and complicated. To understand them mathematicians use a beautiful trick, turning tables into surfaces.</p>
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<p><em>Thanks to <a href="http://www.maths.bris.ac.uk/~maxcu/">Corinna Ulcigrai</a> of the University of Bristol for her help with this article.</em></p><p><a href="https://plus.maths.org/content/billiards-donuts" target="_blank">read more</a></p>https://plus.maths.org/content/billiards-donuts#commentsBMC2014chaosdynamical systemgeodesictopologytorusTue, 27 May 2014 09:40:45 +0000mf3446083 at https://plus.maths.org/contentMaths in a minute: The bridges of Königsberg
https://plus.maths.org/content/maths-minute-bridges-konigsberg
<p>In the eighteenth century the city we now know as Kaliningrad was
called Königsberg and it was part of Prussia. Like many other great
cities Königsberg was divided by a river, called the Pregel. It contained two
islands and there were seven bridges linking the various land masses. A
famous
puzzle at the time was to find a walk through
the city that crossed every bridge exactly once. Many people claimed
they had found such a walk but when asked to reproduce it no one was able to.<p><a href="https://plus.maths.org/content/maths-minute-bridges-konigsberg" target="_blank">read more</a></p>https://plus.maths.org/content/maths-minute-bridges-konigsberg#commentsBridges of Konigsbergcreativitygraph theorytopologyWed, 20 Nov 2013 09:05:11 +0000mf3445969 at https://plus.maths.org/contentHappy birthday, London Underground!
https://plus.maths.org/content/happy-birthday-london-underground
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/9_jan_2013_-_1356/icon-1.jpg?1357739789" /> </div>
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<p>The London Underground turns 150 today! It's probably the most famous rail network in the world and much of that fame is due to the iconic London Underground map. But what makes this map so special?</p>
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<div class="rightimage"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/news/2012/rubber/tube.jpg" width="300" height="199" alt="Tube map"/><p style="width:300px;"> Part of the London Underground map. See the full map <a href="http://www.tfl.gov.uk/assets/downloads/standard-tube-map.pdf">here</a>.</p><p><a href="https://plus.maths.org/content/happy-birthday-london-underground" target="_blank">read more</a></p>https://plus.maths.org/content/happy-birthday-london-underground#commentsBridges of Konigsberggraph theorytopologyWed, 09 Jan 2013 13:55:38 +0000mf3445853 at https://plus.maths.org/contentThe shape of things to come
https://plus.maths.org/content/shape-things-come
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The Plus Team </div>
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Progress in pure mathematics has its own tempo. Major questions may remain open for decades, even centuries, and once an answer has been found, it can take a collaborative effort of many mathematicians in the field to check
that it is correct. The <em>New Contexts for Stable Homotopy Theory</em> programme, held at the Institute in 2002, is a prime example of how its research programmes can benefit researchers and its lead to landmark results. </div>
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<p><em>This article is part of a <a href="https://plus.maths.org/content/isaac-newton-institute">series</a> celebrating the 20th birthday of the <a href="http://www.newton.ac.uk/">Isaac Newton Institute</a> in Cambridge. The Institute is a place where leading mathematicians from around the world can come together for weeks or months at a time to indulge in what they like doing best: thinking about maths and exchanging ideas without the distractions and duties that come with their normal working lives.<p><a href="https://plus.maths.org/content/shape-things-come" target="_blank">read more</a></p>https://plus.maths.org/content/shape-things-come#commentshomotopyNewton InstitutetopologyThu, 19 Jul 2012 08:20:32 +0000mf3445439 at https://plus.maths.org/contentRubber data
https://plus.maths.org/content/rubber-data
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Data, data, data — 21st century life provides tons of it. It's paradise for researchers, or at least it would be if we knew how to make sense of it all. This year's AAAS annual meeting in Vancouver
devoted plenty of time to the question of how to understand large amounts of data. And there's one method we
particularly liked. It's based on the kind of idea that gave us the London tube map. </div>
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<p>Data, data, data — 21st century life provides tonnes of it. It's paradise for researchers, or at least it would be if we knew how to make sense of it all. This year's <a href="http://www.aaas.org/meetings/2012/">AAAS annual meeting in Vancouver</a>
devoted plenty of time to the question of how to understand large amounts of data. And there's one method we
particularly liked.</p>
<div class="rightimage"><img src="https://plus.maths.org/content/sites/plus.maths.org/files/news/2012/rubber/tube.jpg" width="300" height="199" alt="Tube map"/><p style="width:300px;"> The London Underground map.</p><p><a href="https://plus.maths.org/content/rubber-data" target="_blank">read more</a></p>https://plus.maths.org/content/rubber-data#commentsdata analysisdata miningtopologyWed, 07 Mar 2012 10:36:30 +0000mf3445669 at https://plus.maths.org/contentMaths in a minute: The fundamental group
https://plus.maths.org/content/maths-minute-fundamental-group
<p>Topologists famously think that a doughnut is the same as a coffee cup because one can be deformed into the other without tearing or cutting. In other words, topology doesn't care about exact measurements of quantities like lengths, angles and areas. Instead, it looks only at the overall shape of an object, considering two objects to be the same as long as you can morph one into the other without breaking it. But how do you work with such a slippery concept?</p><p><a href="https://plus.maths.org/content/maths-minute-fundamental-group" target="_blank">read more</a></p>https://plus.maths.org/content/maths-minute-fundamental-group#commentsfundamental groupgroup theorytopologyMon, 11 Apr 2011 09:18:34 +0000mf3445465 at https://plus.maths.org/contentThe Abel Prize 2011 goes to John Milnor
https://plus.maths.org/content/abel-prize-2011-goes-john-milnor
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<p>The Abel Prize 2011 goes to John Willard Milnor of Stony Brook University, New York for "pioneering discoveries in topology, geometry and algebra".</p>
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<p>The Abel Prize 2011 goes to <a href="http://www.math.sunysb.edu/~jack/">John Willard Milnor</a> of Stony Brook University, New York for "pioneering discoveries in topology, geometry and algebra". The Abel Prize is one of the most important international prizes in mathematics. It's awarded annually by the <a href="http://english.dnva.no/">Norwegian Academy of Science and Letters</a> and carries a prize money of around £650,000.</p><p><a href="https://plus.maths.org/content/abel-prize-2011-goes-john-milnor" target="_blank">read more</a></p>https://plus.maths.org/content/abel-prize-2011-goes-john-milnor#commentsmathematical realityAbel prizedifferential topologydynamical systemJulia setknotknot theoryMandelbrot settopologyWed, 23 Mar 2011 11:05:49 +0000mf3445456 at https://plus.maths.org/contentWinding numbers: Topography and topology II
https://plus.maths.org/content/winding-numbers-topography-and-topology-ii
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Ian Short </div>
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<p>This is the second in a series of two articles in which Ian Short looks at topology using topographical features of maps. Find out about Jordan curves and winding numbers with the help of hermits, lighthouses and drunken sailors.</p>
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<p><em>This is the second in a series of two articles in which we look at topology using topographical features of maps. You may wish to read the <a href="https://plus.maths.org/content/dividing-walls-topology-and-topography-i">first article</a> on dividing walls, although the two articles are largely independent.</em></p>
<h3>The Jordan Curve Theorem</h3>
<p>A hermit hires some builders to construct a wall around his house. The wall they construct is shown in figure 1. Does the wall actually surround the house?</p><p><a href="https://plus.maths.org/content/winding-numbers-topography-and-topology-ii" target="_blank">read more</a></p>https://plus.maths.org/content/winding-numbers-topography-and-topology-ii#commentsjordan curvetopologywinding numberWed, 23 Mar 2011 10:00:00 +0000mf3445449 at https://plus.maths.org/contentDividing Walls: Topology and topography I
https://plus.maths.org/content/dividing-walls-topology-and-topography-i
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Ian Short </div>
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<p>Journey to distant islands to discover if topology can overcome topography and bring peace to rival towns.</p>
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<p><em>Topography is a branch of geography concerned with the natural and constructed features on the surface of land, such as mountains, lakes, roads, and buildings. Topology is a branch of mathematics concerned with the distortion of shapes. In this, the first of two articles, Ian Short explores topological problems using topography.</em></p><p><a href="https://plus.maths.org/content/dividing-walls-topology-and-topography-i" target="_blank">read more</a></p>https://plus.maths.org/content/dividing-walls-topology-and-topography-i#commentstopographytopologyThu, 24 Feb 2011 15:43:26 +0000Rachel5423 at https://plus.maths.org/contentExotic spheres, or why 4-dimensional space is a crazy place
https://plus.maths.org/content/richard-elwes
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Richard Elwes </div>
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<p>The world we live in is strictly 3-dimensional: up/down, left/right, and forwards/backwards, these are the only ways to move. For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like? Or might it even be true that we already inhabit such a space, that our 3-dimensional home is no more than a slice through a higher dimensional realm, just as a slice through a 3-dimensional cube produces a 2-dimensional square?</p> </div>
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<p>The world we live in is strictly 3-dimensional: up/down, left/right, and forwards/backwards, these are the only ways to move. For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like? Or might it even be true, as some have suggested, that we already inhabit such a space, that our 3-dimensional home is no more than a slice through a higher dimensional realm, just as a slice through a 3-dimensional cube produces a 2-dimensional square?</p><p><a href="https://plus.maths.org/content/richard-elwes" target="_blank">read more</a></p>https://plus.maths.org/content/richard-elwes#commentsmathematical realitydifferential topologyfractalgeometryPoincare Conjecturesmooth Poincare conjecturetopologyWed, 12 Jan 2011 11:03:17 +0000mf3445399 at https://plus.maths.org/contentKnitting by numbers
https://plus.maths.org/content/knitting-numbers
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Lucinda Mathews </div>
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<b>Lucinda Mathews</b> visualises tricky surfaces </div>
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<p><i>This article is the runner-up in the general public category of the <a href="https://plus.maths.org/content/iplusi-new-writers-award-2009"><i>Plus</i> new writers award 2009</a>.</i></p>
<p>Recently I became friends with a keen knitter. She makes such beautiful creations, and knitting looked so meditative, that I decided to take it up myself. What I hadn't expected to end up thinking about during my new hobby was mathematics.</p><p><a href="https://plus.maths.org/content/knitting-numbers" target="_blank">read more</a></p>https://plus.maths.org/content/knitting-numbers#commentshyperbolic geometryklein bottlemobius strippermutationPlus new writers award 2009topologySun, 31 May 2009 23:00:00 +0000plusadmin2365 at https://plus.maths.org/content