geometry
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en Why the world needs pi
http://plus.maths.org/content/why-world-needs-pi
<p>If you had a sloppy maths teacher at school you might have grown up with the idea that the number <img src="/MI/db5bbeff4f0f83f3378282d9737865b4/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> is equal to <img src="/MI/db5bbeff4f0f83f3378282d9737865b4/images/img-0002.png" alt="$22/7.$" style="vertical-align:-5px;
width:36px;
height:18px" class="math gen" /> Now that is completely wrong. Writing those numbers out in decimal gives <img src="/MI/db5bbeff4f0f83f3378282d9737865b4/images/img-0003.png" alt="$22/7 = 3.142...$" style="vertical-align:-5px;
width:106px;
height:18px" class="math gen" /> while <img src="/MI/db5bbeff4f0f83f3378282d9737865b4/images/img-0004.png" alt="$\pi = 3.141...$" style="vertical-align:-1px;
width:83px;
height:13px" class="math gen" />. There’s a difference in the third decimal place after the decimal point! </p>
<div class="rightimage" ><img src="/sites/plus.maths.org/files/articles/2012/budd/pi.jpg" alt="Pi" width="350" height="191" /><p style="width: 350px;">How accurately do we need to know the value of π?</p><p><a href="http://plus.maths.org/content/why-world-needs-pi" target="_blank">read more</a></p>http://plus.maths.org/content/why-world-needs-pi#commentscalculating digits of piFP-top-storygeometryWed, 29 Oct 2014 15:47:53 +0000mf3446221 at http://plus.maths.org/contentFive Martin Gardner eye-openers involving squares and cubes
http://plus.maths.org/content/five-martin-gardner-eye-openers-involving-squares-and-cubes
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Colm Mulcahy </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/20_oct_2014_-_1104/gardner_icon.jpg?1413799485" /> </div>
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<p>This week would have been the 100th birthday of Martin Gardner, who is deservedly credited with turning on several generations of people worldwide to the pleasures of maths! To mark the occasion here are some favourite puzzles that, apart from being fun, also lead to some serious maths.</p>
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<p>American man of letters and numbers <a href="http://www.martin-gardner.org">Martin Gardner</a> was born 100 years ago, on 21 October 1914. Of the hundred or so books he left us, more than half are about mathematics and physics in some form. For over fifty years he championed puzzles and recreational mathematics, most famously in the <em>Scientific American</em> column he wrote regularly from 1957 to 1981. He's deservedly credited with turning on several generations of people worldwide to the pleasures of maths and creative problem solving in general.</p><p><a href="http://plus.maths.org/content/five-martin-gardner-eye-openers-involving-squares-and-cubes" target="_blank">read more</a></p>http://plus.maths.org/content/five-martin-gardner-eye-openers-involving-squares-and-cubes#commentsFP-top-storygeometrypuzzlerecreational gameMon, 20 Oct 2014 08:50:22 +0000mf3446209 at http://plus.maths.org/contentMaking a right angle the Maya way
http://plus.maths.org/content/making-right-angle-maya-way
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John C. D. Diamantopoulos and Cynthia J. Huffman </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/11_sep_2014_-_1221/maya_icon.jpg?1410434486" /> </div>
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<p>The Mayan civilisation brought forth many great things — including this clever way of making a right angle.</p>
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<div class="rightimage" style="width: 300px;"><img src="http://plus.maths.org./content/sites/plus.maths.org/files/articles/2014/maya/maya_small.jpg" width="300" height="200" alt="The great Mayan pyramid of Kukulcan"/><p>The great Mayan pyramid of Kukulcan "El Castillo" as seen from the Platform of the Eagles and Jaguars, Chichen Itza, Mexico. </p><p><a href="http://plus.maths.org/content/making-right-angle-maya-way" target="_blank">read more</a></p>http://plus.maths.org/content/making-right-angle-maya-way#commentsFP-carouselgeometryhistory of mathematicsFri, 12 Sep 2014 11:30:11 +0000mf3446181 at http://plus.maths.org/contentMaryam Mirzakhani: counting curves
http://plus.maths.org/content/mm
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://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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<img src="http://plus.maths.org/content/sites/plus.maths.org/files/news/2014/Seoul/f1.jpg" width="300" height="331" alt="MM"/>
<p></p><p><a href="http://plus.maths.org/content/mm" target="_blank">read more</a></p>http://plus.maths.org/content/mm#commentsfields medalFields Medal 2014geometryICM 2014topologyWed, 13 Aug 2014 00:24:46 +0000mf3446156 at http://plus.maths.org/contentKissing the curve
http://plus.maths.org/content/kissing-curve
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Rachel Thomas </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/4/23_jul_2014_-_1717/icon.jpg?1406132260" /> </div>
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<p>From a smile to a line drawing by Picasso, curves bring great beauty to our world. But how curvy is a curve?</p>
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From a smile to a line drawing by Picasso, curves bring great beauty to our world. Curves are also a beautiful and important part of mathematics and understanding curvature can shed light not just on beautiful shapes but can even reveal the undulations of space-time itself.
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If I ask you to think of a curve, you might think of a circle, a spiral, a sine wave, or some intricate curving squiggle. Almost certainly you won't think of a flat, straight line, and you won't think of a line of jagged spikes and sharp corners.
</p><p><a href="http://plus.maths.org/content/kissing-curve" target="_blank">read more</a></p>http://plus.maths.org/content/kissing-curve#commentscurvaturegeometryThu, 24 Jul 2014 09:02:16 +0000Rachel6111 at http://plus.maths.org/contentBlink and you'll miss it: The free kick in football (part I)
http://plus.maths.org/content/free-kick-football-blink-and-youll-miss-it
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Ken Bray </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/5_jun_2014_-_1717/icon-12.jpg?1401985048" /> </div>
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<p>Free kicks will deliver much of the drama in the football world cup this summer. But how should strikers approach them and how does the design on the ball impact on its behaviour in flight? Maths can give us answers...</p>
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<p>On 12 June the World Cup kicks off in Brazil where 32 teams will compete for the greatest prize in
football. A total of 64 games will be played up to the Final and much skill will be displayed by the
game's elite players. Can science add anything to the mix? The complexity of ninety minutes' play
rules out any possibility of simulating an entire game; but there are some events which are so
fleeting and where the intentions of the players are so specific, that scientific analysis can be fruitful.</p><p><a href="http://plus.maths.org/content/free-kick-football-blink-and-youll-miss-it" target="_blank">read more</a></p>http://plus.maths.org/content/free-kick-football-blink-and-youll-miss-it#commentsfootballgeometrymathematics in sportphysicsThu, 12 Jun 2014 09:34:23 +0000mf3446112 at http://plus.maths.org/contentCircles rolling on circles
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Yutaka Nishiyama </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/16_may_2014_-_1228/icon_coin.jpg?1400239687" /> </div>
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<p>Imagine a circle with radius 1 cm rolling completely along the circumference of a circle with radius 4 cm. How many rotations did the smaller circle make? Be prepared for a surprise!</p>
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<p>How many revolutions will the smaller coin make when rolling around the bigger one?</p>
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<p>Imagine a circle with radius 1 cm rolling completely along the circumference of a circle with radius 4 cm. How many rotations did the smaller circle make? </p><p><a href="http://plus.maths.org/content/circles-rolling-circles" target="_blank">read more</a></p>http://plus.maths.org/content/circles-rolling-circles#commentscircular motiongeometryFri, 16 May 2014 10:46:05 +0000mf3446098 at http://plus.maths.org/contentPolar power
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Marianne Freiberger </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/31_mar_2014_-_1013/icon.jpg?1396257196" /> </div>
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<p>Like spirals and flowers? Then you'll love polar coordinates and the pretty pictures they allow you to draw!</p>
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<p>Cartesian coordinates. </p><p><a href="http://plus.maths.org/content/polar-power" target="_blank">read more</a></p>http://plus.maths.org/content/polar-power#commentsArchimedean spiralgeometrylogarithmic spiralpolar coordinatessystemThu, 03 Apr 2014 15:46:05 +0000mf3446072 at http://plus.maths.org/contentMaths in a minute: Triangle central
http://plus.maths.org/content/maths-minute-triangle-central
<p>How do you balance a cardboard cut-out of a triangle on a pencil? Trial and error is one way, but maths can save you lots of bending down and picking it up. Take the pencil and a ruler and connect the mid-point of each side to the opposite corner. You'll find that the three lines intersect in a single point, which lies exactly a third of the way from the midpoint of each side to the opposite vertex. That point, called the <em>centroid</em>, is the centre of mass of the triangle.<p><a href="http://plus.maths.org/content/maths-minute-triangle-central" target="_blank">read more</a></p>http://plus.maths.org/content/maths-minute-triangle-central#commentsgeometryWed, 06 Nov 2013 09:46:11 +0000Rachel5967 at http://plus.maths.org/content3D printing mathematics
http://plus.maths.org/content/3d-printing
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Saul Schleimer and Henry Segerman </div>
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<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/13_sep_2013_-_1234/print_icon.jpg?1379072049" /> </div>
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<p>Saul Schleimer and Henry Segerman show off some of their beautiful 3D printed mathematical structures.</p>
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<p>When learning about existing mathematics, and especially when trying
to produce new mathematics, we spend a lot of time thinking about
examples. How do parts of the example interact with each other? What
are the regularities and symmetries? Does it come in a family of
examples, or does it live on its own? In many cases, the first thing
to do is to try and draw a picture. We are both geometric topologists,
working mostly with two and three-dimensional objects. As such, two-dimensional pictures are important currency in our field.<p><a href="http://plus.maths.org/content/3d-printing" target="_blank">read more</a></p>http://plus.maths.org/content/3d-printing#commentsgeometrymathematics and artmobius strippolytopevisualisationWed, 23 Oct 2013 08:18:02 +0000mf3445939 at http://plus.maths.org/contentMaths in a minute: Not always 180
http://plus.maths.org/content/maths-minute-strange-geometries
<p>Over 2000 years ago the Greek mathematician <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Euclid.html">Euclid</a> came up with a list of five postulates on which he thought geometry should be built. One of them, the fifth, was equivalent to a statement we are all familiar with: that the angles in a triangle add up to 180 degrees. However, this postulate did not seem as obvious as the other four on Euclid's list, so mathematicians attempted to deduce it from them: to show that a geometry obeying the first four postulates would necessarily obey the fifth.<p><a href="http://plus.maths.org/content/maths-minute-strange-geometries" target="_blank">read more</a></p>http://plus.maths.org/content/maths-minute-strange-geometries#commentsEuclidean geometrygeometryhyperbolic geometryspherical geometryWed, 03 Jul 2013 12:47:01 +0000mf3445921 at http://plus.maths.org/contentWatch out, it's behind you!
http://plus.maths.org/content/watch-out-its-behind-you
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<p>The <em>Plus</em> team's vehicle of choice is the bicycle, so we're particularly pleased about an announcement that hit the news this month: a clever car mirror that eliminates the dreaded blind spot has been given a patent in the US. The mirror was designed by the mathematician Andrew Hicks, of Drexel University, after years of puzzling over the problem.</p>
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<p>The <em>Plus</em> team's vehicle of choice is the bicycle, so we're intrigued by an announcement that hit the news this month: a clever car mirror that eliminates the dreaded blind spot has been given a patent in the US. The mirror was designed by the mathematician <a href="http://www.drexel.edu/math/contact/facultyDirectory/AndrewHicks/">Andrew Hicks</a>, of Drexel University, after years of puzzling over the problem.</p><p><a href="http://plus.maths.org/content/watch-out-its-behind-you" target="_blank">read more</a></p>http://plus.maths.org/content/watch-out-its-behind-you#commentsgeometryopticsreflectionFri, 29 Jun 2012 15:12:41 +0000mf3445725 at http://plus.maths.org/contentBridges, string art and Bézier curves
http://plus.maths.org/content/bridges-string-art-and-bezier-curves
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Renan Gross </div>
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The Jerusalem Chords Bridge, Israel, was built to make way for the city's light rail train
system. Its design took into consideration more than just utility — it is a work of
art, designed as a monument. Its beauty rests not only in the visual appearance of its criss-cross
cables, but also in the mathematics that lies behind it. So let's take a deeper look at it. </div>
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<h3>The Jerusalem Chords Bridge</h3>
<p>The Jerusalem Chords Bridge, Israel, was built to make way for the city's light rail train
system. However, its design took into consideration more than just utility — it is a work of
art, designed as a monument. Its beauty rests not only in the visual appearance of its criss-cross
cables, but also in the mathematics that lies behind it. Let us take a deeper look into these
chords.</p><p><a href="http://plus.maths.org/content/bridges-string-art-and-bezier-curves" target="_blank">read more</a></p>http://plus.maths.org/content/bridges-string-art-and-bezier-curves#commentsarchitectureBezier curveengineeringgeometrymathematics and artparabolaMon, 05 Mar 2012 09:31:51 +0000mf3445654 at http://plus.maths.org/contentMaths behind the rainbow
http://plus.maths.org/content/rainbows
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Marianne Freiberger </div>
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<p>The only good thing about a wash-out summer is that you get to see lots of rainbows. Keats complained that a mathematical explanation of these marvels of nature robs them of their magic, conquering "all mysteries by rule and line". But rainbow geometry is just as elegant as the rainbows themselves.</p>
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<p>When the great mathematician <a href="http://www.gap-system.org/~history/Biographies/Newton.html">Isaac Newton</a> explained the <a href="http://en.wikipedia.org/wiki/Isaac_Newton#Optics">colours of the rainbow</a> with refraction the poet <a href="http://en.wikipedia.org/wiki/John_Keats">John Keats</a> was horrified. Keats complained (through poetry of course) that a mathematical explanation robbed these marvels of nature of their magic, conquering <a href="http://en.wikipedia.org/wiki/Rainbow#Literature">"all mysteries by rule and line"</a>.<p><a href="http://plus.maths.org/content/rainbows" target="_blank">read more</a></p>http://plus.maths.org/content/rainbows#commentsEuclidean geometrygeometryrefractionrefractive indexsnell's lawtrigonometryFri, 21 Oct 2011 08:34:47 +0000mf3445558 at http://plus.maths.org/contentMeet the gyroid
http://plus.maths.org/content/meet-gyroid
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Adam G. Weyhaupt </div>
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What do butterflies, ketchup, microcellular structures, and plastics have in common? It's a curious minimal surface called the <em>gyroid</em>. </div>
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<p>What do butterflies, ketchup, microcellular structures, and plastics have in common?
It's a curious minimal surface called the <em>gyroid</em>. </p>
<div class="centreimage"><img src="http://plus.maths.org./content/sites/plus.maths.org/files/articles/2011/gyroid/gyroid_2views_0.jpg" width="600" height="275" alt="Gyroid"/><p style="width: 600px; margin: auto">Figure 1: Two views of a section of the gyroid surface. </p><p><a href="http://plus.maths.org/content/meet-gyroid" target="_blank">read more</a></p>http://plus.maths.org/content/meet-gyroid#commentsgeometrygyroidminimal surfaceMon, 12 Sep 2011 10:39:33 +0000mf3445547 at http://plus.maths.org/content