tiling
https://plus.maths.org/content/taxonomy/term/322
enMaths in a minute: Tiling troubles
https://plus.maths.org/content/maths-minute-tiling-troubles
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<p>Why there are only three regular polygons you can tile a wall with.</p>
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<p>Out of all the regular polygons there are only three you can use to tile
a wall with: the square, the equilateral triangle, and the regular
hexagon. All the others just won't fit together.</p>
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<p>Trying to fit pentagons around a point.</p><p><a href="https://plus.maths.org/content/maths-minute-tiling-troubles" target="_blank">read more</a></p>https://plus.maths.org/content/maths-minute-tiling-troubles#commentspolygontilingTue, 25 Aug 2015 09:53:20 +0000mf3446423 at https://plus.maths.org/contentA five that fits
https://plus.maths.org/content/five-fits
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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/5/13_aug_2015_-_1435/penta_icon.png?1439472932" /> </div>
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<p>A brand-new pentagon to tile your wall with.</p>
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<p>Mathematicians have found a new type of pentagon you can tile the plane with. It's a nice result because pentagons are famously awkward when it comes to tiling. The most natural pentagon to think about is the <em>regular</em> one, whose sides all have the same length. But you won't ever get this one to tile your bathroom wall. If you try to fit several copies around a point so that they all meet at a corner, you will find that three leave a gap, and four create an overlap:</p><p><a href="https://plus.maths.org/content/five-fits" target="_blank">read more</a></p>https://plus.maths.org/content/five-fits#commentsaperiodic tilingfive-fold tiling problemgeometrytilingFri, 14 Aug 2015 15:29:14 +0000mf3446420 at https://plus.maths.org/contentSecrets from a bathroom floor
https://plus.maths.org/content/secrets-bathroom-floor
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Josefina Alvarez and Cesar L. Garcia </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/issue52/features/alvarez/icon.jpg?1251759600" /> </div>
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Tilings have adorned buildings from ancient Rome to the Islamic world, from Victorian England to colonial Mexico. But while it sometimes seems free from worldly limitations, tiling is a very precise art, where not much can be left to chance. We can push and turn and wiggle, but if the maths is not right, it isn't going to tile. <b>Josefina Alvarez</b> and <b>Cesar L. Garcia</b> investigate. </div>
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<div class="pub_date">September 2009</div>
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<p>Tilings have adorned buildings from ancient Rome to the Islamic world, from Victorian England to colonial Mexico. In general, the word <i>tiling</i> refers to any pattern that covers a flat surface, like a painting on a canvas, using non-overlapping repetitions of one or more shapes, so that the design does not leave any empty spaces. Contemporary tilings can be found in African-American
quilts, Indonesian batiks, molas from the San Blas de Cuna Islands, and Aboriginal paintings.</p><p><a href="https://plus.maths.org/content/secrets-bathroom-floor" target="_blank">read more</a></p>https://plus.maths.org/content/secrets-bathroom-floor#comments52geometrypolygontesselationtilingMon, 31 Aug 2009 23:00:00 +0000plusadmin2367 at https://plus.maths.org/contentThe trouble with five
https://plus.maths.org/content/trouble-five
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Craig Kaplan </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/issue45/features/kaplan/icon.jpg?1196467200" /> </div>
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Squares do it, triangles do it, even hexagons do it — but pentagons don't. They just won't fit together to tile a flat surface. So are there <i>any</i> tilings based on fiveness? <b>Craig Kaplan</b> takes us through the five-fold tiling problem and uncovers some interesting designs in the process. </div>
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<div class="pub_date">December 2007</div>
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<p>We are all familiar with the simple ways of tiling the plane by equilateral triangles, squares, or hexagons. These are the three <i>regular</i> tilings: each is made up of identical copies of a <i>regular polygon</i> — a shape whose sides all have the same length and angles between them — and adjacent tiles share whole edges, that is, we never have part of a tile's edge overlapping part of
another tile's edge.</p>
<p><!-- FILE: include/centrefig.html --></p><p><a href="https://plus.maths.org/content/trouble-five" target="_blank">read more</a></p>https://plus.maths.org/content/trouble-five#comments45Euclidean geometryfive-fold tiling problemgeometryhyperbolic geometrypenrose tilingspherical geometrysymmetrytilingSat, 01 Dec 2007 00:00:00 +0000plusadmin2319 at https://plus.maths.org/contentNon-Euclidean geometry and Indra's pearls
https://plus.maths.org/content/non-euclidean-geometry-and-indras-pearls
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Caroline Series and David Wright </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/issue43/features/serieswright/icon.jpg?1180652400" /> </div>
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If you've ever redecorated a bathroom, you'll know that there are only so many ways in which you can tile a flat plane. But once you move into the curved world of hyperbolic geometry, possibilities become endless and the most amazing fractal structures ensue. <b>Caroline Series</b> and <b>David Wright</b> give a short introduction to the maths behind their beautiful images. </div>
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<div class="pub_date">June 2007</div>
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<p>Many people will have seen and been amazed by the beauty and intricacy of fractals like the one shown below. This particular fractal is known as the <a href="http://mathworld.wolfram.com/ApollonianGasket.html">Apollonian gasket</a> and consists of a complicated arrangement of tangent circles. [Click on the image to see this fractal evolve in a movie created by David Wright.]</p><p><a href="https://plus.maths.org/content/non-euclidean-geometry-and-indras-pearls" target="_blank">read more</a></p>https://plus.maths.org/content/non-euclidean-geometry-and-indras-pearls#comments43chaoscomputer graphicsfractalgeometryhyperbolic geometrytilingThu, 31 May 2007 23:00:00 +0000plusadmin2311 at https://plus.maths.org/contentPrize specimens
https://plus.maths.org/content/prize-specimens
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Mark Wainwright </div>
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<img class="imagefield imagefield-field_abs_img" width="130" height="130" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue13/features/eternity/icon.jpg?978307200" /> </div>
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Last October, two mathematicians won £1m when it was revealed that they were the first to solve the Eternity jigsaw puzzle. It had taken them six months and a generous helping of mathematical analysis. <b>Mark Wainwright</b> meets the pair and finds out how they did it. </div>
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<div class="pub_date">January 2001</div>
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<p>Alex Selby and Oliver Riordan, two mathematicians, with the help of a couple of computers, have shared a £1m prize by solving the "Eternity" puzzle. The puzzle was like an enormously difficult jigsaw. There were 209 pieces, all different, but all made from equilateral triangles and half-triangles, as in the example on the left.<p><a href="https://plus.maths.org/content/prize-specimens" target="_blank">read more</a></p>https://plus.maths.org/content/prize-specimens#comments13bayes theoremcomputer searcheternity gamegrid problemspacking problemsplane geometryprobabilitytilingMon, 01 Jan 2001 00:00:00 +0000plusadmin2175 at https://plus.maths.org/contentIn space, do all roads lead to home?
https://plus.maths.org/content/space-do-all-roads-lead-home
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Janna Levin </div>
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<img class="imagefield imagefield-field_abs_img" width="130" height="130" alt="" src="https://plus.maths.org/content/sites/plus.maths.org/files/issue10/features/topology/icon.jpg?946684800" /> </div>
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Is the Universe finite, with an edge, or infinite, with no edges? Or is it even stranger: finite but with no edges? It sounds far-fetched but the mathematical theory of topology makes it possible, and nobody yet knows the truth. <strong>Janna Levin</strong> tells us more. </div>
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<div class="pub_date">January 2000</div>
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<div class="leftimage" style="width: 150px;"><img src="/issue10/features/topology/walking.gif" alt="Figure 1: Getting nowhere" width="150" height="100" />
<p>Figure 1: Getting nowhere</p><p><a href="https://plus.maths.org/content/space-do-all-roads-lead-home" target="_blank">read more</a></p>https://plus.maths.org/content/space-do-all-roads-lead-home#comments10compact universeconnectednesscosmologycurvatureklein bottlemobius striptilingtopologySat, 01 Jan 2000 00:00:00 +0000plusadmin2164 at https://plus.maths.org/content