Euclidean geometry
http://plus.maths.org/content/taxonomy/term/791
enMaths 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/contentMaths behind the rainbow
http://plus.maths.org/content/rainbows
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Marianne Freiberger </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<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_oct_2011_-_1710/icon.jpg?1318349428" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<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>
</div>
</div>
</div>
<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/contentThe trouble with five
http://plus.maths.org/content/trouble-five
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Craig Kaplan </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/issue45/features/kaplan/icon.jpg?1196467200" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
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>
</div>
</div>
<div class="pub_date">December 2007</div>
<!-- plusimport -->
<br clear="all" />
<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="http://plus.maths.org/content/trouble-five" target="_blank">read more</a></p>http://plus.maths.org/content/trouble-five#comments45Euclidean geometryfive-fold tiling problemgeometryhyperbolic geometrypenrose tilingspherical geometrysymmetrytilingSat, 01 Dec 2007 00:00:00 +0000plusadmin2319 at http://plus.maths.org/contentEditorial
http://plus.maths.org/content/pluschat-21
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="98" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/1/May%2019%2C%202010/plusbirthdaycakeSmall.jpg?1274279790" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<p>The <em>Plus</em> anniversary year — A word from the editors</p>
</div>
</div>
</div>
<div class="pub_date">December 2007</div>
<!-- plusimport -->
<!-- #include virtual="/include/gifd_here_box.html" -->
<h2>This issue's <i>Plus</i>chat topics</h2>
<ul>
<li><a href="#blurb">The <i>Plus</i> anniversary year</a> — A word from the editors;</li>
<li><a href="#plus10000"><i>Plus</i> 10,000</a> — The best maths ever.</li>
</ul>
<a name="blurb" id="blurb"></a>
<h3>The <i>Plus</i> anniversary year — A word from the editors</h3>
<p>This is the last issue of 2007 and, sadly, it's time to wrap up <i>Plus</i>'s tenth birthday party.<p><a href="http://plus.maths.org/content/pluschat-21" target="_blank">read more</a></p>45Al-KhwarizmialgebraeditorialEuclid's ElementsEuclidean geometrygeometrygolden ratiohyperbolic geometryirrational numbernumber systemplus birthdaypythagoras' theoremZeno's paradoxesSat, 01 Dec 2007 00:00:00 +0000plusadmin4901 at http://plus.maths.org/contentWe must know, we will know
http://plus.maths.org/content/we-must-know-we-will-know
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Rebecca Morris </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/issue41/features/morris/icon.jpg?1164931200" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<b>Runner up in the general public category</b>. Great minds spark controversy. This is something you'd expect to hear about a great philosopher or artist, but not about a mathematician. Get ready to bin your stereotypes as <b>Rebecca Morris</b> describes some controversial ideas of the great mathematician David Hilbert. </div>
</div>
</div>
<div class="pub_date">December 2006</div>
<!-- plusimport -->
<br clear="all" />
<p style="color:purple;"><b><i>This article is a runner-up in the general public category of the Plus new writers award 2006.</i></b></p>
<p><i>"It seems that there was a mathematician who had become a novelist. 'Why did he do that?' people in Göttingen marvelled. 'How can a man who was a mathematician write novels?' 'But that is completely simple,' Hilbert said. 'He did not have enough imagination for mathematics, but he had enough for novels.' "</i></p><p><a href="http://plus.maths.org/content/we-must-know-we-will-know" target="_blank">read more</a></p>http://plus.maths.org/content/we-must-know-we-will-know#comments41axiomEuclidean geometryGödel's Incompleteness Theoremhilbert problemshistory of mathematicslogicphilosophy of mathematicsFri, 01 Dec 2006 00:00:00 +0000plusadmin2295 at http://plus.maths.org/contentMathematical mysteries: Strange Geometries
http://plus.maths.org/content/mathematical-mysteries-strange-geometries
<div class="field field-type-text field-field-author">
<div class="field-items">
<div class="field-item odd">
Helen Joyce </div>
</div>
</div>
<div class="field field-type-filefield field-field-abs-img">
<div class="field-items">
<div class="field-item odd">
<img class="imagefield imagefield-field_abs_img" width="100" height="100" alt="" src="http://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/5/30%20Jun%202010%20-%2016%3A30/mystery.gif?1277911831" /> </div>
</div>
</div>
<div class="field field-type-text field-field-abs-txt">
<div class="field-items">
<div class="field-item odd">
<p>The famous mathematician Euclid is credited with being the first person to axiomatise the geometry of the world we live in - that is, to describe the geometric rules which govern it. Based on these axioms, he proved theorems - some of the earliest uses of proof in the history of mathematics.</p>
</div>
</div>
</div>
<div class="pub_date">Jan 2002</div>
<!-- plusimport -->
<h2>Euclidean Geometry</h2>
<p>The famous mathematician Euclid is credited with being the first person to axiomatise the geometry of the world we live in - that is, to describe the geometric rules which govern it. Based on these axioms, he proved theorems - some of the earliest uses of proof in the history of mathematics. Euclid's work is discussed in detail in <a href="/issue7/features/proof1/index.html">The Origins
of Proof</a>, from Issue 7 of <i>Plus</i>.</p><p><a href="http://plus.maths.org/content/mathematical-mysteries-strange-geometries" target="_blank">read more</a></p>http://plus.maths.org/content/mathematical-mysteries-strange-geometries#comments18curvaturecurvature of spaceescherEuclid's ElementsEuclidean geometryflatnesshyperbolic geometryMathematical mysteriesMercator projectionspherical geometrytrigonometrySat, 01 Dec 2001 00:00:00 +0000plusadmin4754 at http://plus.maths.org/content