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https://plus.maths.org/content
enIntroducing Andrew Wiles
https://plus.maths.org/content/introducing-andrew-wiles
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon-15.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Andrew Wiles is a mathematical legend. In 1993, after years of working in secret, he <a href="/content/very-old-problem-turns-20">announced a proof of Fermat's last theorem</a>, which had been taunting mathematicians for centuries.</p>
<p>In these two short videos, filmed at the <a href="http://www.heidelberg-laureate-forum.org">Heidelberg Laureate Forum</a>, Wiles talks about what it's like to do mathematics and how it's all about being creative, why he worked in secret and what it feels like solving a famous problem like Fermat's: "It's what we live for!"</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/KaVytLupxmo" frameborder="0" allowfullscreen></iframe>
<iframe width="560" height="315" src="https://www.youtube.com/embed/aGko3eEPq8o" frameborder="0" allowfullscreen></iframe></div></div></div>Thu, 22 Sep 2016 14:21:33 +0000mf3446684 at https://plus.maths.org/contenthttps://plus.maths.org/content/introducing-andrew-wiles#commentsPolitical tides
https://plus.maths.org/content/political-tides
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Moon_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 320px;"><img src="/content/sites/plus.maths.org/files/news/2016/tides/carswell.jpg" alt="Carswell" width="320" height="213" />
<p>Douglas Carswell. Photo: <a href="https://www.flickr.com/people/spunter/">Steve Punter</a>, <a href="https://creativecommons.org/licenses/by-sa/2.0/">CC BY-SA 2.0</a></p>
</div>
<p>The politician <a href="http://www.douglascarswell.com">Douglas Carswell</a> (UKIP) has <a href="https://www.buzzfeed.com/tomchivers/people-in-this-country-have-had-enough-of-experts?utm_term=.rlLJ14Ga1#.tiG0P1GrP">recently been arguing</a> with the scientist Paul Nightingale about what causes the tides on Earth. Carswell said it was the gravitational influence of the Sun and Nightingale said it was the Moon. The debate came up in the context of Brexit. Using the tides as an analogy, Nightingale had argued that trade relationships with the EU were more important than those with more distant giants like China.</p>
<p>Giving Carswell the benefit of the doubt (after all, hundreds of years' worth of expert opinion might be wrong) we decided to do a back-of-the-envelope calculation to see who's right.</p>
<p>Let’s write <img src="/MI/5959a05d2f1aa4a49a2aac73e7568a91/images/img-0001.png" alt="$m_ E$" style="vertical-align:-2px;
width:25px;
height:9px" class="math gen" /> for the mass of the Earth, <img src="/MI/5959a05d2f1aa4a49a2aac73e7568a91/images/img-0002.png" alt="$m_ M$" style="vertical-align:-2px;
width:28px;
height:9px" class="math gen" /> for the mass of the Moon, and <img src="/MI/5959a05d2f1aa4a49a2aac73e7568a91/images/img-0003.png" alt="$d_ M$" style="vertical-align:-2px;
width:22px;
height:13px" class="math gen" /> for the distance between Earth and Moon, measured centre to centre. Newton’s universal law of gravitation tells us that the gravitational force <img src="/MI/5959a05d2f1aa4a49a2aac73e7568a91/images/img-0004.png" alt="$F$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> between the two is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/5959a05d2f1aa4a49a2aac73e7568a91/images/img-0005.png" alt="\[ F = G\frac{m_ Em_ M}{d_ M^2}, \]" style="width:111px;
height:35px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>
where <img src="/MI/c37c9cd0564c713b944ec9bc897d1f4a/images/img-0001.png" alt="$G$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> is the gravitational constant. </p>
<p>However, the tidal force the Moon exerts on the Earth is calculated by working out the ratio between its gravitational pull on the near side of the Earth and its pull on the far side. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2016/tides/tides.png" alt="Earth and Moon" width="300" height="211" />
<p style="width: 300px;">Diagram not to scale.</p>
</div>
<p>To get the distance between the near side of the Earth to the centre of the Moon, we need to subtract the radius of the Earth, <img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0001.png" alt="$r_ E$" style="vertical-align:-2px;
width:17px;
height:9px" class="math gen" />, from <img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0002.png" alt="$d_ E.$" style="vertical-align:-2px;
width:23px;
height:13px" class="math gen" /> The gravitational pull of the Moon on the near side of the Earth is then </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0003.png" alt="\[ F_{near} = G\frac{m_ Em_ M}{(d_ M-r_ E)^2}. \]" style="width:166px;
height:34px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Similarly, to get the distance between the far side of the Earth to the centre of the Moon, we need to add the radius of the Earth, <img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0001.png" alt="$r_ E$" style="vertical-align:-2px;
width:17px;
height:9px" class="math gen" />, to <img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0002.png" alt="$d_ E.$" style="vertical-align:-2px;
width:23px;
height:13px" class="math gen" /> The gravitational pull of the Moon on the far side of the Earth is then </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0004.png" alt="\[ F_{far} = G\frac{m_ Em_ M}{(d_ M+r_ E)^2}. \]" style="width:159px;
height:34px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>The ratio between the two forces is </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0005.png" alt="\[ \frac{F_{near}}{F_{far}} = \frac{Gm_ Em_ M}{(d_ M-r_ E)^2}\times \frac{(d_ M+r_ E)^2 }{Gm_ Em_ M} = \frac{(d_ M+r_ E)^2 }{(d_ M-r_ E)^2}. \]" style="width:371px;
height:41px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Now is the time to look up some values. A quick internet search suggests we use the following: </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0006.png" alt="\[ d_ M = 384,400km, \]" style="width:131px;
height:15px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0007.png" alt="\[ r_ E=6371km. \]" style="width:102px;
height:15px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Substituting these into our equation for the ratio gives </p><table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0008.png" alt="\[ \frac{F_{near}}{F_{far}} =\frac{(384,400+6371)^2}{(384,400-6371)^2}=1.068. \]" style="width:262px;
height:41px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>This means that the gravitational pull of the Moon is <img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0009.png" alt="$6.8\% $" style="vertical-align:-1px;
width:34px;
height:15px" class="math gen" /> stronger on the near side of the Earth than it is on the far side. </p><p>Now let’s do the same for the Sun, writing <img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0010.png" alt="$d_ S$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> for the distance between Earth and Sun, centre to centre. </p><p>By the same argument as above the ratio between the gravitational influence on either side of the Earth is </p><table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0011.png" alt="\[ \frac{F_{near}}{F_{far}} = \frac{(d_ S+r_ E)^2 }{(d_ S-r_ E)^2}. \]" style="width:149px;
height:41px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>The value of <img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0010.png" alt="$d_ S$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> can be taken as </p><table id="a0000000009" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0012.png" alt="\[ d_ S = 1.5 \times 10^{8}km, \]" style="width:134px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> so we get </p><table id="a0000000010" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0013.png" alt="\[ \frac{F_{far}}{F_{near}} = \frac{(1.5 \times 10^{8}+6371)^2 }{(1.5 \times 10^{8}-6371)^2}=1.00017 \]" style="width:283px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>This means that the gravitational pull of the Sun is only <img src="/MI/e8ec13490dae6209b66755278a6f0fd0/images/img-0014.png" alt="$0.017\% $" style="vertical-align:-1px;
width:50px;
height:15px" class="math gen" /> stronger on the near side of the Earth than it is on the far side. </p><p>We therefore conclude that Nightingale is right. The ratio is far bigger for the Moon than it is for the Sun, so the tides we see are due to the Moon. </p></div></div></div>Thu, 22 Sep 2016 10:44:54 +0000mf3446683 at https://plus.maths.org/contenthttps://plus.maths.org/content/political-tides#commentsMaths in a minute: Einstein's general theory of relativity
https://plus.maths.org/content/maths-minute-einsteins-general-theory-relativity
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/gr_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width:300px;"><img src="/content/sites/plus.maths.org/files/articles/2016/blocktime/sun_web.jpg" alt="the sun"><p>What would happen? (Image <a href="http://www.nasa.gov/sites/default/files/thumbnails/image/faintyoung.jpg">NASA</a>)</p></div>
<p>
What would happen if the Sun suddenly exploded? This was the question puzzling Albert Einstein, shortly after his discovery in 1905 of his special theory of relativity with its famous equation <em>e=mc<sup>2</sup></em>. Since the Sun is so far away it takes light eight minutes to travel to Earth, so we wouldn't know about such an explosion straight away. For eight glorious minutes we'd be completely oblivious to the terrible thing that had happened.
</p><p>
But what about gravity? The Earth moves in an ellipse around the Sun, due to the Sun's gravity. If the Sun wasn't there, it would move off in a straight line. Einstein's puzzle was: when that would happen – straight away, or after eight minutes?
</p><p>
In the beginning of twentieth century Isaac Newton's theory of gravity (which had been around for nearly 250 years – you can read more about it <a href="https://plus.maths.org/content/how-does-gravity-work">here</a>) was considered the bedrock of science. It was an incredibly successful theory, explaining the movement of planets and moons and accurately predicting the passage of comets.
</p><p>
According to Newton's theory, the Earth should know immediately that the Sun had disappeared. But Einstein had shown in his special theory of relativity (you can read more about it <a href="/content/whats-so-special-about-special-relativity">here</a>) that nothing can travel faster than the speed of light – not even the effects of gravity. The speed of light was the speed limit of the Universe. And as the light from the explosion would take eight minutes to reach the Earth, the change in the gravity must take at least the same amount of time to be felt by the Earth.
</p>
<div class="leftimage" style="max-width: 350px;"><img src="/issue45/features/berman/gravity.jpg" alt="Massive bodies warp spacetime. Image courtesy <a href='http://www.nasa.gov'>NASA</a>." width="350" height="257" />
<p>Gravity is the manifestation of
the curvature of space and time. Image courtesy <a href='http://www.nasa.gov'>NASA</a>.</p>
</div>
<p>
Einstein's solution to this puzzle was to completely change the way we understood the Universe. Up until 1915, people thought of space as an inert stage on which the laws of physics play out. We could throw in some stars or some planets and they would move around on this stage. But Einstein's revolutionary realisation was that space isn't as passive as that. It is dynamic and responds to what's happening within it. If you put something heavy in space – let's say a planet like Earth – then space around it gives a little. When something else moves close to the planet – say the Moon – if feels this dent in space and rolls around the planet like a marble rolling in a bowl. This is what we call gravity.
</p><p>
The presence of a heavy object like the Earth, moreover, doesn't only affect space: it affects time as well. Einstein realised that space and time are not separate entities, but instead just different aspects of something called <em>spacetime</em>. This fabric of the Universe dances with matter as galaxies orbit and spiral. Stars and planets move, causing spacetime to bend in their wake, causing other stars and planets to move, causing spacetime to bend in their wake. And so on. This was Einstein's great insight – gravity is the manifestation of the curvature of spacetime.
</p><p>
In 1915 Einstein published these discoveries as his general theory of relativity. It is now known as <em>the</em> theory of gravity, superseding Newton's. We know that general relativity is true because Einstein's theory made a number of important predictions that Newton's theory didn't. One of them is that light is bent by gravity. This was observed in the 1919 eclipse of the Sun, providing stunning proof of Einstein's theory and creating newspaper headlines that were the beginning of Einstein's public fame.
</p><p>
<em>This article is based on a number of Plus articles including <a href="/content/einstein-relativity">Einstein and relativity</a> and <a href="/content/what-general-relativity">What is general relativity?</a>. You can read more about Einstein and his general theory of relativity and the maths behind it in <a href="/content/celebrating-general-relativity">our collection of articles and videos</a></em>.
</p>
<hr />
<h3>About this article</h3>
<div class="rightimage" style="width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p><em>This article is part of our <a href="/content/stuff-happens-physics-events"><em>Stuff happens: the physics of events</em> project</a>, run <a href="/content/stuff-happens-physics-events#fqxi">in collaboration with FQXi</a>.</em></p>
</div></div></div>Mon, 12 Sep 2016 14:00:37 +0000Rachel6639 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-einsteins-general-theory-relativity#commentsMagic 19: Solution
https://plus.maths.org/content/magic-19-solution
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Here are 19 dots arranged in a hexagon. Your task is to label the dots with the numbers 1 to 19 so that each set of three dots that lie along a straight-line segment add up to 22.</p>
<p>Happy puzzling!</p>
<p>(<a href="/content/sites/plus.maths.org/files/puzzle/2016/hexagon2.pdf">Download the grid</a> to print and scribble on.)</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/puzzle/2016/hexagon2.png" alt="Magic 19" width="500" height="467" />
<p style="width: 500px;"></p>
</div>
<h3>Solution</h3>
<p>Here's one solutions. If you have found others, or a good strategy for solving the puzzle, please let us know!</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/puzzle/2016/hexagon_solution.png" alt="Magic 19" width="500" height="481" />
<p style="width: 500px;"></p>
</div>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/puzzle/2016/brette.png" alt="Magic 19" width="500" height="481" />
<p style="width: 500px;"></p>
</div>
<p><a href="/content/magic-19">Back to main puzzle page</a></p></div></div></div>Mon, 05 Sep 2016 09:02:19 +0000mf3446675 at https://plus.maths.org/contenthttps://plus.maths.org/content/magic-19-solution#commentsMagic 19
https://plus.maths.org/content/magic-19
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/19_fronticon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Here are 19 dots arranged in a hexagon. Your task is to label the dots with the numbers 1 to 19 so that each set of three dots that lie along a straight-line segment add up to 22.</p>
<p>Happy puzzling!</p>
<p>(<a href="/content/sites/plus.maths.org/files/puzzle/2016/hexagon2.pdf">Download the grid</a> to print and scribble on.)</p>
<div class="centre image"><img src="/content/sites/plus.maths.org/files/puzzle/2016/hexagon2.png" alt="Magic 19" width="500" />
</div></div></div></div><div class="field field-name-field-sol-link field-type-node-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/content/magic-19-solution">Magic 19: Solution</a></div></div></div>Thu, 01 Sep 2016 14:30:16 +0000mf3446674 at https://plus.maths.org/contenthttps://plus.maths.org/content/magic-19#commentsCalculating the multiverse
https://plus.maths.org/content/node/6673
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon-2-2-1.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/packages/2016/banff/image.jpg" alt="Fred Adams" width="200" height="300" />
<p>Fred Adams.</p>
</div>
<p>Could there be many other universes beside our own? Astrophysicist <a href="https://lsa.umich.edu/physics/people/faculty/fca.html">Fred Adams</a> believes there could: if one universe managed to spring into existence, then why not others? At the <a href="content/physics-what-happens-and-whos-listening">FQXi international conference</a> in Banff, Canada, he told us how he is using knowledge about our own Universe to calculate how many of those other universes could be similar to our own. </p>
</div></div></div><div class="field field-name-field-remote-encl field-type-file field-label-inline clearfix clearfix"><div class="field-label">Podcast download link: </div><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="Audio icon" title="audio/mpeg" src="/content/modules/file/icons/audio-x-generic.png" /> <a href="https://plus.maths.org/content/sites/plus.maths.org/files/packages/2016/banff/Fred_Adams.mp3" type="audio/mpeg; length=10524430">Fred_Adams.mp3</a></span></div></div></div>Wed, 24 Aug 2016 09:57:39 +0000mf3446673 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/6673#commentsCalculating the multiverse
https://plus.maths.org/content/calculating-multiverse
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon-2-2.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/packages/2016/banff/image.jpg" alt="Fred Adams" width="200" height="300" />
<p>Fred Adams.</p>
</div>
<p>Could there be many other universes beside our own? Astrophysicist <a href="https://lsa.umich.edu/physics/people/faculty/fca.html">Fred Adams</a> believes there could: if one universe managed to spring into existence, then why not others? At the <a href="content/physics-what-happens-and-whos-listening">FQXi international conference</a> in Banff, Canada, he told us how he is using knowledge about our own Universe to calculate how many of those other universes could be similar to our own. </p>
<p><a href="/content/sites/plus.maths.org/files/packages/2016/banff/Fred_Adams.mp3">Listen to our interview with Fred Adams</a>.</p></div></div></div>Mon, 22 Aug 2016 16:03:50 +0000mf3446671 at https://plus.maths.org/contenthttps://plus.maths.org/content/calculating-multiverse#commentsWhat's happening and who's listening?
https://plus.maths.org/content/physics-what-happens-and-whos-listening
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/bang_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>What do we mean when we say that something "has happened"?
It's quite hard to put your finger on it even in ordinary life, but for physicists the question of what constitutes an "event" is even more subtle. And it's intimately linked to another conundrum: what or who counts as an observer who witnesses events, and what is the role of observers in physics.</p>
<p> In the first video below physicist <a href="http://scipp.ucsc.edu/~aguirre/Home.html">Anthony Aguirre</a> briefly explores the difficulties with the notions of events and observers, and explains why the <a href="http://fqxi.org">Foundational Questions
Institute</a> has organised a whole <a href="http://fqxi.org/conference/2016?dest=banff">conference</a> to investigate them. In the second video physicist <a href="https://www.preposterousuniverse.com/self.html">Sean Carroll</a> goes into a little more detail, looking at the mysterious quantum wave function, explaining why thinking about these things matters, and why our intuition needs to become more quantum.</p>
<p><em>To find out more about some basic features of quantum mechanics, see <a href="/content/schrodingers-equation-what-does-it-mean">this</a> article. And to find out more about the notion of time in physics (or lack of it), see <a href="/content/what-time-0">this</a> article. We'll be bringing you more reports from the conference in the near future.</em></p>
<iframe width="480" height="270" src="https://www.youtube.com/embed/Jp-FEWqh508" frameborder="0" allowfullscreen></iframe>
<iframe width="480" height="270" src="https://www.youtube.com/embed/8nlUEGCUp-Q" frameborder="0" allowfullscreen></iframe></div></div></div>Sun, 21 Aug 2016 14:25:58 +0000mf3446667 at https://plus.maths.org/contenthttps://plus.maths.org/content/physics-what-happens-and-whos-listening#commentsLife's dirty secrets
https://plus.maths.org/content/lifes-dirty-secrets
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/chick_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightshoutout">You can listen to Davies' talk in a <a href="http://fqxi.org/community/podcast/2016.08.19">podcast</a> on the FQXi community website.</div>
<p>Sometimes even scientists are in denial. We're in Canada at the
moment, at a <a href="http://fqxi.org/conference/2016?dest=banff">conference</a> organised by the <a href="http://fqxi.org">Foundational Questions
Institute</a> investigating some deep questions in physics that lie on the boundary to
philosophy. The first session here was called <em>dirty
secrets</em>. It looked at uncomfortable problems with
existing theories that often get brushed under the carpet.
Physicist <a href="http://cosmos.asu.edu">Paul Davies</a>
explored one of the biggest conundrums of them all:
life. Everyone knows, and is happy to admit, that we don't understand
its origin, but, as Davies pointed out, there are other mysteries hiding in the closet. </p>
<div class="leftimage" width="332px"><img src="/content/sites/plus.maths.org/files/news/2016/life/chick.jpg" alt="bird" width="254px" height="312" /><p>How did <em>that</em> happen? </p></div>
<p>Why is this of interest to theoretical physicists, who spend their time grappling with mathematical equations rather than test tubes in the lab? Because physics investigates the fundamental
building blocks of nature which, after all, give rise to life, though nobody understands how. The question is whether we need new physical theories to explain life. It'll be a while until we can answer this, so for now let's throw open the closet doors and look at some of life's dirty secrets.</p>
<p><strong>Secret 1: We don't know that all life is the same </strong>
</br> It is commonly assumed that all life originated from a common
ancestor (see, for example, <a href="/content/luca-lived-beneath-ocean">this recent news story</a>). But we don't
actually have proof that this is really the case. The life we have
studied in detail so far certainly looks like it has evolved from one
common root, but it's not impossible that there are "aliens under
our noses". There's still plenty to explore in the realm of microbes,
and we might possibly discover a life form which doesn't fit in the common ancestor
picture.</p>
<p><strong>Secret 2: Life cannot be completely understood using
"ordinary" biology.</strong>
</br>
We need to add in quantum theory. Naively you might think that
quantum physics, which describes the smallest building blocks of
matter, isn't required when you talk about bigger organisms. It's
certainly the case that to describe the physical behaviour of something
macroscopic, say a billiard ball, you don't need to resort
to quantum theory: the classical physics of Newton is
enough. However, some recent advances in biology have shown that
quantum effects do play a role. It's been shown, for example, that
European robins use quantum processes to navigate their annual migration (see
<a href="/content/learning-quantum-physics-birds"><em>Flying home with
quantum physics</em></a>). Quantum processes also play a role in
photosynthesis within plants (see <a
href="/content/shining-light-solar-energy"><em>Shining a light on
solar energy</em></a>). And there are other examples too. Just how fundamental quantum effects are in biology is unclear, but it seems that we can't ignore them. </p>
<div class="rightimage" width="250px"><img src="/sites/plus.maths.org/files/news/2010/birds/bird.jpg" alt="bird" width="250px" height="249" /><p>Some birds use a quantum compass to navigate. </p></div>
<p><strong>Secret 3: We have not made life in the lab, or are anywhere
near doing it.</strong>
</br>
There have been experiments that at first sight may appear to have
created life. In 1952 <a
href="https://en.wikipedia.org/wiki/Harold_Urey">Harold Urey</a> and
<a href="https://en.wikipedia.org/wiki/Stanley_Miller">Stanley
Miller</a> conducted an <a
href="https://en.wikipedia.org/wiki/Miller–Urey_experiment">experiment</a>
that involved recreating the conditions that were thought to be
present when the Earth was young. With the help of electric sparks
Ulay and Miller were able to produce different amino acids, which
are the building blocks of life.</p>
<p>But that's just it — they are only building blocks, not life
itself. As Davies pointed out, being able to produce a brick
doesn't mean you're able to build a magnificent city like New
York. There's a "stupendous gap" between being able to make the
building blocks of something and making the whole
thing. Other experiments that proclaim to have produced life are mere
re-engineering of existing life, said Davies.</p>
<p><strong>Secret 4: Most mesoscopic biophysics is "make-believe".</strong>
</br>
<em>Mesoscopic</em> means of intermediate size, and the
make-believe quote comes from <a href="http://www.northwestern.edu/newscenter/stories/2011/07/jonathan-widom-obituary.html">John Widom</a>.
We do understand
a lot about the structure of DNA and of chromosomes, and many textbooks will have you
believe that we understand everything "upwards" from there, to the
level of cells and organisms. However, there is a huge gap in the
middle. For example, the DNA that exists inside cells needs to be
folded up to fit inside the nucleus and that folding isn't really understood
at all. </p>
<p><strong>Secret 5: Survival of the fittest doesn't alone explain
biological complexity.</strong>
</br>
Davies doesn't say that this Darwinian principle is wrong, but only
that it's incomplete. Natural selection can only act on what is
already there: the fittest organisms survive and pass on
their genes, thereby giving rise to evolution. But where did those
organisms and life forms come from in the first place? Inspired by the
title of a <a
href="https://www.amazon.ca/Arrival-Fittest-Solving-Evolutions-Greatest/dp/1591846463">book</a>
by Andreas Wagner, Davies pointed out that we don't just need to worry
about the survival, but also about the <em>arrival</em> of the
fittest.</p>
<p><strong>Secret 6: There's little reason to believe that life occurred
more than once in the Universe.</strong>
</br>
Current thinking is that there's nothing special about life on Earth
and that it should arise wherever conditions are similar to those on
Earth — as <a
href="https://en.wikipedia.org/wiki/Christian_de_Duve">Christian de
Duve</a> put it, life is a "cosmic imperative". But Davies thinks that
this idea isn't based on any evidence. Since we don't know the
mechanism by which life appeared, he said, we can't estimate the odds
of it appearing elsewhere. You can't estimate the odds of an unknown
process. Not long ago, in the 1960s and 1970s, the
prevailing opinion was indeed very different. People
thought life such an improbable
accident of nature, they didn't believe it could have happened twice in the
Universe. At that time, Davies said, professing an interest in in
extraterrestrial life was akin to professing an interest in
fairies. </p></div></div></div>Sat, 20 Aug 2016 22:16:01 +0000mf3446666 at https://plus.maths.org/contenthttps://plus.maths.org/content/lifes-dirty-secrets#commentsThe Basel problem
https://plus.maths.org/content/basel-problem
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon-14.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article is part of <a href="/content/five-eulers-best-0">Five of Euler's best</a>. Click <a href="/content/five-eulers-best-0">here</a> to read the other four problems featured in this series, which is based on a talk given by the mathematician <a href="http://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/">Günter M. Ziegler</a> at the European Congress of Mathematics 2016.</em></p>
<hr/>
<p>This problem is the one that’s most obviously mathematical in appearance. It asks for the result of the sum </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0001.png" alt="\[ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+... \]" style="width:229px;
height:35px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>What are those numbers? If you look closely at the numbers in the denominators of the individual terms, </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0002.png" alt="\[ 1, 4, 9, 16, 25, 36, ..., \]" style="width:137px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>you might spot that they are the <em>square numbers</em> you get by squaring the natural numbers </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0003.png" alt="\[ 1=1^2, \]" style="width:49px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0004.png" alt="\[ 4=2^2, \]" style="width:50px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0005.png" alt="\[ 9=3^3 \]" style="width:44px;
height:15px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0006.png" alt="\[ 16=4^2, \]" style="width:58px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0007.png" alt="\[ 25=5^2, \]" style="width:59px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><table id="a0000000009" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0008.png" alt="\[ 36=6^2, \]" style="width:59px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>so it’s easy to see how to continue the sum. The next term would be </p><table id="a0000000010" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0009.png" alt="\[ \frac{1}{7^2}=\frac{1}{49}, \]" style="width:65px;
height:35px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> the one after </p><table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/95d76d498e8ec56840b80609f12d6bdd/images/img-0010.png" alt="\[ \frac{1}{8^2}=\frac{1}{64}, \]" style="width:65px;
height:35px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>and so on. </p><p>
This also means that the sum has infinitely many terms, so asking for
its result might seem non-sensical at first. However, it's possible for an
infinite sum to <em>converge</em> to a particular value: as you add more
terms, the results inch
closer and closer to that limiting value, getting arbitrarily close to it as
you add more terms (see <a href="/content/when-things-get-weird-infinite-sums">this article</a> for more on converging sums). What the problem is asking for is the limiting
value of the sum above.
</p>
<div class="rightimage" style="width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Euler/leibniz.jpg" alt="Leibniz" width="250" height="316" />
<p>Gottfried Wilhelm Leibniz (1646 - 1716).</p>
</div>
<p>The problem wasn't posed by Euler himself, rather it had already been
around since at least 1644, and several well-known mathematicians had
worked on it. They included three members of the <a
href="/content/news-world-maths-mathematical-moments-mathematical-bernoullis-basel">Bernoulli family</a>, which produced no
fewer than eight great mathematicians and also contained an erstwhile private tutor of
Euler's (<a href="http://www-history.mcs.st-and.ac.uk/Biographies/Bernoulli_Johann.html">Johann Bernoulli</a>). The great Leibniz, who we already met above, had also tried his hand at the problem but failed to find an exact solution. Soon the problem, which was named after the home town of the Bernoullis, became notorious.</p>
<p>A naive way of attacking the problem is to add up as many terms as you can and see what you get. The problem with the Basel sum, however, is that it converges very slowly. Adding up the first 1000 terms (which would be an extremely arduous task by hand) only gives you a result that is correct in the first two decimal places, but differs in the third. Good approximations, therefore, required some mathematical trickery. "So much work has been done on the [sum] that it seems hardly likely that anything new may still turn up," wrote Euler in a letter. "I, too, in spite of repeated efforts, could achieve nothing more than approximate values ..."</p>
<p>Yet in 1735 Euler succeeded. His truly amazing achievement connects to a number nobody had expected to crop up in the context. The number <img src="/MI/cc19bd4f2bedb84d4b0d8df0817185ab/images/img-0001.png" alt="$\pi ,$" style="vertical-align:-3px;
width:14px;
height:10px" class="math gen" /> which is the ratio of the circumference of a circle to its diameter: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/cc19bd4f2bedb84d4b0d8df0817185ab/images/img-0002.png" alt="\[ 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}+... = \frac{\pi ^2}{6}. \]" style="width:277px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>
The maths that led Euler to this result is a bit too hard to present
here, but it's accessible to A level students. You can read about it
in
<a href="/content/infinite-series-surprises"><em>An infinite series of
surprises.</a></em>.</p>
<p>The five problems presented in <a href="/content/five-eulers-best">this series</a> are just the very small tip of the very large ice berg of Euler's work. You can find out a little bit more in our articles <a href="/content/os/issue42/features/wilson/index"><em>"Read Euler, read Euler, he is the master of us all"</em></a> and <a href="/content/frugal-nature-euler-and-calculus-variations"><em>Euler and the calculus of variations</em></a>. Alternatively there are many books about Euler, and if you're really brave you can dig into his original papers at the <a href="http://eulerarchive.maa.org">Euler archive</a>.</p>
<hr/>
<h3>About this article</h3>
<p>This article is part of <a href="/content/five-eulers-best-0">Five of Euler's best</a>. Click <a href="/content/five-eulers-best-0">here</a> to read the other four problems featured in this series, which is based on a talk given by the mathematician <a href="http://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/">Günter M. Ziegler</a> at the European Congress of Mathematics 2016.</p>
<p><a href="http://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/">Günter M. Ziegler</a> is professor of Mathematics at Freie Universität Berlin, where he also directs the Mathematics Media Office of the <a href="https://dmv.mathematik.de">German Mathematical Society DMV</a>. His books include <a href="http://www.springer.com/us/book/9783642008566"><em>Proofs from THE BOOK</em></a> (with Martin Aigner) and <a href="https://www.crcpress.com/Do-I-Count-Stories-from-Mathematics/Ziegler/p/book/9781466564916"><em>Do I count? Stories from Mathematics.</em></a>.</p>
</p>
<p><p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. She really enjoyed Ziegler's talk about Euler at the European Congress of Mathematics in Berlin in July 2016, on which this article is based.</p></div></div></div>Fri, 05 Aug 2016 16:27:22 +0000mf3446661 at https://plus.maths.org/contenthttps://plus.maths.org/content/basel-problem#comments