plus.maths.org
https://plus.maths.org/content
enCalculating 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/VP4S4AvUc8k" frameborder="0" allowfullscreen></iframe>
<iframe width="480" height="270" src="https://www.youtube.com/embed/GlWy9JNoV7Y" 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#commentsEuler's polyhedron formula
https://plus.maths.org/content/eulers-polyhedron-formula-2
<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/polyhedron_icon.png" 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>Think of a triangle. Now think of a
square, followed by a pentagon, a hexagon, and so on. These shapes are
called <em>polygons</em>. "Poly" is the Greek word for "many" and "gon"
is the Greek word for "angle".</p>
<p>Now move up a dimension. Think of a cube, a pyramid, or perhaps an octahedron. These are all <em>polyhedra</em>
("hedra" is the Greek word for "base"). A polyhedron is an object made
up of a number of flat polygonal <em>faces</em>. The sides of the faces are
called <em>edges</em> and the corners of the polyhedron are called
<em> vertices</em>.</p>
<div class="centreimage"><img src="/issue43/features/kirk/Solids.jpg" alt="The Platonic solids" width="600" height="142" />
<p>The Platonic solids are examples of polyhedra. From left to right we have the tetrahedon with four faces, the cube with six faces, the octahedron with eight faces, the dodecahedron with twelve faces, and the icosahedron with twenty faces.</p>
</div>
<p>Now imagine counting the number <em>V</em> of vertices, the number
<em>E</em> of edges and the number <em>F</em> of faces of a
polyhedron. It turns out that, as long as your polyhedron is <em><a href="https://en.wikipedia.org/wiki/Convex">convex</a></em> (has no sticky out bits) and has no holes running through it, the number of vertices minus the number of edges
plus the number of faces,</p>
<p><em>V</em> - <em>E</em> + <em> F</em>,</p>
<p>is always equal to 2. Your polyhedron could be a cube, an octahedron, something more intricate like the <a href="http://mathworld.wolfram.com/GreatRhombicosidodecahedron.html">great rhombicosidodecahedron</a> in the picture below, or even something much more irregular. That's a truly amazing result.</p>
<div class="rightimage" style="width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Euler/polyhedron.png" alt="great rhombicosidodecahedron" width="250" height="246" />
<p>A great rhombicosidodecahedron. Image from the <a href="http://demonstrations.wolfram.com/PolyhedraSpheresAndCylinders/">Wolfram Demonstrations Project</a>, created by <a href="http://demonstrations.wolfram.com/author.html?author=Russell%20Towle">Russell Towle</a> , reproduced under <a href="https://creativecommons.org/licenses/by-sa/3.0/">CC BY-SA 3.0</a>.</p>
</div>
<p>The expression</p>
<p><em>V</em> - <em>E</em> + <em> F</em> = 2</p>
<p>is known as Euler's polyhedron formula. Euler wasn't the first to
discover the formula. That honour goes to the French mathematician
<a href="http://www-history.mcs.st-and.ac.uk/Biographies/Descartes.html">René Descartes</a> who already wrote about it around 1630. After his death in Sweden in 1650, Descartes'
papers were taken to France, but the boat carrying them sunk in the river Seine. The papers remained at the bottom of the river for three days, but thankfully could be dried after they had been fished out. Another famous mathematician, <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Leibniz.html">Gottfried Wilhelm von Leibniz</a>, copied Descartes' notes on the formula around 1675. Thereafter Descartes' manuscript disappeared completely and Leibniz' copy was lost until someone rediscovered it in a cupboard at the Royal Library of Hanover in 1860. Quite a dramatic history for a mathematical formula.</p>
<p>While Descartes may have discovered the formula first, it was Euler who had a crucial insight. When you are
looking at the polyhedron formula, precise measurements don't matter:
you don't need to know at what angle to faces of a polyhedron meet, or
how long their sides are. The polyhedron formula belongs to the world
of topology, just as the <a href="/content/bridges-k-nigsberg">bridges of Königsberg</a> problem does. It tells us
something, not about individual objects with certain dimensions, but
about the nature of space.</p>
<p>Proving the polyhedron formula isn't actually that hard. You can read a proof <a href="/content/eulers-polyhedron-formula">here on <em>Plus</em></a>, or find a total of twenty proofs on the <a href="https://www.ics.uci.edu/~eppstein/junkyard/euler/">Geometry Junkyard</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 16:18:28 +0000mf3446660 at https://plus.maths.org/contenthttps://plus.maths.org/content/eulers-polyhedron-formula-2#commentsThe 36 officers problem
https://plus.maths.org/content/36-officers-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/officer_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"><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 looks like a puzzle you might find in a Sunday
paper. Imagine there's a war. You're in command of an army that
consists of six regiments, each containing six officers of six different
ranks. Can you arrange the officers in a 6x6 square so that each row and
each column of the square holds only one officer from each regiment
and only one officer from each rank?</p>
<p>Just as any Sunday puzzler Euler had a go at the problem, but
without success. "After we have put a lot of thought into finding a
solution, we have to admit that such an arrangement is impossible,
though we can't give a rigorous demonstration of this" he
<a
href="http://eulerarchive.maa.org/docs/originals/E530_intro.pdf">wrote
in 1782</a>. A proof that the problem is impossible to solve didn't
come along until nearly 130 years later. The French mathematician <a
href="https://en.wikipedia.org/wiki/Gaston_Tarry">Gaston Terry</a>
provided it in 1901.</p>
<p>If you've ever come across Latin
squares, then the 36 officers problem might have reminded you of
them. A <em>Latin square</em> is a square array of symbols (for example numbers
or letters) in which every
symbol occurs just once in every row and column. If you combine two
Latin squares of the same size but with different symbols, you get what's called a
<em>Graeco-Latin square</em> (also called an Euler square). It contains pairs of symbols so that every
member of a pair occurs exactly once in each row and column, and so
that every possible pair appears exactly once on the array.</p>
<center>
<table border="0">
<tr>
<td><!-- FILE: include/centrefig.html -->
<div class="centreimage"><img src="/issue38/features/aiden/table9.gif" alt="A Latin square with numbers" width="174" height="181" /></div>
<!-- END OF FILE: include/centrefig.html --></td>
<td>+</td>
<td><!-- FILE: include/centrefig.html -->
<div class="centreimage"><img src="/issue38/features/aiden/table10a.gif" alt="A different Latin square with letters" width="175" height="174" /></div>
<!-- END OF FILE: include/centrefig.html --></td>
<td>=</td>
<td><!-- FILE: include/rightfig.html -->
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Euler/table10b.gif" alt="A Euler square" width="175" height="175" /></div>
<!-- END OF FILE: include/rightfig.html --></td>
</tr>
</table>
</center>
<p>Euler realised that a solution of the 36
officers problem would give us a Graeco-Latin 6x6 square. The pairs in
this case represent an officer's rank and regiment. That's unlucky: if
their had been five regiments and ranks, or seven regiments and ranks,
then the problem could have been solved. Euler was aware of this too
and speculated that Graeco-Latin squares are impossible if the number
of cells on the side of square (the <em>order</em> of the square) is
of the form 4<em>k</em> + 2 for a whole number <em>k</em> (6 = 4x1
+2). It wasn't until 1960 that he was proved wrong. The mathematicians Bose,
Shrikhande and Parker enlisted the help of computers to prove that Graeco-Latin squares exist for all orders except 2 and 6.
</p>
<p>So why do we mention what at first sight looks like an Euler fail rather than an Euler
success? Because as usual with Euler, there's a lot more to his work
on the problem than a bit of speculation. The 36 officer problem inspired
him to a lot of work on Latin and Graeco-Latin squares, including on
problems akin to modern-day sudoku (see <a href="/content/anything-square-magic-squares-sudoku">this article</a>
for more on sudoku and other types of interesting squares). It
contributed important work to an area of maths called
<em>combinatorics</em> which, as you may have guessed, is about
combining objects in ways that satisfy particular constraints, and
has many applications in the real world.</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 16:14:11 +0000mf3446659 at https://plus.maths.org/contenthttps://plus.maths.org/content/36-officers-problem#commentsThe knight's tour
https://plus.maths.org/content/knights-tour
<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/knight_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"><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>The second question in this <a href="/content/five-eulers-best-0">series</a> is also about finding a path, but this time on a
chess board. To phrase it in Euler's own words,</p>
<p><em>"I found myself one day in a company where, on the occasion of a game of chess, someone proposed this question:
To move with a knight through all the squares of a chess board,
without ever moving two times to the same square, and beginning with a
given square."</em></p>
<div class="rightimage" style="width: 300px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Euler/knight.jpg" alt="Knight's tour" width="300" height="250" />
<p>A knight's tour as it appears in Euler's paper. Euler just numbered the squares in the order in which they're traversed, but we have traced the tour in blue. From paper E309 on the <a href="http://eulerarchive.maa.org">Euler archive</a>.</p>
</div>
<p>Euler called this a "curious problem that does not submit to any
kind of analysis". But analyse it he did, and he was the first person
to do so methodically. He came up with a method for constructing knight's tours which, as he claimed, allows you to discover as many tours as you like. Because "although their number isn't infinite, it is so great that we will never exhaust it." </p>
<p>These statements are contradictory by modern mathematical standards. If you can discover as many tours as you like, then surely their number is infinite. But we know what Euler means: the number of tours is so huge, it might as well be infinite. To really try and count them you need to use a computer. In 1996 the mathematicians Martin Löbbing
and Ingo Wegener did just this. They counted only those tours for which the end square is just one knight's hop away from the starting square, that is, tours that can be turned into a loop. They claimed that <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r5/pdf"> there were 33,439,123,484,294 of them</a> (in fact, their aim wasn't so
much to count the number of tours, but to demonstrate the usefulness of
a particular enumeration method).</p>
<p> It was <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v3i1r5/comment">soon pointed
out to them</a>, however, that the result can't be right: the number of
knight's tours must be divisible by 4 which 33,439,123,484,294
isn't. The apparently correct result is 13,267,364,410,532. That's more
than the diameter of the solar system measured in kilometres. As
for knight's tours that can't be turned into a loop, a number computed by Alexander Chernov in 2014 is 19,591,828,170,979,904. People don't seem to be entirely certain as to whether it's correct though.</p>
<p>If you read French then you can read Euler's original article on the <a href="http://eulerarchive.maa.org">Euler archive (article E309)</a>. For an easier ride, read about Euler's methods for constructing
knight's tours in this <a
href="http://eulerarchive.maa.org/hedi/HEDI-2006-04.pdf">MAA online
article</a>. There's more about variations of the classical knight's tour
problem on the <a
href="http://www-history.mcs.st-and.ac.uk/Projects/MacQuarrie/Chapters/Ch3.html">MacTutor
History of Mathematics 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 16:04:49 +0000mf3446658 at https://plus.maths.org/contenthttps://plus.maths.org/content/knights-tour#commentsThe bridges of Königsberg
https://plus.maths.org/content/bridges-k-nigsberg
<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/bridges_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"><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>We'll start with a favourite problems of ours, which we have often
visited on <em>Plus</em> (we've even made a <a href="/content/bridges-konigsberg-movie">movie</a> of it). In the eighteenth century citizens of the Prussian city of Königsberg (now Kaliningrad) had set themselves a puzzle. Königsberg was divided by a river, called the Pregel, which contained two
islands with seven bridges linking the various land masses. The puzzle
was to find a walk through
the city that crossed every bridge exactly once. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2014/bridges_networks/konigsberg_bridges.png" alt="bridges of Königsberg" width="302" height="238" />
<p style="width: 302px;">Can you find a path that crosses every bridge once and only once? Image: <a href="http://en.wikipedia.org/wiki/File:Konigsberg_bridges.png">Bogdan Giuşcă</a>.</p>
</div>
<p>Euler realised that this type of problem required a new way of thinking. In a <a href="http://eulerarchive.maa.org/correspondence/correspondents/Marinoni.html">letter</a> he wrote to the Italian mathematician and astronomer
<a href="https://de.wikipedia.org/wiki/Johann_Jakob_Marinoni">Giovanni Jacopo Marioni</a> Euler said,</p>
<p><em>This question is so banal, but seemed to me worthy of attention
in that geometry, nor algebra, nor even the art of counting was
sufficient to solve it. In view of this, it occurred to me to wonder
whether it belonged to the geometry of position
which Leibniz had once so much longed for. And so, after some
deliberation, I obtained a simple, yet completely established, rule
with whose help one can immediately decide for all examples of this
kind, with any number of bridges in any arrangement, whether such a
round trip is possible, or not....</em></p>
<p>By <em>geometry of position</em> Euler meant a geometry that isn't
concerned with precise measurements of lengths, angles or areas. It's
what's today called <em>topology</em>. In the Königsberg
problem the exact lay-out of
the city doesn't matter. The only thing that is important is how
things are connected. Bearing this in mind, you can turn the messy map
of the town into a neat network (also called a <em>graph</em>), with
dots representing land masses and links between them the bridges.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/blog/112013/bridges.jpg" alt="" width="677" height="159" /><p style="width: 677px;">Transforming the problem. Image: <a href="http://en.wikipedia.org/wiki/File:Konigsberg_bridges.png">Bogdan Giuşcă</a>.
</p></div>
<p>Euler made a crucial observation: if a path through this network is
going to cross every link exactly once, then each node within the path
must have an
even number of links attached to it. That's because whenever you enter
the node by one link, you need to leave it by another, so the node
needs two links if you visit it once, four if you visit it twice, and
so on. The only nodes that can have an odd number of links attached to
them are the nodes where the walk starts and ends (if they are
distinct).</p>
<p>This immediately tells us that the path we are looking for doesn't
exist in the Königsberg problem. All nodes have an odd number of links
attached to them. The beauty of this argument, as Euler suggested in his letter, is that it works for
any network, no matter how large or complex. A path that crosses every
link exactly once only exists if all, or all but two, nodes have an
even number of links attached to them. The converse is also true
(though Euler didn't deliver a rigorous proof for this): if all, or all but two, nodes have an
even number of links attached to them, then a path that crosses every
link exactly once exists.</p>
<p>Euler's thoughts about the Königsberg problem marked the beginning
of an area of maths called <em>graph theory</em>, which you might also
call <em>network theory</em>. And since we're surrounded by networks,
be they social network, transport networks, or the internet, network
theory plays an important part in modern mathematics (see <a
href="/content/taxonomy/term/288">here</a> for articles about networks).</p>
<p>If you know Latin, you can read Euler's original paper on the bridges of Königsberg on 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 15:57:38 +0000mf3446657 at https://plus.maths.org/contenthttps://plus.maths.org/content/bridges-k-nigsberg#commentsFive of Euler's best
https://plus.maths.org/content/five-eulers-best-0
<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/euler_icon_0.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: 250px;"><img src="/issue43/features/kirk/euler.jpg" alt="Leonhard Euler" width="250" height="291" />
<p>Leonhard Euler, 1707 - 1783</p>
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<p>
<a href="http://www-history.mcs.st-and.ac.uk/Biographies/Euler.html">Leonhard Euler</a> is probably the most prolific mathematician of all
time. Born in 1707 in Basel, Switzerland, Euler spent much of his life
in
Berlin. Mathematicians of Berlin are proud of
this heritage, which is why it's no surprise that Euler featured in
the <a href="/content/european-congress-mathematics-2016">European Congress of Mathematics</a>, which took place in that
beautiful city last month. <a href="http://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/">Günter M. Ziegler</a>, a mathematician at the Freie Universität Berlin and champion of public engagement with maths, delivered a lecture on
five famous problems that are associated with Euler. </p>
<p>The beauty of
these "five strikes of genius", as Ziegler called them, is that you don't have to be a
mathematician to appreciate them: their solution may be tricky, but
the problems themselves are easily explained and fun. Which is why
we've decided to recap them in this article.</p>
<p>We won't go into Euler's biography here (you can find much interesting
information on the <a
href="http://www-history.mcs.st-and.ac.uk/Biographies/Euler.html">MacTutor
History of Maths archive</a> and in a variety of books written on Euler). Suffice to say that Euler also spent a lot of time in the
Russian city of St Petersburg, where he eventually died in 1783, that
he produced more than half of his work after having gone
blind, and that he had 13 children. Euler claimed that <a
href="http://www-history.mcs.st-and.ac.uk/Biographies/Euler.html">"he
made some of his greatest mathematical discoveries while holding a
baby in his arms with other children playing round his
feet</a>". Sadly, only five of his children survived
their childhood years. </p>
<p>Leaving Euler's biography to one side, let's turn to the five famous problems. You can read them in sequence or skip to your favourite one.</p>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/bridges_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/bridges-k-nigsberg">The bridges of Königsberg</a> </strong> — Can you find a path through on this city map that crosses every bridge exactly once? Euler's answer to this problem started off the filed of graph theory. </p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/knight_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/knights-tour">The knight's tour</a> </strong> — Can you move a knight on a chessboard so that it visits every square exactly once? Euler was one of the first to analyse this problem systematically, but some questions about it are still open today.</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/officer_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/36-officers-problem">The 36 officers problem</a> </strong> — Euler may not have cracked this problem completely, but it led to a lot of important work, including on what we today know as sudoku.</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/polyhedron_icon.png" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/eulers-polyhedron-formula-2">Euler's polyhedron formula</a> </strong> — This surprising result about 3D shapes tells us something deep about the nature of space.</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon-14.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/basel-problem">The Basel problem</a> </strong> — Here's an infinite sum that puzzled quite a few famous mathematicians until Euler found the surprising answer.</p></div>
<h3>About these articles</h3>
<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>
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<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 15:54:09 +0000mf3446656 at https://plus.maths.org/contenthttps://plus.maths.org/content/five-eulers-best-0#comments