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enPhantom jams
https://plus.maths.org/content/phantom-jams
<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/traffic_miami_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 and 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>Traffic jams are a pain, especially when they happen for no apparent reason. Even when a road is only moderately full and there are no obstacles up ahead, traffic can bunch up, slow down, and even come to a complete halt. Traffic researchers have long been interested in such <em>phantom jams </em> — and mathematical models can tell us why they happen.</p>
<div class="rightimage" style="max-width: 320px;"><img src="/content/sites/plus.maths.org/files/articles/2019/traffic/traffic_miami.jpg" alt="Traffic" width="320" height="240" />
<p>Rush hour on IS 95 in Florida. Photo: <a href="https://commons.wikimedia.org/wiki/File:Miami_traffic_jam,_I-95_North_rush_hour.jpg">B137</a>, <a href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4,0</a>.</p>
</div>
<p>
You can build a model based on the simple assumption that drivers vary their speed according to the distance between themselves and the vehicle in front.
If the distance decreases because the car in front brakes, a driver will slow down.
If the distance increases, a driver will speed up until they reach the speed they desire (or the speed dictated to them by a speed limit). This is exactly what happens in reality.</p>
<p>To keep things simple let's also assume that we have a number of cars travelling in a single lane around a ring. This is less realistic but means we can assume that the number of cars on the road stays the same, which makes for easier maths.</p>
<p>It's relatively straight-forward to capture this situation using mathematical equations that express the acceleration of each car in terms of the headway in front of them and their current speed (see <a href="/content/phantom-jams#maths">below</a> for some details). Given an initial speed and distribution of cars, the solutions to the equations will tell you how fast the cars will be travelling at any given time. </p>
<p>This toy model is so simple, you'd expect the traffic to settle down into an even flow going round and round the ring. Car-following models like this do indeed predict just that: there is a <em>uniform flow equilibrium</em>, which is stable in the sense that little variations in speed won't knock the equilibrium off kilter. That's exactly the kind of situation we'd want in real life, but we all know that in reality things can play out very differently. </p>
<h3>Chain reaction</h3>
<p>One crucial thing the model is missing is the fact that drivers aren't machines: while driving along they might be fiddling with the radio, talking to the kids on the back seat, or arguing with the satnav. In short, drivers don't react immediately to a change in the distance to the car in front. A little delay in their reaction could make quite a difference to how hard they need to brake.</p>
<p>
We can add this fact into the model by including a reaction time delay parameter. When you do this, the dynamical system does indeed change its overall nature: apart from the stable uniform flow behaviour, the equations now also admit a stop-and-go wave that can persistently travel down the road in the opposite direction of the traffic flow, even if the density of traffic isn't all that bad. The crucial ingredient here are the drivers' delayed reactions to even minor disruptions of traffic, such as a lorry ahead changing lanes. It's those delays that can trigger the stop-and-go wave. Our simple model has therefore delivered an explanation for phantom jams that previously wasn't obvious.</p>
<p>Traffic jam waves spreading backwards through the traffic have been predicted by other mathematical traffic models too. They have even been observed in a real-life experiment which reconstructs the simple set-up we described above: it had 22 cars travelling around a ring at a constant speed of roughly 30kmh and even though traffic started out flowing smoothly, jam waves were soon triggered by little variations in speed. </p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/Suugn-p5C1M" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
<p>The good thing about mathematical models of traffic is that they can not only explain baffling phenomena, but also help us decide how to avoid them. The simple model above suggests that keeping your reaction time to a minimum can do more to avoid jams than you might think — perhaps driverless cars will help in this context. But models can also be used to figure out how things such as variable speed limits may help guide traffic out of a jam and back into uniform flow. The roads of the future should, and hopefully will, be built on maths.</p>
<a name="maths"></a>
<h3>A traffic model</h3>
<p>Suppose there are <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> cars and write <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0002.png" alt="$v_ i(t)$" style="vertical-align:-4px;
width:30px;
height:18px" class="math gen" /> for the velocity of car <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0003.png" alt="$i$" style="vertical-align:0px;
width:5px;
height:11px" class="math gen" /> at time <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0004.png" alt="$t$" style="vertical-align:0px;
width:6px;
height:10px" class="math gen" />. Write <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0005.png" alt="$h_ i(t)$" style="vertical-align:-4px;
width:31px;
height:18px" class="math gen" /> for the distance between car <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0006.png" alt="$i+1$" style="vertical-align:-1px;
width:33px;
height:13px" class="math gen" /> and car <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0003.png" alt="$i$" style="vertical-align:0px;
width:5px;
height:11px" class="math gen" /> at time <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0004.png" alt="$t$" style="vertical-align:0px;
width:6px;
height:10px" class="math gen" /> so <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0007.png" alt="$h_ i$" style="vertical-align:-2px;
width:13px;
height:13px" class="math gen" /> is the headway of car <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0003.png" alt="$i$" style="vertical-align:0px;
width:5px;
height:11px" class="math gen" />. Then <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0005.png" alt="$h_ i(t)$" style="vertical-align:-4px;
width:31px;
height:18px" class="math gen" /> should satisfy </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0008.png" alt="\[ dh_ i(t)/dt = v_{i+1}(t)-v_ i(t), \]" style="width:193px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> capturing how the headway between two cars changes as a result of the cars’ changes in velocity. In an <em>optimal velocity model</em> the acceleration <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0009.png" alt="$dv_ i(t)/dt$" style="vertical-align:-4px;
width:63px;
height:18px" class="math gen" /> of car <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0003.png" alt="$i$" style="vertical-align:0px;
width:5px;
height:11px" class="math gen" /> is described by the differential equation </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0010.png" alt="\[ dv_ i(t)/dt =\alpha \left(V(h_ i(t))-v_ i(t)\right), \]" style="width:233px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> where <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0011.png" alt="$\alpha $" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> is a parameter called the <em>sensitivity</em> and <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0012.png" alt="$V(h_ i(t))$" style="vertical-align:-4px;
width:57px;
height:18px" class="math gen" /> is a function that represents the <em>optimal velocity</em> for a given amount of headway. <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0013.png" alt="$V(h_ i)$" style="vertical-align:-4px;
width:39px;
height:18px" class="math gen" /> could be any type of function that satisfies certain realistic conditions. For example, as the headway <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0007.png" alt="$h_ i$" style="vertical-align:-2px;
width:13px;
height:13px" class="math gen" /> in front of a car increases, the optimal velocity <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0013.png" alt="$V(h_ i)$" style="vertical-align:-4px;
width:39px;
height:18px" class="math gen" /> should also increase until it reaches the legal speed limit, reflecting that drivers want to speed up when they can. </p><p>The equation above shows that if the velocity <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0014.png" alt="$v_ i$" style="vertical-align:-2px;
width:12px;
height:9px" class="math gen" /> of car <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0003.png" alt="$i$" style="vertical-align:0px;
width:5px;
height:11px" class="math gen" /> is smaller than the optimal velocity <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0013.png" alt="$V(h_ i)$" style="vertical-align:-4px;
width:39px;
height:18px" class="math gen" /> then the driver will accelerate, if it is larger then the driver will brake, and if the two are equal the drive will do neither. </p><p>(In both of the equations above, if <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0015.png" alt="$i=n$" style="vertical-align:0px;
width:37px;
height:11px" class="math gen" /> we take <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0006.png" alt="$i+1$" style="vertical-align:-1px;
width:33px;
height:13px" class="math gen" /> to be <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0016.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" />, reflecting the fact that cars are travelling around a ring.) </p><p>As it stands, the second equation means that drivers react to a change in their headway instantaneously by breaking or speeding up. But we can also include a reaction time <img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0017.png" alt="$T$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" /> to get the equation </p><p><img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0018.png" alt="$dv_ i(t)/dt =\alpha \left(V(h_ i(t-T))-v_ i(t)\right).$" style="vertical-align:-4px;
width:264px;
height:18px" class="math gen" /><img src="/MI/d5221bb551beb5038d1eb0eede5f0e7a/images/img-0019.png" alt="$$" style="vertical-align:0px;
width:1px;
height:1px" class="math gen" /></p><p>You can find out more in <a href="http://www-personal.umich.edu/~orosz/articles/PREpublished.pdf">this paper</a> by Gábor Orosz, R. Eddie Wilson, and Bernd Krauskopf or <a href="https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2006.1660">this one</a> by Gábor Orosz and Gábor Stépán.</p>
<hr/>
<h3>About this article</h3><div class="rightimage" style="max-width: 150px;"><img src="/content/sites/plus.maths.org/files/articles/2019/dating/cover.jpg" alt="Book cover" width="150" height="210" /><p></p>
</div>
<p>This article is based on a chapter from the new book <em><a href="https://amzn.to/2G6v65L">Understanding numbers</a></em> by the <em>Plus</em> Editors <a href="/content/people/index.html#rachel">Rachel Thomas</a> and <a href="/content/people/index.html#marianne">Marianne Freiberger</a>. </p>
</div></div></div>Thu, 18 Apr 2019 15:53:10 +0000Marianne7190 at https://plus.maths.org/contenthttps://plus.maths.org/content/phantom-jams#commentsThe power of snooker
https://plus.maths.org/content/power-snooker
<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_12.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">Wim Hordijk</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:350px"><img src="/content/sites/plus.maths.org/files/articles/2019/snooker/snooker.png"><p>Snookered... (Photo by <a href="https://nl.wikipedia.org/wiki/Snooker_(spelsituatie)#/media/File:Snookered_on_two_reds.jpg">Florian Albrecht</a> – <a href="https://creativecommons.org/licenses/by-sa/3.0">CC BY-SA 3.0</a>)</p></div>
<p>A particular mathematical relationship known as a <em>power law</em> has been observed in many day-to-day situations, from the frequencies in which words are used in natural languages to the connectivity distribution in Facebook friendship networks. As it turns out, though, such a power law can also be found in snooker statistics. And if the amazing <a href="https://en.wikipedia.org/wiki/Ronnie_O%27Sullivan">Ronnie O'Sullivan</a> continues to produce centuries at the same rate, the mathematical correspondence will be even better!
</p>
<br class="brclear"/>
<h3>Power laws</h3>
<p>In mathematics, we say a given value <em>x</em> is <em>raised to the power k</em> if it is multiplied by itself <em>k</em> times. Common examples where such mathematical power functions are used are in the calculation of the area of a square that has sides of length <em>x</em>, which is equal to <em>x<sup>2</sup></em>, or the volume of a cube with sides of length <em>x</em>, which is <em>x<sup>3</sup></em>.
</p>
<div class="leftimage" style="max-width: 350px">
<img src="/content/sites/plus.maths.org/files/articles/2019/snooker/volume.jpg" width="500" alt="Volume graph">
<p>The volume (<em>y</em>, vertical axis) of a cube against the length of its sides (<em>x</em>, horizontal axis). The open circles represent actual measurements on different cubes. The solid line represents the mathematical relationship <em>y=x<sup>3</sup></em>, which forms a perfect fit to the observed data.</p>
</div>
<p>A <em>power law</em> is a mathematical relationship between two variables, say <em>x</em> and <em>y</em>, such that the value of one is directly related to the value of the other raised to a certain power, for example, <em>y=x<sup>a</sup></em> for some fixed value of <em>a</em>. This value <em>a</em> is called the <em>exponent</em> of the power law. For example, if we were to measure the volume (<em>y</em>) of many cubes with different lengths of their sides (<em>x</em>), we would find a perfectly fitting power law with an exponent <em>a</em>=3.
</p>
<p>Power law relationships are found in many common systems and processes, both natural and man-made. For example, power laws are observed in the frequency distribution of the magnitudes of earthquakes, or the sizes of cities in a given country. They also show up in the distribution of connections in networks like Facebook friendships, movie star co-appearances, or the national electricity grid. And they occur in the frequencies of words used in natural languages, or even in computer code. (You can read more about power laws in <a href="/content/statistics-language">language</a> and <a href="/content/maths-minute-power-networks">networks</a> on <em>Plus</em>.)</p>
<p>In the example above, the data is plotted on a <em>linear</em> scale. In other words, each point along an axis represents a value that is larger than the previous point by a fixed amount. However, it is also possible to plot the same data on a non-linear scale. An example of such a non-linear scale is a <em>logarithmic scale</em>, which is based on orders of magnitude rather than a linear increase. This means that the value represented by each point along an axis is the value of the previous point <em>multiplied</em> by a fixed amount.</p>
<div class="rightimage" style="max-width:350px">
<img src="/content/sites/plus.maths.org/files/articles/2019/snooker/logvolume.jpg" width="500" alt="Volume on a log-log scale">
<p>The volume (<em>y</em>, vertical axis) of a cube against the length of its sides (<em>x</em>, horizontal axis) using a logarithmic scale on both axes. The data points (open circles) now fall along a straight line representing a power law with exponent <em>a</em>=3.</p>
</div>
<p>An example of this can be found in the way that the strength of an earthquake is indicated. A magnitude three (M3) earthquake releases a certain amount of energy, which translates into the amount of shaking we feel. However, a magnitude four (M4) earthquake releases an amount of energy that is ten times larger than an M3 earthquake. Similarly, an M5 earthquake is again ten times as strong as an M4 earthquake, and thus one hundred times stronger than an M3 earthquake.</p>
<p>The figure to the right shows the same data for the volume of a cube, but using such a logarithmic scale on both axes, resulting in a so-called <em>log-log plot</em>. Mathematically, a logarithmic function is the inverse of a power function. As a consequence, a power law shows up as a straight line in a log-log plot, as this figure illustrates.</p>
<br class="brclear"/>
<h3>Finding the power</h3>
<p>Unfortunately, most real-world data does not behave as nicely as the volume of a cube. In reality, there is always some noise in the data due to imprecise measurements, random fluctuations, or missing data. An example is given in the figure below, which shows a log-log plot of the frequencies of words used in English (as measured over a large collection of different texts) against their rank, where the most frequent word ("the") has rank one, the second most frequent word ("be") rank two, and so on.</p>
<p>As the figure shows, the data points do not fall along a straight line perfectly, but still reasonably well. The solid straight line represents a power law that best fits the given data, which can be calculated using a standard statistical technique known as a <em>regression analysis</em>. (You can read more about regression analysis <a href="/content/maths-minute-linear-regression">here</a>.) This best fit results in an exponent of <em>a</em>=-0.92. The exponent is negative in this case, as the word frequency decreases with increasing rank.</p>
<div class="centreimage" style="max-width:500px">
<img src="/content/sites/plus.maths.org/files/articles/2019/snooker/language.jpg" width="500" alt="log-log plot of word ranks in English">
<p>The frequency of words used in English (vertical axis) against their rank (horizontal axis) using a logarithmic scale on both axes. The data points (open circles) closely follow a straight line, although not exactly.</p>
</div>
<p>So, the frequency of a word as used in the English language is roughly proportional to its rank raised to the power <em>a</em>=-0.92. It may be difficult to imagine what it means to multiply a number by itself -0.92 times, but mathematically this is perfectly well defined. In other words, the exponent <em>a</em> in a power law does not always need to be a positive whole number.</p>
<p>Finally, the regression analysis that calculates the exponent that gives the best fit also provides a measure of accuracy, that is, how closely the data falls along a straight line. In the case of the volumes of cubes, as we saw above, the accuracy (or <em>fit</em>) is obviously 100%, as the data points fall exactly on the line. However, for the word frequencies the accuracy is slightly less: 98%. Still pretty good.</p>
<p>As these examples have shown, to find out whether a given data set follows a power law, we can simply present the data in a log-log plot, and calculate how closely it falls along a straight line. So let's give this a try with some snooker statistics.</p>
<h3>Power laws in snooker statistics</h3>
<p>Consider the <a href="http://snookerinfo.webs.com/100centuries" target="_blank">ranking list</a> of all professional snooker players who have made at least 100 centuries throughout their career. A <em>century break</em> is a score of at least 100 points within one visit to the table (i.e., without missing a shot). There are currently 68 players who have scored at least a "century of centuries". The table below shows the ten highest ranked players according to this statistic, of course topped by the amazing Ronnie O'Sullivan who recently scored his 1000<sup>th</sup> century, and still going strong!</p>
<div class="rightimage" style="max-width:350px">
<img src="/content/sites/plus.maths.org/files/articles/2019/snooker/ronnie.jpg" alt="Ronnie O'Sullivan">
<p>Ronnie O'Sullivan in action at the snooker table. (Image: <a href="https://commons.wikimedia.org/wiki/File:Ronnie_O%E2%80%99Sullivan_at_Snooker_German_Masters_(DerHexer)_2015-02-06_08.jpg" target="_blank">DerHexer, CC-BY-SA 4.0</a>).</p>
</div>
<table class="data-table">
<tr><th>Rank</th><th>Player</th><th>Centuries</th></tr>
<tr><td>1</td><td>Ronnie O'Sullivan</td><td align="right">1008</td></tr>
<tr><td>2</td><td>Stephen Hendry</td><td align="right">775</td></tr>
<tr><td>3</td><td>John Higgins</td><td align="right">750</td>
<tr><td>4</td><td>Neil Robertson</td><td align="right">636</td></tr>
<tr><td>5</td><td>Judd Trump</td><td align="right">602</td></tr>
<tr><td>6</td><td>Mark Selby</td><td align="right">577</td></tr>
<tr><td>7</td><td>Ding Junhui</td><td align="right">501</td></tr>
<tr><td>8</td><td>Marco Fu</td><td align="right">493</td></tr>
<tr><td>9</td><td>Shaun Murphy</td><td align="right">479</td></tr>
<tr><td>10</td><td>Mark Williams</td><td align="right">464</td></tr>
</table>
<p>Now, if we plot the number of career centuries of these 68 players against their rank in this list, but in a log-log plot, we get the result as shown in the next figure. The straight line represents a power law that gives the best fit to the given data, resulting in an exponent of <em>a</em>=-0.63.</p>
<div class="centreimage" style="max-width:500px">
<img src="/content/sites/plus.maths.org/files/articles/2019/snooker/centuries.jpg" width="500" alt="Log-log plot of career centuries versus rank">
<p>The number of career centuries (vertical axis) against the rank (horizontal axis). The straight line is a fitted power law with exponent <em>a</em>=-0.63.</p>
</div>
<p>Note that, as with the word frequencies data, the fit it is not perfect, especially not at the top of the ranking (i.e., the data points in the top left of the plot). This is often the case, especially with ranking data that is still in the making. However, according to the regression analysis, the current fit still has an accuracy of 95%. And if Ronnie O'Sullivan produces a few more centuries, the fit will be even better!</p>
<p>In a similar way, we can look at the number of <a href="https://en.wikipedia.org/wiki/List_of_snooker_players_by_number_of_ranking_titles" target="_blank">ranking titles</a> of each player (the number of tournaments they've won which count towards the snooker world rankings). There are currently 26 players who have obtained at least three ranking titles throughout their career. The table below shows the ten highest ranked players according to this statistic.</p>
<table class="data-table">
<tr><th>Rank</th><th>Player</th><th>Titles</th></tr>
<tr><td>1</td><td>Ronnie O'Sullivan</td><td align="right">36</td></tr>
<tr><td>2</td><td>Stephen Hendry</td><td align="right">36</td></tr>
<tr><td>3</td><td>John Higgins</td><td align="right">30</td>
<tr><td>4</td><td>Steve Davis</td><td align="right">28</td></tr>
<tr><td>5</td><td>Mark Williams</td><td align="right">22</td></tr>
<tr><td>6</td><td>Neil Robertson</td><td align="right">16</td></tr>
<tr><td>7</td><td>Mark Selby</td><td align="right">15</td></tr>
<tr><td>8</td><td>Ding Junhui</td><td align="right">13</td></tr>
<tr><td>9</td><td>Judd Trump</td><td align="right">10</td></tr>
<tr><td>10</td><td>Jimmy White</td><td align="right">10</td></tr>
</table>
<p>If we plot the number of ranking titles of these 26 players against their rank in the list, again in a log-log plot, we get the result as shown in the figure below. The straight line once more represents a power law that gives the best fit to the data, resulting in an exponent of <em>a</em>=-0.96. The accuracy of the fit is slightly less in this case (92%), mostly due to the top two players (Ronnie O'Sullivan and Stephen Hendry) having exactly the same number of ranking titles. If we would rank them together as a joint number 1, the fit would actually be excellent.</p>
<div class="centreimage" style="max-width:500px">
<img src="/content/sites/plus.maths.org/files/articles/2019/snooker/rankingtitles.jpg" width="500" alt="Log-log plot of titles versus rank of the player">
<p>The number of titles (vertical axis) against the rank (horizontal axis). The straight line is a fitted power law.</p>
</div>
<p>Finally, if we add up the number of ranking titles of all players from the same country, we get the list shown in the following table</p>.
<table class="data-table">
<tr><th>Rank</th><th>Country</th><th>Titles</th></tr>
<tr><td>1</td><td>England</td><td align="right">170</td></tr>
<tr><td>2</td><td>Scotland</td><td align="right">78</td></tr>
<tr><td>3</td><td>Wales</td><td align="right">37</td>
<tr><td>4</td><td>Australia</td><td align="right">16</td></tr>
<tr><td>5</td><td>China</td><td align="right">14</td></tr>
<tr><td>6</td><td>Northern Ireland</td><td align="right">8</td></tr>
<tr><td>7</td><td>Republic of Ireland</td><td align="right">7</td></tr>
<tr><td>8</td><td>Thailand</td><td align="right">4</td></tr>
<tr><td>9</td><td>Hong Kong</td><td align="right">3</td></tr>
<tr><td>10</td><td>Canada</td><td align="right">3</td></tr>
</table>
<p>Plotting the number of titles per country against the rank in the
list, as a log-log plot, gives the result as shown in the next
figure. The straight line shows the power law that gives the best fit,
with an exponent of <em>a</em>=-2.12. The accuracy is much better again in this case: 96%.</p>
<div class="centreimage" style="max-width:500px">
<img src="/content/sites/plus.maths.org/files/articles/2019/snooker/countries.jpg" width="500" alt="Number of titlse by country against the rank">
<p>The number of titles per country (vertical axis) against the rank (horizontal axis). The straight line is a fitted power law.</p>
</div>
<p>Snooker statistics seem to follow mathematical power laws, just like natural languages, earthquake occurrences, and many types of natural and man-made networks. Why should this be so? Scientists are still arguing about the significance of the occurrence of power laws. On the one hand, there is no particular reason to expect such behavior in many of these situations. On the other hand, the phenomenon seems to be so common that perhaps it has little meaning after all. Either way, it is interesting to see that snooker statistics do indeed follow precise mathematical laws with a high degree of accuracy. I'm curious to see if this year's world championships snooker, with its usual display of power shots, will also produce some additional power laws.</p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2016/altruism/wim.jpg" alt="Wim Hordijk" width="200" height="200" /></div>
<p>Wim Hordijk is a computer scientist currently on a fellowship at the <a href="http://ias.uva.nl/">Institute for Advanced Study</a> of the University of Amsterdam, The Netherlands. He has worked on many research and computing projects all over the world, mostly focusing on questions related to evolution and the origin of life. More information about his research can be found on his <a href="http://www.worldwidewanderings.net">website</a>.</p></div></div></div>Tue, 16 Apr 2019 14:43:06 +0000Rachel7193 at https://plus.maths.org/contenthttps://plus.maths.org/content/power-snooker#commentsMaths in a minute: The Sydney Opera House
https://plus.maths.org/content/maths-minute-sydney-opera-house
<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/sydney_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">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>On 29 January 1957, when Jørn Utzon's sail-like sketches were announced as the winning design for the Sydney Opera House, Utzon had a problem — he didn't know exactly how he would build them. The problem still wasn't solved two years later when construction began on 2 March 1959. </p>
<div class="rightimage" style="max-width: 320px;"><img src="/content/sites/plus.maths.org/files/articles/2019/sydney/sydney.jpg" alt="Sydney Opera House" width="320" height="211"/>
<p>The Sydney Opera House. Photo: <a href="https://commons.wikimedia.org/wiki/File:Sydney_Opera_House,_botanic_gardens_1.jpg">AdamJ.W.C.</a>, <a href="https://creativecommons.org/licenses/by-sa/2.5/deed.en">CC BY-SA 2.5</a>.</p>
</div>
<p>Inspired by the harbour location, the young Danish architect had envisioned a series of sweeping curved shells. But in order to build these shells the shapes had to be described mathematically to accurately calculate all the loads and stresses on the building. When asked by the engineers to specify the curves Utzon bent a ruler to trace the curves he wanted. Over the next four years they tried various mathematical forms — ellipses, parabolas — to try to capture Utzon's design.</p>
<p>Finally, in October 1961, Utzon found the "key to the shells": every sail was formed from a wedge cut from a single sphere (and its mirror image). This ingenious solution not only provided a mathematical description of the roof of his design, it also solved all the problems of constructing such a complex structure. Previously each shell appeared to be different, and it would have been almost impossible, both in terms of engineering and finance, to construct these huge bespoke parts. Instead, with shells having a common spherical geometry (based on a sphere with radius of 246 feet), they could instead be constructed by standard parts that could be mass produced and then assembled.</p>
<p>Describing the design mathematically made it possible to create one of the most iconic buildings in the world. The mathematics of the design also exemplified Utzon's artistic vision, providing "full harmony between all the shapes in this fantastic complex".
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2019/sydney/shells.jpg" alt="Spherical shells" width="640" height="479"/>
<p style="max-width: 640px;">The key to the shells, Sydney Opera House. </p>
</div>
<!-- Image in public domain -->
<p> It is now standard practise for large buildings to be modelled mathematically in three-dimensions on a computer, so that the effect of any small changes in architectural design, or practical construction (such as moving the position of piping or electrics), immediately cascade through the model of the building. This highlights any unforeseen clashes, and allows for the most economic use of materials and people during the construction process. Mathematical models also help make sure that buildings are energy efficient, ideally suited for their purpose and improve people's lives — the people using the buildings as well as those admiring their forms from the outside.</p>
<p>To find out more about maths and architecture, see</p>
<ul><li><a href="/content/maths-minute-st-pauls-dome">Maths in a minute: St Paul's dome</a></li>
<li><a href="gcontent/how-velodrome-found-its-form">How the velodrome found its form</a></li>
<li><a href="/content/perfect-buildings-maths-modern-architecture">Perfect buildings: The maths of modern architecture</a></li>
<li><a href="/content/catenary-goes-wembley">The catenary goes to Wembley</a> (video)</li>
<li><a href="/content/node/7127">Stadium maths</a> (podcast).</li></ul>
<hr/>
<h3>About this article</h3><div class="rightimage" style="max-width: 150px;"><img src="/content/sites/plus.maths.org/files/articles/2019/dating/cover.jpg" alt="Book cover" width="150" height="210" /><p></p>
</div>
<p>This article is based on a chapter from the new book <em><a href="https://amzn.to/2G6v65L">Understanding numbers</a></em> by the <em>Plus</em> Editors <a href="/content/people/index.html#rachel">Rachel Thomas</a> and <a href="/content/people/index.html#marianne">Marianne Freiberger</a>. </p>
</div></div></div>Fri, 12 Apr 2019 12:39:05 +0000Marianne7189 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-sydney-opera-house#commentsPhenomenal physics
https://plus.maths.org/content/theory-practice
<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/maria_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">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>In the old days the job of physicists was to explain the things we observe in the world around us. In modern physics explanation often comes before observation. The famous Higgs boson is an example. Physicists realised that something like the Higgs must exist for their theory to make sense; they went looking for it and, a few decades later, <a href="/content/higgs">finally discovered it</a>. </p>
<div class="rightimage" style="max-width: 319px;"><img src="/content/sites/plus.maths.org/files/articles/2019/Ubiali/mariau2.jpg" alt="Maria Ubiali" width="319" height="182"/>
<p>Maria Ubiali. You can see a video of our interview with her <a href="/content/theory-practice#video">below</a>. </p>
</div>
<p>But given a theory written, as always, in the language of maths, it's not always clear what kind of experiments you should perform in order to confirm or debunk it. This is where <em>phenomenologists</em> come in. <a href="http://www.damtp.cam.ac.uk/people/mu227/">Maria Ubiali</a>, of the University of Cambridge, is one of them. Her job is to work out what observable consequences our theories of nature might have. In particular, she focusses on the <em><a href="/content/physics-elementary-particles">standard model of particle physics</a></em> which describes the fundamental particles of nature.</p>
<p>"What I really do is to formulate theoretical predictions, to give numbers to the experimentalists," says Ubiali. "They verify what I give them to see if the standard model is the law that correctly describes nature, or if there is something more than that."</p>
<h3>What can we predict?</h3>
<p>A good example of the interplay between theory and experiment is our evolution of the understanding of the proton. When protons were discovered a century ago, scientists thought they were elementary, indivisible particles. In the 1960s it was realised that protons were actually built of three smaller particles, which we now call <em>quarks</em>, bound together by some force carrying particles called <em>gluons</em>. And today our picture of protons is even more complicated: "There is an interesting life inside the proton. Inside this tiny radius of the proton quarks keep creating and annihilating in pairs, quarks and antiquarks, with gluons radiating all the time off the quarks and antiquarks."</p>
<p>At high energies (when quark/antiquark pairs are more likely to pop in and out of existence) you might think of a proton as more like a swarm of particles: three quarks, and a crowd of gluons and quark/antiquark pairs. The structure of the proton is then described in statistical terms capturing how the properties of the proton, such as its momentum, are distributed across the swarm. </p>
<h3>Machine learning</h3>
<div class="leftimage" style="max-width: 350px;"><img src="/issue29/features/kalmus/UniverseKit.jpg" alt="Acme universe building kit" width="350" height="350" /><p>What are the fundamental building blocks of matter?</p></div>
<p>Theoretical predictions for processes involving elementary particles such that electrons or photons can be determined from the standard model by solving the relevant equations. "What is interesting about the structure of the proton is that it can't be determined by using first principles," says Ubiali. For the mathematical description of the protons, called quantum chromodynamics (QCD), the equations cannot be completely solved. "That's why we need experimental data to infer the structure of the proton in terms of its elementary constituents."</p>
<p>
The problem is that experiments like the <a href="https://home.cern/science/accelerators/large-hadron-collider">Large Hadron Collider</a> (LHC) produce a huge amount of data. Trailing through this data to see what it might teach us about the structure of the proton is a near impossible task. Which is why Ubiali and her colleagues harness the power of computers. In particular, they use something called <em>machine learning</em> (find out more <a href="/content/what-machine-learning">here</a>) to find patterns in these large data sets and then use this information to fine tune their mathematical description of the proton.</p>
<p>"That's how we apply machine learning, so we have a much better determination of the [proton's structure]," says Ubiali. This data-driven description not only gives the best prediction of the structure, but also the uncertainty associated with this best prediction. "We have more realistic estimates of the uncertainty, of what we still don't know."</p>
<h3>Beyond the standard model</h3>
<p>"The success of the standard model is extraordinary," says Ubiali. Recent comparisons of the standard model's predictions to experimental data show an amazing agreement over a wide range of scales (from 10<sup>-4</sup> to 10<sup>11</sup> picobarns – a measurement of area). However we know that the standard model cannot be a complete description of nature as there are many mysteries – such as why <a href="/content/mysterious-neutrinos">neutrinos have mass</a> and the nature of <a href="g/content/maths-minute-dark-matter">dark matter</a> – that the standard model cannot solve.</p>
<p>Ubiali says that part of her work involves stretching the standard model to the maximum precision possible in order to experimentally observe any deviations from the predictions. And then she also works on some particular models that go beyond the standard model.</p>
<p>"As a theorist I'm really interested to see what is beyond the standard model. I'm particularly interested in the paradigm shift we are living in. When the LHC was turned on we were all expecting to see some spectacular deviation from our predictions because we would reach the energy [of the collisions] when something would break down. But that didn't happen."</p>
<p>"This could be a source of depression, or a source of excitement. For me it is a source of excitement - it means that nature will give us some more subtle hints about deviations."</p>
<a name="video"></a>
<p><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/XPwyp0dh43Y" width="560"></iframe></p></div></div></div>Tue, 09 Apr 2019 11:39:52 +0000Marianne7191 at https://plus.maths.org/contenthttps://plus.maths.org/content/theory-practice#commentsLooking for love?
https://plus.maths.org/content/looking-love
<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_34.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 and 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>Are you looking for love? Then in today's world you're almost certainly looking for love online. Dating websites and apps are now a common way to look for a hook-up as well as for a life partner, rather than just relying on our social circles in the physical world.</p>
<div class="rightimage" style="max-width: 315px;"><img src="/issue48/features/billingham/frog1.jpg" alt="A frog" width="315" height="259" /><p>Who is your prince(ss)?</p>
</div>
<p>Dating apps rely on mathematics to link you up with potential dates — whether it's by shared interests based on surveys, compatibility based on personality tests, proximity or even automatic profiling produced from your use of social media. But some of these services wear their maths more proudly on their sleeves than others: <a href="https://www.okcupid.com/home">OkCupid</a> was started by four maths students from Harvard in 2004 and has the sales pitch "we use maths to find you dates".</p>
<p>OKCupid's matching algorithm relies on users answering multiple-choice questions that vary from "Did you delete Facebook?" to "Do you want to have children?" Users can answer as many questions as they want, and for each they also record how they'd like their potential match to answer. The user also indicates how important each question is to them personally: from irrelevant to mandatory.</p>
<p>If you're one of those users then each of your potential dates is given a percentage score based on how they answered the questions you have in common. The importance you give each question changes the <em>weight</em> of that question in the calculations: your rating of a question's importance decides how many points that question will contribute to your potential match's score. Similarly, you are also given a score that rates you from a potential match's point of view.
</p>
<h3>What's your score?</h3>
<p>To make it easy to see how the scores are allocated, we'll follow the example Christian Rudder, one of the founders of OkCupid, used in <a href="https://www.youtube.com/watch?v=m9PiPlRuy6E">his explanation of the algorithm</a>. Suppose you and your potential love, call them X, have just two questions in common: "How messy are you?" and "Do you like to be the centre of attention?". Here is how you both answered:</p>
<table class="data-table"><th></th><th>Messy?</th><th>Answer you will accept from the other</th><th>Importance of question</th>
<tr><td>You</td><td>Not at all</td><td>Not at all</td><td>Very important</td></tr>
<tr><td>X</td><td>Not at all</td><td>Average</td><td>A little important</td></tr><th></th><th>Centre of attention?</th><th>Answer you will accept from the other</th><th>Importance of question</th>
<tr><td>You</td><td>No</td><td>No</td><td>A little</td></tr>
<tr><td>X</td><td>Yes</td><td>No</td><td>Somewhat important</td></tr></table>
<p>The following table shows how many points a questions scores depending on how important you rated it:
<table class="data-table"><tr><td>Irrelevant</td><td>0</td></tr>
<tr><td>A little important </td><td>1</td></tr>
<tr><td>Somewhat important </td><td>10</td></tr>
<tr><td>Very important </td><td>50</td></tr>
<tr><td>Mandatory</td><td>250</td></tr></table><p>
<p>From this we see that for the first question X scores 50/50 in your eyes (they gave the answer you wanted to a question you rated as very important) and you scored 0/1 points in their eyes (you didn't give the answer they wanted to a question they rated a little important). Similarly, for the second question X scores 0/1 in your eyes and you scored 10/10 points in their eyes. Combining these scores, X scores 50/51 or 98% in your eyes, and you scored 10/11 or 91% in their eyes. You can see how this procedure works if you and X have more than two questions in common.
</p>
<h3>Do you match?</h3>
<p><p>The final step of the algorithm, calculating the <em>match percentage</em>, is to combine the score your potential match has in your eyes and the score you have in your potential match’s eyes in a type of average. But rather than the most familiar average — where we add <img src="/MI/6d0af7cc7a3f597a5767861db75a4668/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> things up and then divide the answer by <img src="/MI/6d0af7cc7a3f597a5767861db75a4668/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />, called the <em>arithmetic mean</em> — the algorithm uses something like the <em>geometric mean</em> — taking the <img src="/MI/6d0af7cc7a3f597a5767861db75a4668/images/img-0002.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> root of a product of <img src="/MI/6d0af7cc7a3f597a5767861db75a4668/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> numbers. Geometric means are particularly useful when the quantities you are averaging might be quite different, perhaps even measuring different properties. </p>
</p>
<p><p>The match percentage is based on the scores you each gave each other based on the <img src="/MI/2068e29f220cce4b19819a480552c6fa/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> questions you both answered. These scores are multiplied and the <img src="/MI/2068e29f220cce4b19819a480552c6fa/images/img-0002.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> root is taken of the product. If you have two questions in common, with scores of 98% for your potential match and 91% for you, the match percentage is given by </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/2068e29f220cce4b19819a480552c6fa/images/img-0003.png" alt="\[ \sqrt {0.98 \times 0.91} \approx 0.94, \]" style="width:148px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> which translates to 94%. </p></p>
<p><p>If your satisfaction scores were based on three questions, we would instead have taken the cube root and your match percentage would have been: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/00077ec590a876eacd2a499be589a94f/images/img-0001.png" alt="\[ \sqrt [3]{0.98 \times 0.91} \approx 0.96, \]" style="width:147px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> which translates to 96%. </p><p>We’d have taken the fourth root if you’d had four questions in common: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/00077ec590a876eacd2a499be589a94f/images/img-0002.png" alt="\[ \sqrt [4]{0.98 \times 0.91} \approx 0.97, \]" style="width:147px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> which translates to 97%. </p><p>The nature of this mathematical operation ensures that the more questions you answer — the higher the value of <img src="/MI/00077ec590a876eacd2a499be589a94f/images/img-0003.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> in the <img src="/MI/00077ec590a876eacd2a499be589a94f/images/img-0004.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> root — the higher your match percentage will be. </p></p>
<p>The site then uses the match percentage to suggest potential dates: the higher your match percentage with a person, the higher up your profile will appear in their results and their profile in yours. </p>
<p>The match percentage in OKCupid is calculated only for the questions that you and your potential date both answered. So it's worth answering a bunch of questions — the more you answer the wider your pool of questions, increasing the chance that you will have questions in common with other users. </p><p>
In 2012 a maths PhD student, Chris McKinlay, famously "hacked" the OkCupid algorithm. He didn't hack into their machines, instead he applied his maths skills to the process of choosing which questions to answer (and he answered these honestly), and what importance to give them, to increase his match percentages dramatically. He went from just 100 matches over 90%, to being the top match for more than 30,000 women. This didn't automatically make these matches successful relationships, it just meant he got a chance to be higher in other people's results and get that, up until that point, elusive first date. It took 88 of these first dates to meet his true love, and they were engaged by the time the story hit the press in 2014. Reader, he married them.</p>
<hr/>
<h3>About this article</h3> <div class="rightimage" style="max-width: 150px;"><img src="/content/sites/plus.maths.org/files/articles/2019/dating/cover.jpg" alt="Book cover" width="150" height="210" /><p></p>
</div>
<p>This article is based on a chapter from the new book <em><a href="https://amzn.to/2G6v65L">Understanding numbers</a></em> by the <em>Plus</em> Editors <a href="/content/people/index.html#rachel">Rachel Thomas</a> and <a href="/content/people/index.html#marianne">Marianne Freiberger</a>. The book will be published on April 11, 2019!</p>
</div></div></div>Thu, 04 Apr 2019 09:52:23 +0000Marianne7188 at https://plus.maths.org/contenthttps://plus.maths.org/content/looking-love#comments'The eternal golden braid: Gödel Escher Bach'
https://plus.maths.org/content/eternal-golden-braid-godel-escher-bach
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Can artificial intelligence be creative? One way to find out is to subject it to a <em>Turing test</em>: if we can't distinguish a computer's creation from a human one, then there are grounds for saying the answer is yes.</p>
<div class="rightimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/reviews/2019/marcus_du_sautoy_barbican_090319_536.jpg" alt="" width="400" height="266"/><p>Marcus Du Sautoy performing at the Barbican. Photo: Mark Allan/Barbican.</p>
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<p>In his show <a href="https://www.barbican.org.uk/whats-on/2019/event/the-eternal-golden-braid-godel-escher-bach"><em> The eternal golden braid: Gödel Escher Bach</em></a> at the Barbican in London Marcus Du Sautoy performed just such a test, using the audience as judges. The show is part of an on-going project called <a href="https://www.barbican.org.uk/whats-on/2019/series/strange-loops"><em>Strange loops</em></a> which celebrates the 40th anniversary of the intellectual cult classic <a href="https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach"><em>Gödel, Escher, Bach</em></a>. </p>
<p>The blurb of the show promised a computerised exploration of the music of <a href="https://en.wikipedia.org/wiki/Johann_Sebastian_Bach">Johann Sebastian Bach</a>. Since I know very little about music in general, and nothing about Bach in particular, I took along <a href="https://www.qmul.ac.uk/maths/profiles/bandtlowo.html">Oscar Bandtlow</a>, a mathematician at Queen Mary, University of London, who is interested in the connections between maths and music and has a deep knowledge of Bach.</p>
<p>Neither Oscar nor I have read <em>Gödel, Escher, Bach</em>, but that didn't matter because Du Sautoy explained the basics. The book, written by Douglas Hofstadter, isn't about how maths, music and art relate to each other, as the title might suggest, but about abstract structures that appear in all three. Hofstadter calls these structures <em>strange loops</em>: processes that appear to involve moving up though a hierarchy of levels but then, paradoxically, leave you where you started. MC Escher's famous lithograph <em><a href="https://en.wikipedia.org/wiki/Ascending_and_Descending">Ascending and descending</a></em> is a great illustration of such a strange loop. The mathematician Kurt Gödel <a href="/content/goumldel-and-limits-logic">showed that such loops exist in any sufficiently complex mathematical system</a>, and they also appear in music, more below.</p>
<p> Hofstadter suggests that strange loops are what creates consciousness. Loosely speaking,
the neurons and synapses within our brain form the bottom level of a hierarchy in which things like feelings, ideas, hopes, etc, emerge at higher levels. Consciousness arises because the hierarchy can twist back on itself: higher levels can influence the lower levels that determine them. Feelings and ideas can have a real physical impact, and this tangling of hierarchies gives us a sense of self.</p>
<p>In his show Du Sautoy weaves these lofty ideas into an entertaining and informative package, including live music by harpsichordist Mahan Esfahani and an ensemble from the London Contemporary Orchestra, and audience feedback. He shows Escher paintings and introduces the <em><a href="https://en.wikipedia.org/wiki/Shepard_tone">Shepard tone</a></em>, a musical strange loop which seems to continually ascend in pitch without getting higher. A percussive version of the tone, which we hadn't come across before, is even weirder (see below). Du Sautoy also finds strange loops in the music of Bach (see <a href="/content/topology-music-m-bius-strip">this article</a> to find out more) and even attempts an explanation of the proof of Gödel's result involving strange loops in maths (see <a href="/content/goumldel-and-limits-logic">here</a>).</p>
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<p>Interspersed are clips of an interview with Hofstadter himself, talking about strange loops and telling anecdotes about the birth of his famous book. The initial inspiration came from Gödel's work, apparently, Escher was added because Hofstadter's father said the book needed pictures, and Bach came in even later. For those in the audience who already knew the book, those snippets would have made up for the familiarity with the rest of the content.
</p>
<p>Du Sautoy's skill as a presenter and entertainer is beyond doubt. He gets across abstract and difficult concepts, he is warm, funny and has impeccable timing. His passion is infectious and despite clearly being a very clever man, he never comes across as arrogant. Had the show only consisted of him talking, that would probably have been good enough.</p>
<p>But there was also music and audience participation. For the Turing test of creativity computational artist Parag K Mital created a <a href="/content/what-machine-learning">machine learning</a> algorithm that composes music in the style of Bach. Esfahani then played a piece by Bach in which gaps had been filled by the algorithm. The audience held up coloured cards to say which passages they thought were human-made and which machine-made. Those cards were, appropriately, being read by another machine learning algorithm hooked to a camera and the images displayed on a screen for everyone to see. </p>
<p> Although the results weren't properly analysed to see if the machine had passed the Turing test overall, this was a fun bit of experimentation that let everyone see for themselves whether the algorithm had fooled them (Esfahani played the piece a second time and the correct answers were shown). At the end of the show an ensemble from the London Contemporary Orchestra played a piece composed out of machine and human-made passages and the audience was able, by holding up their cards again, to influence the mood of the piece. </p>
<p>In the pub after the show I asked Oscar if he had been tricked by the algorithm. Even though he knows the Bach piece involved particularly well he was still a little unsure at times. We both felt that some passages were very clearly machine-made, though it's hard to pin down exactly why we were so sure on those. The harpsichordist's view struck a chord (sorry) with both of us: Esfahani found the machine-made passages harder to learn because they lacked a direction and instrumental or textural mirroring. With the machine-made passages he had to look at every note, but with Bach he didn't.</p>
<p>Du Sautoy also tested us on human versus machine-made paintings (see the video below for an example) and poetry (try <a href="http://botpoet.com">Bot or not</a>, it's fun).
Again, a certain
lack of edge, a blandness, tended to give the algorithms away. </p>
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<p>Overall the algorithms didn't quite fool us, though it's hard to argue they never will. To be truly creative, though, they would have to do more than imitate styles. Machine learning might not be the right kind of AI to do this, I thought, since it works by spotting patterns in vast amounts of example data (eg Bach's music or paintings by Rembrandt) and then reproducing them. But then, humans are also pattern spotting machines. We might not start from scratch every time we look at something because, as Oscar pointed out, evolution and collective cultural memory have done some of the work for us. But ultimately we learn by recognising patterns and create by putting them together. Why should a machine not be able to do that? </p>
<p>We were not the only ones discussing the show over a beer. At the table next to us a group of what appeared to be young computer scientists were deep in discussion, and the pub nearest to the Barbican was rammed with audience members looking equally animated. It was hard to ascertain the make-up of the audience, but whatever their background, they seemed happy and intrigued.</p>
<p>My favourite thought to take away was something Du Sautoy said right at the end. Whatever AI will turn out to be in the future, getting it to do art will have more than just commercial and entertainment value. It may also help us understand what it's like to be an AI — to see into the soul of a robot.</p>
<p><a href="https://www.barbican.org.uk/whats-on/2019/series/strange-loops"><em>Strange loops</em></a> is still going on at the Barbican. On April 17 you can see Du Sautoy and Joana Seguro discuss <em>Gödel, Escher, Bach</em> in <a href="https://www.barbican.org.uk/whats-on/2019/event/marcus-du-sautoy-three-strange-loops"><em>Three strange loops</em></a>. In addition, <a href="https://www.barbican.org.uk/whats-on/2019/event/behind-a-facade-of-order"><em>Behind a façade of order</em></a> is an algorithmic installation which explores what MC Escher might have done with today's technology. It's on until May 5. </p>
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<h3>About the author</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. She went to see <em> The eternal golden braid: Gödel Escher Bach</em> at the Barbican with <a href="https://www.qmul.ac.uk/maths/profiles/bandtlowo.html">Oscar Bandtlow</a>, Senior Lecturer of Applied Mathematics at Queen Mary, University of London, on March 9, 2019. </p></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">Review by </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div>Mon, 01 Apr 2019 15:16:27 +0000Marianne7184 at https://plus.maths.org/contenthttps://plus.maths.org/content/eternal-golden-braid-godel-escher-bach#commentsRay Goldstein: Synchronised swimming
https://plus.maths.org/content/ray-goldstein-synchronised-swimming
<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/gioldstein_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>How do little green algae manage to trot, canter and even do a perfect breaststroke when they haven't got a brain? Ray Goldstein explains what the little micro-swimmers have taught us about the maths of synchronisation, and how these insights even shed light on the evolution of multi-celled organisms from single cells.</p>
<p>You can also read <a href="/content/synchronised-swimming">this article</a> about Goldstein's work.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/PbnL5KELqF8" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe></div></div></div>Thu, 28 Mar 2019 15:24:34 +0000Marianne7187 at https://plus.maths.org/contenthttps://plus.maths.org/content/ray-goldstein-synchronised-swimming#commentsEnter the BSHM maths competition for schools!
https://plus.maths.org/content/enter-bshm-maths-competition-schools
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Have you ever wondered what the world would be like without mathematics? And who are the people who make new mathematics and how they do it?</p>
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<p>Who is your favourite mathematician of all time?</p>
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<p>This competition, organised by the <a href="http://www.bshm.ac.uk">British Society for the History of Mathematics</a>, is your chance to explore how mathematics has developed and achieved its status and who were the most important mathematicians in history who contributed to it. This year we would like you to concentrate on choosing one mathematician who has, in your opinion, been the most important person, your favourite, and to make the case for your choice — to explain his/her mathematics and to show their importance or what you think was special about it and them.</p><p>
The British Society for the History of Mathematics (BSHM) believes that understanding where mathematics comes from and who has contributed to the development of mathematical ideas is an important part of understanding mathematics today. BSHM, working with <em>Plus</em>, invites secondary school students to explore this question and communicate their findings for a wide audience (age 16 upwards).</p>
<p>You could write an article (maximum 1500 words), make a short video (maximum ten minutes) or a multi-media project (maximum ten minutes).</p>
<p>The competition is open to all young people aged 11 to 15 and 16 to 19 who are in secondary education. A number of monetary prizes will be awarded, depending upon the quality and the number of entries. The maximum prize will be £100.</p>
<p>The deadline for entries is Friday, 1st September 2019. All the info about how to submit your entry and where to ask questions is on the <a href="http://www.bshm.ac.uk/plus">BSHM website</a>.</p><p>
Winners will be notified to collect their prizes in London, at the Society's Gresham College meeting on the 23rd October 2019, and the recording of this will be posted on the BSHM website, with a link given also from <em>Plus</em>.</p>
<p>Good luck!</p>
</div></div></div><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/people_icon_0.jpg" width="100" height="100" alt="" /></div></div></div>Thu, 28 Mar 2019 12:26:43 +0000Marianne7186 at https://plus.maths.org/contenthttps://plus.maths.org/content/enter-bshm-maths-competition-schools#commentsA geometry for strings
https://plus.maths.org/content/geometry-strings
<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_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"><p>String theory suggests that the fundamental particles of nature aren't like points, but are tiny little strings. But when there is a smallest length, the ordinary geometry we learn about at school doesn't work any longer.
In this video David Berman of Queen Mary, University of London, talks about generalising geometry to fit string theory.</p>
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<p>This interview is part of our <em><a href="/content/researching-unknown">Researching the unknown</a></em> series featuring physicists from <a href="https://www.qmul.ac.uk/spa/">Queen Mary, University of London</a>.</p>
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<p ></p>
</div></div></div></div>Wed, 27 Mar 2019 15:49:24 +0000Marianne7185 at https://plus.maths.org/contenthttps://plus.maths.org/content/geometry-strings#commentsSynchronised swimming
https://plus.maths.org/content/synchronised-swimming
<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/volvox_darkfield_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>In a tank in an underground laboratory in Cambridge a little green alga is executing a powerful breaststroke. It has a single cell for a body and two hair-like filaments for arms. In another tank algae with four filaments mimic a horse's trot, canter and gallop. Yet others have hundreds of filaments performing a Hawaiian dance to propel their parent cell forward. Amazingly, they manage to do all this even though they haven't got a mind, a will, or anything resembling a brain. </p>
<p>
The algae are called <em>Volvocales</em>. The lineage contains organisms made up of just one cell as well as ones made up from thousands of cells, which is partly what makes it interesting: it can shed light on why and how multi-celled organisms, including ourselves, evolved from single-celled ones in the first place. Because Volvocales are also easy to find in rivers and ponds, and to grow in the lab, they have long been model organisms in biology.</p>
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But the lab in question belongs, not to the University of Cambridge's biology department, but to the maths department. "From the point of view of physics and mathematics, Volvocales are interesting because they have a beautiful high degree of symmetry [they are spherical] and they have a fascinating means of locomotion," says <a href="http://www.damtp.cam.ac.uk/user/gold/">Ray Goldstein</a> FRS, Schlumberger Professor of Complex Physical Systems at the Department of Applied Mathematics and Theoretical Physics. "So they are models to study many problems in biological physics and fluid mechanics. Altogether they are model organisms that make everyone happy."</p><p>
One thing that makes Volvocales particularly interesting is that their arms, called flagella, are nearly identical to hair-like filaments called <em>cilia</em> that can be found in the human body. "In the back of your brain, in your kidneys, in your reproductive system, cilia are everywhere," says Goldstein. "An egg goes down a fallopian tube because cilia induce a flow. In the human body you find them executing [something like a Mexican wave]."</p><p>
"When I first learned about this I thought, this is an important problem to work on. Nobody really understood how the synchronicity occurs, but it's essential for the functioning [of the cilia]. If they were just randomly beating, there would be a much less efficient [fluid] flow." (You can see a video interview with Goldstein <A href="/content/little-green-algae-and-quest-synchronisation#video">below</a>.)</p>
<h3>Tic toc</h3><p>
The flagella of a Volvocale are easier to look at than cilia inside someone's body, so they are a good place to start studying synchronisation. Goldstein and his team started with the simplest member of the lineage - called <a href="https://en.wikipedia.org/wiki/Chlamydomonas"><em>Chlamydomonas</em></a> – which consists of a single cell with two flagella attached and is the one that does the breaststroke.</p>
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<p><em>Chlamydomonas</em> held in place but still free to move its flagella. Photo: Marco Polin, Idan Tuval, Raymond E. Goldstein.</p>
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<p>
"[To look at<em> Chlamydomonas</em>] we used a technique that comes from the world of in vitro fertilisation," says Goldstein. "We use glass capillaries that have tiny little tips to hold the organism, like you would an egg in IVF." Rather than having to chase it round the container, the researchers could watch it while being held in one place, beating its flagella in vain.</p><p>
Nobody knows exactly how <em>Chlamydomonas</em> beats its individual flagella in a regular fashion, but beat they do. The regularity means that you can treat each flagellum as an oscillator: something that moves periodically like a pendulum. One mechanism that allows pendulums to synchronise has been known since the 17th century. Synchronisation of pendulum clocks was of huge interest back then, as <a href="/content/longitude-problem">accurate time keeping was essential for navigation</a> at sea. "<a href="https://en.wikipedia.org/wiki/Christiaan_Huygens">Christiaan Huygens</a> observed that two pendulum clocks mounted on a flexible support, like a chair or a wall, eventually come into sync," says Goldstein. "He deduced that it was vibrations in the wall or chair that pushed on the two pendulums to make them synchronise."
</p><p>In the case of <em>Chlamydomonas</em> there isn't a wall or a chair, but there is the water in which it lives. A beating flagellum will set the water in motion, the water will then push on the other flagellum, which reacts, and so on. Since the 1950s scientists have suspected that it's the motion of the water that causes the flagella to synchronise.</p>
What these scientists were lacking, however, was the technology to properly observe the micro-swimmers, which has only been around since relatively recently. Goldstein and his colleagues used high-speed video microscopy to observe the tiny organisms, together with a clever trick. If you imagine yourself doing a breaststroke while being held in place by giant pipette, you'll notice that your elbow will trace an elliptical shape (like a squashed circle). The same goes for a point on a flagellum: it also traces out an elliptical shape. If something is moving around on an ellipse, then you can describe its location at any given moment in time by an angle (see the figure below). Measuring this phase angle over time for each flagellum told the researchers exactly when, and for how long, the flagella were in sync.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2019/Goldstein/angle2.jpg" alt="A phase angle" width="350" height="226"/>
<p style="max-width: 350px;">Each point on an ellipse is given by an angle.</p>
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<h3>The power of water</h3>
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<h3>
Run and turn</h3><p>
The data showed that the two flagella of a single <em>Chlamydomonas</em> are in sync only 85% of the time — that's when the cell is doing a breaststroke, swimming straight ahead. The fact that the flagella are out of sync the rest of the time means that, overall, <em>Chlamydomonas</em> is performing a run and turn motion, with straight runs interrupted by sharp turns. Using further experiments Goldstein and his team established that, all in all, <em>Chlamydomonas</em> is performing what mathematicians call a <a href="https://en.wikipedia.org/wiki/Random_walk">random walk</a>.</p><p>
This way of moving may have evolved for a good reason. If you are looking for food, or perhaps trying to evade a predator, then always swimming straight ahead isn't your best option. You'll miss all the food that swims pass to your sides and a predator will have an easy job of snapping you up.
"This is the same thing that bacteria do. And with bacteria it's established, without a shadow of a doubt, that it is part of a search strategy." </p></div>
<p>The data showed that the two flagella of a single <em>Chlamydomonas</em> are in sync only 85% of the time, possibly for good reason (see the box). To address the question of synchronisation the researchers turned to maths. In 2008 a German team of researchers had developed a <a href="http://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/FDSE/NiedermayerEckhardtLenz08.pdf">simplified mathematical model</a> of how the flagella interact with the water around them and, through the water, with each other. In the model the flagella are represented by small spheres that move around an elliptical orbit. This simplification may seem drastic, but studies of how water flows around <em>Chlamydomonas</em> have shown that it's justified.</p><p>
"Rather than considering the entire [flagellum] just replace it with the little sphere going around in an orbit," says Goldstein. "You can calculate the fluid mechanics of this a lot more easily. You have two of these spheres talking to each other [through the fluid] and you imagine that there is a little spring controlling how far they are from their centres. Then you turn the mathematical crank, and, amazingly, you end up with an equation that was developed decades ago by <a href="https://en.wikipedia.org/wiki/Robert_Adler">Robert Adler</a> to describe coupled oscillators in electrical engineering."</p><p>
The team estimated the parameters in the model using their cleverly extracted data and came up with an interesting result. "We established that the interaction between the two flagella was of a magnitude that is consistent with the idea that it was fluid mechanical coupling," says Goldstein. This was the first experimental support for the idea that fluid flow alone can cause the flagella to synchronise.</p>
<h3>Enter the mutants</h3><p>
But only because the dynamics of the water can cause the flagella to sync, doesn't mean it is the only factor play. After studying the theory Goldstein and his team returned to experiment, this time looking at mutants: organisms that are just like <em>Chlamydomonas</em> in many ways, but have some deficiency or another. "If you look [at Volvocales] very carefully, using electron microscopy, you discover that there is a whole set of filamentary connections [between the flagella] hidden from view beneath the cell membrane," explains Goldstein. One type of mutant he and his team looked at was missing exactly those connections.</p><p>
"It turns out that almost 100% of the time [in this mutant] there is no synchrony at all between the two flagella," explains Goldstein. "The only cases in which there is occur when [the flagella] are very close together. This led us to think that it's not quite as simple as just fluid mechanical coupling; that probably there are [filaments] playing a role too."</p><p>
More complex organisms that naturally have more than two flagella, and that do have the filamentary connections, confirmed this. "One of them, called <em><a href="https://en.wikipedia.org/wiki/Tetraselmis">Tetraselmis</a></em>, has four flagella and a beat pattern that is remarkable," says Goldstein. "The patterns of beats are in a one -to-one correspondence to [that of] terrestrial quadrupeds like horses: you can have different types of gallops and canters, and so on. The fact that there is a lot of study of the cross-connection inside the cells allowed us to see whether the symmetry of the beating reflects the symmetry of the connections — and it does."</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2019/Goldstein/4_quadri_combo_annotated.gif" alt="Different gaits" width="400" height="400"/>
<p style="max-width: 400px;">Algae going through their paces. The one showing the transverse gallop is Tetraselmis. Movies: Kirsty Wan and Raymond E. Goldstein, see <a href="https://www.pnas.org/content/113/20/E2784/tab-figures-data">this paper</a>.</p>
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<p>
"So the picture we finally have is that, yes, fluid mechanical interaction can synchronise flagella, but the internal connections play an equally important role. So it's a combination of the two."</p><h3>
Swimming for evolution</h3><p>
The work of Goldstein and his colleagues has done more than shed light on <em>Chlamydomonas</em>. It has also formed an experimental and mathematical basis for studying other hair-like appendices, including the cilia inside the human body. "In our body there are situations in which one has large cells with many cilia very close together, with one cell separated by a long distance from the next cell. What we have come to understand is that the synchrony that is observed within one of those cells is hydrodynamically driven, as is the synch observed between it and the next cell along. But when one has, as in fallopian tubes, carpets of cilia there can also be cross-connections between the bases. So [in this case] it's always a combination of the two."</p>
<div class="leftimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/articles/2019/Goldstein/volvox_darkfield_big.jpg" alt="Volvox" width="400" height="266"/>
<p>This photo shows a multi-celled Volvocale called <em>Volvox</em>.</p>
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<p>
It turns out that flagella also play a role in that crucial evolutionary step from single to multi-celled organisms. Two things every living thing needs to accomplish are nutrient uptake and waste removal, which in Volvocales happens through the surface of the body. Because of the spherical shape of that surface it's easy to calculate the rate at which Volvocales achieve both.</p><p>
"As one gets to larger and larger organisms, had these organisms had no flagella and no ability to stir the fluid [they live in], they would have reached a bottle neck where their metabolic needs would have exceeded what could come in from the surroundings by diffusion." Having the flagella is what allows Volvocales to bypass this evolutionary bottleneck and become larger. "This taught us that the fluid mechanical flows play a role in the evolution from single to multi-celled organisms."</p><p>
Results like these give essential glimpses into the workings of evolution that ultimately resulted in the dazzling variety of life we see today. Who knew that little green algae have so much to give?</p><hr/>
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<h3>About this article</h3>
<p><a href="http://www.damtp.cam.ac.uk/user/gold/">Ray Goldstein</a> FRS, is Schlumberger Professor of Complex Physical Systems at the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. He talked to the <em>Plus</em> team in March 2019.</p><p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>.</p>
</div></div></div>Fri, 22 Mar 2019 16:21:33 +0000Marianne7177 at https://plus.maths.org/contenthttps://plus.maths.org/content/synchronised-swimming#comments