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enMaths in a minute: Polar coordinates
https://plus.maths.org/content/maths-minute-polar-coordinates
<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_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: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2014/polar/cart.jpg" alt="Cartesian coordinates" width="200" height="172" />
<p>Cartesian coordinates. </p>
</div>
<p>How do you locate a point on the plane? One way of doing this, which you’re probably familiar with, it to use the Cartesian coordinate system. Draw two perpendicular axes and describe the location of a point <img src="/MI/235c09ce55162cd34ef26f41682ee1e1/images/img-0001.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> by two coordinates <img src="/MI/235c09ce55162cd34ef26f41682ee1e1/images/img-0002.png" alt="$(x_0,y_0)$" style="vertical-align:-4px;
width:50px;
height:18px" class="math gen" />. To find <img src="/MI/235c09ce55162cd34ef26f41682ee1e1/images/img-0001.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> you start at the point <img src="/MI/235c09ce55162cd34ef26f41682ee1e1/images/img-0003.png" alt="$(0,0)$" style="vertical-align:-4px;
width:34px;
height:18px" class="math gen" /> and walk a distance <img src="/MI/235c09ce55162cd34ef26f41682ee1e1/images/img-0004.png" alt="$x_0$" style="vertical-align:-2px;
width:15px;
height:9px" class="math gen" /> along the horizontal axis and a distance <img src="/MI/235c09ce55162cd34ef26f41682ee1e1/images/img-0005.png" alt="$y_0$" style="vertical-align:-3px;
width:14px;
height:10px" class="math gen" /> along the vertical axis. </p>
<p>But there’s another way of locating points on the plane, which is very nice too. To each point <img src="/MI/c85ac92f26fcd5aabdaf4913f0dc2baf/images/img-0001.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> assign the pair of numbers <img src="/MI/c85ac92f26fcd5aabdaf4913f0dc2baf/images/img-0002.png" alt="$(r, \theta )$" style="vertical-align:-4px;
width:33px;
height:18px" class="math gen" />, where <img src="/MI/c85ac92f26fcd5aabdaf4913f0dc2baf/images/img-0003.png" alt="$r$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> is the distance from <img src="/MI/c85ac92f26fcd5aabdaf4913f0dc2baf/images/img-0004.png" alt="$(0,0)$" style="vertical-align:-4px;
width:34px;
height:18px" class="math gen" /> to <img src="/MI/c85ac92f26fcd5aabdaf4913f0dc2baf/images/img-0001.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> along a straight radial line, and <img src="/MI/c85ac92f26fcd5aabdaf4913f0dc2baf/images/img-0005.png" alt="$\theta $" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> is the angle formed by that radial line and the positive <img src="/MI/c85ac92f26fcd5aabdaf4913f0dc2baf/images/img-0006.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />-axis, measured anti-clockwise from the <img src="/MI/c85ac92f26fcd5aabdaf4913f0dc2baf/images/img-0006.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />-axis to the line. These new coordinates are called <em>polar coordinates</em>, because you treat the crossing point of the axes as a pole from which everything radiates out. </p>
<p>In the image below, click on the point and drag it around to see how its polar coordinates <img src="/MI/1666df21bb7fd75338988bd913cd0e60/images/img-0001.png" alt="$(r, \theta )$" style="vertical-align:-4px;
width:33px;
height:17px" class="math gen" /> change (degrees are measured in radians). </p>
<p><iframe scrolling="no" src="https://www.geogebratube.org/material/iframe/id/WM7XYwxJ/width/616/height/418/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/preferhtml5" width="616px" height="418px" style="border:0px;"> </iframe></p>
<p>Some shapes that are hard to describe in Cartesian coordinates are easier to describe using polar coordinates. For example, think of a circle of radius <img src="/MI/af7e6d8827c287d1d6219812c9692373/images/img-0001.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> centred on the point <img src="/MI/af7e6d8827c287d1d6219812c9692373/images/img-0002.png" alt="$(0,0)$" style="vertical-align:-4px;
width:34px;
height:18px" class="math gen" />. It is made up of all the points that lie a distance <img src="/MI/af7e6d8827c287d1d6219812c9692373/images/img-0001.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> from <img src="/MI/af7e6d8827c287d1d6219812c9692373/images/img-0002.png" alt="$(0,0)$" style="vertical-align:-4px;
width:34px;
height:18px" class="math gen" />. In polar coordinates these are all the points with coordinates <img src="/MI/af7e6d8827c287d1d6219812c9692373/images/img-0003.png" alt="$(2,\theta )$" style="vertical-align:-4px;
width:34px;
height:18px" class="math gen" />, where <img src="/MI/af7e6d8827c287d1d6219812c9692373/images/img-0004.png" alt="$\theta $" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> can take any value at all. </p>
<p>In Cartesian coordinates this circle is a little harder to describe. It is made up of all points with coordinates <img src="/MI/d5ca00e9b3fab4c6a3694b0916a5cd82/images/img-0001.png" alt="$(x,y)$" style="vertical-align:-4px;
width:36px;
height:18px" class="math gen" /> where </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/d5ca00e9b3fab4c6a3694b0916a5cd82/images/img-0002.png" alt="\[ x^2+y^2=2^2=4. \]" style="width:125px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> (This follows from Pythagoras’ theorem.) </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/polar/angle.png" alt="Circle" width="280" height="157" />
<p style="max-width: 350px;">A circle. It is made up of all points whose Cartesian coordinates (<em>x</em>, <em>y</em>) satisfy <em>x</em><sup>2</sup>+<em>y</em><sup>2</sup>= 4 and whose polar coordinates (<em>r</em>, <em>θ</em>) satisfy <em>r</em>=2. The points shown has Cartesian coordinates (√2, √2) and polar coordinates (2,45), with the angle measured in degrees. </p>
</div>
<!-- Image made by MF -->
<p>Another nice example comes from looking at all the points whose first polar coordinate is equal to the second polar coordinate. In other words, all points of the form <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0001.png" alt="$(r,r).$" style="vertical-align:-4px;
width:37px;
height:18px" class="math gen" /> As <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0002.png" alt="$r$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> grows, the point <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0003.png" alt="$(r,r)$" style="vertical-align:-4px;
width:32px;
height:18px" class="math gen" /> moves further out from the point <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0004.png" alt="$(0,0).$" style="vertical-align:-4px;
width:39px;
height:18px" class="math gen" /> The angle <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0005.png" alt="$\theta =r$" style="vertical-align:0px;
width:38px;
height:11px" class="math gen" /> grows at the same rate. As <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0002.png" alt="$r$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> becomes larger, the angle <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0006.png" alt="$\theta = r$" style="vertical-align:0px;
width:38px;
height:11px" class="math gen" /> turns round and round the point <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0007.png" alt="$(0,0)$" style="vertical-align:-4px;
width:34px;
height:18px" class="math gen" />. The result is an <em>Archimedean spiral</em>. The movie below shows the points with coordinates <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0003.png" alt="$(r,r)$" style="vertical-align:-4px;
width:32px;
height:18px" class="math gen" />, as <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0002.png" alt="$r$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> grows from <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0008.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> to <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0009.png" alt="$20\pi $" style="vertical-align:0px;
width:26px;
height:12px" class="math gen" /> (which corresponds to ten full turns of <img src="/MI/dc11a698ad61cd981dc9e81adf42c1c4/images/img-0005.png" alt="$\theta =r$" style="vertical-align:0px;
width:38px;
height:11px" class="math gen" />). It’s a lot harder to describe such an Archimedean spiral in Cartesian coordinates! </p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/QjcRG0D1HxE?rel=0" frameborder="0" allowfullscreen></iframe>
<p>Finally, we look at points whose second polar coordinate <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0001.png" alt="$r$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> is equal to <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0002.png" alt="$r=e^{\theta /5}$" style="vertical-align:0px;
width:57px;
height:15px" class="math gen" />, where <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0003.png" alt="$\theta $" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> is the second polar coordinate and <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0004.png" alt="$e \approx 2.718$" style="vertical-align:0px;
width:67px;
height:13px" class="math gen" /> is the base of the natural logarithm. In this case, the first coordinate <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0002.png" alt="$r=e^{\theta /5}$" style="vertical-align:0px;
width:57px;
height:15px" class="math gen" /> (the distance from the corresponding point to <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0005.png" alt="$(0,0)$" style="vertical-align:-4px;
width:34px;
height:18px" class="math gen" />) grows faster than the second coordinate <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0003.png" alt="$\theta $" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> (the angle). The result is a spiral whose turns aren’t as tight as that of an Archimedean spiral — it’s an example of a <em>logarithmic spiral</em>. The movie below shows the points with coordinates <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0006.png" alt="$(e^{\theta /5},\theta )$" style="vertical-align:-4px;
width:54px;
height:19px" class="math gen" />, as <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0003.png" alt="$\theta $" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> grows from <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0007.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> to <img src="/MI/faf0066af3c9e48abea4235e77521b8c/images/img-0008.png" alt="$8\pi $" style="vertical-align:0px;
width:18px;
height:12px" class="math gen" /> (corresponding to four full turns). </p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/ypuuqoPLoZs?rel=0" frameborder="0" allowfullscreen></iframe>
<p>See <a href="/content/polar-power"><em>Polar power</em></a> for more about Archimedean and logarithmic spirals, as well as other interesting shapes you can draw with polar coordinates.</p></div></div></div>Fri, 23 Jun 2017 11:21:46 +0000Marianne6848 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-polar-coordinates#commentsFermat's last theorem
https://plus.maths.org/content/fermats-last-theorem
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_15.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage"><iframe width="400" height="225" src="https://www.youtube.com/embed/aGko3eEPq8o?rel=0" frameborder="0" allowfullscreen></iframe></div><p>Fermat's last theorem is one of the most beguiling results in mathematics. In 1637 mathematician Pierre de Fermat wrote into the margin of his maths textbook that he had found a "marvellous proof" for the result, which the margin was too narrow to contain.
</p><p>
If you look at the theorem you can see why Fermat might have thought that he found an elegant proof: the theorem is easy to explain, even to primary school students. But the proof turned out to be elusive even to the most talented mathematicians. It wasn't until over 350 years after Fermat's scribble that Andrew Wiles announced a proof, after years of working in secrecy and using mathematical machinery that goes well beyond the theorem's humble appearance. When Wiles announced the proof at the Isaac Newton Institute in Cambridge the atmosphere, according to eye witnesses, was electric.</p>
<p>This selection of articles and videos explores the theorem, some of the maths used in the proof, Andrew Wiles' monumental effort to find it, and also some other relevant bits of maths. Enjoy!</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/5/20_jun_2013_-_1534/icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/very-old-problem-turns-20">A vey old problem turns 20</a> — This article, published on the 20th anniversary of the announcement of the proof, gives a brief overview of Fermat's last theorem and the battle for its resolution. </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/notes_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/andrew-wiles-what-does-if-feel-do-maths">Andrew Wiles: What does it feel like to do maths?</a> — These two videos, and accompanying article, document a fascinating encounter we had with Wiles at the Heidelberg Laureate Forum in 2016. Wiles talks about what it was like to finally prove such an important result and what it feels like to do maths in general — it's a bit like composing a symphony!</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/5/7_jul_2015_-_1731/fermat_icon.png" alt="" width="100" height="100" /> </div><p><a href=/content/fermats-last-theorem-and-andrew-wiles">Fermat's last theorem and Andrew Wiles</a> — This article looks at the maths behind the proof in a little more detail and contains some beautiful thoughts of Wiles himself about his quest. (The image shows a portrait of Fermat.) </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/5/2_nov_2015_-_0943/ram_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/ramanujan">Ramanujan surprises again</a> — In 2015 two mathematician discovered a manuscript by another mathematical legend, Srinivasa Ramanujan, and found that he too had been working on Fermat's last theorem. Wiles is in good company indeed! </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/4/19_aug_2014_-_0201/icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/very-old-question-very-latest-maths-fields-medal-lecture-manjul-bhargava">Answers on a donut: the Fields medal lecture of Manjul Bhargava</a> — In 2014 Manjul Bhargava won the Fields medal, one of the highest honours in maths, for work that's related to the maths behind Fermat's last theorem. </p></div></div></div></div>Fri, 23 Jun 2017 10:13:56 +0000Marianne6851 at https://plus.maths.org/contenthttps://plus.maths.org/content/fermats-last-theorem#commentsConform to the norm
https://plus.maths.org/content/conform-norm
<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/ducks-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>A sure sign that I've adapted to England, my country of residence but
not birth, is that I find it impossible to jump
queues. The mere idea causes me
pain. That's despite the fact that queue jumping would save me time and
usually draws no more than a few tuts for punishment (another very
English behavioural trait). So why do I always patiently wait my turn?</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/norms/queue.jpg" alt="Queue" width="350" height="263" />
<p>Nobody likes queueing, but most of us do it.</p>
</div>
<!-- Image from fotolia.com -->
<p>The answer is that I've <em>internalised</em> the elaborate social norm English people have built around queueing. I don't so much follow it because I believe in the practical virtues of queues (though I do), but because I feel emotionally compelled to conform.
We're all born with this capacity to internalise social norms, and it's easy to see why: norms are the oil that greases cooperative societies. Without them there would be chaos.</p>
<p>In evolutionary terms, however, there's a missing link. Life didn't start with an innate desire to stick to social rules. Our very early ancestors were out for their own survival. Since evolution through natural
selection favours the fittest, it's hard to see how polite individuals who are happy to follow collective rules at their own expense would have survived. The same puzzle surrounds the evolution of any form of cooperation and of <a href="/content/does-it-pay-be-nice-maths-altruism-part-i">altruism</a>. How did our ability to internalise social norms take hold?
</p>
<p>A recent <a href="http://www.pnas.org/content/114/23/6068.abstract?sid=bcc20cfa-8275-478a-b906-84078464f2cb">study</a> addresses this puzzle using a
favourite tool in the area: <em>game theory</em>. The idea is to look at life as a game in which each individual can make certain moves directed by clear rules. Individuals follow specific strategies, and those who do better are more likely to produce offspring than those who don't do so well. If you can describe all this in precise mathematical terms, you can use a computer to simulate thousands of generations and see which strategies die out and which thrive. A strategy that looks silly at first sight, such as sticking to norms that offer no immediate price or punishment, may turn out to be beneficial in the long run and over the generations become entrenched. </p>
<p><a href="http://www.tiem.utk.edu/~gavrila/">Sergey Gavrilets</a> and <a href="http://www.des.ucdavis.edu/faculty/Richerson/Richerson.htm">Peter
Richerson's</a> game theoretical model involves lots of groups of hypothetical individuals. During its life time each individual has a number of chances (40 to be precise) to join in some collective action (such as forming a queue) or to refuse (to jump the queue). Since social norms aren't likely to take hold unless they are policed, individuals also get the chance to punish those who didn't take part.</p>
<p>Each time a social action comes around an individual is faced with four choices: take part and punish non-conformists, take part and don’t punish them, don’t take part and punish, and don’t take part and don’t punish. Gavrilets and Richerson describe the four choices using two variables: the variable <img src="/MI/35ff48e664c265db9b5f9514ac3f8fa6/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> describes the choice of taking part and can be either <img src="/MI/35ff48e664c265db9b5f9514ac3f8fa6/images/img-0002.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> (don’t take part) or <img src="/MI/35ff48e664c265db9b5f9514ac3f8fa6/images/img-0003.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" /> (take part). The variable <img src="/MI/35ff48e664c265db9b5f9514ac3f8fa6/images/img-0004.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> describes the choice of punishment and can also be either <img src="/MI/35ff48e664c265db9b5f9514ac3f8fa6/images/img-0002.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> (don’t punish) or <img src="/MI/35ff48e664c265db9b5f9514ac3f8fa6/images/img-0003.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" /> (punish). </p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/norms/ducks.jpg" alt="Rubber ducks" width="350" height="234" />
<p>What's the price of not taking part?</p>
</div>
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<p> Each of these four possibilities comes with a net pay-off for the individual that picked it. That's just as in real life: not taking part and not punishing norm-breakers might give you some advantage (barge to the front of the queue and grab the best piece) but a punishment (being beaten up) might diminish your gain. Your net pay-off then depends on how the advantage of dodging the norm compares to the disadvantage of punishment. Gavrilets and Richerson worked out a formula that determines the pay-off <img src="/MI/e2038b49e7a1acc3e42e3d868a42a4c8/images/img-0001.png" alt="$p(x,y)$" style="vertical-align:-4px;
width:46px;
height:18px" class="math gen" /> for each of the four possibilities. It depends on a number of other parameters, for example a number measuring the benefit and cost of a social action, and a number measuring the cost involved in monitoring and punishing rule-breakers — we won't go into the details here, but you can see <a href="http://www.pnas.org/content/114/23/6068.abstract?sid=bcc20cfa-8275-478a-b906-84078464f2cb">their paper</a> to find out more.
</p>
<p>Individuals can change their values of <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0002.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> during their life times, but there’s also a parameter they don’t choose themselves: that’s their innate ability to internalise social norms, which depends on their genes. It is measured by an <em>internalisation parameter</em> <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0003.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />, which lies between <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0004.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> (not at all able to internalise norms) and <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0005.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" /> (very able to internalise norms). </p><p>When an individual changes its values for <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0002.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> (whether they take part in a social action, and punish norm-breakers, respectively) they don’t make their new choices randomly. Instead, they choose the values that maximise the expression </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0006.png" alt="\begin{equation} u(x,y) = (1-n) \times p(x,y) + n \times (v_1x + v_2y).\end{equation}" style="width:325px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>1</span>)</span></td>
</tr>
</table><p>The parameters <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0007.png" alt="$v_1$" style="vertical-align:-2px;
width:14px;
height:9px" class="math gen" /> and <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0008.png" alt="$v_2$" style="vertical-align:-2px;
width:14px;
height:9px" class="math gen" /> are fixed numbers greater than or equal to zero, which measure the maximum value you can derive by taking part in an action (<img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0009.png" alt="$x=1$" style="vertical-align:0px;
width:38px;
height:12px" class="math gen" />) and punishing (<img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0010.png" alt="$y=1$" style="vertical-align:-3px;
width:38px;
height:15px" class="math gen" />) respectively. </p><p>You can see that if an individual has <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0011.png" alt="$n=0$" style="vertical-align:0px;
width:40px;
height:12px" class="math gen" />, so it’s not able at all to internalise norms, then equation 1 becomes </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0012.png" alt="\[ u(x,y) = p(x,y). \]" style="width:121px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> As you would expect, this individual will choose values of <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0002.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> that maximise its own pay-off <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0013.png" alt="$p(x,y)$" style="vertical-align:-4px;
width:46px;
height:18px" class="math gen" /> without worrying about others. </p><p>If, on the other hand, an individual has <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0014.png" alt="$n=1$" style="vertical-align:0px;
width:39px;
height:12px" class="math gen" />, then equation 1 becomes </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0015.png" alt="\[ u(x,y) = (v_1x + v_2y). \]" style="width:157px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>In this case the individual doesn’t care about their own pay-off and will always choose <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0009.png" alt="$x=1$" style="vertical-align:0px;
width:38px;
height:12px" class="math gen" /> and <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0010.png" alt="$y=1$" style="vertical-align:-3px;
width:38px;
height:15px" class="math gen" />: take part in the action and punish. Someone with a value of <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0003.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> in between <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0004.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> and <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0005.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" /> will choose whatever combination of <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0002.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> makes <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0016.png" alt="$u(x,y)$" style="vertical-align:-4px;
width:46px;
height:18px" class="math gen" /> as large as possible (though the model also accounts for the possibility that individuals might occasionally get this optimisation wrong). The function <img src="/MI/17059a0e99b1e8d56d579173dc53261c/images/img-0017.png" alt="$u$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is called a <em>utility function</em> because it measures how much benefit — utility — individuals get out of their action. </p>
<p> Gavrilets and Richerson started their society off with individuals all having very low values of the internalisation parameter <img src="/MI/57aa287884bcca4f8de39133e0b08aab/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />, between <img src="/MI/57aa287884bcca4f8de39133e0b08aab/images/img-0002.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> and <img src="/MI/57aa287884bcca4f8de39133e0b08aab/images/img-0003.png" alt="$0.05$" style="vertical-align:0px;
width:29px;
height:13px" class="math gen" />. They then set the games rolling, with groups of individuals facing 40 opportunities to take part in social actions. After each social action a proportion of all individuals (chosen randomly) choose new values for <img src="/MI/57aa287884bcca4f8de39133e0b08aab/images/img-0004.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and <img src="/MI/57aa287884bcca4f8de39133e0b08aab/images/img-0005.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> in a way that maximises their utility function. After the 40 social actions their life comes to an end. Individuals then reproduce according to how well they did during their lives. Groups whose social actions were successful are more likely to survive. In each surviving group, the fittest individuals (where success and fitness are also measured by mathematical formulas) are more likely to produce offspring. When individuals reproduce they pass their internalisation parameter on to their children, though there’s also a small chance that the parameter mutates to a different value. </p>
<div class="rightimage" style="max-width: 346px;"><img src="/content/sites/plus.maths.org/files/articles/2014/gametheory/fotolia_60884518_xs.jpg" alt="" width="346" height="346"/>
<p>Game theory looks at life as a game in which each individual can make certain moves directed by clear rules.</p>
</div>
<p>The two researchers ran computer simulations of their society for 10,000 generations. Since their model depends on various parameters (measuring cost and benefit, etc, as mentioned above) they had various scenarios to play with. For example, some parameter combinations mimic situations in which groups overcome challenges posed by their environment, while others include competition between groups. The simulations were run for every combination of parameters and showed that the internalisation parameter <img src="/MI/e87616ba51886fc0d44e89ef95d01a33/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> could evolve to larger values over the generations. Social norms can win the evolutionary game. </p>
<p>The figure below shows the results for one particular combination of parameter values. You can see in the bottom graph that, to start with, the internalisation parameter <img src="/MI/3657933ef287e5cd387c5d94dca82768/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> is low and most individuals behave selfishly (<img src="/MI/3657933ef287e5cd387c5d94dca82768/images/img-0002.png" alt="$x=y=0$" style="vertical-align:-3px;
width:70px;
height:15px" class="math gen" />), represented by the pink line in the top graph. Then, at a time just before the 2500th generation, there’s a dramatic jump in <img src="/MI/3657933ef287e5cd387c5d94dca82768/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />. At roughly the same time selfish behaviour drops dramatically and two types of behaviour become wide-spread: taking part in social actions and punishing (<img src="/MI/3657933ef287e5cd387c5d94dca82768/images/img-0003.png" alt="$x=y=1$" style="vertical-align:-3px;
width:69px;
height:15px" class="math gen" />, green line) and taking part in social actions and not punishing (<img src="/MI/3657933ef287e5cd387c5d94dca82768/images/img-0004.png" alt="$x=1$" style="vertical-align:0px;
width:38px;
height:12px" class="math gen" /> and <img src="/MI/3657933ef287e5cd387c5d94dca82768/images/img-0005.png" alt="$y=0$" style="vertical-align:-3px;
width:39px;
height:15px" class="math gen" />, blue line). </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/norms/Figure.jpg" alt="Black holes" width="296" height="425" />
<p>The top image shows how the proportion of individuals using different combinations of <em>x</em> and <em>y</em> changes over the generations. For example, the pink line represents the proportion of individuals using <em>x</em>=0 and <em>y</em>=0, the blue line represents the proportion of individuals using <em>x</em>=1 and <em>y</em>=0, and so on. The bottom image shows how the internalisation parameter <em>n</em> changes over the generations. The intensity of the black colour is proportional to the number of individuals with the corresponding value for <em>n</em> present at a given time. The red line shows the mean value of <em>n</em>. (These plots correspond to a particular combination of parameter values.) Figure from <a href="http://www.pnas.org/content/114/23/6068.abstract?sid=bcc20cfa-8275-478a-b906-84078464f2cb"><em>Collective action and the evolution of social norm internalization</em></a> by Gavrilets and Richerson, published in <em>PNAS</em>.</p>
</div>
<p>Many of the simulations had intermediate values of <img src="/MI/8f44fa37c063b81a5577d3977d187a7b/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />, but there were also societies that contained a small proportion of individuals with high values: heroes, local leaders, or perhaps fanatics who are willing to make huge sacrifices without regard for their own well-being. Similarly, there can be a small proportion of individuals
with a very low value of <img src="/MI/8f44fa37c063b81a5577d3977d187a7b/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />, who are completely unable to internalise social norms: psychopaths, perhaps. </p>
<p> As all game theoretical models, this one greatly simplifies our behaviour and the world we live in. It does show, however, that it's theoretically possible for the ability to internalise norms to evolve in a society where it was initially very low.
Gavrilets and Richerson argue that this ability was "likely a crucial step on the path to large-scale human cooperation".
</p>
<p>In our highly evolved society social norms of all shapes and colours have emerged. Some are obviously useful, some appear downright silly. Kate Fox's book <a href="https://www.hodder.co.uk/books/detail.page?isbn=9780340818862"><em>Watching the English</em></a> has an eye-opening collection, particularly for immigrants like me. But even if some norms are strange, we at least have an excuse for our meek observance of them: it's instinctive.</p></div></div></div>Fri, 23 Jun 2017 09:44:40 +0000Marianne6846 at https://plus.maths.org/contenthttps://plus.maths.org/content/conform-norm#commentsCitizen science: The statistics of language
https://plus.maths.org/content/statistics-language
<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/magnets_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">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: 354px;"><img src="/content/sites/plus.maths.org/files/articles/2017/Zipf/book_tree.jpg" alt="Book shelf" width="354px" height="289" /><p>Language is life!</p></div>
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<p>Our ability to learn, use, and process language is something that sets us apart from other animals. Language is used for effective communication, but also allows us to express our creativity through literature, poetry, and song. However, our use of language follows strict mathematical principles as well. One of the best known of these is <em>Zipf's law</em>.</p>
<h3>Zipf's law in theory</h3>
<p>If you count the frequency with which different words are used in written or spoken texts, and you rank them from most frequent to least frequent, you will find that (on average) the most frequent word will occur roughly twice as often as the second-most frequent word, three times as often as the third-most frequent word, and so on. In other words, the frequency <img src="/MI/6e6ecc066389ebbb05b5c7691e82ed23/images/img-0001.png" alt="$f(r)$" style="vertical-align:-4px;
width:28px;
height:18px" class="math gen" /> of a word is inversely proportional to its <em>rank</em> (place on the list) <img src="/MI/6e6ecc066389ebbb05b5c7691e82ed23/images/img-0002.png" alt="$r$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" />: <img src="/MI/6e6ecc066389ebbb05b5c7691e82ed23/images/img-0003.png" alt="$f(r) \propto 1/r.$" style="vertical-align:-4px;
width:79px;
height:18px" class="math gen" /> </p>
<p>The American linguist <a href="https://en.wikipedia.org/wiki/George_Kingsley_Zipf">George Zipf</a> described and popularised this phenomenon already more than 70 years ago. He actually used the more general relationship
</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/6031af47a86261d7d52b6af49696de48/images/img-0001.png" alt="\begin{equation} f(r) \propto 1/r^{a},\end{equation}" style="width:88px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>1</span>)</span></td>
</tr>
</table><p>arguing that, for word frequencies, the parameter <img src="/MI/6031af47a86261d7d52b6af49696de48/images/img-0002.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> will be close to <img src="/MI/6031af47a86261d7d52b6af49696de48/images/img-0003.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" />. </p>
<p>The above equation is an instance of a so-called <em>power law</em>. One of the characteristics of a power law is that if you visualise it in a <em>log-log</em> plot, it follows a linear relationship. A log-log graph plots the logarithms of both sides of an equation against each other. That this results in a linear relationship for a power law can be shown easily by taking the logarithm on both sides of equation (1) and slightly rewriting it.</p>
<p>First, let’s write equation (1) as a proper equality: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0001.png" alt="\begin{equation} f(r) = \frac{C}{r^ a},\end{equation}" style="width:76px;
height:33px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>2</span>)</span></td>
</tr>
</table><p> where <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0002.png" alt="$C$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> is some constant. For example, if <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0002.png" alt="$C$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> is equal to the frequency <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0003.png" alt="$f(1)$" style="vertical-align:-4px;
width:28px;
height:18px" class="math gen" /> of the most frequent word (i.e., the word of rank <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0004.png" alt="$r=1$" style="vertical-align:0px;
width:37px;
height:12px" class="math gen" />), and <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0005.png" alt="$a=1$" style="vertical-align:0px;
width:38px;
height:12px" class="math gen" /> (as Zipf argued), then we get the sequence <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0006.png" alt="$f(1)=C,$" style="vertical-align:-4px;
width:68px;
height:18px" class="math gen" /> <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0007.png" alt="$f(2)=f(1)/2,$" style="vertical-align:-4px;
width:103px;
height:18px" class="math gen" /> <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0008.png" alt="$f(3)=f(1)/3$" style="vertical-align:-4px;
width:98px;
height:18px" class="math gen" />, and so on, as described above. </p><p>Next, we can take the logarithm on both sides of equation (2) and rewrite it as follows: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0009.png" alt="\begin{equation} \log {(f(r))} = \log {(C/r^ a)} = \log {(C)} - a\log {(r)}.\end{equation}" style="width:317px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>3</span>)</span></td>
</tr>
</table><p>Remembering that the general equation of a straight line is <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0010.png" alt="$y=mx+b$" style="vertical-align:-3px;
width:82px;
height:14px" class="math gen" />, where <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0011.png" alt="$m$" style="vertical-align:0px;
width:14px;
height:7px" class="math gen" /> is the slope of the line, we see that the logarithm of the frequency, <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0012.png" alt="$\log {(f(r))}$" style="vertical-align:-4px;
width:66px;
height:18px" class="math gen" />, does indeed follow a linear relationship in terms of the logarithm of the rank, <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0013.png" alt="$\log {(r)}$" style="vertical-align:-4px;
width:43px;
height:18px" class="math gen" />. In this case the slope is <img src="/MI/04338f24af13e8bdaa8701ed1b4f8e3c/images/img-0014.png" alt="$-a$" style="vertical-align:0px;
width:21px;
height:7px" class="math gen" />. </p>
<p>This phenomenon, as observed in word frequencies, is known as <em>Zipf's law</em>. Interestingly, though, the same phenomenon has also been observed in many other areas. For example, it occurs in areas closely related to language such as music or computer code, but also in completely unrelated systems such as sizes of cities or connections in networks like the internet or the power grid. It even shows up in snooker statistics (see below)! The main difference, though, is that the values for the parameter <img src="/MI/c590c23b1108fac0a78365ae4859c031/images/img-0001.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> in the corresponding power law (i.e., the slope of the linear log-relationship) can be quite different for these different systems. (You can find out more about power laws in <a href="/content/tags/power-law">these <em>Plus</em> articles</a>.)</p>
<h3>Zipf's law in practice</h3>
<p>As good citizen scientists, we of course want to verify Zipf's law for ourselves, and perhaps even analyse some of our own favourite texts (see <a href="/content/citizen-science-public-databases-and-bit-maths">this article</a> for our first venture into the world of citizen science). Luckily this is very easy to do.</p>
<p>To verify the general phenomenon, a <a href="http://www.wordfrequency.info/free.asp?s=y">list of the 5000 most frequent words</a> (together with their actual frequencies) from the <em>Corpus of Contemporary American English</em> (COCA) can be used. The COCA is a 450-million-word collection of contemporary English texts, spanning many different literature categories and authors.</p>
<p>Taking the 50 most frequent words from this list and plotting their frequencies <img src="/MI/b6aba06bbe6bf2d93730a42fff14c657/images/img-0001.png" alt="$f(r)$" style="vertical-align:-4px;
width:28px;
height:18px" class="math gen" /> against their rank <img src="/MI/b6aba06bbe6bf2d93730a42fff14c657/images/img-0002.png" alt="$r$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> in a log-log plot results in a graph as shown in figure 1. To check how closely this graph follows a linear relationship, and to estimate the parameter value of the corresponding power law, a <a href="/content/maths-minute-linear-regression">linear regression</a> can be performed on the logarithms of these frequencies and ranks. Such a regression finds the straight line that best fits the data. The straight line in figure 1 represents the result, giving a slope of <img src="/MI/ee6db9a05f41c9763d71bdcb32a8088e/images/img-0001.png" alt="$-0.922$" style="vertical-align:0px;
width:49px;
height:12px" class="math gen" />, i.e., a value of <img src="/MI/ee6db9a05f41c9763d71bdcb32a8088e/images/img-0002.png" alt="$a=0.922$" style="vertical-align:0px;
width:68px;
height:12px" class="math gen" /> in the corresponding power law. This is indeed a value close to <img src="/MI/ee6db9a05f41c9763d71bdcb32a8088e/images/img-0003.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" />, as Zipf originally argued. Furthermore, according to the regression statistics, this linear relationship can explain 98% of the observed frequency distribution (for those who know some statistics, <img src="/MI/ee6db9a05f41c9763d71bdcb32a8088e/images/img-0004.png" alt="$R^2=0.98$" style="vertical-align:0px;
width:71px;
height:14px" class="math gen" />), and is thus an excellent fit.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/Zipf/COCA.jpg" alt="Zipf's law for COCA" width="500" height="500" />
<p style="width: 500px;">Figure 1: Distribution of word frequencies in the COCA database. The frequencies of words (vertical axis) against their rank (horizontal axis) in a log-log plot for the 50 most frequent words in the Corpus of Contemporary American English (COCA). The straight line represents a linear regression estimate of the corresponding power law, resulting in a parameter value of <em>a</em>=0.922.
</p>
</div>
<!-- image made by author -->
<a name="regression"></a>
<p>To perform the linear regression analysis, the <a href="https://www.r-project.org"><font style="font-family:Courier New;" >R</font> program</a> was used. This software can be downloaded and installed for free, and runs on all platforms (Windows, MacOS, and Linux). To see the short <font style="font-family:Courier New;" >R</font> script I wrote for the regression analysis and to produce figure 1, click <a href="/content/r-code-linear-regression">here</a>. You can modify it to analyse your own data. </p>
<p>But how to get frequency data for your own favorite texts, or even for your own writings? For that we can use another free tool, the <a href="http://1.1o1.in/en/webtools/semantic-depth">semantic depth analyzer</a>, which includes a nice example of Zipf's law occurring in a large (and also free) database of books. After clicking on the link "This gadget" on that web page, a small app opens up that allows you to paste in any text, for which it will calculate the word frequencies. You can then replace the frequency data in the above-mentioned <font style="font-family:Courier New;" >R</font> script with your own data, and simply re-run the script.</p>
<p>I did this for one of my personal favorite books, <em>The Origin of Species</em> by Charles Darwin. The results are shown in figure 2. The estimated parameter value for the corresponding power law is <img src="/MI/81362da3e226708f20289efe92e0747d/images/img-0001.png" alt="$a=0.829$" style="vertical-align:0px;
width:68px;
height:12px" class="math gen" />, a little less close to <img src="/MI/81362da3e226708f20289efe92e0747d/images/img-0002.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" /> than for the COCA database, but the fit of the straight line is even better: <img src="/MI/81362da3e226708f20289efe92e0747d/images/img-0003.png" alt="$R^2=0.99.$" style="vertical-align:0px;
width:75px;
height:14px" class="math gen" />
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/Zipf/Darwin.jpg" alt="Zipf's law for Darwin" width="500" height="500" />
<p style="width: 500px;">Figure 2: Distribution of word frequencies in <em>The Origin of Species</em>. The estimated parameter value for the corresponding power law is <em>a</em>=0.829.
</p>
</div>
<!-- image made by author -->
<p>As this example shows, there can be differences between the "average" word frequency distribution (as represented by the COCA database) and that of an individual book or author, even though both follow a power law. Clearly there are differences in the parameter value of the respective power laws, such as <img src="/MI/cc3689f3ba572c52bd000714ff218c0c/images/img-0001.png" alt="$0.922$" style="vertical-align:0px;
width:37px;
height:12px" class="math gen" /> for the COCA database and <img src="/MI/cc3689f3ba572c52bd000714ff218c0c/images/img-0002.png" alt="$0.829$" style="vertical-align:0px;
width:37px;
height:12px" class="math gen" /> for <em>The Origin of Species</em>. However, and perhaps more importantly, there are also differences between the actual ranks <img src="/MI/619995a3af76b168a18e2793802551e7/images/img-0001.png" alt="$r$" style="vertical-align:-1px;
width:8px;
height:9px" class="math gen" /> of specific words.</p>
<p>For example, the first five words in the COCA ranking are "the", "be", "and", "of", and "a" (in that order). However, the first five words in the ranking of Darwin's book are "the", "of", "and", "in", and "to". The most frequent word ("the") is the same for both, but "of" appears in second place for Darwin instead of fourth for COCA, and "be" (in second place for COCA) does not even appear in Darwin's top five. I will return to these individual differences shortly.</p>
<p>If you are interested in learning more about the intricacies (and sometimes controversies) of word frequencies and Zipf's law, <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4176592/">this insightful review paper</a> provides more details. It also presents a nice overview of what has been done so far to try and explain why language does (or even should) follow Zipf's law at all, and what is still missing and left to do for future research.</p>
<h3>Zipf's law in snooker</h3>
<p>As a fun little exercise, having just watched the world championships snooker, I wondered whether Zipf's law might also show up in snooker statistics. A quick search turned up this <a href="http://snookerinfo.webs.com/100centuries">ranking list of career centuries</a>. If you are familiar with snooker, you will know that a century break is a score of at least 100 points within one visit to the table (i.e., without missing a shot). The "career centuries ranking" lists all players who have made at least 100 centuries throughout their career (during official play, that is, not including practice or exhibition games). Of course this list is topped by the amazing <a href="https://en.wikipedia.org/wiki/Ronnie_O%27Sullivan">Ronnie "The Rocket" O'Sullivan</a> with close to 900 career centuries (and counting!).</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/Zipf/Ronnie.jpg" alt="Ronnie O'Sullivan" width="350px" height="266" /><p>Ronnie O'Sullivan. Image: <a href="https://commons.wikimedia.org/wiki/File:Ronnie_O’Sullivan_at_Snooker_German_Masters_(DerHexer)_2015-02-06_08.jpg">DerHexer</a>, <a href="https://creativecommons.org/licenses/by-sa/4.0/legalcode">CC BY-SA 4.0.</a></p></div>
<p>Plotting these statistics in a log-log plot and fitting a power law gives the results as shown in figure 3. As the graph shows, the match is not quite as close as for the word frequencies above, but with <img src="/MI/581de3a2ea54ad9f3491887a1eb11e3c/images/img-0001.png" alt="$R^2=0.95$" style="vertical-align:0px;
width:71px;
height:14px" class="math gen" /> it is still a fairly good fit. Especially for the highest and lowest ranks the discrepancies between the actual data and the fitted power law are somewhat larger, which is often the case with these kinds of statistics. The estimated parameter for the power law is <img src="/MI/2cb1f07d08129e517b85400a25bf9722/images/img-0001.png" alt="$a=0.594$" style="vertical-align:0px;
width:68px;
height:13px" class="math gen" />, which is quite different from that for the word frequencies. But it is interesting, and somewhat surprising, to see that Zipf's law indeed shows up in snooker statistics as well!</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/Zipf/centuries.jpg" alt="Zipf's law for Darwin" width="500" height="500" />
<p style="width: 500px;">Figure 3: Distribution of career century breaks in snooker. The estimated parameter value for the corresponding power law is <em>a</em>=0.594.
</p>
</div>
<!-- image made by author -->
<h3>Making your words count</h3><p>
As was already indicated above, there can be differences in actual word frequency distributions between individual pieces of text or authors. This brings up the question of whether the specific frequency distribution of a given text can say something about its author.</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/Zipf/magnets.jpg" alt="Book shelf" width="350px" height="232" /><p></p></div>
<!-- Image from fotolia.com -->
<p>Indeed, how authors express their thoughts reveals much about their character, according to psychologist <a href="http://www.secretlifeofpronouns.com/index.php">James Pennebaker</a>. In particular, an author's use of so-called <em>function words</em> (such as pronouns, articles, and a few other categories of words that, on their own, convey little meaning) is apparently directly linked to their social and psychological states. Simply put, your choice of words says something about your personality.</p>
<p>Pennebaker and colleagues have developed a <a href="https://liwc.wpengine.com">sophisticated computer program</a> (unfortunately not free, though) that collects statistics about an author's use of words in specific categories (such as function words). With this software they then analysed thousands of books, blogs, and speeches, and were able to link an author's specific word use to their personality, honesty, social skills, and intentions. This connection had already been discovered earlier, but with this new software tool it has become possible to investigate it in much more detail and on a much larger scale, firmly establishing the link between linguistics and psychology.</p>
<p>The language of statistics is not always easy to understand. But statistical analysis provides a useful and versatile tool. And this mathematical language — as Zipf, Pennebaker, and others have shown — can in turn be used to analyse natural language as well, making your words count by counting your words: the statistics of language.</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 Konrad Lorenz Institute in Klosterneuburg, Austria. 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>Wed, 07 Jun 2017 14:07:44 +0000Marianne6839 at https://plus.maths.org/contenthttps://plus.maths.org/content/statistics-language#comments'The mathematics lover's companion'
https://plus.maths.org/content/mathematics-lovers-companion
<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="leftimage" style="width: 150px; border: 1px;"><img src="/content/sites/plus.maths.org/files/reviews/2017/cover_scheinerman.png" alt="cover" width="150" height="226" /></div><h2>The mathematics lover's companion: Masterpieces for everyone</h2>
<h3>by Edward Scheinerman</h3>
<p>This book is a well-organised and friendly exploration of some of the most beautiful and significant results in mathematics. It's a collection of important mathematical theorems and concepts that everyone can understand and learn to appreciate, irrespective of their scientific or mathematical background.</p>
<p>The book is divided into twenty-three chapters, each falling under one of three main categories: number, shape and uncertainty. There is also a preface that includes Scheinerman's suggestions on how this book should be read, as well as a prelude describing the concepts of theorem and proof in mathematics. The first twelve chapters are devoted to numbers, and they range from concepts such as irrational, Fibonacci, transfinite and imaginary numbers to specific well-known mathematical constants such as π and Euler's <em>e</em>. The next six chapters study shapes in mathematics, starting with familiar Euclidean triangles and circles and progressing towards more exotic and perhaps less intuitive shapes such as the Sierpinski triangle, which exists between dimensions, and the hyperbolic plane, which fits entirely within a disk. The final five chapters deal with uncertainty, and they include chaos theory, probabilities and a fascinating game called Newcomb's paradox, which manages to incorporate the effects of free will into a mathematical problem.</p>
<p>The subtitle "Masterpieces for everyone" is appropriately chosen, because the book truly contains mathematical masterpieces that everyone can understand. The only prerequisite is high school algebra. And given the thematic variability of the chapters and their self-containment, it is easy for readers to find something in this book that they will enjoy reading about, and even skip chapters that they might find less interesting. From my experience reading this book, though, I would recommend against this strategy. Even if one of the topics might seem unappealing to a reader, Scheinerman often includes real-world applications, witty comments and interesting asides that might prove the reader's expectations quite wrong. For example, the chapter on the irrationality of the square root of two might sound too abstract to some readers or all too familiar to others, but it includes a beautiful application to instrument tuning and the connection between irrational numbers and harmonic sounds, which many might find interesting and surprising.</p>
<p>Even though the basic goal of this book is to present mathematical masterpieces in a way that everyone can understand, it aspires to do much more than that. Its ultimate purpose is to guide the reader from being able to simply follow the results presented to learning to appreciate their significance and mathematical elegance. Naturally, most mathematicians and people who already have a good understanding of mathematics have an edge here. If you fall into this category and already have a deeper appreciation of mathematics, this book is still a very enjoyable read because it will remind you why you fell in love with mathematics in the first place.</p>
<p> But where Scheinerman's skills as an educator truly shine is in guiding readers who have an interest in mathematics but not necessarily much experience with it. Sadly mathematics is often taught in a dry and uninspiring way at school, usually with an emphasis on memorising formulas and solving equations without any real motivation. This book focuses instead on the analytical thinking approaches and intuition behind famous mathematical results. Its very first chapter is a <a href="/content/maths-minute-how-many-primes">proof by contradiction that there exist infinitely many prime numbers</a>. This deceptively simple proof relies entirely on logic and high-school algebra, which is surprising for a theorem that Scheinerman (as well many other mathematicians) consider one of the most significant and beautiful results in all of mathematics.</p>
<p> As Scheinerman points out in the preface, however, it can be difficult to teach someone the joy of something until they experience it themselves. With that in mind, he offers various short problems throughout the book for the readers to solve, and of course provides all the answers in subsequent pages. The result is a book which can easily transform into a series of intellectually challenging mathematical exercises and puzzles, all done under the guidance of someone who clearly has a passion for mathematics and perhaps an even bigger passion for teaching it.</p>
<p><em>The mathematics lover's companion</em> lives up to its title. Readers already well-versed in mathematics will be taken back to the first time they were left in awe by Euler's equation or their first attempt to wrap their head around all the different kinds of infinity — but Scheinerman's humorous and witty style, as well plenty of enjoyable asides, will make reading the book worthwhile. To readers who are interested in mathematics but not yet very familiar with it, this book
offers a fun tour into the world of mathematics.</p>
<dl> <dt><strong>Book details:</strong></dt>
<dd><em>The mathematics lover's companion: Masterpieces for everyone</em></dd> <dd> Edward R. Scheinerman</dd> <dd>hardback — 296 pages</dd>
<dd> Yale University Press (2017)</dd>
<dd> ISBN 978-0300223002</dd>
</dl>
<hr><h3>About the author</h3>
<p>Veni Karamitsou is a PhD student in applied mathematics at the University of Cambridge. She is a member of the <a href="http://www.damtp.cam.ac.uk/research/dd/">Disease Dynamics Group</a>, and is currently working on modelling how the influenza virus evolves and spreads.</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">Venetia Karamitsou</div></div></div>Wed, 07 Jun 2017 11:22:09 +0000Marianne6842 at https://plus.maths.org/contenthttps://plus.maths.org/content/mathematics-lovers-companion#commentsLIGO detects a new gravitational wave
https://plus.maths.org/content/ligo-detects-new-gravitational-wave
<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/black_holes_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/waves/black_holes.jpg" alt="Black holes" width="350" height="280" />
<p>A dancing duo of black holes. <em>Image: LIGO/Caltech/MIT/Sonoma State (Aurore Simonnet).</em></p>
</div>
<p>Physicists are celebrating the detection of a new gravitational wave that was caused by the collision of two black holes around 3 billion light years away from Earth. The wave was detected by the <a href="https://www.ligo.caltech.edu">Laser Interferometer Gravitational-Wave Observatory</a> (LIGO), an international collaboration of researchers from all around the globe. It is only the third gravitational wave to have been observed, and the discovery adds to a milestone in the history of physics. "We have opened our ears to the Universe in a waveband in which we have been 100% deaf up to now," says <a href="http://www.damtp.cam.ac.uk/people/us248/">Ulrich Sperhake</a> of the University of Cambridge, who is involved with the LIGO project.</p>
<h3>Spacetime shivers</h3>
<p>Gravitational waves are ripples in the fabric of spacetime. According to Einstein's <a href="/content/maths-minute-einsteins-general-theory-relativity">general theory of relativity</a>, spacetime is not a static stage on which physics unfolds, but a dynamic entity that is warped by massive objects moving within it. The force of gravity is the result of such distortion of spacetime: the deformation caused by a massive object, such as our Sun, diverts the path of other objects, causing them to orbit the more massive body or hurtle towards it.</p>
<p>According to general relativity a collision of two massive objects, such as two black holes, should cause spacetime to ripple just as the water of a pond might ripple as fish cavort within it. Einstein himself suggested that such ripples — gravitational waves — might exist, but it wasn't until 2016 that they were <a href=/content/spacetime-does-ripple">first detected by LIGO</a>. The third detection heralds a whole catalogue of observations which, researchers hope, will unlock important information about black holes and help us solidify our theories. "The fact that we have now seen three gravitational waves in a year gives us hope that, as we improve the detector, we will see hundreds in the next ten years or so," says <a href="http://www.damtp.cam.ac.uk/people/cjm96/">Chris Moore</a>, also of the University of Cambridge.</p>
<h3>Massive objects, tiny signals</h3>
<div class="leftimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/waves/relativity.jpg" alt="General relativity" width="350" height="197" />
<p>A massive object warps the fabric of spacetime, visualised in this artist's impression by the green grid. <em>Image: T. Pyle/Caltech/MIT/LIGO Lab.</em></p>
</div>
<p>The latest collision involved two black holes of masses 19 and 32 times that of the Sun. Their merger formed a new black hole of 49 solar masses and converted the remaining two solar masses into gravitational wave energy. The resulting signal, detected by LIGO on 4 January 2017, was tiny. It lasted about a tenth of a second and distorted space by only 0.000,000,000,000,000,001 metres. The signal was measured by LIGO's two detectors in Washington and Louisiana. Each detector is L-shaped, with 4km long legs either side. At the end of each leg is a mirror with a laser passing back and forth, which measures the distance between the mirrors and can therefore pick up any distortion caused by a gravitational wave.</p>
<p>The data collected by LIGO isn't as straightforward as a photo you might take with a telescope, however. Any signal is hidden within an abundance of noise, and can only be picked out using careful analysis with theoretical insights leading the way.</p>
<h3>New insights</h3>
<p>This and future detections of gravitational waves will shed light on the nature of our Universe and on the theories we use to describe it. "One special feature of this particular [observation] is that it's the furthest one we have seen so far," says <a href="http://www.damtp.cam.ac.uk/people/ma748/">Michalis Agathos</a>, of the University of Cambridge. "The gravitational waves have travelled longer from their source to us, and this allows us to perform tests on the propagation of gravitational waves that are more precise than what we have done so far." The tests have already confirmed a prediction of general relativity, that all gravitational waves travel at the speed of light, a fact which rules out alternative theories of gravity in which this isn't the case.</p><div class="rightimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/waves/Ligo.jpg" alt="LIGO" width="350" height="235" />
<p>The LIGO Laboratory operates two detector sites, one near Hanford in eastern Washington, and another near Livingston, Louisiana. This photo shows the Livingston detector site.</p>
</div>
<p>Gravitational waves will also tell us more about the mysterious objects that cause them: black holes. "There are many things we would like to find out," says Moore. "Are heavy black holes more common than light black holes? Does the number of black hole mergers change as the Universe gets older?" A catalogue of gravitational wave detections would help us answer these questions. In fact, two of the three detections made so far involve black holes that are more massive than was expected, suggesting the population of massive black holes might be larger than was at first thought. In the future researchers also hope to observe collisions involving neutron stars, to be able to draw conclusions about their nature and abundance.</p>
<p>For the moment, however, physicists involved with the LIGO project reflect on their latest success. "It's a bit like winning the FA cup final," says Sperhake. "You've scored a goal, but when you lift the cup you haven't quite digested how significant a moment in time this is going to be regarded as ten or fifteen years from now. It hasn't quite sunk in yet."</p>
<p><em><em><em><em><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/FGC_DM7ZgAk?rel=0" width="560"></iframe></em></em></em></em></p>
<p><em><em><em><em>Simulation of the binary black-hole coalescence detected by LIGO.<br />
<em>Credit: Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)</em></em></em></em></em></p></div></div></div>Mon, 05 Jun 2017 11:42:21 +0000Marianne6845 at https://plus.maths.org/contenthttps://plus.maths.org/content/ligo-detects-new-gravitational-wave#commentsThe beauty of maths is in the brain of the beholder
https://plus.maths.org/content/beauty-maths-brain-beholder
<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/E8_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">Josefina Alvarez</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>Maths is the language that explains our physical world, what the mathematician
and physicist <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Wigner.html">Eugene Wigner</a> called "the unreasonable
effectiveness of mathematics in the natural sciences". It
is also the lightning that, sometimes unnoticed, precedes the thunder of
technological innovations. Yes, mathematics serves and touches upon the
advances of practically every field. Still, as a discipline unto itself,
mathematics is an extraordinary example of the human pursuit of knowledge,
versatile and capable of unifying areas that appear very different. The
mathematician <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Atiyah.html">Michael F. Atiyah</a> referred to these qualities in an <a href="https://pdfs.semanticscholar.org/c5de/3785b3633de14b65ba11dd32ac3ca5ea2f42.pdf">address</a> to
the London Mathematical Society titled <em>The unity of mathematics</em>. In
his own words, "The aspect of mathematics which fascinates
me most is the rich interaction between its different branches, the
unexpected links, the surprises, ..." .</p>
<div class="rightimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/articles/2017/Beauty/E8.png" alt="E8" width="400px" height="402" /><p>Some visualisations of mathematical objects, like this one of the <em><a href="/content/solving-symmetry">exceptional Lie group E8</a></em>, are undoubtedly beautiful. But is maths by itself beautiful? (Image <a href="https://commons.wikimedia.org/wiki/File:E8Petrie.svg">Jgmoxness</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>.)</p></div>
<p>But mathematicians and mathematics aficionados also talk about the
<em>beauty of mathematics</em>. For instance, the philosopher and mathematician
<a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Russell.html">Bertrand Russell</a> affirmed that "mathematics, rightly viewed, possesses not
only truth, but supreme beauty". Now, beauty is an slippery concept, much
more subjective and difficult to frame than, say, applicability within maths or in
other fields. To be sure, whatever beauty might be, mathematicians find an aesthetic value in the power of abstraction that mathematics has and in
the connections it finds.</p>
<p> But how is this mathematical beauty similar to, or
different from, the beauty we ascribe to a work of art? There
are frequent references to mathematics as an art. In an article published
in <em>The Mathematical Intelligencer</em> with the title <em><a href="http://www.forthelukeofmath.com/documents/borel.pdf">Mathematics: art and
science</a></em>, the mathematician <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Borel_Armand.html">Armand Borel</a> has this to say: "... mathematics
is an extremely complex creation which displays so many essential traits in
common with art and experimental and theoretical sciences that it has to be
regarded as all three at the same time, and thus must be differentiated from
all three as well."</p>
<p>So, as to how and why mathematics is connected to art or to other endeavours, the answer is typically ambiguous. It seems that questions of this
nature are destined to remain a part of the mathematical folklore.
Still, perhaps unknown to most mathematicians, something has been happening somewhere else.</p>
<h3>A brain revolution</h3>
<p>As early as 1909, an anatomist, <a href="https://en.wikipedia.org/wiki/Korbinian_Brodmann">Korbinian Brodmann</a>, divided the cortex
of the human brain into 47 areas, now called <a href="https://en.wikipedia.org/wiki/Brodmann_area"><em>Brodmann areas</em></a>, according to the structure and organisation of the cells. Later, as researchers
began to understand the functions of different cells in the cortex, they were
astounded by the fairly close relation emerging between certain Brodmann
areas and the location of specific cell functions. Today, much is known about
a wide range of these functions, showing the prodigious complexity of our
brain. For instance, there seem to be about three thousand interconnected neurons, controlling most of our breathing and involving about 65 types of neurons!</p>
<div class="leftimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2017/Beauty/Brodman.png" alt="Brodmann areas" width="250px" height="425" /><p>Brodmann areas in the brain.</p></div>
<!-- image in public domain -->
<p>Besides what we might call the physical functions, researchers are
understanding the locations of what we might call intellectual functions.
For example, an <a href="http://www.sciencedirect.com/science/article/pii/S1053811910013017">article</a> published in 2011 in the journal <em>NeuroImage</em>,
discussed the findings of a study locating the brain areas needed for
numbers and calculations. Coincidentally, a <a href="journals.plos.org/plosone/article?id=10.1371/journal.pone.0021852">study</a> published the same year in
the journal <em>PLoS ONE</em>, presented evidence towards a brain-based theory
of beauty. Several studies had already shown how beauty, as related to
visual, auditory and moral experience, was connected to activity
observed in a specific region of what researchers call the <em>emotional brain</em>.</p>
<p>Based on all these findings, a team of two neurobiologists, <a href="http://www.vislab.ucl.ac.uk">Semir Zeki</a> and
<a href="http://www.ucl.ac.uk/~ucgajpr/">John Paul Romaya</a>, a physicist, <a href="http://www.sissa.it/app/members.php?ID=175">Dionigi M. T. Benincasa</a>, and the mathematician
<a href="http://www-history.mcs.st-and.ac.uk/Biographies/Atiyah.html">Michael F. Atiyah</a> conjectured that the perception of mathematical beauty
should excite the same parts of the emotional brain, roughly described by a collection of Brodmann areas. <a href="http://journal.frontiersin.org/article/10.3389/fnhum.2014.00068/full">Their study</a>, published
in the journal <em>Frontiers of human neuroscience</em> in 2014, seems to
confirm their conjecture, indeed placing mathematical beauty, as perceived
by trained mathematicians, in the same areas previously identified with
other manifestations of beauty.</p>
<p>As part of the study participants (mathematicians and non-mathematicians) were presented with sixty mathematical formulas and asked to rank them according to three categories; ugly, neutral and beautiful. The expression that was most consistently ranked as beautiful by the mathematicians was Euler’s equation, </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/ff7390f581fc819d56ccbcbd87f68041/images/img-0001.png" alt="\[ e^{i\pi }+1=0, \]" style="width:85px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>and the one most consistently ranked as ugly by the mathematicians was the following, extremely convoluted, formula representing the reciprocal of <img src="/MI/ff7390f581fc819d56ccbcbd87f68041/images/img-0002.png" alt="$\pi $" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> as an infinite sum: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/ff7390f581fc819d56ccbcbd87f68041/images/img-0003.png" alt="\[ \frac{1}{\pi } = \frac{2\sqrt {2}}{9801} \sum _{k=0}^{\infty } \frac{(4k)! (1103 + 26390k)}{(k)!^4 396^{4k}}. \]" style="width:261px;
height:46px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
The formula is due to the mathematician <a href="/content/happy-birthday-ramanujan">Srinivasa Ramanujan</a>. His collaborator <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Hardy.html">Godfrey Hardy</a> said that the formula had to be true,
because no one could possibly concoct something so complicated.</p>
<p> The choices that mathematicians made in the study
seem to point to simplicity as an attribute of the mathematically
beautiful.</p>
<h3>Beauty and understanding</h3>
<p>Although the perception of mathematical beauty might have the same
characteristics, neurologically speaking, as the perception of beauty
elicited by other sources, some differences remain.</p>
<p>To find beauty in a particular piece of music, we do not need to understand
the intricacies of the composition. Likewise, we can feel the beauty of a
painting or a sculpture at a "gut level",
without thinking explicitly about the technical aspects.</p>
<p>However, the authors of the study speak at length of the
difficulty of separating the perception of mathematical beauty from the
understanding of the underlying mathematics. The subjects recruited for
their study were sixteen postgraduate or postdoctoral students of
mathematics, and twelve non-mathematicians. Among the mathematicians there was a strong correlation between understanding and perception of beauty, but it wasn't a perfect correlation: some formulas were rated ugly even though the mathematicians understood them well. Unsurprisingly, the non-mathematicians generally didn't understand the formulas very well, but they still rated some as beautiful. The authors suggest that the non-mathematician probably liked something about the formal qualities of the equations, for example their symmetrical distribution — this is something that will have to be investigated in a future study.
</p>
<p>Finally, the tool used in this and other studies is a form of magnetic
resonance imaging, fMRI, the letter f standing for "functional". The American chemist <a href="https://en.wikipedia.org/wiki/Paul_Lauterbur">Paul
Lauterbur</a> and the English physicist <a href="https://en.wikipedia.org/wiki/Peter_Mansfield">Peter Mansfield</a>
were awarded the <a href="http://www.nobelprize.org/nobel_prizes/medicine/laureates/2003/">2003 Nobel Prize in Physiology or Medicine</a> for their
development of MRI. In particular, Mansfield was credited with introducing its mathematical formalism. A nice way for our article to come full
circle: the unreasonable effectiveness of maths allows us to observe our mental world and help us understand the nature of mathematics itself.</p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="max-width: 148px;"><img src="/content/sites/plus.maths.org/files/articles/2015/traffic/josefinaalvarez3.jpg" alt="Josefina" width="148" height="168" />
<p></p>
</div>
<p>Josefina Alvarez first learnt about the study discussed in this article from a <a href="https://www.nytimes.com/2017/04/15/opinion/sunday/the-worlds-most-beautiful-mathematical-equation.html?action=click&pgtype=Homepage&clickSource=story-heading&module=opinion-c-col-left-region®ion=opinion-c-col-left-region&WT.nav=opinion-c-col-left-region&_r=1">piece in the <em>New York Times</em></a>, via her husband Larry Hughes.</p>
<p>Josefina (Lolina) Alvarez was born in Spain. She earned a doctorate in mathematics from the University of Buenos Aires in Argentina, and is currently professor emeritus of mathematics at New Mexico State University, in the United States. Her long time interest is to communicate mathematics to general audiences. Lolina lives in Santa Fe, New Mexico with husband Larry and dog Lily. She walks every week many kilometres along the beautiful trails of Northern New Mexico.
For more on her work, visit <a href="http://www.math.nmsu.edu/~jalvarez/vita.pdf">this website</a>.</p></div></div></div>Tue, 30 May 2017 07:50:17 +0000Marianne6838 at https://plus.maths.org/contenthttps://plus.maths.org/content/beauty-maths-brain-beholder#commentsMaths in a minute: Cellular automata
https://plus.maths.org/content/maths-minute-cellular-automata
<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/CA_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>
The name "cellular automaton" may sound a bit frightening, but the concept is actually quite simple. Think of a grid on the plane, for example a square grid or a honeycomb, in which each individual cell (each little square or hexagon) has one of two colours, say black or white. At each time step (say every second or every minute) the cells change colour in a way that depends on what colour the neighbouring cells are.</p>
<p>For instance, in the honeycomb example above, which shows only three states separated by two time steps, a cell changes colour if at least four of its neighbours are the opposite colour. You could carry on evolving the grid indefinitely, changing the colours according to the rules at each time step.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/lizards/honeycomb.jpg" alt="Cellular automaton" width="600" height="321" />
<p style="width: 600px;"></p>
</div>
<!-- image made by MF -->
<p>More generally, cellular automata can be defined in any dimension (you could have just a one-dimensional row of cells, or a three-dimensional grid of cells, for example), they can involve more than one colour, and they can also involve an element of chance.</p>
<p>Cellular automata are capable of amazingly complex behaviour. Even simple rules can give us patterns that evolve chaotically, leaving us no hope of ever predicting them accurately. But they can also produce stable patterns that change little over time, or patterns that look like they couldn't possibly be the result of mindless interactions between neighbours, but involve some grand overall design (technically, cellular automata can exhibit <em>self-organisation</em> and <em><a href="/content/outer-space-emergence">emergence</a></em>).</p>
<p>The video below shows the evolution of a cellular automaton in which the individual cells are so small, you can hardly see them (they're like the pixels on a computer screen). You can see how spiral patterns emerge over time. These actually resemble spiral patterns found in nature. Find out more in the article <em><a href="/content/spontaneous-spirals">Spontaneous spirals</a></em> by Wim Hordijk, who also made the movie.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/-WGB2RaGVfA" frameborder="0" allowfullscreen></iframe>
<p>Cellular automata are used to simulate processes in nature (for example the pattern formation on animal skins). Theoretical computer scientists like them because they can represent a kind of universal computing machines. And some even wonder whether the whole Universe is a cellular automaton. You can find out more in these Plus articles:</p><ul>
<li><a href="https://plus.maths.org/content/games-life-and-game-life"><em>Games, life and the game of life</em></a></li><li>
<a href="/content/matrix-simulating-world-part-ii-cellular-automata"><em>Matrix: Simulating the world</em></a></li><li><em>
<a href="/content/spontaneous-spirals">Spontaneous spirals</a></em></li><li>
<a href="/content/spotting-lizards"><em>Spotting lizards</em>.</a></li></ul></div></div></div>Tue, 16 May 2017 10:19:29 +0000Marianne6836 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-cellular-automata#commentsIntroducing Florence Nightingale
https://plus.maths.org/content/introducing-florence-nightingale
<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.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width: 200px;"><img src="/sites/plus.maths.org/files/articles/2010/nightingale/Nightingale_young.jpg" alt="Florence Nightingale" width="200px" height="259" /><p>A young Florence Nightingale</p></div>
<!-- Image from Wikipedia, which says that it is in the public domain -->
<p>Florence Nightingale is most famous for her role as a nurse. During the Crimean war, which raged in Turkey between 1853 and 1856, she worked at the military hospitals where British troops were treated. She became known as the "lady with the lamp", who made her rounds at night to look after injured soldiers. After the war, Florence Nightingale pushed through a reform of military hospitals, which until then had been dirty and disorganised. It was a major achievement: women in Victorian Britain were not expected to do this sort of work, and Nightingale had to fight hard to be taken seriously by the authorities.</p>
<p>Although Florence Nightingale is famous as a nurse, the main tool she used in her campaign to reform hospitals was statistics. Nightingale had been shocked by the conditions she'd found in the military hospitals in Turkey: there were no blankets, beds, furniture, food, or cooking utensils, and there were rats and fleas everywhere. Nightingale was unhappy about the appalling lack of cleanliness and hygiene, but also about the fact that no one had properly organised the medical records. Even the number of deaths was not accurate; hundreds of men had been buried, but their deaths were not recorded.</p>
<div class="leftshoutout">Read a <a href="http://plus.maths.org/content/florence-nightingale-compassionate-statistician">longer version of this article</a> by Eileen Magnello!</div>
<p>Nightingale carefully recorded statistics such as numbers and causes of deaths, and found that the unsanitary conditions, which could lead to diseases such as cholera and typhoid, killed more soldiers than actual war-wounds. Her conclusion was shocking: "our soldiers are enlisted to die in barracks", she wrote.</p>
<p>After the war Nightingale set about persuading people that a hospital reform was necessary, using her statistics and the help of the statistician William Farr. It can be hard to get people to look at and understand long lists of numbers — and this is where one of Nightingale's brightest ideas came in. Like other statisticians at the time, Nightingale realised that the best way to get across statistical information is to use pictures. She invented what are called <em>polar area graphs</em> — you can see an example below.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/Nightingale/polar_area.jpg" alt="Florence Nightingale" width="640px" height="402" /><p style="max-width: 640px;">Example of a polar area diagram.</p></div>
<!-- Image from Wikipedia, which says that it is in the public domain -->
<p>Such a diagram is cut into twelve equal angles, with each slice representing one month of the year. Each colour represents a cause of death and the area of each coloured wedge, measured from the centre, is in proportion to the number of soldiers that died from that cause — so the larger the wedge, the more soldiers died. The blue outer wedges represent the deaths from contagious diseases, such as cholera and typhus. The central red wedges show the deaths from wounds. The black wedges in between represent deaths from all other causes.</p>
<p>Since the blue wedges are so much larger than the others, one glance at the diagram shows that diseases were the greatest killers. If this rate had continued, and troops had not been replaced frequently, then disease alone would have killed the entire British Army in the Crimea. Nightingale's graph showed just how many soldiers died needlessly during the Crimean War, and was used as a tool to persuade the government and medical profession that deaths could be prevented with better sanitation in military, and also in civilian, hospitals.</p>
<p>Nightingale's knack with statistics not only brought about hospital reforms, but also led to other mathematicians honouring her for her contribution to the subject. In 1858 William Farr nominated her as the first woman to be elected a Fellow of the Statistical Society of London. In the same year she was elected to the Statistical Congress, and she was made an honorary foreign member of the American Statistical Association in 1874. She published a total of 200 books, reports and pamphlets during her lifetime.</p>
<p>And her legacy continues today. Open any newspaper and you will find lots of graphs, pie charts, histograms or other visual methods used to bring across statistical information. Today the medical profession relies on statistics more than ever: without careful records and analyses of experiments it's impossible to tell what makes people sick and how to cure them.</p>
<hr />
<h3>About this article</h3>
<p>This article is based on <a href="http://plus.maths.org/content/florence-nightingale-compassionate-statistician"><em>The compassionate statistician</em></a> by Eileen Magnello. </p>
</div></div></div>Sat, 13 May 2017 11:09:50 +0000Marianne6835 at https://plus.maths.org/contenthttps://plus.maths.org/content/introducing-florence-nightingale#commentsSpotting lizards
https://plus.maths.org/content/spotting-lizards
<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/lizard_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/lizards/lizard_nature.jpg" alt="Lizard" width="350" height="202" />
<p>A baby ocellated lizard (top) and an adult one (bottom). Image reprinted by permission from Macmillan Publishers Ltd: <em>Nature</em> (doi:10.1038/nature22031), copyright (2017)</p>
</div>
<p>The ocellated lizard is a beautiful creature. When it's young it's
brown with white dots, but as it gets older its scales become black
and green. Individual scales continue to switch colour, until the
simple
polka dot pattern turns into an intricate
black-and-green labyrinth. It's an amazing transition —
but how does the lizard do it?</p>
<p>The answer seems to come from some relatively straight-forward
maths. The lizard's scales are roughly hexagonal, so imagine a
honeycomb pattern in which each individual hexagon is either black or
green. Now imagine that periodically each hexagon looks at
its six neighbours and changes its colour depending on what the
colours of these neighbours are. For example, in the pictures below
each hexagon changes its colour if four or more of
its neighbours are not of its own colour. As time goes by, scales
keep switching colours and new patterns evolve. In our example,
which shows three time steps, small
islands of one colour surrounded by the other colour
gradually disappear, until only bands of colours remain. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2017/lizards/honeycomb.jpg" alt="Cellular automaton" width="600" height="321" />
<p style="width: 600px;"></p>
</div>
<!-- image made by MF -->
<p>This kind of set-up — a pattern in which cells update their colour
at each time step depending on the colour of their neighbours at a
previous time step — is called a <em>cellular
automaton</em>. Mathematicians and computer scientists have enjoyed
playing with cellular automata since the 1940s, creating all kinds of weird and wonderful examples. Because
cellular automata are good at creating patterns, they've also
been used to simulate processes that happen in nature (see, for
example, <a href="/content/spontaneous-spirals">this article</a>), but so far nobody has found a biological
system that actually <em>is</em> a cellular automaton.</p>
<p>It seems like the ocellated lizard might provide the first example. In a
recent study a team of mathematicians and geneticists (including the
Fields medallist <a href="/content/new-phase-mathematics">Stanislav Smirnov</a>) watched three
male lizards as they grew from little hatchlings into three to
four-year-old adults. Carefully counting lizard spots, the team
tried to find out whether a scale flipping colour at a particular time
depends on the colours of its direct neighbours at a previous
time step. Their work strongly suggests that it does, which means that
the lizard's scales do indeed behave like a cellular automaton,
albeit with a difference to our example above: scales aren't sure to
change colour as soon as their neighbours display a certain colour
configuration. Instead, they change colour with a certain probability,
which depends on the neighbouring colours: the more neighbours of a
scale have the same colour as the original scale, the more likely the original
scale is to change colour (note that that's the opposite behaviour to our example above). The lizard's scales seem to form what's
called a <em>probabilistic cellular automaton</em>.</p>
<p>The first video below shoes the evolution of the scales on a real lizard's back and the bottom shows the cellular automaton. You can see the similarity. The team didn't just rely on videos though, but used several mathematical checks to make sure their conclusions are justified.</p>
<video width="600" height="450" poster="/content/sites/plus.maths.org/files/news/2017/lizards/Real_still.png" controls>
<source src="/content/sites/plus.maths.org/files/news/2017/lizards/Real.mp4.mp4" type="video/mp4">
<source src="/content/sites/plus.maths.org/files/news/2017/lizards/Real.webmsd.webm" type="video/webm">
Your browser does not support the video tag.
<a href="/content/sites/plus.maths.org/files/news/2017/lizards/Real.mp4.mp4">Direct link</a>
</video>
<video width="600" height="450" poster="/content/sites/plus.maths.org/files/news/2017/lizards/CA_still.png" controls>
<source src="/content/sites/plus.maths.org/files/news/2017/lizards/Ca.mp4.mp4" type="video/mp4">
<source src="/content/sites/plus.maths.org/files/news/2017/lizards/Ca.webmsd.webm" type="video/webm">
Your browser does not support the video tag.
<a href="/content/sites/plus.maths.org/files/news/2017/lizards/Ca.mp4.mp4">Direct link</a>
</video>
<p style="width: 600px; font-size: small; color: purple;">The pattern on an ocellated lizard (top) and the cellular automaton (bottom). Videos used by permission from Macmillan Publishers Ltd: <em>Nature</em> (doi:10.1038/nature22031), copyright (2017)</p>
<p>But how do the scales do that? After all, individual skin cells
don't know what scale they're part of, they don't have binoculars to
see what
colour a neighbouring scale is, or a mind to decide whether to switch
colour or not. This is where it gets really interesting. In the 1950s
the famous mathematician and code breaker <a href="/content/alan-turing-ahead-his-time">Alan Turing</a> came up with a
model for how animal patterning might arise that doesn't involve skin
cells having to "know" anything.
The general idea is that an
animal's skin contains two chemicals, which diffuse through the skin, just like milk poured into
coffee will diffuse, and also interact with each other. Using a set of equations Turing described how the concentrations of the two chemicals very over the animal's skin and over time. He found that, if the
chemicals are also linked to the production of pigments in the
skin, their behaviour can give rise to the spots and stripes we commonly see on
animals (see <a href="/content/how-leopard-got-its-spots">here</a> for
a detailed explanation of Turing's model).</p>
<p>A crucial feature of
Turing's model, and other models inspired
by his work, is that it's continuous: the concentrations of the
chemicals change gradually, rather than suddenly, over time and the skin of the animal. At first sight, this couldn't be more
different from the lizard-inspired cellular automaton, in which individual
scales flip colour instantaneously and have sharply
defined edges. A cellular automaton is a <em>discrete</em> object. Is this at odds with
Turing's well-known theory?</p>
<div class="leftimage" style="width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2017/lizards/lizard.jpg" alt="Lizard" width="350" height="232" />
<p>An ocellated lizard. </p>
</div>
<!-- Image from fotolia -->
<p>The answer is no. The new study shows that if you take
into account the varying thickness of the individual scales, that is, you
treat the skin as a 3D object rather than a 2D surface, then the
continuous equations start to "notice" the boundaries between
different scales and behave in a particular way. Mathematical analyses show that a single skin
scale can rapidly take on a uniform colour, which can
quickly turn into one of two extremes (green or black). These
extremes
depend on the states of neighbour scales. Thus, the continuous model
naturally gives rise
to a discrete cellular automaton.</p>
<p>This is also what makes the lizard cellular automaton
unique. Usually,
cellular automata are used to simulate pattern formation. The space on which the patterns form (say an animal skin) is imagined to be made up of many tiny pixels (just like a computer screen), and the cellular automation uses these pixels as its cells. Because these cells are very small compared to the features of the patterns that emerge, the patterns look nice and continuous, rather than pixelated. In this case, a continuous reality is simulated by a discrete
system. </p>
<p>In the ocellated lizard's case, however, things are fundamentally different. The cellular automaton arises from a continuous system,
described by Turing's equations. It's not a simulation tool and its
cells aren't microscopic, but
clearly visible and of a size comparable to the features of the
pattern. This, so the team behind the study argue,
provides evidence that cellular automata aren't just computational tools, but can exist in living things.</p>
<p>Their research has been published in the journal <em><a href="http://www.nature.com/nature/journal/v544/n7649/full/nature22031.html">Nature</a></em>.</p></div></div></div>Thu, 11 May 2017 11:45:40 +0000Marianne6834 at https://plus.maths.org/contenthttps://plus.maths.org/content/spotting-lizards#comments