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https://plus.maths.org/content
enGiving numbers meaning
https://plus.maths.org/content/david-spiegelhalter-wins-faraday-prize
<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/dacids_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-hidden"><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>David Spiegelhalter, also known as Professor Risk and a good friend of <em>Plus</em>, has won this year's <a href="https://royalsociety.org/grants-schemes-awards/awards/michael-faraday-prize/">Michael Faraday Prize and Lecture </a>for communicating key concepts from statistics and probability theory to the public. The Prize and Lecture are awarded annually by the Royal Society to scientists or engineers whose expertise in communicating scientific ideas in lay terms is exemplary. And in Spiegelhalter's case, they have been awarded in particular for work relating to the COVID-19 pandemic. </p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2020/spiegelhalter/dacids.jpg" alt="David Spiegelhalter" width="350px" height="300" /><p>David Spiegelhalter. Photo: Ilan Goodman</p></div>
<!-- David sent the pic saying to credit Iian. -->
<p>"It's a great honour, it really is," says Spiegelhalter. "If you look down the list of people who have won the Prize in the past, it includes all the people in science communication I admire greatly. And being in the same list as David Attenborough is something we can all just dream of."</p>
<h3>Giving numbers meaning</h3>
<p>Spiegelhalter's job during the pandemic has been about helping people interpret the statistics and urging caution. "[My media work] has been explaining basic stuff about what the numbers actually mean — for example what is a COVID death? — and the caution we should have about making naive interpretations of the numbers that we hear."</p>
<p>With a pandemic the statistics have meaning for each of us individually, but they also have a global character. Spiegelhalter's work has spanned that spectrum, explaining the risk that COVID-19 poses to individuals, especially how quickly it grows with age, but also looking at those international comparisons that held us in thrall at the beginning of the pandemic. "[There is a] real difficulty in evaluating the effect of interventions, such as face masks, and therefore in making comparisons between countries, and in particular saying why there have been differences between countries," he explains. "In a way, a lot of what I have done has been critiquing claims that others have made."</p>
<div class="leftshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</div><p>Crucially, Spiegelhalter's critiques don't imply opinions on policy. "I don't comment on what the correct policy is," he says. "Policies need to be political decisions that are politically accountable. It's not a scientist's, and least of all a statistician's job to recommend what the policy should be. I think that many other scientists should stick more to the science and less to the advocacy."</p>
<p>Spiegelhalter's work has been about putting simple mathematical ideas in context, rather than explaining sophisticated concepts and methods. "Proportions are the main mathematical insight [one needs to understand]," he explains. "But I have also [explained] trends, exponential growth, the idea of doubling times, rates in the population, and so on."</p>
<h3>What we still need to learn</h3>
<p>By now most of us probably have some grip on the central ideas of the statistics of disease. The famous <a href="/content/maths-minute-r0-and-herd-immunity">reproduction number <em>R</em></a>, and things like the <a href="/content/epidemic-growth-rate">growth rate of a disease</a>, are now household concepts. But Spiegelhalter says that there are also still things that aren't well understood by the public.</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2020/Gog/sirsys-p9_0.png" alt="Output from a SIR model" width="350" height="263" /><p>This is the typical output from a simple epidemiological model (find out more <a href="/content/how-can-maths-fight-pandemic">here</a>). The number of susceptible people is shown in blue, the number of infected in green, and the number of recovered in red. Such models are always based on specific assumptions, which need to be communicated when model predictions are presented.</p></div><!-- Image in public domain -->
<p>"One problem are model-based projections," he says. "No matter how many times somebody says that these represent only possible scenarios, afterwards critics will say, 'this didn't happen, and therefore everything was wrong.' But scenarios are just possible futures under various assumptions, and those assumptions may very well soon prove to be wrong." This doesn't mean that modelling is pointless. Good mathematical models offer the best way of exploring what might happen — but one has to keep in mind that they don't deliver sure-fire predictions. (You can find out more about epidemiological modelling <a href="/content/how-can-maths-fight-pandemic">here</a>.)</p>
<p>Another area Spiegelhalter thinks is still poorly understood are <a href="/content/maths-minute-false-positives">false positives and false negatives</a> of diagnostic tests. "This [area is] is very counter-intuitive - the idea that a test may appear to be rather accurate, in terms of its sensitivity and specificity, and yet can make a lot of false claims. There should be a lot more education on that."</p>
<h3>Getting the message right...or wrong</h3>
<p>Some of the lack of understanding, Spiegelhalter believes, is down to bad communication, something of he admits he's been guilty of himself. "I've made various mistakes and been misunderstood," he says. "For example, about six months ago I said that your risk of dying from COVID-19 is the same as your risk of dying this year." Many people thought this meant that COVID-19 makes no difference to your baseline risk of dying this year, when really what it means is that your baseline risk is doubled. "It hadn't crossed my mind that anyone would interpret it like that. That just shows I didn't test my message well enough, so it was my fault."</p>
<p>Spiegelhalter has also been roundly critical of the government briefings. "At the briefings the communication is clearly not in the hands of the scientists," he says. "These are political communications to justify the actions that are being proposed. For example, at the briefing announcing the second lockdown there were some awful graphs, which were rushed through and which were making things look rather frightening, but which were out of date even by the time they were shown. These only represented possible worst-case scenarios anyway."</p>
<p>As he stresses again, Spiegelhalter's criticism here doesn't mean he's against a second lockdown, but simply that information needs to be communicated carefully. Especially when model predictions are concerned, he believes that both government and some media coverage have neglected the first rule of communication: that no prediction should be presented without stating clearly the underlying assumptions, and that you shouldn't pick a single scenario out of various possible alternatives in order to make a point.</p>
<h3>Data, interpretation, and misinformation</h3>
<p>Government communications are one thing, but what about the actual data put out by the Office of National Statistics and Public Health England? Spiegelhalter believes that trust in these figures is still high and that the two authorities have done a great job at making them so easily available. It has allowed thousands of people to become "armchair statisticians", engaging with real-life statistics from scratch for the first time. And the importance of the data makes the case for better education.</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2020/UnlockingHE/covid19_test.jpg" alt="Covid-19 test" width="350" height="233" /><p>Greater data literacy would help people navigate the data jungle, for example gaining a better understanding of false negatives and positives of tests.</p>
</div><!-- Image from Adobe Stock, originally used in Going back to uni during a pandemic -->
<p>"There's a huge need for what you might call data literacy among the whole population, and that means schools, politicians, the media, and the general public," Spiegelhalter says. "That's something I feel deeply committed to, and I hope that it'll be a long-term impact of this crisis."</p>
<p>Journalists in particular face a difficult task when it comes to reporting statistics, and not just about COVID-19. All statistics regarding our health are important to report, and make great headlines, but research papers can be hard to read, and can even contain misleading claims.</p>
<p>Helping journalists is one of Spiegelhalter's aims. He works with the <a href="https://www.sciencemediacentre.org/" target="blank">Science Media Centre</a>, and has also just released an exciting new app designed with his colleagues at the Winton Centre. <a href="https://realrisk.wintoncentre.uk/" target="blank">RealRisk</a> allows anyone needing to communicate research findings to put in measures of <em>relative risk</em> they have seen in a research paper (such as "your risk of cancer is doubled when eating this particular food") and get clear representations of the corresponding <em>absolute risk</em> ("when eating this food your chance of getting cancer is increased from x% to y%"), which is what you really need to know to properly understand the risk. (To understand why, note that the scary-sounding doubling of a risk can correspond to a tiny increase if your baseline risk is small to start with — two times 0.00001 is still only 0.00002. You can find out more in <a href="/content/understanding-uncertainty-2845-ways-spinning-risk-0">this article</a>.)</p>
<p>"I'm really delighted with the app," says Spiegelhalter. "We are currently making short videos explaining some related issues, such as correlation not implying causation, and these will be out soon."</p>
<p>Coming back to COVID-19, one thing that's clear is that many challenges still await. One example is some people's fear of new vaccines, often fired up by misinformation. Spiegelhalter has no time for conspiracy theorists (people who cherry pick their evidence aren't worth engaging with, he says) but he is aware that there's a body of people open to influence from many directions. Here he believes in what the psychologist <a href="https://www.psychol.cam.ac.uk/people/sander-van-der-linden" target="blank">Sander van der Linden</a> has called <em>pre-bunking</em>. "You have to provide an alternative to [false] narratives that is interesting and engaging, but also balanced and reasonable," he says. "[You need to] pre-empt misunderstandings, and that means getting in there hard, and getting in there now, with a counter-campaign. We need forceful, energetic, and pro-active countering of misinformation."</p>
<p>With hopes for a vaccine on the rise, there's a good chance you'll soon hear Spiegelhalter in the media, answering related questions in his calm and reassuring way. Or if you don't want to wait until then, you can watch his Faraday Lecture on the Royal Society website.</p>
</div></div></div>Tue, 24 Nov 2020 14:43:27 +0000Marianne7364 at https://plus.maths.org/contenthttps://plus.maths.org/content/david-spiegelhalter-wins-faraday-prize#commentsMaths in a minute: Triangular numbers
https://plus.maths.org/content/maths-minute-triangular-numbers
<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/triangle_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Zoheir Barka</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 triangular number is a number that can be represented by a pattern of dots arranged in an equilateral triangle with the same number of dots on each side. </p>
<p>For example:</p>
<div class="centreimage"><img alt="Triangular numbers" src="/content/sites/plus.maths.org/files/articles/2020/triangular/triangular.png" style="max-width: 401px; height: 199px;" />
<p style="max-width: 401px;"></p>
</div><!-- image made by MF -->
<p>The first triangular number is 1, the second is 3, the third is 6, the fourth 10, the fifth 15, and so on. </p>
<p>You can see that each triangle comes from the one before by adding a row of dots on the bottom which has one more dot than the previous bottom row. This means that the <img src="/MI/4941c19aa57ec57a7960caef4dd5929f/images/img-0001.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> triangular number <img src="/MI/4941c19aa57ec57a7960caef4dd5929f/images/img-0002.png" alt="$T_ n$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> is equal to </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/4941c19aa57ec57a7960caef4dd5929f/images/img-0003.png" alt="\[ T_ n=1+2+ 3+...+n. \]" style="width:175px;
height:14px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>There’s also another way we can calculate the <img src="/MI/4941c19aa57ec57a7960caef4dd5929f/images/img-0001.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> triangular number. Take two copies of the dot pattern representing the <img src="/MI/4941c19aa57ec57a7960caef4dd5929f/images/img-0001.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> triangular number and arrange them so that they form a rectangular dot pattern.</p>
<div class="centreimage"><img alt="Triangular numbers" src="/content/sites/plus.maths.org/files/articles/2020/triangular/rectangl2.png" style="max-width: 250px; height: 201px;" />
<p style="max-width: 250px;"></p>
</div><!-- image made by MF -->
<p> This rectangular pattern will have <img src="/MI/6963e1a4d2570369a0f95928c5a214b7/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> dots on the shorter side and <img src="/MI/6963e1a4d2570369a0f95928c5a214b7/images/img-0002.png" alt="$n+1$" style="vertical-align:-2px;
width:37px;
height:14px" class="math gen" /> dots on the longer side, which means that the rectangular pattern contains <img src="/MI/6963e1a4d2570369a0f95928c5a214b7/images/img-0003.png" alt="$n(n+1)$" style="vertical-align:-4px;
width:59px;
height:18px" class="math gen" /> dots in total. And since the original triangular dot pattern constitutes exactly half of the rectangular pattern, we know that the <img src="/MI/6963e1a4d2570369a0f95928c5a214b7/images/img-0004.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> triangular number <img src="/MI/6963e1a4d2570369a0f95928c5a214b7/images/img-0005.png" alt="$T_ n$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/6963e1a4d2570369a0f95928c5a214b7/images/img-0006.png" alt="\[ T_ n=\frac{n(n+1)}{2}. \]" style="width:110px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>Note that with this consideration we have proved the formula for the summation of <img src="/MI/defa98f9a5235e8584eaec890e69002e/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> natural numbers, namely </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/defa98f9a5235e8584eaec890e69002e/images/img-0002.png" alt="\[ 1+2+ 3+...+n=\frac{n(n+1)}{2}. \]" style="width:220px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>Triangular numbers have lots of interesting properties. For example, the sum of consecutive triangular numbers is a <em>square number</em>. You can see this by arranging the triangular dot patterns representing the <img src="/MI/115ab01c4e9b97ef16dfb87fb76ef5aa/images/img-0001.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> and <img src="/MI/115ab01c4e9b97ef16dfb87fb76ef5aa/images/img-0002.png" alt="$(n+1)st$" style="vertical-align:-4px;
width:63px;
height:18px" class="math gen" /> triangular numbers to form a square which has <img src="/MI/115ab01c4e9b97ef16dfb87fb76ef5aa/images/img-0003.png" alt="$n+1$" style="vertical-align:-2px;
width:37px;
height:14px" class="math gen" /> dots to a side:</p>
<div class="centreimage"><img alt="Triangular numbers" src="/content/sites/plus.maths.org/files/articles/2020/triangular/square.png" style="max-width: 200px; height: 198px;" />
<p style="max-width: 200px;"></p>
</div><!-- image made by MF -->
<p>Alternatively, you can see this using the formulas for the consecutive triangular numbers <img src="/MI/3b78e41e0551dee8fb0e520d27c2adfc/images/img-0001.png" alt="$T_ n$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> and <img src="/MI/3b78e41e0551dee8fb0e520d27c2adfc/images/img-0002.png" alt="$T_{n+1}$" style="vertical-align:-4px;
width:34px;
height:15px" class="math gen" />: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/3b78e41e0551dee8fb0e520d27c2adfc/images/img-0003.png" alt="\[ T_ n+T_{n+1} =\frac{n(n+1)}{2}+\frac{(n+1)(n+2)}{2}=\frac{(n+1)(2n+2)}{2}=(n+1)^2. \]" style="width:512px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>What is more, alternating triangular numbers (1, 6, 15, ...) are also <a href="https://en.wikipedia.org/wiki/Hexagonal_number">hexagonal numbers</a> (numbers formed from a hexagonal dot pattern) and
every even <a href="/content/bizarre-definition-perfect">perfect number</a> is a triangular number. </p>
<p>Triangular numbers also come up in real life. For example, a network of <img src="/MI/344280a8a2e85550df4cad751f3a1253/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> computers in which every computer is connected to every other computer requires <img src="/MI/344280a8a2e85550df4cad751f3a1253/images/img-0002.png" alt="$T_{n-1}$" style="vertical-align:-2px;
width:34px;
height:13px" class="math gen" /> connections. And if in sports you are playing a <em>round robin tournament</em>, in which each team plays each other team exactly once, then the number of matches you need for <img src="/MI/1fe9c3ef6de6df40295670ab6a5f5973/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> teams is <img src="/MI/1fe9c3ef6de6df40295670ab6a5f5973/images/img-0002.png" alt="$T_{n-1}.$" style="vertical-align:-2px;
width:39px;
height:13px" class="math gen" /> These two results are equivalent to the handshake problem we have <a href="/content/maths-minute-shake-solve">explored </a> on <em>Plus</em> before. </p>
<p><em>We would like to thank Zoheir Barka who sent us the first draft of this article. We will publish a lovely article by Barka about triangular numbers soon. In the mean time, you can read Barka's article about beautiful patterns in multiplication tables <a href="/content/hidden-beauty-multiplication-tables">here</a>.</em></p></div></div></div>Tue, 24 Nov 2020 11:03:17 +0000Marianne7367 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-triangular-numbers#commentsPhysics in a minute: The double slit experiment
https://plus.maths.org/content/physics-minute-double-slit-experiment-0
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/interference_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>One of the most famous experiments in physics is the double
slit experiment. It demonstrates, with unparalleled strangeness, that
little particles of matter have something of a wave about them, and suggests that the very act of
observing a particle has a dramatic effect on its behaviour.</p>
<p>To start off, imagine a wall with two
slits in it. Imagine throwing tennis balls at the wall. Some will
bounce off the wall, but some will travel through the slits. If there's
another wall behind the first, the tennis balls that have travelled
through the slits will hit it. If you mark all the spots where a ball
has hit the second wall, what do you expect to see? That's right. Two
strips of marks roughly the same shape as the slits.</p>
<p>In the image below, the first wall is shown from the top, and the
second wall is shown from the front.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/double_balls.png" alt="Double slit" width="350" height="303" />
<p style="max-width: 350px;">The pattern you get from particles.</p>
</div>
<!-- Image made by MF -->
<p>Now imagine shining a light (of a single colour, that is, of a
single wavelength) at a wall with two slits (where the distance between the slits is roughly the same as the light's wavelength). In the image below, we show the
light wave and the wall from the top. The blue lines
represent the peaks of the wave. As the wave passes though both
slits, it essentially splits into two new waves, each spreading out from one of the slits. These two waves then interfere with each other. At some points, where a peak meets a trough, they will cancel each other out. And at
others, where peak meets peak (that's where the blue curves cross in the diagram), they will reinforce each other. Places where the waves reinforce each other give the brightest
light. When the light meets a second wall placed behind the first, you will
see a stripy pattern, called an <em>interference pattern</em>. The bright
stripes come from the waves reinforcing each other.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/double_waves.png" alt="Double slit" width="350" height="323" />
<p style="max-width: 350px;">An interference pattern.</p>
</div>
<!-- Image made by MF -->
<p>Here is a picture of a real interference pattern. There are more
stripes because the picture captures more detail than our diagram. (For the sake of correctness, we should say that the image also shows a <em>diffraction pattern</em>, which you would get from a single slit, but we won't go into this here, and you don't need to think about it.)</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/interference.jpg" alt="Double slit" width="558" height="240" />
<p style="max-width: 558px;">Image: <a href="https://commons.wikimedia.org/wiki/File:Single_slit_and_double_slit2.jpg">Jordgette</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>.</p>
</div>
<p>Now let's go into the quantum realm. Imagine firing electrons at
our wall with the two slits, but block one of those slits off for the moment. You'll find that some of the electrons will pass through the open slit and strike the second wall just as tennis balls would: the spots they arrive at form a strip roughly the same shape as the slit.</p>
<p>Now open the second slit. You'd expect two rectangular strips on the second wall, as with the tennis balls, but what you actually see is very different: the spots where electrons hit build up to replicate the
interference pattern from a wave.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/electrons.png" alt="Double slit" width="350" height="298" />
<p style="max-width: 350px;"></p>
</div>
<!-- Image made by MF -->
<p>Here is an image of a real double slit experiment with electrons. The individual pictures show the pattern you get on the second wall as more and more electrons are fired. The result is a stripy interference pattern.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/contextuality/electrons2.jpg" alt="Double slit" width="350" height="205" />
<p style="max-width: 350px;">Image: <a href="https://commons.wikimedia.org/wiki/File:Double-slit_experiment_results_Tanamura_four.jpg">Dr. Tonomura and Belsazar</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a></p>
</div>
<p> How can this be?</p>
<p>One possibility might be that the electrons somehow interfere with each other, so they don't arrive in the same places they would if they were alone. However, the interference pattern
remains even when you fire the electrons one by one, so that they have
no chance of interfering. Strangely, each individual electron contributes one dot to an overall pattern that looks like the interference pattern of a wave. </p>
<p> Could it
be that each electrons somehow splits, passes through both slits at once,
interferes with itself, and then recombines to meet the second screen as a single, localised particle?</p>
<p>To find out, you might place a detector by the slits, to see which
slit an electron passes through. And that's the really weird bit. If
you do that, then the pattern on the detector screen turns into the
particle pattern of two strips, as seen in the first picture above! The interference pattern disappears. Somehow, the very act of
looking makes sure that the electrons travel like well-behaved
little tennis balls. It's as if they knew they were being spied on and decided not to be caught in the act of performing weird quantum shenanigans.</p>
<p>What does the experiment tell us? It suggests that what we call "particles", such as electrons, somehow combine characteristics of particles and characteristics of waves. That's the famous <em>wave particle duality</em> of quantum mechanics. It also suggests that the act of observing, of measuring, a quantum system has a profound effect on the system. The question of exactly how that happens constitutes the <em>measurement problem</em> of quantum mechanics.</p>
<hr><h3>Further reading</h3>
<ul><li>For an extremely gentle introduction to some of the strange aspects of quantum mechanics, read <a href="/content/watch-and-learn"><em>Watch and learn</em></a>. </li>
<li>For a gentle introduction to quantum mechanics, read <a href="/content/ridiculously-brief-introduction-quantum-mechanics"><em>A ridiculously short introduction to some very basic quantum mechanics</em></a>.</li>
<li>For a more detailed, but still reasonably gentle, introduction to quantum mechanics, read <a href="/content/schrodinger-1"><em>Schrödinger's equation — what is it?</em></li>
<p><em>Originally published on 05/02/2017.</em></p></div></div></div>Thu, 19 Nov 2020 12:37:46 +0000Marianne6774 at https://plus.maths.org/contenthttps://plus.maths.org/content/physics-minute-double-slit-experiment-0#commentsUniverse Unravelled
https://plus.maths.org/content/universe-unravelled
<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_18.png" 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: 400px;"><img src="/content/sites/plus.maths.org/files/news/2020/disco/cp_uu2.jpg" alt="Universe Unravelled" /><p></p>
</div>
<p>If you noticed that things have been a little quiet on <em>Plus</em> for the last 18 months or so, we can finally reveal why: we have been working with the Discovery Channel and the <a href="http://www.ctc.cam.ac.uk/">Stephen Hawking Centre for Theoretical Cosmology</a> on a TV documentary exploring the history and mysteries of our Universe. And now we're proud to announce its launch.</p>
<p>The <em>Universe Unravelled</em> series premieres on <a href="https://www.discoveryplus.co.uk/show/universe-unravelled-with-the-stephen-hawking-centre?idp=Sonic&responseCode=212">Discovery+</a> in November 2020, coinciding with the UK launch of this new digital platform. It's aimed at anyone who is curious about the Universe we live in, with no previous knowledge of cosmology required. In over 20 short episodes the series explores what we already know about the Universe, what cosmologists are working on right now, and what they hope to find out in the future.</p>
<p>The series comes at an exciting time in the development of cosmology and relativity. Over the last few decades, the field evolved from a niche subject on the fringes of science to a fully-fledged precision science. The discovery of the <a href="/content/cosmic-afterglow">cosmic microwave background</a> — left-over radiation from the Big Bang — in the 1960s, and the first observation of <a href="/content/listening-ripples-spacetime">gravitational waves</a> in 2016 were milestones in this context. Both provide powerful observational tools with which to probe our Universe and test our theories.
</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2020/disco/potter_room.jpg" alt="Filming at the Centre for Mathematical Sciences in Cambridge" width="350" height="248" /><p>Some of the CTC team being filmed at the Centre for Mathematical Sciences in Cambridge. Photo: Rachel Thomas.</p>
</div>
<p>Stephen Hawking's career spanned much of this golden age of cosmology. In 2007 he founded the <a href="http://ctc.cam.ac.uk">Stephen Hawking Centre for Theoretical Cosmology</a> (CTC) within the University of Cambridge, which now enables researchers of all career stages to carry on his legacy. They develop and test mathematical theories that describe the history of our Universe, and investigate the two events that shake its basic fabric most violently, revealing its secrets in the process: the Hot Big Bang and the collisions and mergers of black holes. Underlying this work is a theory which recently celebrated its 100th birthday: Einstein's <a href="/content/maths-minute-einsteins-general-theory-relativity">general theory of relativity</a>.</p>
<p><em>Universe Unravelled</em> explores cutting-edge topics in cosmology and extreme gravity in a way that's accessible to everyone. It describes how massive objects warp the fabric of spacetime and how they can collapse under their own gravity to form <a href="/content/maths-minute-black-holes">black holes</a>. It explores how these black holes can send gravitational waves rippling across spacetime, and what happens if you were to fall into a black hole. It also explores the violent explosion that marked the beginning of our Universe, and how the Universe expanded from this initial Big Bang, forming all the structures we observe today – galaxies, stars and planets. It also probes the mysteries that still puzzle cosmologists, such as dark energy and dark matter. And it features some stunning graphics, some produced in collaboration with Intel's Advanced Visualization team.</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2020/disco/amelia.jpg" alt="Amelia Drew" width="350" height="254" /><p>CTC researcher Amelia Drew explaining her work for the camera. Photo: Rachel Thomas.</p>
</div>
<p>The series features 17 CTC researchers explaining these mind-blowing concepts, together with members of the <a href="https://www.kicc.cam.ac.uk/">Kavli Institute of Cosmology</a>, Cambridge. It offers a glimpse of what it's like to work at the cutting edge of cosmology: confronting sophisticated mathematics with observational data, employing some of the world's fastest supercomputers, and even daring to challenge Einstein's highly successful theory in an attempt to explain what has so far defied explanation. Viewers not only learn about the deepest secrets of our Universe, but also find out about the everyday life of students and staff at a world-leading research centre.</p>
<p>The series was filmed on-site at the Centre for Mathematical Sciences (home of <em>Plus</em>) and the Institute of Astronomy. In the role of Science Editors the <em>Plus</em> editorial team explained the science to the production team from <a href="https://www.navadastudios.com">Navada Studios</a>, contributed to the script, interviewed the contributors, and managed the project on the Cambridge side. We enjoyed working closely with the CTC team and with Matthew Scott and Beatriz Clemente from Navada Studios, who led the production of the series. Funding for the project was provided by the <a href="https://www.kavlifoundation.org/">Kavli Foundation,</a> an organisation dedicated to advancing science for the benefit of humanity, promoting public understanding of scientific research, and supporting scientists and their work.</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2020/disco/astro.jpg" alt="Filming at the Institute of Astronomy" width="350" height="263" /><p>Blake Sherwin during filming at the Institute of Astronomy. Photo: Rachel Thomas.</p>
</div>
<p>"We are grateful to Discovery and the Kavli Foundation for supporting this unique opportunity to continue Stephen Hawking's vision of reaching out, especially to younger audiences, to inspire curiosity about our Universe and the huge progress currently being made to unveil its secrets," says Paul Shellard, CTC Director. "This was a remarkable collaboration in which we were able to work closely back and forth with the production team, ensuring both viewer interest and scientific accuracy, which we hope provides a great model for future science outreach."</p>
<p>We're really proud of the series and hope that everyone who watches will share in the fascination of our Universe and the amazing work that's being done by researchers at the CTC and around the world. The series will be available from November 2020 on the new Discovery+ service, which can be found <a href="https://www.discoveryplus.co.uk/show/universe-unravelled-with-the-stephen-hawking-centre?idp=Sonic&responseCode=212">here</a>.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/djktnwAPWuo" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</div></div></div>Fri, 13 Nov 2020 12:37:01 +0000Marianne7363 at https://plus.maths.org/contenthttps://plus.maths.org/content/universe-unravelled#commentsThe fingernail problem and metallic numbers
https://plus.maths.org/content/fingernail-problem-and-metallic-numbers
<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/fingernail_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Gokul Rajiv and Yong Zheng Yew </div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In this article we will look at a beautiful geometric problem and its link to a very special family of numbers: <em>metallic numbers</em>, which arise as generalisations of the famous <em>golden ratio</em>. </p>
<p>We looked at metallic numbers in a previous <a href="/content/silver-ratio"><em>Plus</em> article</a>, but here's a quick recap. The golden ratio arises from the following geometric problem. Given a piece of line, divide it into two segments so that the ratio between the entire line and the longer of the two segments equals the ratio between the longer of the two segments and the shorter one. </p>
<div class="centreimage"><img alt="golden ratio" src="/content/sites/plus.maths.org/files/articles/2020/metallic/golden_ratio.jpg" style="max-width: 400px; height: 45px;" />
<p style="max-width: 400px;">We'd like (<em>A</em>+<em>B</em>)/<em>A</em> = <em>A</em>/<em>B</em>. </p>
</div>
<!-- Image provided by authors -->
<p>If <img src="/MI/ca4063d113586b0c7495869c965f8dfc/images/img-0001.png" alt="$A$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" /> is the longer segment and <img src="/MI/ca4063d113586b0c7495869c965f8dfc/images/img-0002.png" alt="$B$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> the shorter segment then we would like <img src="/MI/ca4063d113586b0c7495869c965f8dfc/images/img-0003.png" alt="$(A+B)/A=A/B$" style="vertical-align:-4px;
width:133px;
height:18px" class="math gen" />. When you do this then the ratio in question is the golden ratio, </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/ca4063d113586b0c7495869c965f8dfc/images/img-0004.png" alt="\[ \phi =\frac{1+\sqrt {5}}{2}=1.618033... \]" style="width:185px;
height:37px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>Now we could also try and divide the original segment into <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> larger parts of equal length <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0002.png" alt="$A$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" /> and one smaller part of length <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0003.png" alt="$B$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" />, so that the ratio between the entire line and <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0002.png" alt="$A$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" /> is the same as the ratio between <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0002.png" alt="$A$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" /> and <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0003.png" alt="$B$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" />. The ratio in question is a <em>metallic number</em> and denoted by <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0004.png" alt="$\lambda _ n$" style="vertical-align:-2px;
width:17px;
height:13px" class="math gen" />. For all natural numbers <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0005.png" alt="$n\geq 1$" style="vertical-align:-2px;
width:39px;
height:14px" class="math gen" />, the metallic mean <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0004.png" alt="$\lambda _ n$" style="vertical-align:-2px;
width:17px;
height:13px" class="math gen" /> satisfies </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0006.png" alt="\[ \frac{1}{\lambda _ n-n}=\lambda _ n. \]" style="width:96px;
height:37px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Note that the golden ratio corresponds to <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0007.png" alt="$\lambda _1$" style="vertical-align:-2px;
width:15px;
height:13px" class="math gen" />. For <img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0008.png" alt="$n=2$" style="vertical-align:0px;
width:40px;
height:12px" class="math gen" /> we get the so-called <em>silver ratio</em>, </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/3b745a3a441bd3aeaade085c9a939219/images/img-0009.png" alt="\[ \lambda _2=1+\sqrt {2}=2.414213... \]" style="width:187px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<div class="centreimage"><img alt="silver ratio" src="/content/sites/plus.maths.org/files/articles/2020/metallic/silver_ratio.jpg" style="max-width: 400px; height: 45px;" />
<p style="max-width: 400px;">The silver ratio arises from dividing the initial line like this.</p>
</div>
<!-- Image provided by authors -->
<p>Now let's move on to our geometric problem with its intriguing link to metallic numbers. It starts off quite simply, with a question involving the trimming of fingernails.</p>
<h3>The fingernail problem</h3>
<p>Suppose you want to cut your fingernails but misplaced your only pair of nail clippers. You do, however, own a pair of straight scissors. Could you, with a series of straight cuts, form a smooth, round, shorter fingernail? </p>
<p>Let's analyse this problem by first making a few simplifying assumptions. We could, to begin with, approximate the shape of fingernail with a semi-circle. We will use a semi-circle of radius 1 for convenience: </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/fingernail/figure1.png" alt="semi-circle" width="350" height="203" /><p style="max-width: 350px;">A semi-circle of radius 1.</p>
</div><!-- Image made by MF -->
<p>We also need a fairly straightforward procedure with which to cut the fingernail. For this, we begin by first cutting the circle into a triangle with two equal sides, like so:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/fingernail/figure2.png" alt="triangle" width="350" height="218" /><p style="max-width: 350px;">A semi-circle of radius 1.</p>
</div><!-- Image made by MF -->
<p>We call the bottom line in this figure the <em>base edge</em>. Apart from the base edge, every other side (let's call these other sides <em>exposed edges</em>) has equal length. By Pythagoras' theorem the exposed edges have length <img src="/MI/fe34a52857184efc7ae3bdb4f4cca715/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" />. At every next step, a corner of the shape is clipped off so that the shape has one more exposed edge, and the new exposed edges all have the
same length:</p>
<div class="centreimage"><img alt="Fingernail cutting procedure" src="/content/sites/plus.maths.org/files/articles/2020/fingernail/exposed_edges.png" style="max-width: 350px; height: 585px;" />
<p style="max-width: 400px;">The cutting procedure.</p>
</div>
<!-- Image provided by authors -->
<p>Every successive cut produces a part of a regular polygon with more sides than the one before. If we had an infinite amount of time, we could continue the cutting procedure forever, and the number of sides of the polygon would tend to infinity. It turns out (as with similar problems with full circles, as shown below) that this will eventually produce an arc of a circle. </p>
<div class="centreimage"><img alt="Converging to a circle" src="/content/sites/plus.maths.org/files/articles/2020/fingernail/image2.gif" style="max-width: 288px; height: 288px;" />
<p style="max-width: 288px;">Adding sides to a regular polygon produces a circle in the limit.</p>
</div>
<!-- Image provided by authors -->
<h3>Enter the silver ratio</h3>
<p>How much shorter is the newly cut fingernail compared to the original one? To work this out let's first find the defining features of the circle <img src="/MI/8b6b77da4b90d794f5a19ba696de5a1b/images/img-0001.png" alt="$O$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> whose arc describes your newly cut fingernail. First note that the two end points of the base side of our triangle (shown in red in the diagram below) are points on the final circle. That's because in our procedure these points are never cut off. The two exposed sides of the equilateral triangle are tangent to the final circle.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/fingernail/tangents.png" alt="circle construction" width="300" height="364" /><p style="max-width: 350px;">The end points of the base side are points on the circle <em>O</em>, and the two exposed sides of our triangles are tangent to the circle at these points.</p>
</div><!-- Image made by MF -->
<p>A line that is tangent to a circle at a given point is perpendicular to the radius of the circle which ends at that point. Our construction therefore gives us two radii of <img src="/MI/8b6b77da4b90d794f5a19ba696de5a1b/images/img-0001.png" alt="$O$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" />, shown in green in the diagram below.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/fingernail/radii.png" alt="circle construction" width="300" height="364" /><p style="max-width: 350px;">The green lines are radii of the circle <em>O</em>.</p>
</div><!-- Image made by MF -->
<p>A little inspection of the symmetry in the diagram will convince you that the two radii have the same length as the exposed side of the original equilateral triangle, namely <img src="/MI/fbcccadf70dd38f1f0b39d9cb5aadabb/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" />, and that the two radii meet exactly at the "South pole" of the original circle. The meeting point of two radii of the circle <em>O</em> is of course the centre of <em>O</em>. So we now know that <em>O</em> is centred on the South pole of the original circle and has radius <img src="/MI/51e92e4a097e9b805f20aef0207f56d7/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" />.</p>
<p>Now let's use the base line of the original triangle as the line from which to measure height. The height above this base line of the original fingernail is 1 (the radius of the original circle).
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/fingernail/ratio.png" alt="circle construction" width="300" height="418" /><p style="max-width: 300px;">The ratio of the height of the old fingernail and the height of the new fingernail.</p>
</div><!-- Image made by MF -->
<p>It is easy to see that the height above the base line of the new fingernail is <img src="/MI/d7ce4df2fc74fe75b6789cdba1a94bd5/images/img-0001.png" alt="$\sqrt {2}-1$" style="vertical-align:-2px;
width:48px;
height:17px" class="math gen" />.We now have </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/d7ce4df2fc74fe75b6789cdba1a94bd5/images/img-0002.png" alt="\[ \frac{\mbox{Height old nail}}{\mbox{Height new nail}}=\frac{1}{\sqrt {2}-1}=\sqrt {2}+1. \]" style="width:269px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>As we mentioned above, this number is known as the silver ratio <img src="/MI/07efd4f9e31e93f77e2d6dcdd631cd64/images/img-0001.png" alt="$\lambda _2$" style="vertical-align:-2px;
width:15px;
height:13px" class="math gen" />, and the equation above reflects the fact that </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/07efd4f9e31e93f77e2d6dcdd631cd64/images/img-0002.png" alt="\[ \frac{1}{\lambda _2-2}=\frac{1}{\sqrt {2}-1}=\sqrt {2}+1 =\lambda _2. \]" style="width:241px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>But there is still more to discover... </p>
<h3>A family of circles</h3>
<p>In our problem above we had two circles: the second circle is centred on the South pole of the first and has radius <img src="/MI/1c4795bd43cc6010ee08aaa453e51b6c/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> times that of the first. We can now construct a third circle: as its centre we take the South pole of the second circle and for its radius we multiply the radius of the second circle by <img src="/MI/1c4795bd43cc6010ee08aaa453e51b6c/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" />. In the same way we can construct a fourth circle from the third, a fifth from the fourth, and so on, giving us an infinite family of circles. In this family every successive circle has radius <img src="/MI/1c4795bd43cc6010ee08aaa453e51b6c/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> times the ratio of the previous circle and is centred at the South pole of the previous circle. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/fingernail/fig6.png" alt="Family of circle" width="331" height="451" /><p style="max-width: 350px;">A family of circles.</p>
</div><!-- Image from Desmos interactivity -->
An interesting property of this family of circles stems from the intersections of the various circles with one another. We find that we can draw straight lines through the intersections of all the circles in the family. We call these the <em>lines of intersection</em>.
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/fingernail/family_labelled.png" alt="Lines of intersection" width="331" height="447" /><p style="max-width: 350px;">The lines of intersection associated to our family of circles. The <em>y</em>-intercept is the silver ratio.</p>
</div><!-- Image from Desmos interactivity -->
<a name="silver"></a>
<p>It turns out that these two lines of intersection again involve the silver ratio: the point at which they meet the <img src="/MI/1248bbcacb726480c6628d72c7f5aef4/images/img-0001.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" />-axis (and each other) has coordinates <img src="/MI/1248bbcacb726480c6628d72c7f5aef4/images/img-0002.png" alt="$(0,\lambda _2)$" style="vertical-align:-4px;
width:43px;
height:18px" class="math gen" />. (You can work this out for yourself or see the calculation <a href="/content/lines-intersection-and-silver-ratio">here</a>.)</p>
<p>It seems that the silver ratio is written all over our extension of the fingernail problem! And if you do the calculations, you will find that this seems to hinge on the fact that </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/568472578540d639e2bcfcbd3150b358/images/img-0001.png" alt="\[ \frac{1}{\lambda _2-2}=\frac{1}{\sqrt {2}-1}=\sqrt {2}+1 =\lambda _2. \]" style="width:241px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<h3>Other metallic numbers</h3>
<p>This suggests that we may be able to generalise the construction of our family of circles to get all other metallic numbers too. One thing we could change is the ratio of the radii of successive circles. Let's call this ratio <img src="/MI/9527cc855a2529a717de8be6ecfbd38f/images/img-0001.png" alt="$A$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" />. In the above construction we had <img src="/MI/9527cc855a2529a717de8be6ecfbd38f/images/img-0002.png" alt="$A=\sqrt {2}$" style="vertical-align:-2px;
width:57px;
height:17px" class="math gen" />, which is related to the silver ratio as follows:
</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/5159d1e6ab17b73cb54465f7a64793dd/images/img-0001.png" alt="\[ A=\sqrt {2} = (\sqrt {2}+1)-1=\lambda _2-1. \]" style="width:243px;
height:20px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>So what if, for any other natural number <img src="/MI/5159d1e6ab17b73cb54465f7a64793dd/images/img-0002.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />, we choose this ratio to be <img src="/MI/5159d1e6ab17b73cb54465f7a64793dd/images/img-0003.png" alt="$A_ n=\lambda _ n-(n-1)$" style="vertical-align:-4px;
width:132px;
height:18px" class="math gen" /> by analogy? </p>
<p>The first circle in our family will again be centred at <img src="/MI/5d13946ff06cbb50aefa72e229d83e1d/images/img-0001.png" alt="$(0,0)$" style="vertical-align:-4px;
width:34px;
height:18px" class="math gen" /> and have radius <img src="/MI/5d13946ff06cbb50aefa72e229d83e1d/images/img-0002.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" />. The second will be centred at the South pole of the first, so that’s the point <img src="/MI/5d13946ff06cbb50aefa72e229d83e1d/images/img-0003.png" alt="$(0,-1)$" style="vertical-align:-4px;
width:47px;
height:18px" class="math gen" /> and have radius <img src="/MI/5d13946ff06cbb50aefa72e229d83e1d/images/img-0004.png" alt="$A_ n$" style="vertical-align:-2px;
width:20px;
height:13px" class="math gen" />. The ratio between the height of the top of the first circle above the <img src="/MI/5d13946ff06cbb50aefa72e229d83e1d/images/img-0005.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />-axis and the height of the top of the second circle above the <img src="/MI/5d13946ff06cbb50aefa72e229d83e1d/images/img-0005.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />-axis is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/5d13946ff06cbb50aefa72e229d83e1d/images/img-0006.png" alt="\[ \frac{1}{A-1}=\frac{1}{\lambda _ n-n}=\lambda _ n. \]" style="width:163px;
height:37px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> That’s exactly analogous to the result we got above, where arcs of the two circles represented fingernails. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/fingernail/ratio_general.png" alt="Ratio" width="250" height="314" /><p style="max-width: 250px;">Ratio between the heights of the arcs of the circle above the <em>x</em>-axis.</p>
</div><!-- Image made by MF -->
<p>What about those lines of intersection?
It's possible to show
(see <a href="/content/lines-intersection-and-silver-ratio">here</a>) that for <img src="/MI/7a5486f76d90238c51c2d61d3cb1f99f/images/img-0001.png" alt="$A=\lambda _ n-(n-1)$" style="vertical-align:-4px;
width:123px;
height:18px" class="math gen" /> these meet the <img src="/MI/7a5486f76d90238c51c2d61d3cb1f99f/images/img-0002.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" />-axis at (and each other) at the point <img src="/MI/7a5486f76d90238c51c2d61d3cb1f99f/images/img-0003.png" alt="$(0,\lambda _ n)$" style="vertical-align:-4px;
width:45px;
height:18px" class="math gen" />.
<p>The interactive below lets you play with different values for the ratio <img src="/MI/de83c44434d28de2c2be401191d1be5c/images/img-0001.png" alt="$A$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" />. Move down the left panel to find the slider that varies <img src="/MI/de83c44434d28de2c2be401191d1be5c/images/img-0001.png" alt="$A$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" />. To make the left panel disappear so you can see more of the graph, click on the double arrow at the top right of the panel. To help you find the values corresponding to the metallic numbers, the first four are: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray">
<tr id="a0000000003">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0001.png" alt="$\displaystyle A_1 $" style="vertical-align:-2px; width:18px; height:13px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0002.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0003.png" alt="$\displaystyle \lambda _1-0 $" style="vertical-align:-2px; width:44px; height:14px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0002.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0004.png" alt="$\displaystyle \lambda _1 \approx 1.618 $" style="vertical-align:-2px; width:75px; height:14px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr><tr id="a0000000004">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0005.png" alt="$\displaystyle A_2 $" style="vertical-align:-2px; width:18px; height:13px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0002.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0006.png" alt="$\displaystyle \lambda _2 -1 $" style="vertical-align:-2px; width:43px; height:14px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0002.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0007.png" alt="$\displaystyle \sqrt {2} \approx 1.414 $" style="vertical-align:-1px; width:80px; height:17px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr><tr id="a0000000005">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0008.png" alt="$\displaystyle A_3 $" style="vertical-align:-2px; width:18px; height:13px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0002.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0009.png" alt="$\displaystyle \lambda _3 - 2 $" style="vertical-align:-2px; width:44px; height:14px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0010.png" alt="$\displaystyle \approx $" style="vertical-align:1px; width:11px; height:7px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0011.png" alt="$\displaystyle 3.303 - 2 =1.303 $" style="vertical-align:0px; width:125px; height:12px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr><tr id="a0000000006">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0012.png" alt="$\displaystyle A_4 $" style="vertical-align:-2px; width:18px; height:13px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0002.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0013.png" alt="$\displaystyle \lambda _4 -3 $" style="vertical-align:-2px; width:44px; height:14px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0014.png" alt="$\displaystyle \approx $" style="vertical-align:1px; width:11px; height:7px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="/MI/16d5e675877a9a9936fe054a0029c32c/images/img-0015.png" alt="$\displaystyle 4.236 - 3 =1.236. $" style="vertical-align:0px; width:129px; height:12px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr>
</table>
<iframe height="500" width="600" src="https://www.desmos.com/calculator/y1tmsc66hg" style="border: 1px solid #ccc" frameborder=0></iframe>
<p>Thus, a simple problem about a fingernail opens a door to an infinite collection of very special numbers!</p>
<hr/>
<h3>About the authors</h3>
<p>Gokul Rajiv is a first year computer science and mathematics student at the National University of Singapore and Yong Zheng Yew is a first year engineering student at the Singapore University of Technology and Design. They worked on the fingernail problem when they were at high-school and found it interesting enough to share. </p>
<div class="centreimage"><img alt="authors" src="/content/sites/plus.maths.org/files/articles/2020/metallic/authors_picture.jpg" style="max-width: 400px; height: 299px;" />
<p style="max-width: 400px;"></p>
</div>
<!-- Image provided by authors -->
</div></div></div>Thu, 05 Nov 2020 16:50:51 +0000Marianne7338 at https://plus.maths.org/contenthttps://plus.maths.org/content/fingernail-problem-and-metallic-numbers#commentsClearing the air: Making indoor spaces COVID safe
https://plus.maths.org/content/clearing-air-making-indoor-spaces-covid-safe
<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/vent_icon_0.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>How should offices and other indoor spaces be ventilated to keep people safe from COVID-19? A recent <a href="https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/effects-of-ventilation-on-the-indoor-spread-of-covid19/CF272DAD7C27DC44F6A9393B0519CAE3">paper</a> by mathematicians at the University of Cambridge has some answers — and shows that wearing masks does make a difference.</h2>
<div class="rightshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</div>
<p>"As winter approaches in the northern hemisphere and people start spending more time inside, understanding the role of ventilation is critical to estimating the risk of contracting the virus and helping slow its spread," says <a href="https://www.damtp.cam.ac.uk/person/pfl4">Paul Linden</a>, who wrote the paper along with <a href="https://www.maths.cam.ac.uk/person/rkb29">Rajesh Bhagat</a>, <a href="http://www.damtp.cam.ac.uk/person/sd103">Stuart Dalziel</a>, and <a href="http://www.eng.cam.ac.uk/profiles/msd38">Megan Davies Wykes</a>. Linden has studied the ventilation of buildings since 1990 and is a pioneer in the area. Previously Linden's aim was to make those buildings energy efficient and comfortable for their occupants, but in the light of the pandemic his work has taken on a new dimension. Since April he has been contributing to the Royal Society's <a href="/content/call-action-covid-19"><em>Rapid Assistance in Modelling the Pandemic</em></a> (RAMP) taskforce, which unites scientists and mathematicians in the effort to fight the pandemic.</p>
<h3>Mixing or layering?</h3>
<p>Evidence suggests that the virus which causes COVID-19 is transmitted mostly through larger droplets and smaller aerosols, which we expel when we breathe, talk, cough or laugh. These are hard to monitor, but the work of Linden and his colleagues confirms earlier research which suggests that CO<sub>2</sub> can be used as a proxy. "Small respiratory aerosols containing the virus are transported along with the carbon dioxide produced by breathing, and are carried around a room by ventilation flows," explains Linden. "Insufficient ventilation can lead to high carbon dioxide concentration, which in turn could increase the risk of exposure to the virus."</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2020/ventilation/vent-206929_640.jpg" alt="Vent" width="350" height="233" /><p>What kind of ventilation is safest? Image: <a href="https://pixabay.com/photos/vent-fireplaces-metal-206929/">stux</a>.</p>
</div>
<p>Many modern indoor spaces, such as offices, hospitals, or restaurants, use some sort of ventilation system. The way the ventilation is achieved varies — you could use either natural ventilation driven by wind and heat, or mechanical systems — but there are two main modes of operation. <em>Mixing ventilation</em> aims to keep the air in a space well-mixed and <em>displacement ventilation</em> creates a cooler lower zone and a warmer upper zone, with warm air being extracted through the top part of the room and cooler air supplied through vents near the floor.</p>
<p>When it comes to transmission of COVID-19, it seems obvious that mixing up the air isn't the way you'd want to go, and the work of Linden and his team broadly supports this idea. "[Mixing ventilation] is intended to keep everything airborne and stirred around, and if it works properly then that's what mixing ventilation does," Linden explains. "I think essentially that's not a good idea and there are better strategies you can adapt." Displacement ventilation appears to be such a strategy: the warm and potentially dangerous air we exhale will rise to the ceiling, where it can be extracted.</p>
<p>What Linden and his colleagues also showed, however, is that things aren't quite as simple as they might first seem. Ventilation systems aren't the only things that cause significant air flows in a room. Other factors — people moving around, the opening and closing of doors, and heat emanating from people's bodies or appliances — can also cause the air to move in ways you can't just ignore. This means that even with displacement ventilation you can run into trouble: when there are various heat sources in a room, exhaled breath can get trapped below the warm ceiling layer and breathed in again by others.</p>
<p>The exact behaviour of people's exhaled breath and its role in disease transmission is extremely difficult to predict, so Linden and his colleagues headed to the lab. They did this in the middle of the spring lockdown, when the fluid dynamics laboratory was the only part of the Centre for Mathematical Sciences at the University of Cambridge that was open, and only to a select few.</p>
<h3>Breathing, talking, laughing</h3>
<p>Even when a person is sitting still and holding their breath, their body heat produces a plume of warm air that rises to the ceiling. When the person starts breathing, or opens their mouth to talk, sing, cough or laugh, their exhaled breath produces a second plume. In terms of transmission, it's best for this second plume to be entrained by the main body plume to be carried to the ceiling with it.</p>
<p>Air is invisible, of course, but Linden and his team used an imaging technique that allows you to track warm air. "You can see the change in temperature and density when someone breathes out warm air – it refracts the light and you can measure it,"explains Bhagat.</p>
<p>The images the team produced are shown below. In the left hand image the person is sitting quietly breathing through their nose, in the middle one he is speaking at a normal volume, and in the right one he is laughing. In each image, you can also see the body plume rising up gently — and in each of the three cases, you can see that the exhaled plume isn't absorbed by the body plume.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2020/ventilation/no_mask.jpg" alt="Breathing, talking and laughing without a mask" width="470" height="199" /><p style="max-width: 470px;">Breathing, talking and laughing without a mask. Image from <a href="<a href="https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/effects-of-ventilation-on-the-indoor-spread-of-covid19/CF272DAD7C27DC44F6A9393B0519CAE3"><em>Effects of ventilation on the indoor spread of COVID-19</em></a> by Bhagat, Davies Wykes, Dalziel, and Linden. Journal of Fluid Mechanics, 903, F1.</p>
</div>
<p>This is in contrast to the images you get when the person is wearing a mask, shown below. "One thing we could clearly see is that one of the ways that masks work is by stopping the breath's momentum," said Linden.</p>
<p>"While pretty much all masks will have a certain amount of leakage through the top and sides, it doesn't matter that much, because slowing the momentum of any exhaled contaminants reduces the chance of any direct exchange of aerosols and droplets as the breath remains in the body's thermal plume and is carried upwards towards the ceiling. Additionally, masks stop larger droplets, and a three-layered mask decreases the amount of those contaminants that are recirculated through the room by ventilation."</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2020/ventilation/mask.jpg" alt="Breathing, talking and laughing with a mask" width="481" height="200" /><p style="max-width: 481px;">Breathing, talking and laughing with a mask. Image from <a href="<a href="https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/effects-of-ventilation-on-the-indoor-spread-of-covid19/CF272DAD7C27DC44F6A9393B0519CAE3"><em>Effects of ventilation on the indoor spread of COVID-19</em></a> by Bhagat, Davies Wykes, Dalziel, and Linden. Journal of Fluid Mechanics, 903, F1.</p>
</div>
<h3>Experiments meet mathematics</h3>
<p>Experiments like these are hugely important, but they only form a part of the work of Linden and his team. Equally important are mathematical models that describe the behaviour of gasses and contaminants within them. An interesting example is the <em>Wells-Riley equation</em>, which estimates the expected number <img src="/MI/07b73b05aad2e47e84ce5ee630d54297/images/img-0001.png" alt="$I$" style="vertical-align:0px;
width:9px;
height:11px" class="math gen" /> of people who become infected by sharing a room with people who have an airborne disease:</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/bc1f84f8428c4120c18c6d2366300074/images/img-0001.png" alt="\[ I=S\left(1-exp\left(-\frac{q \Gamma t}{Q}\right) \right) \]" style="width:189px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>Here <img src="/MI/3bee8185912b3064a6195697e45a1474/images/img-0001.png" alt="$S$" style="vertical-align:0px;
width:10px;
height:11px" class="math gen" /> is the number of people in the room who are susceptible to the disease, and, loosely speaking, <img src="/MI/3bee8185912b3064a6195697e45a1474/images/img-0002.png" alt="$\Gamma $" style="vertical-align:0px;
width:10px;
height:11px" class="math gen" /> describes the total rate at which already infected occupants in the room emit infectious material, <img src="/MI/3bee8185912b3064a6195697e45a1474/images/img-0003.png" alt="$q$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> measures the average breathing rate per person, and <img src="/MI/3bee8185912b3064a6195697e45a1474/images/img-0004.png" alt="$t$" style="vertical-align:0px;
width:6px;
height:10px" class="math gen" /> the time period people are sharing the room. The number <img src="/MI/3bee8185912b3064a6195697e45a1474/images/img-0005.png" alt="$Q$" style="vertical-align:-3px;
width:13px;
height:14px" class="math gen" /> measures the ventilation rate of the room the room, that is, the rate at which fresh air enters it. </p><p>A close look at the equation reveals that the larger <img src="/MI/3bee8185912b3064a6195697e45a1474/images/img-0005.png" alt="$Q$" style="vertical-align:-3px;
width:13px;
height:14px" class="math gen" /> (the better the ventilation of the room) the smaller the number <img src="/MI/3bee8185912b3064a6195697e45a1474/images/img-0006.png" alt="$I$" style="vertical-align:0px;
width:9px;
height:11px" class="math gen" /> of people who become infected. The Wells-Riley equation assumes that the ventilation <img src="/MI/3bee8185912b3064a6195697e45a1474/images/img-0005.png" alt="$Q$" style="vertical-align:-3px;
width:13px;
height:14px" class="math gen" /> is uniform across the space, and as Linden and his team have shown, this isn’t usually the case in reality, as air flows created by people and appliances also matter. However, the Wells-Riley equation (along with many other relevant mathematical expressions) can form part of more complex models that describe real life more accurately.</p>
<h3>A warning system</h3>
<p>Linden and his colleagues stress that there's still more work to do on ventilation and COVID-19. For example, it would be good to know more about what happens to people's body plumes and exhaled breath as they move around a room. But the work done so far already carries an important message. Displacement ventilation systems, when properly set up, appear to be the better option, and face coverings are beneficial.</p>
<p>The research also suggests another interesting possibility. Since virus-laden aerosols behave like the CO<sub>2</sub> we breathe out, the CO<sub>2</sub> level in a room could provide a warning system. It can quite easily be measured, and when it's high, the risk of airborne infection is high too. "What we have in mind is something like a traffic light system," explains Linden. "But what we are not clear about yet is [which CO<sub>2</sub> thresholds should mark the boundaries between green, amber and red]. So that's an area that still needs discussion."</p>
<p>For those of us without ventilation systems or ways of measuring CO<sub>2</sub>, the message is simple. When there are several people in a room, open the windows, keep your distance, and wear your masks.</p>
<hr/>
<h3>About this article</h3>
<p>Some of the quotes in this article are taken from <a href="https://www.cam.ac.uk/research/news/many-ventilation-systems-may-increase-risk-of-covid-19-exposure-study-suggests">Many ventilation systems may increase risk of COVID-19 exposure, study suggests</a>, reproduced under a <a href="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License.</a></p></div></div></div>Thu, 05 Nov 2020 15:23:22 +0000Marianne7349 at https://plus.maths.org/contenthttps://plus.maths.org/content/clearing-air-making-indoor-spaces-covid-safe#commentsThe maths of COVID-19
https://plus.maths.org/content/covid-19
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/corona_icon_1.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Maths plays an essential role in fighting COVID-19, which is why the pandemic has featured a lot on <em>Plus</em>. Here is all our coverage at a glance. This page will be continually updated with new content.
<div style="float: left; 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/dacids_icon.jpg" alt="" width="100" height="100" /> </div>
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<p><a href="/content/going-back-uni-during-pandemic">Going back to uni during a pandemic</a> — What can maths tell us about how to make universities safe from COVID-19?</p></div></p>
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<p><a href="/content/epidemic-growth-rate">The growth rate of COVID-19</a> — We all now know about <em>R</em>, the reproduction number of a disease. But sometimes it can be good to consider another number: the growth rate of an epidemic.</p></div>
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<p><a href="/content/problem-combining-r-rates">The problem with combining <em>R</em> ratios</a> — We explore why you need to be extremely careful when combining the reproduction ratios of a disease in different settings, such as hospitals and the community.</p></div>
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<p><a href="/content/maths-minute-r0-and-herd-immunity">Maths in a minute: <em>R<sub>0</sub></em> and herd immunity</a> — What is herd immunity and how does it relate to the basic reproduction number of a disease?</p></div>
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<p><a href="/content/node/7281">The virus podcast</a> — In this podcast we explore the famous curve, talk about how to communicate science in a crisis, and explain the maths of herd immunity in one minute.</p></div>
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<p><a href="/content/how-can-maths-fight-pandemic">How can maths fight a pandemic?</a> — How do mathematical models of COVID-19 work and should we believe them? We talk to Julia Gog, an epidemiologist who has been working flat out to inform the government, to find out more.</p></div>
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<p><a href="/content/call-action-covid-19">A call to action on COVID-19</a> — An urgent call has gone out to the scientific modelling community to help fight against the COVID-19 pandemic. Two Cambridge mathematicians are helping to lead the charge.</p></div>
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Finding a way out of lockdown</a> — Mathematical models can help the nation return to (some sort of) normality. </p></div>
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<p><a href="/content/social-distancing-how-close-safe">
Social distancing: How close is safe?</a> — Mathematicians are investigating how we can keep safe as we emerge from lockdown, by seeing how far virus-carrying droplets can fly in different environments — from buses to supermarkets.</p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/xray_icon.png" alt="" width="100" height="100" /> </div>
<p><a href="/content/artificial-intelligence-takes-covid-19">
Artificial intelligence takes on COVID-19</a> — Mathematicians are helping to develop an AI tool to help with diagnosing COVID-19 and making prognoses for infected patients.</p></div>
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<p><a href="/content/communicating-corona-crisis">Communicating the coronavirus crisis</a> — David Spiegelhalter, expert in risk and evidence communication, tells us how well the UK government has done so far communicating about Covid-19.</p></div>
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<p><a href="/content/taking-pandemic-temperature">Taking the pandemic temperature</a> — How do people in different countries feel about the COVID-19 pandemic and the measures taken by their governments?</p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/corona_icon_0.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/how-best-deal-covid-19">Squashing the curve?</a> — A study published in March by researchers from Imperial College suggested that measures against COVID-19 needed to be much more drastic than was previously hoped.</p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/football_virus_icon.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/how-resolve-premier-league">How to resolve the Premier League</a> — As football leagues around the world have been suspended due to COVID-19, how should the final rankings of teams be decided? Here is a suggestion.</p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/football_icon_0.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/were-football-leagues-fair">Was points per game fair?</a> — With the Premier League finished for this year we check if the solution suggested in the previous article, and other ways the League could have been decided without playing the remaining matches, would have been fair.</p></div></div></div></div>Thu, 05 Nov 2020 14:19:12 +0000Marianne7296 at https://plus.maths.org/contenthttps://plus.maths.org/content/covid-19#commentsCambridge mathematicians win Whitehead Prizes
https://plus.maths.org/content/prize-young-mathematicians
<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_40.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>Every year the <a href="https://www.lms.ac.uk/">London Mathematical Society</a> awards prestigious Whitehead Prizes to mathematicians who are in an early stage of their career. Out of the six Whitehead Prizes awarded in 2020, three went to mathematicians at the University of Cambridge. And since this is also the home of <em>Plus</em>, we took the opportunity to talk to them and find out more about their work.</p>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/bruna._icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/taming-complexity">Maria Bruna: Taming complexity</a> — Maria Bruna won a Whitehead Prize for finding a systematic way of simplifying complex systems. Her methods not only work in biology and medicine, but have also helped the company Dyson improve their vacuum cleaners. </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/krieger_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/dynamic-numbers">Holly Krieger: Dynamic numbers</a> — Holly Krieger won a Whitehead Prize for work that combines arithmetic and dynamical systems, and leads to beautiful fractal shapes like the Mandelbrot set.</p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/henry_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/changing-perspectives">Henry Wilton: Changing perspectives</a> — Henry Wilton won a Whitehead Prize for his work studying the mathematics of symmetry, bringing together geometry and algebra.</p></div></div></div></div>Tue, 27 Oct 2020 16:08:18 +0000Marianne7347 at https://plus.maths.org/contenthttps://plus.maths.org/content/prize-young-mathematicians#commentsChanging perspectives
https://plus.maths.org/content/changing-perspectives
<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/henry_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-hidden"><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 href="https://www.dpmms.cam.ac.uk/~hjrw2/">Henry Wilton</a> is one of this year's winners of a Whitehead Prize, which is awarded annually by the <a href="https://www.lms.ac.uk/">London Mathematical Society</a> to mathematicians who are in an early stage of their career. Wilton works at the University of Cambridge, the home of <em>Plus</em>, so we took the opportunity to talk to him to find out about his work.</p>
<p>"I feel really honoured to be on a list that includes lots of my mathematical heroes and heroines," Wlton told us. "It's a prize with quite a long history — 41 years is exactly my age, so the prize has been around exactly as long as me. And among the [previous prize winners] are some of the mathematicians I have admired most in my life."</p>
<h3>From geometry to algebra and back</h3>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2020/whiteheads/henry.jpg" alt="Henry Wilton" width="350" height="263" /><p>Henry Wilton.</p>
</div>
<p>Wilton's area of research illustrates the curious paths mathematical research can take. For a long time people have been interested in the study of symmetry. Symmetry is immunity to change: if you rotate a square through multiples of 90 degrees around its central point. its appearance remains unchanged, as it does if you reflect it in vertical, horizontal or diagonal lines that run through the central point. A circle can be rotated through any angle you like and reflected in any line that runs through its centre point without changing its appearance. Unlike a square, which has a total of eight symmetries, a circle has infinitely many symmetries.</p>
<p>But symmetries don't just come up when you are dealing with geometric objects. The equation</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/87ee80b7506ac432fc5b33619767ed2f/images/img-0001.png" alt="\[ x^2=y, \]" style="width:52px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>remains the same whether you substitute in <img src="/MI/638a226e5be7d1984711eb01d60f0069/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> or <img src="/MI/638a226e5be7d1984711eb01d60f0069/images/img-0002.png" alt="$-x$" style="vertical-align:0px;
width:21px;
height:7px" class="math gen" />, so we can detect a notion of symmetry here too (find out more in <a href="/content/stubborn-equations">this <em>Plus</em> article</a>). Even when it comes to real-world problems you can often exploit symmetry to find solutions. For example, if in a school there are two teachers who can both teach the same subjects and have the same availability, then you can swap them without disturbing your time-table. Studying the symmetries of an object or problem is often a useful way of understanding it.</p>
<p>"The study of symmetry is called <em>group theory</em>," explains Wilton. "That's a very old topic that goes back about 200 years. But about a hundred years ago there was a shift in the way people thought about groups. Lots of mathematicians stopped thinking about [groups] as symmetries of actual objects and started working with them in a purely algebraic way."</p>
<div class="leftshoutout"><p>See <a href="/content/prize-young-mathematicians">here</a> to find out more about two other 2020 Whitehead Prizes.</p></div>
<p>To see what Wilton means note that you could denote the eight symmetries of a squares by the letters <img src="/MI/3726bf011e7eeaf225a3160bf5335745/images/img-0001.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> to <img src="/MI/3726bf011e7eeaf225a3160bf5335745/images/img-0002.png" alt="$h$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" />. When you follow one symmetry by another, the result is also a symmetry. For example, doing a 90 degree rotation in the same direction twice amounts to doing a rotation through 180 degrees. Thus, the symmetries of the square tell you how any two of your eight letters combine to give you a third — you can then forget about the square altogether and just think about your group of symmetries as a collection of abstract objects denoted by letters that combine in pairs in a specific way. (For a precise definition of a group see <a href="/content/maths-a-minute-groups">this <em>Plus</em> article</a>.)</p>
<p>"[This algebraic way of thinking about groups] is the topic of <em>combinatorial group theory</em>," explains Wilton. "This dominated group theory for much of the twentieth century. Then there was some amazing work in the 1980s by Mikhael Gromov and William Thurston, and others, who remembered that a hundred years ago people were thinking of group theory as studying the symmetry of things — [in particular] geometric objects that you can visualise." This area is called <em>geometric group theory</em>.</p>
<p>Wilton works on the confluence of combinatorial and geometric group theory. "[They are] the same field really, but viewed from different perspectives."</p>
<h3>Tackling the infinite</h3>
<p>Wilton got into this area of research through pure luck, as he puts it, after completing his undergraduate degree at the University of Cambridge. "All I knew was that I really wanted to do [a PhD in] geometry, but there weren't many people working in geometry in Cambridge at the time," he recalls. Luckily, though, there was someone at Imperial College, London, who did, albeit from a group theoretic perspective. That person was Martin Bridson, and he became Wilton's PhD supervisor and guide into the world of geometric group theory.</p>
<div style="float: left; width: 50%; margin-right: 10px;"><iframe scrolling="no" src="https://tube.geogebra.org/material/iframe/id/ABFWp1JC/width/366/height/412/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" width="366px" height="412px" style="border:0px;"> </iframe><p style="width: 400px; font-size: small; color: purple;">This applet rotates a regular 12-gon (use the slider to rotate it). You can see that rotations through a 12th of a circle leave the shape unchanged, so they are symmetries of the 12-gon.</p></div>
<p>One of Wilton's favourite results, which he proved together with Bridson and published in 2015, involves infinite groups of symmetries, like the one of the circle mentioned above. One way of tackling such infinite beasts, which can be a lot more complex than the symmetry group of a circle, is to see if you can break them up into finite chunks in some way. This would be incredibly useful because finite groups a very well understood - in what is perhaps the biggest effort in collective mathematics, the <em>classification of finite simple groups</em> has provided an atlas listing all possible components a finite group can be made up of (find out more <a href="/content/enormous-theorem-classification-finite-simple-groups">here</a>).</p>
<p>When it comes to infinite groups, Wilton and Bridson proved what Wilton calls a "bad news theorem". Loosely speaking, if you have an infinite collection of symmetries of some object, you might try and see if looking at the collection of symmetries of a slice of the object makes things simpler. "If you have an infinite space, maybe you can slice it to get something finite, and that's the kind of thing that you want," explains Wilton. "But what we showed is that there is no way of determining if your infinite collection of symmetries has a finite slice that isn't just completely trivial."</p>
<p>This doesn't mean that given a particular infinite group you can never tell if it's got a finite slice — it may be that there's an algorithm for working out the answer for that particular group. But it does mean that there isn't a general algorithm that will work for every infinite group that you care to throw at it. (For those who know about these things, the question of whether there's a finite slice is <em>undecidable</em> in the sense of Alan Turing's <a href="/content/not-just-matter-time-part-ii"><em>halting problem</em></a>.) Bad news theorems of this nature aren't actually that bad: they give deep insight into the complexity of the objects you are dealing with.</p>
<h3>Young and old</h3>
<p>Wilton thinks his area of research is a great field for young researchers to go into. "It's both a very old and a very young area. It studies things that should have made sense to people a hundred years ago," he says. "But because it went through this long hiatus of looking at things from different points of view it's still a very young area in terms of the [techniques being used]. So it's not an incredibly theory-heavy area and a young person can still get to the coalface and start answering difficult questions quite quickly."</p>
</div></div></div>Tue, 27 Oct 2020 15:42:43 +0000Marianne7346 at https://plus.maths.org/contenthttps://plus.maths.org/content/changing-perspectives#commentsDynamic numbers
https://plus.maths.org/content/dynamic-numbers
<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/krieger_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-hidden"><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 href="https://www.dpmms.cam.ac.uk/~hk439/">Holly Krieger</a> is one of this year's winners of a Whitehead Prize, which is awarded annually by the <a href="https://www.lms.ac.uk/">London Mathematical Society</a> to mathematicians who are in an early stage of their career. Krieger works at the University of Cambridge, the home of <em>Plus</em>, so we took the opportunity to talk to her to find out about her work.</p>
<p>"It's exciting to have my work recognised by my colleagues," Krieger told us. "This work was done in collaboration with Laura DeMarco and Hexi Ye, with lots of input from others through conversations and discussions at conferences. The prize is a nice reminder of the good fortune I have to work with all these people."</p>
<h3>Numbers and change</h3>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2020/whiteheads/krieger.jpg" alt="Holly Krieger" width="350" height="219" /><p>Holly Krieger.</p>
</div>
<p>"Arithmetic dynamics is exactly what you get when you take the two words apart," says Krieger. "Arithmetic relates to those very simple notions you learn about at school: counting, integers, prime numbers and so on. These are the fundamental building blocks of what we think of as abstract mathematics." The term "dynamics" refers to systems that evolve over time. These can often be quite easily described mathematically, but can still exhibit incredibly complex behaviour in the long run.</p>
<p>The two areas may appear quite different, but they actually combine quite naturally. As an example, consider the following basic arithmetic procedure: given any integer, square it and add 1. If you do this to 0, the result is 0+1=1. If you in turn apply the procedure to 1, the answer is 1+1=2. Applying it to 2 gives 4+1=5, and applying it to 5 gives 25+1=26. You could keep going like this forever, at each step in time applying the procedure to the number you got at the previous step. The result is an infinite sequence of integers, starting with</p>
<p>0, 1, 2, 5, 26, 677, 458330, ...</p>
<div class="leftshoutout"><p>See <a href="/content/prize-young-mathematicians">here</a> to find out more about two other 2020 Whitehead Prizes.</p></div>
<p>"Here is a very basic question about this system," explains Krieger. "I can take each number in this sequence and factor it into its prime factors — which prime numbers appear when I do that?" There is no explicit answer to this question, but we can get a statistical handle on how many different primes appear in this way. "It's a relatively sparse set of prime numbers [that appear as factors]," says Krieger. "Because the values [in the sequence] quickly get quite large and they tend not to have small prime factors. It's not an insignificant piece of mathematics to understand these prime factors, even though it's very simple to ask the question."</p>
<h3>Numbers and fractals</h3>
<p>The dynamical system we just described doesn’t exist in isolation. Rather than considering the expression </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0001.png" alt="\[ x^2+1 \]" style="width:44px;
height:17px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>you could consider the expression </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0002.png" alt="\[ x^2+c. \]" style="width:49px;
height:17px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>for some other number <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0003.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" />. For some values of <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0003.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> (such as <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0004.png" alt="$c=1$" style="vertical-align:0px;
width:36px;
height:12px" class="math gen" /> in our example above) the sequence you get when you start with <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0005.png" alt="$x=0$" style="vertical-align:0px;
width:39px;
height:12px" class="math gen" /> exceeds all bounds: it <em>escapes to infinity</em>. For other values of <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0003.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> the sequence always stays within certain bounds: it doesn’t escape to infinity. The most dramatic example of this is occurs when <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0006.png" alt="$c=0$" style="vertical-align:0px;
width:37px;
height:12px" class="math gen" />. In this case, the calculation is simply <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0007.png" alt="$0^2=2$" style="vertical-align:0px;
width:46px;
height:14px" class="math gen" /> so the sequence you end up with consists entirely of 0s. </p><p>Do we know which values of <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0003.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> lead to which kind of behaviour? The answer is yes: escape to infinity is only prevented if <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0008.png" alt="$c=0, -1$" style="vertical-align:-3px;
width:65px;
height:15px" class="math gen" /> or <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0009.png" alt="$-2$" style="vertical-align:0px;
width:20px;
height:12px" class="math gen" />. If we allow <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0003.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> to take non-integer values, then the sequence stays bounded precisely when <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0003.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> lies in the interval <img src="/MI/1dcab50f6b7106e604a6a089c4e3b923/images/img-0010.png" alt="$[-2,1/4].$" style="vertical-align:-5px;
width:64px;
height:18px" class="math gen" /> </p>
<p>We can go further still: we can allow <img src="/MI/fdd96bc37f4db8f290144e32e5aaa0a3/images/img-0001.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> to be a <em>complex number</em>. Each complex number can be represented by a point in the plane (find out more <a href="/content/maths-minute-complex-numbers">here</a>) so we could go and colour in exactly those points in the plane that, when taken as a value for <img src="/MI/916625ecbf4c24d7d7f6efa8c5dedbc4/images/img-0001.png" alt="$c$" style="vertical-align:-1px;
width:7px;
height:9px" class="math gen" />, don't give us a sequence that escapes to infinity. The picture we'd get from colouring in those points would be the famous <em>Mandelbrot set</em>: a beautifully intricate fractal whose shape is intimately related to the behaviour of the dynamical systems that lie behind it.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2020/whiteheads/mandelbrot.jpg" alt="The Mandelbrot set" width="300" height="225" /><p style="max-width: 300px;">The Mandelbrot set. Image by <a href="https://en.wikipedia.org/wiki/File:Mandel_zoom_00_mandelbrot_set.jpg">Wolfgang Beyer</a>.</p>
</div>
<p>The Mandelbrot set, too, plays a major role in Krieger's work. "There are some special values inside the Mandelbrot set," explains Krieger. We already saw above that <em>c</em>=0 results in a dynamical system forever stuck at 0. There are other values of <em>c</em> that lead to similarly limited dynamics (you might want to try <em>c</em>=-1) . "Studying where these <em>post-critically finite</em> points lie in the Mandelbrot set is the type of question that I work on."</p>
<p>You can find out more about the Mandelbrot set and the related dynamical systems in <a href="/content/what-mandelbrot-set">this <em>Plus</em> article</a>.</p>
<h3>Numbers and people</h3>
<p>What drew Krieger to working in arithmetic dynamics? Partly it was the age-old lure of number theory, whose seemingly simple questions often turn out fiendishly hard to prove. "I like the simplicity of the questions, combined with the difficulty and depth of the techniques," she says.</p>
<p>But while number theory is old, arithmetic dynamics is new. "Arithmetic dynamics is a relatively young field, so it has a lot of energy. In the 1990s it was realised that there's a very deep connection [between the theory of dynamical systems] and hard questions in number theory. The question was, 'what can we do with this analogy; is there power there?' This led to an entirely new way of studying dynamical systems. It's that innovation that I liked."</p>
<p>People too played a role in Krieger's decision to go into the field. "The leaders in the field when I went into it were high-calibre mathematicians, doing fantastic things, without even a hint of arrogance or exclusivity. The energy of the people and the atmosphere of the field is what drew me in."</p>
</div></div></div>Tue, 27 Oct 2020 15:25:59 +0000Marianne7345 at https://plus.maths.org/contenthttps://plus.maths.org/content/dynamic-numbers#comments