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enR's not all you need
https://plus.maths.org/content/R-not-all
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/angry_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">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>When it comes to loosening COVID restrictions all eyes are usually trained on the famous <em>R</em> number. But as epidemiologists <a href="https://www.infectiousdisease.cam.ac.uk/directory/jrg20@cam.ac.uk" target="blank">Julia Gog</a> and <a href="https://www.research.manchester.ac.uk/portal/thomas.house.html" target="blank">Thomas House</a> recently explained to us, there's also another important factor to consider alongside <em>R</em>. That's the <em>prevalence </em> of COVID-19 in the population: the proportion of people who currently have the disease. </p>
<div class="rightshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</p></div>
<p>Put simply, if prevalence has been so high that the NHS is in crisis, then opening up might stretch it to breaking point, even if <em>R</em> is less than 1, or would remain so. If, on the other hand, prevalence is very low, we might be able to tolerate a higher value of <em>R</em> as it would not immediately lead to many cases. This is true particularly if prevalence has been low for some time. (For an introduction to <em>R</em> see <a href="/content/maths-minute-r0-and-herd-immunity" target="blank">this article</a>.)</p>
<p> We've illustrated this idea in the schematic plot below. The vertical axis measures prevalence and the horizontal axis measures <em>R</em>. Any point on this plot, such as the one we marked in black right in the middle, corresponds to a situation where we have the value of <em>R</em> that lies directly beneath the point on the horizontal axis, and the value of prevalence that lies directly to the left of the point on the vertical axis.</p>
<div class="centreimage"><img alt="Prevalence versus R" width="600" height="508" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/square/angry_final1_blur.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 600px;">The horizontal axis measures <em>R</em> and the vertical axis measures prevalence. This is a schematic plot only, which is why we have not labelled the axes further.
</p></div><!-- Image made by MF -->
<p>The two dashed horizontal lines correspond to values of the prevalence which mean that the NHS is in crisis (top dashed line) or under considerable pressure (bottom dashed line). These are meant as abstract representations of the general situation, so we haven't indicated the exact values of prevalence they might correspond to. The dashed vertical line corresponds to <em>R</em> being equal to 1.</p>
<p>What do the coloured regions mean? First off, note that our plot is only meant to illustrate the idea that both <em>R</em> and prevalence are important factors, which is why we haven't attached further labels to the axes indicating exactly where one region ends and another begins. </p>
<p><strong> The vertical red strip</strong> on the right represents the situations where <em>R</em> is greater than 1.</p>
<p><strong> The horizontal pink strip</strong> at the top represents a situation where our healthcare capacity is in a perilous situation. Red and pink are situations we want to avoid.
</p>
<p><strong>The green region</strong> in the bottom left corner represents a situation where both prevalence and <em>R</em> are quite low, so lifting restrictions wouldn't land us in the red or pink zones any time soon. </p>
<p><strong>The blue region</strong> represents a situation where the balance between prevalence and <em>R</em> is such that some careful easing of restrictions might be possible without quickly moving into red or pink — there is some breathing room in prevalence, even with <em>R</em> increased a little. </p>
<p> <strong>The yellow region</strong> represents a situation where the NHS is not in crisis and prevalence is not growing, but a small change could push <em>R</em> over 1 and very soon our health service would be at risk. If that happened, we'd need to react quickly — any delay would come at the price of a longer lockdown (see <a href="/content/relaxing-rules" target="blank">this article</a>).
</p>
<p><strong>The narrow horizontal purple strip</strong> under the red represents a situation where, although <em>R</em> is greater than 1, prevalence is essentially zero and so there is no ongoing transmission in the community. There may however be sporadic introductions of the virus, for example through international travellers, which can be detected so that onward transmission can be stopped. This situation would allow us to go about many activities as the risk of infection is low, but it is unstable and requires great care to maintain. Any rise in prevalence would get us into the red zone. To avoid this we'd need to be vigilant, have very good surveillance to pick up introductions of new cases, and take action very quickly.
</p>
<p>"Right now (late February 2021) I think we are in the blue region," says Gog, who is a participant of <a href="https://www.gov.uk/government/organisations/scientific-advisory-group-for-emergencies" target="blank">SAGE</a>, and a member of the <a href="https://www.gov.uk/government/groups/scientific-pandemic-influenza-subgroup-on-modelling" target="blank">SPI-M</a> and <a href="https://maths.org/juniper" target="blank">JUNIPER</a> modelling groups. "We have come down a little bit in prevalence and <em>R</em> is around 0.7 or so across the UK. So we can open something, but not everything — one thing at a time, and then pause to check that we're still OK."</p>
<h3>Why do the regions look like they do?</h3>
<p>Most of the coloured regions in the plot have horizontal or vertical boundaries. These boundaries correspond to threshold values of prevalence or <em>R</em> that are deemed to be important. For example, the left-hand border of the red zone corresponds to <em>R</em>=1, a famously critical value. The top border of the blue region corresponds to a threshold value of prevalence: below it the NHS is under pressure, but above it the NHS is considered to be in crisis.
</p><p>
Exactly where those threshold values lie is a matter of judgment, but this judgment can be informed by sound reasoning and mathematics. For example, <em>R</em>=1 is deemed to be a critical threshold because the maths shows (see <a href="/content/maths-minute-r0-and-herd-immunity" target="blank">here</a>) that the epidemic will shrink when <em>R</em> is less than 1 and grow and when it is greater than 1. That's why it's generally agreed that <em>R</em> greater than 1 isn't a good thing.
</p>
<p>The most interesting boundary is the one separating the blue and yellow regions. Its curvy shape illustrates the interplay between <em>R</em> and prevalence particularly well. Suppose we are somewhere in the top left of the blue region, so the NHS is under pressure. "In this case you don't want to get too close to <em>R</em>=1, because then if you got your estimate of <em>R</em> wrong by, say, 10% you're in big trouble quickly," says Gog. However, if prevalence is low, so we're somewhere near the bottom of the blue region, we can tolerate <em>R</em> being closer to 1. "In that case you have a little bit of a breathing room if we've got it wrong or something new is happening." says Gog. </p>
<p>This is why the right edge of the blue region slopes down gradually, rather than dropping vertically, and extends under the yellow towards the line marking <em>R</em>=1. For lower prevalence, we can risk sailing closer to <em>R</em>=1 as it will take some time for cases to grow again to any given level. Again, mathematical models can inform the exact shape of the boundary between the blue and yellow regions. (For example, converting <em>R</em> into <a href="/content/epidemic-growth-rate" target="blank"><em>growth rate</em></a>, and then using an <a href="/content/compound-infections" target="="blank""><em>exponential model</em></a> to predict the growth in cases, can tell you what combinations of <em>R</em> and prevalence ensure that prevalence doesn't grow above your threshold value too quickly. These combinations can then be part of the blue region.) </p>
<h3>Moving around the plot</h3>
<p>How do we get ourselves from any point in the plot to another, hopefully better, situation? It turns out that, like pawns on a chessboard, we only have a limited number of possible moves we can make. Suppose we are at a given point on the plot and change nothing about our behaviour. Then if we are in a situation where <em>R</em> is less than 1, the epidemic will shrink and prevalence will go down. The value of <em>R</em> won't be affected any time soon — the only way <em>R</em> can change without us changing our behaviour is through people becoming immune by becoming infected, and that takes a long time. This means that doing nothing when <em>R</em> is less than 1 takes us vertically down on the plot, to a lower prevalence. </p>
<p>If we are in a situation where <em>R</em> is greater than 1 then not changing our behaviour will allow the epidemic to grow, and so prevalence goes up. This means we go vertically up in the plot. And the further right we are, the faster upwards we will go.</p>
<div class="centreimage"><img alt="Prevalence versus R" width="600" height="519" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/square/angry_final2_blur.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 600px;">Not changing our behaviour will lead to prevalence going down or up, depending on whether <em>R</em> is less than or greater than 1.</p>
</div><!-- Image made by MF -->
<p>The other thing we can do is loosen restrictions or tighten them up: this leads to a horizontal move on the plot, either to the right or left. "Suppose we are in the top bit of the blue region and reopen some of the things that are currently closed, then that steps us to the right. The big question is always how far." says Gog. "If we are in the red region and decide to lock down, then that brings <em>R</em> down, so that steps us to the left."</p>
<div class="centreimage"><img alt="Prevalence versus R" width="600" height="534" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/square/angry_final3_blur.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 600px;">Loosening restrictions steps us to the right and tightening them steps us to the left.</p>
</div><!-- Image made by MF -->
<p>"So if we are somewhere inside the red region and want to bring prevalence down we can't just step down, that's not an option. We have to go left first to reduce <em>R</em> and then wait for prevalence to come down." Working out how far a change in our behaviour can take us to the left or right, and how long we have to wait for prevalence to go up or down when we do nothing, is exactly the kind of question disease modellers like Gog and House are trying to answer using mathematical models.</p>
<p>
Vaccination is of course another tool in our armoury against COVID-19, so where will it leave us on the plot?
Gog hopes that it might get us to the bottom part of the yellow or blue regions if we are unlocking at a careful pace, paying attention to <em>R</em> and prevalence as we go. We should be able to get to where <em>R</em> is less than 1 and prevalence low. "But what we don't know yet is if vaccination is enough to let that happen and have most/all of our current interventions removed. It will depend on how effective the vaccine is, who is offered vaccination, and who agrees to be vaccinated." The answer will become clearer over the next few months, as we can see the effects of the vaccination programme and also gradual re-openings.
</p>
<p>
As we said above, our plot is only schematic which is why we haven't made explicit where one region ends and another starts (apart from the line corresponding to <em>R</em>=1). The exact locations of boundaries between regions will depend on many factors, such as health care capacity and what proportion of cases end up needing hospital care, which may vary both in space and time. However, the schematic plot serves as quite a good conceptual map illustrating the interaction between <em>R</em> and prevalence.
</p>
<p>One question it raises is where on the plot we'd be happy to settle come, say, the summer? If low prevalence is a goal, then the answer is somewhere in the bottom part. But how far right are we willing to go? Aiming for <em>R</em> less than 1 could mean keeping some restrictions in place longer term, and letting it go above 1 will run the risk of explosive outbreaks and sudden and hard lockdowns. With all that's been going on it's a question most of us probably haven't had time to think about yet. But it's something that public debate will soon need to consider.</p>
<hr /><h3>About this article</h3>
<p><a href="https://www.infectiousdisease.cam.ac.uk/directory/jrg20@cam.ac.uk" target="blank">Julia Gog</a> is Professor of Mathematical Biology at the University of Cambridge. </p><p><a href="https://www.research.manchester.ac.uk/portal/thomas.house.html" target="blank">Thomas House</a> is Reader in Mathematical Statistics at the Department of Mathematics at the University of Manchester. </p><p>
Gog and House are members of the <a href="https://maths.org/juniper/" target="blank">JUNIPER modelling consortium</a> and the modelling group <a href="https://www.gov.uk/government/groups/scientific-pandemic-influenza-subgroup-on-modelling" target="blank">SPI-M</a>, and they contribute to the <a href="https://www.gov.uk/government/organisations/scientific-advisory-group-for-emergencies" target="blank">Scientific Advisory Group for Emergencies</a> (SAGE).</p><p>
</p><p><a href="/content/people/index.html#marianne" target="blank">Marianne Freiberger</a> is Editor of <em>Plus</em>.</p>
<p><em>This article was produced as part of our collaboration with <a href="https://maths.org/juniper/">JUNIPER</a>, the Joint UNIversity Pandemic and Epidemic Response modelling consortium. JUNIPER comprises academics from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster, who are using a range of mathematical and statistical techniques to address pressing question about the control of COVID-19. You can see more content produced with JUNIPER
<a href="/content/juniper">here</a>.</em></p>
<div class="centreimage"><img alt="Juniper logo" width="400" height="83" class="b-lazy" data-src="/content/sites/plus.maths.org/files/packages/2021/Juniper-logos/juniper-light-bg.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 400px;"></p></div></div></div></div>Fri, 26 Feb 2021 12:41:43 +0000Marianne7434 at https://plus.maths.org/contenthttps://plus.maths.org/content/R-not-all#commentsOn the mathematical frontline: Julia Gog
https://plus.maths.org/content/mathematical-frontline-julia-gog
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/julia_small_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width: 350px;"><img alt="Julia Gog" width="350" height="233" class="b-lazy" data-src="/content/sites/plus.maths.org/files/packages/2017/Women/Julia_small.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p>Julia Gog. Photograph by <a href="http://henrykenyonphotography.com">Henry Kenyon</a>.</p>
</div>
<p>Over the last year we have done a lot of reporting on the maths of the COVID-19 pandemic. Behind the maths there are of course people — those mathematicians who make the epidemiological models that do (and sometimes do not do not) inform government policy, who are grappling with the unprecedented challenge of coming to grips with a live pandemic unfolding in front of their eyes. </p>
<p>Our new podcast series, called <em>On the mathematical frontline</em>, is about those people. It explores the maths they do, how they go about it, and the impact it has on their personal lives.</p><p>
The first person we speak to in this new series is <a href="https://www.infectiousdisease.cam.ac.uk/directory/jrg20@cam.ac.uk" target="blank">Julia Gog</a>, Professor of Mathematical Biology at the University of Cambridge, participant of SAGE and member of the epidemic modelling group SPI-M. </p>
<p>Gog is also a founding member of the <a href="https://maths.org/juniper/" target="blank">JUNIPER</a> modelling consortium we are <a href="/content/joining-forces-covid19" target="blank">collaborating</a> with, and which you'll hear more about in the podcast.</p>
<p>So what is it like working on the mathematical frontline? Find out more with Julia Gog!</p>
<p><a href="/content/sites/plus.maths.org/files/podcast/2021/Gog/julia_gog.mp3"><strong>Listen to the podcast</strong></a>
</p><hr /><p><em>The podcast is part of our <a href="/content/joining-forces-covid19" target="blank">collaboration</a> with <a href="https://maths.org/juniper/" target="blank">JUNIPER</a>, the Joint UNIversity Pandemic and Epidemic Response modelling consortium. JUNIPER comprises academics from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster, who are using a range of mathematical and statistical techniques to address pressing questions about the control of COVID-19. You can see more content produced with JUNIPER
<a href="/content/juniper" target="blank">here</a>.</em></p>
<div class="centreimage"><img alt="Juniper logo" width="400" height="83" class="b-lazy" data-src="/content/sites/plus.maths.org/files/packages/2021/Juniper-logos/juniper-light-bg.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 400px;"></p></div></div></div></div><div class="field field-name-field-remote-encl field-type-file field-label-inline clearfix clearfix"><div class="field-label">Podcast download link: </div><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="Audio icon" title="audio/mpeg" src="/content/modules/file/icons/audio-x-generic.png" /> <a href="https://plus.maths.org/content/sites/plus.maths.org/files/podcast/2021/Gog/julia_gog.mp3" type="audio/mpeg; length=20816439">julia_gog.mp3</a></span></div></div></div>Mon, 22 Feb 2021 10:31:58 +0000Marianne7435 at https://plus.maths.org/contenthttps://plus.maths.org/content/mathematical-frontline-julia-gog#commentsJoining forces to fight COVID-19
https://plus.maths.org/content/joining-forces-covid19
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/screenshot_2021-02-18_at_10.13.03.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"><p>One of our earliest COVID memories is talking to epidemiologist <a href="https://www.infectiousdisease.cam.ac.uk/directory/jrg20@cam.ac.uk" target="blank">Julia Gog</a> in mid-March last year.
"Life is not going to be the same for a long time," Gog said back then, pointing to the potential death toll and multiple lockdowns for months, or years, to come. It was then that the gravity of what was about to unfold began to dawn on us.</p>
<div class="rightimage" style="max-width: 350px;"><img alt="JUNIPER members" width="350" height="357" class="b-lazy" data-src="/content/sites/plus.maths.org/files/news/2021/Juniper/junipers.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p>JUNIPER principal and co-investigators. Left to right and top to bottom: E. Brooks-Pollock, H. Christensen, L. Danon, D. De Angelis, L. Dyson, G. Gog, I. Hall, D. Hollingsworth, T. House, C. Jewell, M. Keeling, P. Klepac, TJ McKinley, L. Pellis, J. Read, M. Tildesley. See <a href="https://maths.org/juniper/people" target="blank">this list </a> for affiliations.</p>
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<p>
Nearly one year later, we have worked on multiple articles with Gog and her colleagues, and are now
proud to announce a collaboration with some of the best minds in epidemiology in the country. The <a href="https://maths.org/juniper/" target="blank">Joint Universities Pandemic and Epidemiological Research modelling consortium</a> (JUNIPER), which Gog helped to form, comprises <a href="https://maths.org/juniper/people" target="blank">16 academics</a> from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster. They are using a range of mathematical and statistical techniques to address pressing questions about the control of COVID-19. </p>
<p>JUNIPER members represent experienced research groups from these universities that have been generating predictions, forecasts and insights feeding into the Scientific Pandemic Influenza Group on Modelling (SPI-M) and the Scientific Advisory Group for Emergencies (SAGE), both of whom advise the UK government on scientific matters relating to the UK's response to the pandemic. Members continue to respond to rapid requests from the UK government via SPI-M and SAGE. This includes providing weekly forecast of the reproductive number <em>R</em> and growth rate of COVID-19 in the UK and predictions of the effect interventions have on the spread of the disease.</p>
<p><em>Plus's</em> job in the collaboration with JUNIPER is to communicate the work of JUNIPER members to the general public, policy makers, and whoever else wants or needs to know. As JUNIPER has been coming together over the last few months we already started on this work. You can see a list of existing articles produced in collaboration with JUNIPER members <a href="/content/juniper-news#list">below</a>. We'll be continually reporting on JUNIPER activities (and update the content on <a href="/content/juniper" target="blank">this page</a>), and we're particularly looking forward to JUNIPER's first <a href="http://www.newton.ac.uk/event/ooew03">research meeting on February 23</a>. The meeting is free for anyone to attend, if you are interested then please <a href="http://www.newton.ac.uk/event/ooew03">register here</a>.</p>
<p>"I can already see how brilliantly JUNIPER is working together; how many collaborations are emerging," says Gog. "What we are doing is turbo-charged." One aim of JUNIPER is to react to current issues relevant to COVID-19, such as vaccines, new strains of the virus, or new data sets becoming available. "But JUNIPER also allows us to be a little more forward-looking," says Gog. "As academic modellers we can see topics on the horizon that are still three or four months from [becoming relevant], but are really important, so we need to start work on them right now." As well as focussing on the current pandemic, JUNIPER also aims to help train the next generation of mathematicians to build the UK’s capacity for future epidemics and pandemics.
</p>
<p>"Communicating what we do to non-expert audiences, be they policy makers, civil servants, or the general public, is a key part of JUNIPER's work," says <a href="https://warwick.ac.uk/fac/sci/maths/people/staff/matt_keeling/" target="blank">Matt Keeling</a>, who is Principal Investigator on JUNIPER along with Gog. "By integrating the collaboration with <em>Plus</em> in JUNIPER from the outset we are making sure we achieve that goal. We are very excited about working with <em>Plus</em>."</p>
<p>Here at <em>Plus</em> we are honoured to be able to contribute to JUNIPER. In the many years we have been editing <em>Plus</em>, it's never seemed as important as now to communicate mathematical concepts to as broad an audience as possible.</p>
<p><em><a href="https://www.infectiousdisease.cam.ac.uk/directory/jrg20@cam.ac.uk" target="blank">Julia Gog</a> is Professor of Mathematical Biology at the University of Cambridge. <a href="https://warwick.ac.uk/fac/sci/maths/people/staff/matt_keeling/" target="blank">Matt Keeling</a> is Professor at the University of Warwick, holding a joint position in Mathematics and Life Sciences. He is also the current director of the Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research. Both are Principal Investigators on Juniper, members of the modelling group <a href="https://www.gov.uk/government/groups/scientific-pandemic-influenza-subgroup-on-modelling" target="blank">SPI-M</a> and contributors to <a href="https://www.gov.uk/government/organisations/scientific-advisory-group-for-emergencies" target="blank">SAGE</a></em>.</p>
<div class="centreimage"><img alt="Juniper logo" width="400" height="83" class="b-lazy" data-src="/content/sites/plus.maths.org/files/packages/2021/Juniper-logos/juniper-light-bg.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 400px;"></p></div>
<h3>Content produced so far in collaboration with JUNIPER</h3> <p>See <a href="/content/juniper" target="blank">this page</a> for an updated list of JUNIPER articles and podcasts.</p>
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<div class="leftimage" style="width: 100px;"><img alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/vaccine_icon.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p><a href="/content/covid-19-vaccines-your-questions-answered">The COVID-19 vaccines: Your questions answered</a> — Are they safe? Are they effective? Will they stop the pandemic? Find out with our FAQ informed by experts.</p></div>
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<div class="leftimage" style="width: 100px;"><img alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/closed_icon.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p><a href="/content/relaxing-rules">What's the price for relaxing the rules?</a> — After weeks of tight restrictions we are all longing to go into a lower tier, but this can come at a high price later on.</p></div>
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<div class="leftimage" style="width: 100px;"><img alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/exponentialicon.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<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>
<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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/hops_icon.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<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>
<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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_38.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<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></div></div></div>Thu, 18 Feb 2021 10:37:03 +0000Marianne7432 at https://plus.maths.org/contenthttps://plus.maths.org/content/joining-forces-covid19#commentsWhy study mathematics?
https://plus.maths.org/content/why-study-mathematics-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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_21.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">Author </div><div class="field-items"><div class="field-item even">Vicky Neale</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>A friend of Plus, Vicky Neale, has written a book about why people study maths at university, and to give a taste of what it's like to study a maths degree. We asked her to tell us about her book and why she wrote it...</em></p>
<div class="rightimage"><img width="250px" alt="Why Study Mathematics? by Vicky Neale" class="b-lazy" data-src="/content/sites/plus.maths.org/files/reviews/2021/whystudymaths_vickyneale.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p>
Maths is a versatile subject, with different flavours that appeal to different people with different tastes. It equips graduates with skills that employers value. It's full of fascinating ideas and powerful applications, and the process of understanding a new mathematical concept or solving a problem using maths is enormously satisfying. Whatever your priorities – whether you're looking to help other people, to earn a lot, to explore a creative subject or to make a difference in society – maths has something to offer you. The study of maths is rewarding in and of itself, and it gives you lots of options for the future.
</p><p>
At school, there are standard requirements about what students have to learn: there's a national curriculum. This isn't the case at university. Universities have significant flexibility when it comes to how they organise degree programmes. There's a huge variety of courses – offering differences in mathematical content and emphasis, in teaching style and in assessment methods – that can lead you to a maths degree. This means that it's really important to research the available options before you apply, to find courses that'll suit you.
</p><p>
There's a lot of flexibility about which topics are covered in a maths degree. In the UK, the only topics specifically mentioned in the <a href="https://www.qaa.ac.uk/quality-code/subject-benchmark-statements">QAA benchmark statement</a> (which sets out expectations for maths degrees) are calculus and linear algebra. Beyond that, it's up to individual universities to design appropriate degree courses.
</p><p>
Different people have different mathematical tastes. Some like nothing more than to get their hands on a large data set, to interrogate it in order to see what conclusions they can draw, and to consider the robustness of those conclusions. Others are motivated by a particular application and spend time exploring which mathematical tools can help to answer the questions they find exciting in that area. Others still are fascinated by the beautiful, fundamental questions in this subject (it's surprising just how many fundamental questions there are for which we still don't have complete answers) and devote themselves to curiosity-driven maths.
</p><p>
Within a maths degree, you're likely to have opportunities to experience all of these facets. However, degree programmes do differ in the emphasis they place on each aspect of maths, as well as in the number of options they offer to students, so it's really worth thinking about what style of course will suit you. Are you motivated by the use of maths in industry and other applications, and therefore interested in building a mathematical toolkit for that purpose, or would you relish delving into the background theory of how and why the tools work? You’re not restricted to one or the other: many courses combine elements of both. But when you're researching courses, it can be helpful to consider the extent to which each might be described as "theory-based" or "practice-based", because viewing courses through that lens might help you to focus on the ones that'll suit you.
</p><p>
You might have heard people talking about there being a jump in difficulty from school maths to university maths. While there are certainly differences between the two, this isn't the same as the latter representing a massive step up in difficulty. Some of the differences might be to do with teaching approaches, with the style of questions you're being asked to tackle, or with the amount of independent study you’re being expected to undertake. Others might be due to the fact that you're meeting new topics.
</p><p>
It's important to remember that universities know the move from secondary to higher education involves a transition to new material and new ways of working: degree programmes are designed to support students making this transition. Universities are familiar with the profiles of their incoming students and what they can expect them to know or to be able to do, and they tailor their programmes accordingly. Having said that, you'll still need to adapt to new ways of working, to be willing to persevere and to ask for support when you need it. There's no need to panic, though. Over the course of a three- or four-year maths degree, you'll become increasingly independent and develop your study skills and mathematical sophistication. You don't need to have done all this before the first day of your degree!
</p><p>
Deciding which subjects you'd like to pursue is just one of the many factors you'll need to consider when choosing a course of study. For instance, there are a variety of ways these subjects can be combined to form a degree that suits your interests. And there are many other factors to consider, such as how teaching and assessment are structured, what opportunities you’ll have to interact with staff and your fellow students, and what the career possibilities are.
</p><p>
I wrote <em><a href="https://www.amazon.co.uk/Why-Study-Mathematics-Vicky-Neale/dp/191301911X/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=&sr=">Why Study Mathematics?</a></em> in the hope that it will help you to find out more about maths at university. In Part I (from which the extract above is taken), we explore the practicalities of a maths degree. What's involved in studying a maths degree? What topics might you study? What teaching methods and types of assessment might you encounter? How do you choose between the wide variety of maths degree courses on offer? What makes a good maths student? What careers are open to maths graduates?
</p><p>
In Part II, we look more closely at some of the topics you might study at university, providing a taste of the theoretical underpinnings of maths and offering insight into its diverse applications: in medicine and health care, in digital communication, in engineering, in tackling climate change, and more. One chapter concentrates on topics that are common to pretty much all maths degrees, illustrating the relevance of differential equations to modelling the spread of disease, connections between linear algebra and JPEG image compression, and the surprising use of non-real complex numbers to very real problems in aerodynamics. Other chapters illustrate further aspects of the mathematical sciences, from the use of operational research to facilitate kidney donation to the role of data science in genetics and retail, from compressed sensing to improve medical imaging to non-Euclidean geometry and uncountably infinite sets. Don't worry if you haven’t heard of all these mathematical ideas – these are topics you might meet in a maths degree, not ones you need to understand before you study one!
</p><p>
If you're a student looking to make a decision about university, or you are supporting someone who is making that decision, I hope that this book will give you a clearer picture of why a maths degree is a good option for many people. And if you are the one who's thinking about embarking on this adventure, then I would like to wish you all the very best with your mathematical studies.
</p>
<dl><dt><strong>Book details:</strong></dt>
<dd><em>Why Study Mathematics?</em></dd> <dd>Vicky Neale</dd> <dd>paperback, kindle — 208 pages</dd>
<dd>London Publishing Partnership (2020)</dd>
<dd>ISBN 978-1913019112</dd>
</dl><p><em>You can also watch a short interview with Vicky when she told us about her favourite mathematical moments.</em></p>
<iframe width="560" height="315" frameborder="0" allowfullscreen="" class="b-lazy" data-src="https://www.youtube.com/embed/CnUoAeOWcZY" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="></iframe>
<hr /><h3>About this article</h3>
<div class="rightimage"><img width="300px" height="300px" alt="Vicky Neale" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/Vicky/vicky_photo.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p>Vicky Neale</p></div>
<p><a href="http://people.maths.ox.ac.uk/neale/">Vicky Neale</a> is the Whitehead Lecturer at the Mathematical Institute, University of Oxford, and a Supernumerary Fellow at Balliol College. She teaches pure mathematics to undergraduates, and combines this with work on public engagement with mathematics: she gives public lectures, leads workshops with school students, and has appeared on numerous BBC radio and television programmes. One of her current interests is in using knitting, crochet and other craft to explore mathematical ideas – her two cats enjoy joining in.</p></div></div></div>Wed, 10 Feb 2021 12:07:11 +0000Rachel7426 at https://plus.maths.org/contenthttps://plus.maths.org/content/why-study-mathematics-0#commentsAmaze your friends! Astound your family!
https://plus.maths.org/content/amaze-your-friends-astound-your-family
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_23.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"><p>
Are you a card shark? Or do you fumble your shuffles? Do you want to learn some magic to impress your friends? Then read these articles to uncover the maths, and magic, hidden within a simple pack of cards...
</p>
<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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/magic_icon.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p><a href="/content/magic-shuffling">The magic of shuffling</a> — Want to shuffle like a professional magician? Find out how to shuffle perfectly, imperfectly, and the magic behind it.</p></div>
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<div class="leftimage" style="width: 100px;"><img alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_22.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p><a href="/content/mathematics-shuffling">The mathematics of shuffling</a> — Take a journey into the maths of card shuffling and experience how a mathematician thinks.</p></div>
<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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_19.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p><a href="/content/how-pretend-memorise-pack-cards">How to (pretend to) memorise a pack of cards</a> — Learn a neat way of using maths to pretend to memorise a pack of cards.</p></div>
<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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/sleeve_icon.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p><a href="/content/probably-magic">Probably magic!</a> — When you shuffle a deck of cards chances are the order of cards you produced has never been produced before! Find out why and learn a card trick too!</p></div></div></div></div>Wed, 03 Feb 2021 13:54:49 +0000Rachel7431 at https://plus.maths.org/contenthttps://plus.maths.org/content/amaze-your-friends-astound-your-family#commentsMaths in a minute: Cyclic groups
https://plus.maths.org/content/maths-minute-cyclic-groups
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/clock_icon_2.jpg" width="100" height="99" 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><em>This article is about understanding abstracts objects called <em>groups</em>. If you don't know what they are, then you might want to read <a href="/content/maths-a-minute-groups" target="blank">this brief explanation</a>.</em></p>
<p>Some things go round and round. An example are the hours of a clock. If you start at 12 and add one hour you get to 1, add another hour and you get to 2, and so on, until after twelve additions you're back to where you started.</p>
<p>You could also think of this in terms of rotating the clock face through a twelfth of a circle. One rotation moves 12 to 1, another rotation moves 1 to 2, and so on, until after twelve rotations you get back to where you started.</p>
<p>In each case you have a collection of twelve things: twelve numbers, or twelve rotations (the rotations are through 1/12 of a circle, 2/12 of a circle, 3/12 of a circle, and so on). You also have a way of combining one thing with another to get a third: adding two numbers, or following one rotation by another. And you have one very special thing that lets you generate all the others: you can get to any of the twelve numbers from 1 by repeatedly adding one, and similarly, repeatedly performing rotations through a twelfth of a circle gives you all the other rotations in your collection.</p>
<div class="rightimage" style="width: 300px;"><img src="/issue53/puzzle/confused.jpg" alt="Clock" width="300" height="298" /><p>A clock gives a good example of a cyclic group</div>
<p>If a collection of objects, together with a <em>binary operation</em> that combines two objects to give you a third, adheres to certain rules, then mathematicians call it a <em>group</em> — you can find out more about these rules <a href="/content/maths-a-minute-groups">here</a>. And if a group contains a special object that, through repeated application of the binary operation can generate all other objects in the group, then the group is called a <em>cycling group</em>.</p>
<p>Cyclic groups exist in all sizes. For example, a rotation through half of a circle (180 degrees) generates a cyclic group of size two: you only need to perform the rotation twice to get back to where you started. Similarly, a rotation through a 1/1,000,000 of a circle generates a cyclic group of size 1,000,000. Generally, a rotation through a 1/<em>n</em>th of a circle, where <em>n</em> is any positive integer, generates a cyclic group of order <em>n</em>.</p>
<p>Is there also an infinite cyclic group? We could try making one by taking the number 1, and instead of imagining it to be part of the clock face where things go round and round, we imagine it as the ordinary number 1 on the number line. By repeatedly adding 1s you can get to each of the infinitely many positive integers. So do the positive integers form an infinite cyclic group generated by 1?</p>
<p>The answer is no. That's because the positive integers don't form a group in the first place. According to the <a href="/content/maths-a-minute-groups">definition of a group</a>, a group must contain an <em>identity element</em>, and every other element must have an <em>inverse</em> (see <a href="/content/maths-a-minute-groups">here</a> to find out what that means). For the integers, the identity element is 0, and the inverse of every other element is the negative of that number. So to get a group we must include, not just the positive integers, but also 0 and the negative integers.</p>
<p>Now if we are more generous with the definition of a cyclic group, allowing not just one element but also its inverse to help us get to all the other elements, then the integers are an infinite cyclic group generated by 1 (with the help of its inverse -1). Starting with 1 and -1 we can get to any other integer by repeatedly adding 1s or -1s.</p>
<p>This more general definition is the official definition of a cyclic group: one that can be constructed from just a single element and its inverse using the operation in question (e.g. addition or composing rotations). Note that for finite groups the two definitions coincide because the inverse of the generating element can itself be constructed from that generating element. For example, for the twelve numbers on the clock, the identity element is 12: if you add 12 to any number in this group, the number remains unchanged. The inverse of 1 is 11, because 1+11=12. Since you can get from 1 to 11 by adding 1s, this means that 1 generates its own inverse and is therefore enough to give you the whole group.</p>
<hr/>
<h3>About this article</h3>
<p><em>This article is part of our collaboration with the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more.</em></p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" alt="INI logo" width="400" height="51" />
<p style="max-width: 400px;"></p></div></div></div></div>Mon, 01 Feb 2021 14:22:14 +0000Marianne7416 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-cyclic-groups#commentsMaths in a minute: Representing groups
https://plus.maths.org/content/maths-minute-representing-groups
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icosa_icon.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"><p><em>This article is about understanding abstracts objects called <em>groups</em>. If you don't know what they are, then you might want to read <a href="/content/maths-a-minute-groups" target="blank">this brief explanation</a>.</em></p>
<div style="float: right; width: 50%; margin-right: 10px;"><iframe scrolling="no" width="366px" height="412px" style="border:0px;" class="b-lazy" data-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" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="> </iframe><p style="width: 400px; font-size: small; color: purple;">Move the slider to rotate the 12-gon. Whenever you have rotated through a multiple of 1/12th of the full circle, the 12-gon looks the same.</p></div>
<p>In the introduction to groups we just mentioned we gave two examples of a group. One example came from the 12 hour clock. Here the group consists of the numbers 1 to 12 with the operation of addition <a href="/content/maths-minute-modular-arithmetic"><em>modulo</em> 12</a>, which means that when you have gone around the clock face once, instead of carrying on counting 13, 14, 15, etc, you start from the beginning again, counting 1, 2, 3, etc.</p>
<p>The other example came from a regular 12-gon — that's a shape in the plane with twelve sides that all have the same length, and twelve internal angles that are the same size (see the image on the right). Rotations around the centre point of the 12-gon through angles of 1/12 of a circle, 2/12 of a circle, 3/12th of a circle, etc, up to the rotation through a full circle, leave the 12-gon unchanged so they are <em>symmetries</em> of the 12-gon. These rotational symmetries also form a group. The operation for combining two group elements here is simply to perform one rotation after the other. </p>
<p>It's not too hard to see that our two examples of a group are very similar. Each consists of twelve elements and you can associate to an element of one group exactly one element from the other group and vice versa: the number 1 can be associated to the rotation through 1/12 of a circle, the number 2 to the rotation through 2/12 of a circle, and so on. </p>
<p>This <em>one-to-one correspondence</em> between elements of the two groups respects the operations on the groups. For example, adding the numbers 2 and 5 gives you 7, and combining the rotations associated to 2 and 5, through 2/12th and 5/12th of a circle, gives you the rotation through 7/12th of a circle, which is exactly the rotation associated to the number 7. When to groups behave
the same in this way, we say that they are <a href="https://en.wikipedia.org/wiki/Group_isomorphism"><em>isomorphic</em></a>.</p>
<div class="leftimage" style="max-width: 250px;"><img alt="Sixteenth stellation of icosahedron." width="250" height="240" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/representation/501px-sixteenth_stellation_of_icosahedron.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p>The collection of symmetries of an object form a group. This image shows the sixteenth <a href="https://en.wikipedia.org/wiki/Icosahedron#Stellated_icosahedra">stellation of the icosahedron</a>. Image: <a href="https://commons.wikimedia.org/wiki/File:Sixteenth_stellation_of_icosahedron.png">Jim2k</a>.</p></div>
<p>As you can imagine, this multi-lingual way of talking about groups can result in a bit of a mess. When you study groups as abstracts objects, not linked to a particular example like a clock face or 12-gon, it would be good to agree to describe them using just one type of mathematical object. Luckily it turns out that every group can be written down in terms of matrices — these are arrays of numbers any two of which can be combined in a particular way (see <a href="https://www.mathsisfun.com/algebra/matrix-introduction.html">here</a> to find out more). </p>
<p>As an example, if you place your 12-gon in a co-ordinate system so that its centre lies at the point <img alt="$(0,0)$" style="vertical-align:-4px; width:34px; height:18px" class="math gen b-lazy" data-src="/MI/980295f2abedc75eb468d2f3333358b4/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />, then the clockwise rotation through 1/12th of a circle can be written as the matrix </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7"><tr><td style="width:40%"> </td>
<td><img alt="\[ \left( \begin{array}{cc}cos(\pi /6) & sin(\pi /6) \\ -sin(\pi /6) & cos(\pi /6)\end{array} \right), \]" style="width:200px; height:40px" class="math gen b-lazy" data-src="/MI/980295f2abedc75eb468d2f3333358b4/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr></table><p>where angles are measured in radians. Similarly every other element of the group can be written as a matrix. Isomorphic groups can be represented by the same group of matrices. </p>
<p><em>Representation theory</em> is all about studying groups by representing them as groups of matrices. This not only gives us a unified language in which to talk about groups, but also makes it easier to understand them because matrices are objects that mathematicians know intimately and understand well.</p>
<hr /><h3>About this article</h3>
<p><em>This article is part of our collaboration with the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more.</em></p>
<div class="centreimage"><img alt="INI logo" width="400" height="51" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 400px;"></p></div></div></div></div>Mon, 01 Feb 2021 11:49:58 +0000Marianne7428 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-representing-groups#commentsGhosts in the tiling - continued
https://plus.maths.org/content/ghosts-tiles-continued
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/tricurves_icon_0.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">Tim Lexen</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>In the <a href="/content/ghosts-tiles" target="blank">first part of this article</a> we learnt what tricurves are, had a look at their phantoms, and asked how both shapes might be used to tile the plane. We'll now go on to see how we can tile other shapes.</em></p>
<h3>Filling a Circle</h3>
<p>
We can start with any lens shape, fill it with a tricurve and get two left-over smaller lenses. Each smaller lens can in turn be filled with a tricurve with two left-over smaller lenses, and so on. Thus any lens can be filled with a series of ever-smaller tricurves. This also applies to the circle, which is itself a lens.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/fig_3_levels_of_lenses.jpg" alt="Levels of lenses" width="319" height="148" /><p style="max-width: 200px;">Levels of lenses.</p>
</div>
<p>Filling a circle can be done in various ways. We'll use the convention here of starting with the largest possible tricurve at the top, then working down and out toward the lower half perimeter of the circle. We'll work downward and generally outward from the centre, leaving "remainder" small lenses against the perimeter of the bottom half. These perimeter lenses are then filled with progressively small tricurves, in an infinite series (of ever more of smaller tricurves at each "level"). </p>
<p>Let's look at four approaches, going down only to a certain level of perimeter lenses.</p>
<p>Case A is done symmetrically, with arcs divided by half at each level, as shown below: the first tricurve has arc angles 90°-90°-180°, the second 45°-45°-90°, and the third 22.5-22.5°-45°.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/fig_4_symmetrcial_filling_of_a_circle.jpg" alt="Symmetrical filling of a circle" width="519" height="220" /><p style="max-width: 200px;">Symmetrical filling of a circle.</p>
</div>
<p>Alternatively, we can keep the smallest arc the same at 22.5° for all the main tricurves. This may be done with the small-end tails (with angle also 22.5°) emanating from the same point (Case B, below left), or with the large tricurves alternating (Case C, below right):</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/fig_5_cases_b_and_c.jpg" alt="Cases B and C" width="462" height="228" /><p style="max-width: 200px;">Cases B and C.</p>
</div>
<p>Note these two examples use the same seven main tricurves, just arranged differently. The top tricurve in both cases is 22.5°-157.5°-180°.</p>
<p>Also note that all three above examples have the same number and size of lenses in the lower half perimeter; this happens to be eight 22.5° lenses, but these could be any sizes and number — or combination — that fill the lower 180°. These lenses in turn will eventually be filled with a succession of smaller tricurves.</p>
<p>Case D is a variation of Case B, but with ever-thinner main tricurves. For starters we can use tricurves with a 5° small angle (and of course 5° small arc), so the largest tricurve is 5°-175°-180°: </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/thin_tricurves.jpg" alt="Thin tricurves" width="400" height="353" /><p style="max-width: 200px;">Thin tricurves.</p>
</div>
<p>The figure above doesn't show the 5° perimeter arcs and lenses in the lower half of the circle, since they are hard to distinguish at that size — and we'll make them yet smaller. The figure illustrates the idea, but eventually, as the size of the small arc goes to zero, the number of these main tricurves will go to infinity, and the perimeter lens size goes to zero. So there will be no perimeter lenses to fill. This seems the simplest and most elegant way to fill the circle.</p>
<p>Now, what are the phantoms doing?</p>
<p>
Consider the first three circle filling ways above (Cases A-C). The phantoms for these tricurves develop as shown here: </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/case_a_phantoms.jpg" alt=" Case A phantoms" width="600" height="298" /><p style="max-width: 600px;">Case A phantoms.</p>
</div>
<p>In case A above, with symmetric tricurves, the tracking of the phantoms is somewhat easier, since each phantom is a mirror image of the original, placed on the line of symmetry that runs through the original tricurve. </p>
<p>The phantoms for the tricurves in Cases B and C are shown below:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/phantoms_for_cases_b_and_c.jpg" alt=" Case A phantoms" width="600" height="245" /><p style="max-width: 600px;">Phantoms for cases B and C.</p>
</div>
<p>The resulting shape outline is the same for all three above cases — what is happening here?</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/mysterious_outline_of_phantoms.jpg" alt="Mysterious outline of phantoms" width="400" height="229" /><p style="max-width: 600px;">Mysterious outline of phantoms.</p>
</div>
<p>Note what happens as we continue by filling the circle's perimeter 22.5° lenses with tricurves. Assume for now that we keep filling the successive lenses with ever smaller tricurves that are symmetrical. Using Case A as an example:
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/filling_the_cracks.png" alt="Filling in the cracks " width="400" height="279" /><p style="max-width: 600px;">Filling in the cracks.</p>
</div>
<p>The effect is the same for all three cases A to C. With each level, the phantoms are filling in between bumps on the outside of the new shape, or extending the outer tips upward, at the limit approaching a full half circle with twice the diameter of the original circle. This is true even if the tricurves at any level aren't symmetrical or systematically sized. Thus the phantoms create and fill the same new shape: a symmetrical <a href="https://en.wikipedia.org/wiki/Arbelos"><em>arbelos</em></a> (a region bounded by three semi-circles), upside down.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/resulting_superphantom.jpg" alt="The resulting superphantom" width="400" height="279" /><p style="max-width: 600px;">The resulting superphantom.</p>
</div>
<p>We get the same results when Case D is taken to the limit: as the number of tricurves in the circle approaches infinity, the phantom shape approaches the filled symmetrical arbelos:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/case_d_conversion.jpg" alt="Case D conversion" width="400" height="207" /><p style="max-width: 600px;">Case D conversion.</p></div>
<p>We could presumably do this in reverse, although it would be a little harder: we could start with the symmetric arbelos, fill it with tricurves in an infinite series, make the phantoms of those tricurves, and end up back at the filled circle. This is true regardless of how the circle is filled with tricurves. Since the filled symmetrical arbelos is the union of all the phantoms tricurves in the filled circle, let's call the symmetrical arbelos the <em>superphantom</em> of the circle. And vice versa: it's reversible. Thus the superphantom of a shape is the outline of the union of the phantoms of the infilling tricurves of that shape.</p>
<h3>Back to lenses</h3>
<p>What we did above for a circle we can do for any lens. Let's go back to case B above and look at the
next lens down from the full circle, outlines in orange in the figure below. The superphantom of this red lens (down to the 22.5° level) is shown here:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/sublens_and_its_superphantom.jpg" alt="Sub-lens and its superphantom" width="400" height="296" /><p style="max-width: 600px;">Sub-lens and its superphantom.</p></div>
<p>The lens and superphantom a few more levels down look like this:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/at_another_level.jpg" alt="Sub-lens and its superphantom" width="400" height="297" /><p style="max-width: 600px;">At another level.</p>
</div>
<p>If all the lenses at different levels are pulled out and put in the same orientation, they look like this:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/fig_21_lenses_and_phantom_sets_of_case_b.jpg" alt="lenses and phantom sets of case B" width="624" height="131" /><p style="max-width: 600px;">Lenses and phantom sets of case B.</p>
</div>
<p>There are again a number of ways to fill the lens with tricurves, and again we can fill it below the 22.5° lens level. As we did for the circle, we take it to the limits. In the end, for any lens, the superphantom is a generalised symmetrical arbelos, where the arcs are something less than 180°. This has the three arcs with the same angle as the original lens. The relation of the lens to its superphantom can be laid out as shown below, compared to a similar tricurve and its phantom.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/fig_22_lens_and_its_superphantom.jpg" alt="Lens and its superphantom" width="463" height="229" /><p style="max-width: 600px;">Lens and its superphantom.</p>
</div>
<h3>Subtracting superphantoms</h3>
<p>For this last section, note that a lens and its superphantom can be subtracted from a larger lens and its superphantom. In the sequence below, the original lens (left) has two smaller lenses (middle) removed (right) with attending changes to the superphantom. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/fig_24_subtracting_two_lenses.jpg" alt="Subtracting two lenses" width="624" height="195" /><p style="max-width: 600px;"> Subtracting two lenses.</p>
</div>
<p>This is shown in another example below, where two lenses are subtracted from a circle, and the two generalised symmetrical arbelos are subtracted from the full symmetrical arbelos.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/subtracting_lenses.png" alt="Two lenses subtracted from a circle" width="600" height="284" /><p style="max-width: 600px;">Two lenses subtracted from a circle.</p>
</div>
<p>You might notice that in both the above figures, two smaller lenses were removed from a larger lens, creating a tricurve. The resulting reduced superphantom is also the phantom of that tricurve. It turns out that for any tricurve the superphantom is the same as the phantom!</p>
<p>We started this article with the simple idea that an arc can form the side of a shape. This has led us progressively to lenses, to tricurves, to phantoms, to filled circles, to superphantoms and to arbelos. And we could go further, putting together groups and patterns of circles and arbelos and their superphantoms, as was done for the title image of this article. </p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2021/Tricurves/on_mars.jpg" alt="Tim Lexen" width="200" height="160" /><p>Tim Lexen on Mars.</p>
</div>
<p> Tim Lexen has been a mechanical engineer for 40+ years in various areas of new product R&D. He enjoys elegant solutions, the design process, tinkering, great communication, great stories, and family. He is married with 6 children and 10 grandchildren, and lives in the small town of Cumberland, Wisconsin, USA.</p></div></div></div>Fri, 29 Jan 2021 14:33:08 +0000Marianne7418 at https://plus.maths.org/contenthttps://plus.maths.org/content/ghosts-tiles-continued#commentsThe magic of shuffling
https://plus.maths.org/content/magic-shuffling
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/magic_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">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
How do you shuffle a pack of cards? Our shuffling is very unimpressive, we've even been known to resort to jumbling them around in a big pile on the table. But good news – here is how you shuffle like a professional!
</p>
<h3>How to shuffle perfectly</h3>
<p>
Take a pack of cards and split the pack into two equal halves (we're assuming your pack, like an ordinary 52-card pack, has an even number of cards). Now interleave them perfectly: top card from the first pile, top card from the second pile, next card from the first pile, next card from the second pile and so on. This is exact interleaving of two halves of the pack is called a <em>perfect shuffle</em>.
</p>
<div class="centreimage" style="max-width:560px;">
<iframe width="560" height="305" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" class="b-lazy" data-src="https://www.youtube.com/embed/2TTrHmFC2bM" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="></iframe>
<p>Will Houstoun demonstrating a perfect shuffle (Video: Will Houstoun)</p>
</div>
<p>
After a perfect shuffle, your pack will be shuffled in one of two ways. If the first pile was the top half of the pack, then the top card of your shuffled pack will be the top card of the original pack, and the bottom card of the shuffled pack will be the bottom card of the original pack. This is called an <em>out-shuffle</em>, and it fixes the top and bottom cards of the pack, they remain in their original position, while the other cards are interleaved.
</p>
<p>
If, like us, you are not a magician, you'll have to do this slowly and methodically. If you are a card player, or an excellent magician like Will Houstoun, you can do it almost instantly as you can see in the video!
</p>
<p>If you switch your piles – the bottom half becomes the first pile and the top half the second pile – and then interleave the piles in the same way, you get the other type of perfect shuffle, called an <em>in-shuffle</em>.
</p>
<div style="max-width: 80%; position: relative; left: 10%; border: thin solid grey; background: #CCC CFF; padding: 1em; margin-left: 1em; font-size: 75%; margin-bottom: 1em; ">
<p><strong>An out-shuffle in action...</strong></p>
<p>Suppose your pack consisted of the following 12 cards in this order, top to bottom:</p>
<p>
Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen
</p><p>
You split the pack exactly in half:<br />
A, 2, 3, 4, 5, 6<br />
in the first pile, and<br />
7, 8, 9, 10, J, Q<br />
in the second pile.
</p><p>
Then an out-shuffle would reorder the cards in this way:
</p><p>
A, 7, 2, 8, 3, 9, 4, 10, 5, J, 6, Q
</p><p>
The top card, the Ace, and the bottom card, the Queen, of the original pack are fixed as the top and bottom card of the shuffled pack after an out-shuffle.
</p>
<p><strong>...and an in-shuffle in action</strong></p>
<p>This time we swap the piles so the bottom half of the original pack is in the first pile:<br />
7, 8, 9, 10, J, Q<br />
and the top half of the pack is in the second pile:<br />
A, 2, 3, 4, 5, 6.
</p><p>
Then an in-shuffle would reorder the cards in this way:<br />
7, A, 8, 2, 9, 3, 10, 4, J, 5, Q, 6
</p>
</div>
<br class="brclear" /><h3>What could a magician do with a perfect shuffle?</h3>
<p>
To see a perfect shuffle in action, we were lucky enough to speak with <a href="http://drhoustoun.co.uk/index.html">Will Houstoun</a>, a professional conjuror and the Magician in Residence at the <a href="https://performancescience.ac.uk/">Imperial College London and Royal College of Music Centre for Performance Science</a>. As you'd expect, Houstoun has amazing physical control over the cards as you can see from the video. He can even do a variation of a perfect shuffle where he perfectly interleaves the cards two at a time, rather than one at a time. His control of the cards is incredible!
</p><p>
"Magicians are very interested in any way that you can make something impossible seem to happen," said Houstoun. "And knowing what is likely to happen to a pack of cards when it gets mixed up is part of that." Imagine if you wanted to begin with the cards in a particular order but you wanted people to think that the pack had been thoroughly shuffled. "Not that a magician would ever do such a thing!" Houstoun assured us.
</p><p>
"If you do eight perfect shuffles in a row [of a standard 52 card pack], the out-shuffles where you keep the top and bottom cards the same, then the deck goes back to the original order after eight repetitions of those out-shuffles. Admittedly it would take quite a long time, but if you wanted somebody to think that the deck was well mixed at the beginning of a trick, you could give it eight of these shuffles. [Everyone else] would then be 100% certain that the cards were well mixed, as they'd seen you mix them quite a lot, but in fact you would have your order ready to go." (You can see a mathematical proof of why this works, at the <a href="#proof">end of this article</a>.)
</p>
<h3>What could a magician do with an imperfect shuffle?</h3>
<div class="rightimage" style="width: 250px;"><img alt="cards" width="250" class="b-lazy" data-src="/issue55/features/nishiyama/iStock_cards_web.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p>Our favoured method for an imperfect shuffle!</p></div>
<p>An <em>imperfect shuffle</em> is really what we have in mind when we think of a normal, fair, shuffle. "An imperfect shuffle is the best shuffle possible, as a shuffle's real purpose it to mix up the cards in random fashion," says Houstoun. To do an imperfect shuffle you might split the pack into roughly half and half, and then alternate the cards from the two halves in a more random fashion – with clumps of two or three cards alternating, as well as the occasional single card too. That randomness in how the cards interweave from the two halves is the thing that makes it a fair shuffle.
</p><p>
Imperfect shuffles are interesting to mathematicians too. "<a href="https://en.wikipedia.org/wiki/Persi_Diaconis">Persi Diaconis</a>, who's a statistician as well as a very good magician, did work on the optimum number of shuffles to mix a pack of cards randomly", says Houstoun. Diaconis developed a procedure where you would undercut about a third of the pack (so a third of the pack gets cut from the bottom of the pack and placed on the top), then split the cards approximately in half, and then you give them an imperfect shuffle as described above. "If you repeat this procedure seven times that gives you the best balance between the number of shuffles performed and the impact that has on destroying the orders that could be located in the pack at the end of the process."
</p><p>
This mathematical understanding is useful in practice in places like casinos, where the amount of money they make relies on the randomness they expect to be found in a shuffled pack of cards: "The fairer the shuffle, the better it is for the casino, because they will be able to take advantage of their mathematical edge in the game's construction. That means they’ll win more than the player does." However, everytime the dealer is shuffling the cards, people are not playing cards or laying bets, so the casino has to decide on a process for shuffling that balances the randomness with the amount of bets that they can get out of customers.
</p>
<h3>Perfect magic</h3>
<p>
The essence of an imperfect shuffle is that you don't want to end up with any particular, predictable order in the shuffled pack of cards. When we try to think of a special ordering of a pack of cards, we might think of one where the cards are in face value order, grouped in their suits, as they usually are with a new deck of cards. But one of Houstoun's most amazing card tricks plays with this idea of what orders we should think of as important.
</p>
<div class="centreimage">
<iframe width="640" height="360" frameborder="0" allow="autoplay; fullscreen" allowfullscreen="" class="b-lazy" data-src="https://player.vimeo.com/video/191677714" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="></iframe>
<p><a href="https://vimeo.com/191677714">Will Houstoun and Jim Carter on The Next Great Magician</a> from <a href="https://vimeo.com/user2696089">Will Houstoun</a> on <a href="https://vimeo.com">Vimeo</a>.</p>
</div>
<p>
After Houstoun and his audience have totally mixed up the pack of cards, we're not surprised to see a totally random looking order in the shuffled pack of cards. And the randomness of this order is emphasised by the huge surprise at the end of the trick.
</p><p>
Although Houstoun obviously couldn't divulge the secret behind the magic of the trick, he could tell us what motivated him to develop the trick: "I was thinking about the question of order, and why one thinks some orders are very important and some are not. A mixed up deck of cards, for example, is instantly dismissed. A pack of cards in new deck order is something people think is important. I thought it would be fun to do a trick that explores how an order that is seemingly unimportant is actually precise and is interesting in its own particular way."
</p>
<p>We may have started out learning how to shuffle perfection, but actually it's imperfection that we really want whether we are playing cards at home against a friend, or betting against the house at a casino. As you'll see below, as well as magic there is a great deal of maths hidden within a card shuffle – something you can find out more about in the next article, <a href="/content/mathematics-shuffling"><em>The mathematics of shuffling</em>.
</a></p>
<hr /><a name="proof" id="proof"></a><h3>Show me the maths! Why eight out-shuffles is the same as none...</h3>
<p>
We were intrigued when Houstoun told us that doing eight out-shuffles of a standard 52 card pack returned you to the order of your original pack! There are various ways you can mathematically prove this, but the way we found most straightforward was explained by mathematician and magician <a href="https://www.youtube.com/watch?v=FUDv1QR_N4w">Tori Noquez</a>.
</p><p>
What we are interested in here is the order of the cards in the original pack, so we're going to forget about the value and suit of the card, and instead label all the cards with their original position in the pack. And, because it makes the maths easier, we're going to start by labelling the top card with a 0, the second card with a 1, and so on. So assuming we have a standard 52 card pack (the maths works for any even-number-sized pack) we'll label the cards:
</p><p>
0, 1, 2, 3, ..., 49, 50, 51
</p><p>
where 0 was the top card of the original pack, and 51 the bottom card.
</p><p>
An out-shuffle splits the cards into two halves, the top half in the first pile (0, 1, ..., 25) and the bottom half in the second pile (26, 27,..., 51), and interleaves them to give the following order:
</p><p>
0, 26, 1, 27, 2, ..., 24, 50, 25, 51.
</p>
<table class="data-table"><tr><th><strong>Label</strong> = position in original pack</th>
<th>Position in shuffled pack</th></tr><tr><td>0
</td><td>
0
</td></tr><tr><td>
1
</td><td>
2
</td></tr><tr><td>
2
</td><td>
4
</td></tr><tr><td>
...
</td><td>
...
</td></tr><tr><td>
25
</td><td>
50
</td></tr><tr><td>
26
</td><td>
1
</td></tr><tr><td>
...
</td><td>
...
</td></tr><tr><td>
50
</td><td>
49
</td></tr><tr><td>
51
</td><td>
51
</td></tr></table><p>
The top card, with the label 0, remains at the top, position 0 in the shuffled pack. The bottom card with the label 51, remains at the bottom, position 51 in the shuffled pack. So we are really only interested in what happens for the rest of the cards, labelled 1 to 50, during an out-shuffle.
</p><p>
The cards in the first pile, the top half of the pack labelled <img alt="$x$" style="vertical-align:0px; width:9px; height:7px" class="math gen b-lazy" data-src="/MI/885de729a28eac793956a8507dd53b0a/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> with <img alt="$1 \leq x \leq 25$" style="vertical-align:-2px; width:77px; height:15px" class="math gen b-lazy" data-src="/MI/885de729a28eac793956a8507dd53b0a/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />, are moved to the position <img alt="$2x$" style="vertical-align:0px; width:17px; height:12px" class="math gen b-lazy" data-src="/MI/885de729a28eac793956a8507dd53b0a/images/img-0003.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> in the shuffled pack. The card labelled 1 is moved to position 2 in the shuffled pack. The card labelled 2 is moved to position 4 in the shuffled pack. All the way to the card labelled 25, which is moved to position 50. </p><p>
The cards in the second pile, the bottom half of the pack labelled <img alt="$x$" style="vertical-align:0px; width:9px; height:7px" class="math gen b-lazy" data-src="/MI/1362131b6944e418dd53dee0d5f76b48/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> with <img alt="$26 \leq x \leq 50$" style="vertical-align:-2px; width:86px; height:15px" class="math gen b-lazy" data-src="/MI/1362131b6944e418dd53dee0d5f76b48/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />, are moved to the position <img alt="$2x-51$" style="vertical-align:0px; width:53px; height:13px" class="math gen b-lazy" data-src="/MI/1362131b6944e418dd53dee0d5f76b48/images/img-0003.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> in the new pack. The card labelled 26 is moved to position 1 (<img alt="$=2\times 26-51$" style="vertical-align:0px; width:97px; height:13px" class="math gen b-lazy" data-src="/MI/1362131b6944e418dd53dee0d5f76b48/images/img-0004.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />)in the shuffled pack. All the way to the card labelled 50 which is moved to position 49 (<img alt="$=2\times 50-51$" style="vertical-align:0px; width:97px; height:13px" class="math gen b-lazy" data-src="/MI/1362131b6944e418dd53dee0d5f76b48/images/img-0005.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />) in the shuffled pack. </p><p>
Using modular arithmetic (see <a href="/content/maths-minute-modular-arithmetic">here</a> for a quick introduction to modular arithmetic) you can describe what happens to all the cards labelled 1 to 50 in the pack with just one rule: a card with label <img alt="$x$" style="vertical-align:0px; width:9px; height:7px" class="math gen b-lazy" data-src="/MI/4997a2d0d43d77c976910af8ea74a51e/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> is moved to the position <img alt="$2x \pmod{51}$" style="vertical-align:-4px; width:99px; height:18px" class="math gen b-lazy" data-src="/MI/4997a2d0d43d77c976910af8ea74a51e/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> in the shuffled pack. </p><p></p><p> So for a card labelled <img alt="$x$" style="vertical-align:0px; width:9px; height:7px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> where <img alt="$1 \leq x \leq 50$" style="vertical-align:-2px; width:77px; height:15px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />: </p><p>one out-shuffle takes <img alt="$x$" style="vertical-align:0px; width:9px; height:7px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to <img alt="$2x \pmod{51}$" style="vertical-align:-4px; width:99px; height:18px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0003.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />; </p><p>two out-shuffles takes <img alt="$x$" style="vertical-align:0px; width:9px; height:7px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to <img alt="$2^2 x \pmod{51}$" style="vertical-align:-4px; width:106px; height:18px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0004.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />; </p><p>and so on. In general, <img alt="$k$" style="vertical-align:0px; width:8px; height:11px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0005.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> out-shuffles takes the card labelled <img alt="$x$" style="vertical-align:0px; width:9px; height:7px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to position <img alt="$2^ k x \pmod{51}$" style="vertical-align:-4px; width:107px; height:18px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0006.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />. To return to the original order of the pack, we are looking for the number <img alt="$k$" style="vertical-align:0px; width:8px; height:11px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0005.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> of out-shuffles that returns the card labelled <img alt="$x$" style="vertical-align:0px; width:9px; height:7px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to its original position, that is: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7"><tr><td style="width:40%"> </td>
<td><img alt="\[ 2^ k x \equiv x \pmod{51} \]" style="width:139px; height:19px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0007.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr></table><p> So we are looking for the number <img alt="$k$" style="vertical-align:0px; width:8px; height:11px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0005.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> that gives us </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7"><tr><td style="width:40%"> </td>
<td><img alt="\[ 2^ k \equiv 1 \pmod{51}. \]" style="width:133px; height:19px" class="math gen b-lazy" data-src="/MI/4edf4e2c3296bc24fc61028c5b943ad4/images/img-0008.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr></table><p> The first eight powers of two are: </p><p>
</p><table class="data-table"><tr><th>
<img alt="$k$" style="vertical-align:0px; width:8px; height:11px" class="math gen b-lazy" data-src="/MI/506aa37b2be0537bfb810f1b52281713/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></th><th>
<img alt="$2^ k$" style="vertical-align:0px; width:15px; height:14px" class="math gen b-lazy" data-src="/MI/a3e403eb03018cf926de6772e9e97fce/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></th><th>
<img alt="$2^ k \pmod{51}$" style="vertical-align:-4px; width:98px; height:18px" class="math gen b-lazy" data-src="/MI/c21d3e356d04be0a32de3d1fc497af3f/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></th></tr><tr><td>
1
</td><td>
2
</td><td>
2
</td></tr><tr><td>
2
</td><td>
4
</td><td>
4
</td></tr><tr><td>
3
</td><td>
8
</td><td>
8
</td></tr><tr><td>
4
</td><td>
16
</td><td>
16
</td></tr><tr><td>
5
</td><td>
32
</td><td>
32
</td></tr><tr><td>
6
</td><td>
64
</td><td>
13
</td></tr><tr><td>
7
</td><td>
128
</td><td>
26
</td></tr><tr><td>
8
</td><td>
256
</td><td>
1
</td></tr></table><p>
Therefore eight out-shuffles gives us back the original pack order, and this is the smallest number of out-shuffles that can do that.
</p><p> The same mathematics works for any size pack, <img alt="$N$" style="vertical-align:0px; width:15px; height:11px" class="math gen b-lazy" data-src="/MI/fbe196376117e42dc6797e5855685208/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> (where <img alt="$N$" style="vertical-align:0px; width:15px; height:11px" class="math gen b-lazy" data-src="/MI/fbe196376117e42dc6797e5855685208/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> is assumed to be even so you can split the pack exactly in half). To return the original pack order, you are looking for a number <img alt="$k$" style="vertical-align:0px; width:8px; height:11px" class="math gen b-lazy" data-src="/MI/fbe196376117e42dc6797e5855685208/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> such that </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7"><tr><td style="width:40%"> </td>
<td><img alt="\[ 2^ k \equiv 1 \pmod{N-1}. \]" style="width:161px; height:19px" class="math gen b-lazy" data-src="/MI/fbe196376117e42dc6797e5855685208/images/img-0003.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr></table> Then that number <img alt="$k$" style="vertical-align:0px; width:8px; height:11px" class="math gen b-lazy" data-src="/MI/fbe196376117e42dc6797e5855685208/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> of out-shuffles will return your pack of size <img alt="$N$" style="vertical-align:0px; width:15px; height:11px" class="math gen b-lazy" data-src="/MI/fbe196376117e42dc6797e5855685208/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to the original pack order.
<p><em>You can find our more in the next article, <a href="/content/mathematics-shuffling">The mathematics of shuffling</a></em></p>
<hr /><h3>About this article</h3>
<div class="rightimage"><img width="200" height="200" alt="Will Houstoun" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/shuffling/willhoustoun_web.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p>Will Houstoun</p></div>
<p>
<a href="http://www.drhoustoun.com">Will Houstoun</a> is an international award-winning magician and instructor, with a PhD in the history of magic education. He specialises in consultancy for media and advertising as well as instruction on both the technical and historic aspects of magic. Will is the Magician in Residence at the Imperial College London/Royal College of Music Centre for Performance Science.
</p>
<p><a href="https://plus.maths.org/content/people/index.html#rachel">Rachel Thomas</a> is Editor of <em>Plus</em>.</p>
<p><em>This article is part of our collaboration with the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more.</em></p>
<div class="centreimage"><img alt="INI logo" width="400" height="51" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 400px;"></p></div></div></div></div>Fri, 29 Jan 2021 13:55:17 +0000Rachel7380 at https://plus.maths.org/contenthttps://plus.maths.org/content/magic-shuffling#commentsThe mathematics of shuffling
https://plus.maths.org/content/mathematics-shuffling
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_22.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">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
In a previous article, <a href="/content/magic-shuffling"><em>The magic of shuffling</em></a>, we found out what a magician can do with a card shuffle. But what might a mathematician do when shuffling cards? To find out we asked <a href="https://research-repository.uwa.edu.au/en/persons/cheryl-praeger">Cheryl Praeger</a>, from the University of Western Australia. It was Praeger who first intrigued us in card shuffling with her fascinating talk at the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> last year. (You can watch the lecture online <a href="http://www.newton.ac.uk/seminar/20200317160017001">here</a>.) Praeger works in <em>group theory</em>, an area of mathematics that is motivated by symmetry. We spoke to her last year after she returned to Australia, and she revealed the fascinating insights group theory brings to understanding shuffling cards.
</p>
<h3>What could a mathematician do with a perfect shuffle?</h3>
<p>
A mathematician thinks of the cards in a pack, not in terms of their face value and suit, but in terms of their position in the original pack. So if you have a pack of 52 cards, label each card 0 to 51 by its position (the top card is labelled 0, rather than 1, as this makes the maths simpler). "I might want to know where each of these cards might lie after different combinations of the shuffles," says Praeger. "So each outcome is like a reordering, a <em>permutation</em>, of the cards in the pack. I want to know how many different permutations there are. And perhaps what sort of structure this whole set of reorderings, this <em>permutation group</em>, has."
</p><p>
The first things we can do is count all the possible permutations you can create by shuffling the cards. If we start with a pack of <em>k</em> cards, then we know there are <em>k</em> different positions in the shuffled pack to place the top card. Then there's only <em>k-1</em> slots left for the second card, <em>k-2</em> slots for the third card, and so on. The total number of possible reorderings is <em>k x (k-1) x (k-2) x ... x 2 x 1</em>, otherwise known as the <em>factorial</em> of <em>k</em>, written <em>k!</em>. The collection of all possible permutations of a pack of <em>k</em> cards is called the symmetric group – and the size of the symmetric group for a pack of <em>k</em> cards is <em>k!</em>. For example, for a pack consisting of 6 cards, there are 6! = 6x5x4x3x2x1 = 720 possible permutations in the symmetric group.
</p>
<div style="max-width: 40%; float: right; border: thin solid grey; background: #CCC CFF; padding: 0.5em; margin-left: 1em; font-size: 75%; margin-bottom: 1em; ">
<p><strong>An out-shuffle in action...</strong></p>
<p>Suppose your pack consisted of the following 12 cards in this order, top to bottom:</p>
<p>
Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen
</p><p>
You split the pack exactly in half:<br />
A, 2, 3, 4, 5, 6<br />
in the first pile, and<br />
7, 8, 9, 10, J, Q<br />
in the second pile.
</p><p>
Then an out-shuffle would reorder the cards in this way:
</p><p>
A, 7, 2, 8, 3, 9, 4, 10, 5, J, 6, Q
</p><p>
The top card, the Ace, and the bottom card, the Queen, of the original pack are fixed as the top and bottom card of the shuffled pack after an out-shuffle.
</p>
<p><strong>...and an in-shuffle in action</strong></p>
<p>This time we swap the piles so the bottom half of the original pack is in the first pile:<br />
7, 8, 9, 10, J, Q<br />
and the top half of the pack is in the second pile:<br />
A, 2, 3, 4, 5, 6.
</p><p>
Then an in-shuffle would reorder the cards in this way:<br />
7, A, 8, 2, 9, 3, 10, 4, J, 5, Q, 6
</p>
</div>
<p>
Praeger, however, is interested in just those permutations of the cards that you can get by using <em>perfect shuffles</em>, when a pack of cards is exactly split in half and the cards from each half perfectly interleaved one by one. There are two types of perfect shuffles: an out-shuffle, where the top and bottom cards are fixed, and an in-shuffle, where the top and bottom cards are moved in by one position in the shuffled pack. You could do just one in-shuffle, or, say, three in-shuffles followed by an out-shuffle, or, perhaps, eight out-shuffles, one after the other. Any combination of the two types of perfect shuffles is allowed.
</p><p>
The set of permutations of the pack of cards that you get from all the possible combinations of in- and out-shuffles is called the <em>shuffle group</em>. This doesn't include every possible permutation of the pack of cards; the shuffle group is smaller than the symmetric group. For example a pack of six cards has 720 permutations in the symmetric group, but only 24 possible permutations in the shuffle group. But both groups share an important mathematical property.
</p>
<div class="centreimage" style="max-width:560px;">
<iframe width="560" height="305" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" class="b-lazy" data-src="https://www.youtube.com/embed/2TTrHmFC2bM" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="></iframe>
<p>Will Houstoun, the magician we spoke to in the <a href="/content/magic-shuffling">previous article</a>, demonstrating a perfect shuffle (Video: Will Houstoun)</p>
</div>
<h3>The shuffle group</h3>
<p>
Both the shuffle group and the symmetric group have what is called a <em>group</em> structure. In maths, a <em>group</em> is a collection of objects that combine in a specific way that satisfies four rules (you can read the details <a href="/content/maths-a-minute-groups">here</a>). In our shuffle group , for example, the objects are shuffles generated from some combination of the two types of perfect shuffles. The shuffle group is <em>closed</em>: if you combine any two shuffles from the group, the resulting shuffle will then also be a combination of perfect shuffles, and so is also in the shuffle group. And every shuffle has an <em>inverse</em> – so that a shuffle, combined with its inverse, effectively leaves the pack of cards unchanged. For example, we saw in the <a href="/content/magic-shuffling">previous article</a> that, for a pack of 52 cards, a single out-shuffle, combined with 7 more out-shuffles gives you back your original pack order. So the inverse of a single out-shuffle is a combination of 7 out-shuffles. A shuffle group, for any evenly sized pack of cards, and the larger symmetric group, satisfy all the rules of being a mathematical group.
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<p>
The key tool of a group theorist is to break a finite group down into its basic building blocks, called the simple groups. "Typically you can split a group up into two parts, so that each of the parts is a group, and you can continue to do this" says Praeger. "It really is just like a tree branching and branching and branching, until you reach a stage where it is not possible to split it any more, and that’s like getting to the leaves of the tree. And when you find the leaves of the tree, the groups that you've reached are what we call the simple ones, because they can’t be divided any further." (You can read more about simple groups <a href="/content/enormous-theorem-classification-finite-simple-groups">here</a>.)
</p><p>
Studying a group in this way reveals lots of useful information. For example you can tell how big the group is by multiplying together the sizes of all the simple groups on the leaves of that tree. "[Breaking a group down into its simple groups] is a very useful tool and it’s one of the first things a group theorist is going to look for."
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<h3>Central symmetry</h3>
<p>
An intriguing early result about shuffle groups was proven by the mathematicians <a href="http://statweb.stanford.edu/~cgates/PERSI/papers/83_05_shuffles.pdf">Persi Diaconis, Ronald Graham and William Kantor</a> in the 1980s. "A very beautiful property they saw and explained is what they called <em>central symmetry</em>," says Praeger. "You think of the top and the bottom card in the pack as being associated. After any sequence of perfect shuffles – suppose the top card went down to the third position, and then the bottom card would come up to the third last position – there's this sort of symmetry, that whatever one of the pair does, the other mimics it." So, if after a sequence of perfect shuffles the top card ends up a certain number of cards from the top, you know that the bottom card ends up same number of cards from the bottom.
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<div class="leftimage" style="max-width:300px"><img width="300px" height="364px" alt="shuffled pack of cards" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/shuffling/order.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p>
The same thing happens for every such symmetrical pair of cards in the pack: the pairs located the same distance around the centre of the pack are linked, and always get moved together by the in- and out-shuffles. "The pairs get shuffled around together, it's like they’ve got some sort of invisible bonds connecting them. Whereas a completely random shuffle, not one of these perfect ones, wouldn't have that property at all." This central symmetry property puts some real restrictions on the possible permutations of a pack of cards you can have in your shuffle group.
</p><p>
Diaconis, Graham and Kantor showed that the shuffle group of a pack having central symmetry implies that its simple groups are very restricted. One impact of this is on the size of the shuffle groups, which depends on the sizes of the associated simple groups. For most sizes of packs of cards the shuffle groups would be enormous (relative to the size of the pack, despite being much smaller than the full symmetric group). And typically the size of the shuffle group grows exponentially with the size of the pack.
</p><p>
If you have a pack of 6 cards, you have 24 possible permutations in its shuffle group. A pack of 10 cards has 1,920 permutations in its shuffle group. And a pack of 26 cards has over 25 trillion possible permutations in its shuffle group. These numbers illustrate that the size of the shuffle group grows exponentially as you increase the size of your pack of cards.
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<p>
But an amazing thing happens when the size of the pack is a power of 2, something mathematicians call the <em>power case</em>. A 14 card pack has over 320,000 permutations in the shuffle group. But the shuffle group for a 16 card pack, and remember 16 is 2<sup>4</sup>, has just 64 possible permutations! A pack of 30 cards has quadrillions (10<sup>15</sup>) of possible permutations using combinations of in- and out- shuffles, but a 32 card pack, and again remember 32 is 2<sup>5</sup>, has just 160 possible permutations in the shuffle group. When the size of the pack is a power of 2, it gives a dramatically smaller shuffle group.
</p><p>
"The power case was incredible," says Praeger. "In that case the size of the shuffle group is exponentially smaller than the typical size." She explains that in the power case the only building blocks of the shuffle groups are something called <em>cyclic groups</em>, which are the smallest simple groups. The building blocks being so much smaller means the shuffle group is dramatically smaller too: "There's nothing big about it at all!" (You can find out more about cyclic groups <a href="/content/maths-minute-cyclic-groups" target="blank">here</a>.)
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<p>
</p><h3>Many hands make interesting work</h3>
<p>
It turns out a similar power case is true for <em>many-handed</em> perfect shuffles, something Praeger investigated with her colleagues Carmen Amarra and Luke Morgan in 2019. When they were visiting with her research group, Praeger remembered a piece of paper covered with data from long ago hidden at the bottom of a drawer: "It had been sitting in my drawer for decades!" Now was a perfect time to investigate.
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<p>
The data was from computer experiments conducted by Kent Morrison and John Cannon in the 1980s for <em>many-handed shuffles</em>. Imagine that instead of dividing up the pack of cards into two equal piles, you instead divided it up into, say, three equal piles, and then perfectly interleaved those three piles. Or you divided the pack up into four piles, or seven, or 11 piles, and then perfectly shuffled the equal piles.
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<div class="rightimage" style="max-width:300px"><img width="350px" height="280px" alt="shuffled pack of cards" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/shuffling/splitdeck.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></div>
<p>
"But then there were a lot of questions that arose," says Praeger. "You divide your pack of cards into <em>k</em> equal piles – and then what?" You could pick up the cards one at a time from the top of each pile in the order that the piles were split from the deck, then that might be like an out-shuffle. But what if you decided to start with the second pile, and then go to the 7th pile, and then the 15th pile, and then the 27th pile! There were now lots of possible orders in which you picked up your cards from the piles – what counted as a perfect shuffle when you were using many piles?
</p><p>
Praeger and her colleagues decided to think about a many-handed shuffle in a new way: they considered not only how many piles you split the pack into, but also how to specify the allowable orderings of the piles which determined the way you were alternating the cards in your shuffle. They labelled the top pile from the pack with 0, the second pile 1, and so on to the <em>k</em>th pile labelled with <em>k-1</em>. Then they could investigate the impact of the order in which you chose cards from the piles, on the reordering of the cards in the final shuffle.
</p><p>
With this new approach, Praeger, Amerra and Morgan proved something that had been suspected: that in some cases, something similar to the power case happened in this many-handed shuffling (you can read their <a href="https://arxiv.org/abs/1908.05128">paper here</a>). "In my work with Carmen and Luke, when we were splitting up the cards into <em>k</em> piles of length <em>n</em>, we observed that there's a similar special case when the size of the piles is a power of the number of piles. So it's like the cards in the standard case being a power of 2: again you got a very, very small shuffle group."
</p><p>
Mathematicians had observed this power case for a many-handed shuffle, along with some experimental evidence that the shuffle groups seemed enormously larger outside of the power case. That the shuffle groups would be gigantic in all cases except the power case, for a many-handed shuffle, was stated in a conjecture by Morrison and another mathematician, Steve Medvedoff. Praeger and her colleagues were able to use their new approach to prove this conjecture about the non-power case for a lot of the many-handed possibilities. But many open questions still remain for proving this conjecture more generally, to cover the case of dividing the pack into an arbitrary number of piles for a many-handed shuffle.
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<h3>Open questions and new horizons</h3>
<p>
Also, thanks to their new approach, Praeger, Carmen and Luke were able to discover some fascinating new results. Their new approach considered the permutations of the piles of cards when shuffling your pack. In particular, they discovered that the shuffle groups preserved many properties of the permutations of the piles that you allowed. If you had what is called a <em>transitive</em> permutation group of the piles in your allowed shuffles (which means you could get from one pile to any other pile using the permutations you were allowing of the piles in your many-handed shuffles) then your resulting shuffle group would also be transitive. There was a combination of shuffles you could use to move any card to any other position in the pack. This transitive property was just the first of some of the group properties that they found the shuffle groups preserved. And again, many open questions remain.
</p><p>
The mathematics of card shuffling is a great example of how mathematicians often work. In this case they started with perfect shuffles, arising from standard card games, where the deck is split into two piles, with two types of perfect shuffles - the in and out shuffle. Then, after exploring the maths of that setting, they generalised the setup to see how far these results can be extended: splitting the pack into three, four, or indeed any number of equal piles. They may have started with a simple pack of cards, at a table, playing with friends, but, as always, their eye is inevitably drawn to the mathematical horizon.
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<hr /><h3>About this article</h3>
<div class="rightimage"><img width="300px" alt="Cheryl Praeger" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/shuffling/cheryl_web.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p>Cheryl Praeger</p></div>
<p><a href="https://research-repository.uwa.edu.au/en/persons/cheryl-praeger">Cheryl Praeger</a> is an Australian mathematician with a passion for the mathematics of symmetry, as measured mathematically by groups. Her research focuses on permutation groups and the symmetrical structures they act on, along with algorithmic and computational questions about groups. She is Emeritus Professor of Mathematics at the University of Western Australia and is so far the only pure mathematician to receive the Australian Prime Minister’s Prize for Science.</p>
<p>Rachel Thomas is Editor of <em>Plus</em>.</p>
<p><em>This article is part of our collaboration with the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more.</em></p>
<div class="centreimage"><img alt="INI logo" width="400" height="51" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p style="max-width: 400px;"></p></div></div></div></div>Fri, 29 Jan 2021 12:56:05 +0000Rachel7414 at https://plus.maths.org/contenthttps://plus.maths.org/content/mathematics-shuffling#comments