plus.maths.org
https://plus.maths.org/content
enMaths in a minute: The knapsack problem
https://plus.maths.org/content/maths-minute-knapsack-problem
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/knapsack_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Imagine you're going to see your friends on the other side of the world and you've bought them so many presents, you can't fit them all into your luggage — some have to be left behind. You decide to pack a combination of presents that is highest in value to your friends, but doesn't exceed the weight limit imposed by the airline. How do you find that combination?</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2017/knapsack/knapsack.png" alt="Knapsack" width="350" height="303" />
<p>The knapsack problem. Image: <a href="https://commons.wikimedia.org/wiki/File:Knapsack.svg">Dake</a>, <a href="https://creativecommons.org/licenses/by-sa/2.5/deed.en">CC BY-SA 2.5</a>.</p>
</div>
<p>The obvious thing to do is to try all possible combinations, calculate their value and weight, and pick one that is below the weight limit but maximises value. This is fine if there are only a few items, but gets hard when many are involved: there are simply too many combinations to try out and your plane is leaving soon. The question is whether there is an algorithm
— a recipe for finding a solution — which works for any number of items and doesn't take too long even when many items are involved. </p>
<p>This kind of problem interests computer scientists because creating algorithms, mechanical procedures a computer can execute, is what computer science is all about. Computer scientists have a way of measuring the complexity of a problem in terms of how fast the time it takes to solve the problem grows with the size of the input. For example, suppose the size of the input (the number of presents) is <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0001.png" alt="$n.$" style="vertical-align:0px;
width:14px;
height:7px" class="math gen" /> If the time it takes to solve the problem grows roughly like <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0002.png" alt="$n,$" style="vertical-align:-3px;
width:14px;
height:10px" class="math gen" /> or <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0003.png" alt="$n^2,$" style="vertical-align:-3px;
width:21px;
height:17px" class="math gen" /> or <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0004.png" alt="$n^3,$" style="vertical-align:-3px;
width:21px;
height:17px" class="math gen" /> or <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0005.png" alt="$n^ k$" style="vertical-align:0px;
width:17px;
height:14px" class="math gen" /> for any other natural number <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0006.png" alt="$k,$" style="vertical-align:-3px;
width:12px;
height:14px" class="math gen" /> as <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0007.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> gets larger, then the problem is deemed "easy". The number <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0005.png" alt="$n^ k$" style="vertical-align:0px;
width:17px;
height:14px" class="math gen" /> could still grow fast with <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0007.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> if <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0008.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> is large, of course, but this kind of <em>polynomial growth</em>, as it's called, is nowhere near as explosive as exponential growth, described, for example, by the expression <img src="/MI/83273fded2de608f3b76089e7f10d053/images/img-0009.png" alt="$2^ n$" style="vertical-align:0px;
width:16px;
height:12px" class="math gen" />. (See <a href="https://en.wikipedia.org/wiki/Time_complexity#Polynomial_time">here</a> for a more technical definition of so-called <em>polynomial time algorithms</em>.)</p>
<p>
So does such a polynomial time algorithm exist for our problem?
There's a slightly easier <em>decision version</em> of the problem. Rather than asking "what is the optimal combination of presents that stays under the weight limit but maximises value", we ask "given a particular value <em>V</em>, is there a combination that stays within the weight limit and has a value exceeding <em>V</em>?". This problem has the nice property that given a potential solution (a combination of presents) it's very easy to check that it's correct: you only need to add up the weights and values, and you can do that in polynomial time. However, whether or not a polynomial time algorithm exists to solve the problem from scratch isn't known. This puts the decision version of the knapsack problem into a class of problems called <em>NP</em>. </p>
<div class="leftimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/articles/2017/knapsack/NP_0.png" alt="Complexity classes" width="400" height="250" />
<p>The hierarchy of complexity classes. The class <em>P</em> contains all those problems that can be solved in polynomial time. The class <em>NP</em> contains all those problem for which a potential solution can be checked in polynomial time. It's possible that the <em>NP</em> class is equal to the <em>P</em> class, though nobody knows if it is, which is why there are two diagrams: one for the case that <em>P=NP</em> and one for the case that it isn't. <em>NP</em> complete problems are within the <em>NP</em> class, but particularly hard, and <em>NP</em> hard problems are at least as hard as <em>NP</em> complete ones.
Image: <a href="https://commons.wikimedia.org/wiki/File:P_np_np-complete_np-hard.svg">Behnam Esfahbod,</a> <a href="https://creativecommons.org/licenses/by-sa/3.0/">CC BY-SA 3.0</a>.</p>
</div>
<p> The decision version of the knapsack problem also has another amazing feature: if you do find a polynomial time algorithm to solve the decision knapsack problem, then you'll be able to derive a polynomial time algorithm for <em>every</em> problem in the <em>NP</em> class, which would be quite something. This feature means that the decision version of the knapsack problem belongs to a subclass of <em>NP</em> problems called <em>NP complete</em>. (Confused? See the diagram on the left.)</p>
<p>And the full knapsack problem? Well, it's at least as hard to solve as the decision version of the problem.
Even given a potential solution, there's no known polynomial time algorithm that can tell you whether the solution is correct. But as for the decision version of the problem, if you do find a polynomial time algorithm to solve the full knapsack problem, then you'll be able to derive a polynomial time algorithm for every problem in the <em>NP</em> class.
The knapsack problem is a so-called <em>NP hard</em> problem. </p>
<p>
Optimisation problems such as the knapsack problem crop up in real life all the time. Luckily there are efficient algorithms which, while not necessarily giving you the optimal solution, can give you a very good approximation for it. Thus, the question of whether the knapsack problem can be solved in polynomial time isn't that interesting for practical purposes, rather it's something that enthuses theorists. When it comes to delivering presents, however, our advice is not to buy too many in the first place.</p>
<p>
You can find out more about the complexity of algorithms on <em><a href="/content/tags/complexity">Plus</a></em>.
</p></div></div></div>Thu, 13 Apr 2017 13:31:10 +0000mf3446822 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-knapsack-problem#commentsWomen of mathematics
https://plus.maths.org/content/women
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/women_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: 250px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/hands_maths.jpg" alt="Hands" width="250" height="575" />
<p></p>
</div>
<p>These articles and videos accompany the <em><a href="http://womeninmath.net">Women of
Mathematics</a></em> photo exhibition, which celebrates female
mathematicians from institutions throughout Europe. The exhibition was
launched in Berlin in the summer of 2016 and featured thirteen female
mathematicians. Another six women, all mathematicians at the
University of Cambridge, have now contributed their portraits
too. We
took the opportunity to interview these women about their work and
life, and you can read or watch the interviews by clicking on the links below.</p>
<p>The portraits will be on display in the <a
href="https://www.newton.ac.uk">Isaac Newton Institute</a> and the
<a href="http://www.cms.cam.ac.uk">Centre for Mathematical
Sciences</a>, both in Cambridge, from Tuesday 25th April 2017, and in
the <a href="http://moore.libraries.cam.ac.uk">Betty and Gordon Moore Library</a> following the exhibition.</p>
<p>If you would like to attend the exhibition opening on Tuesday, April 25th 2017, which will feature talks from prominent mathematicians at the University of Cambridge and a panel discussion on issues affecting women in mathematics, then please <a href="https://www.eventbrite.co.uk/e/women-of-mathematics-cambridge-exhibition-opening-tickets-32299228863">register here</a>.</p>
<p>Photographs by <a href="http://henrykenyonphotography.com">Henry Kenyon</a>.</p>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Natasha_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/women-mathematics-natalia-berloff">Natalia Berloff</a> </strong> — Natalia Berloff is a professor of applied mathematics. It was a problem in network theory that lured her into the exciting world of maths when she was ten years old.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Nilanjana_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/nilanjana-datta">Nilanjana Datta</a> </strong> — Nilanjana Datta works in quantum information theory. She loves how mathematics can describe nature simply and elegantly.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Anne_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/women-mathematics-anne-christine-davis">Anne-Christine Davis</a> </strong> — Anne-Christine Davis is a professor of theoretical physics whose long career has seen attitudes towards women change for the better. She had to put up with quite a lot at the start.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Julia_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/womem-mathematics-julia-gog">Julia Gog</a> </strong> — Julia Gog is a mathematical biologist, helping to understand how infectious diseases spread. One of her favourite eureka moments came while she was playing a computer game.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Holly_icon.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/women-mathematics-holly-krieger">Holly Krieger</a> </strong> — Holly Krieger works in dynamical systems theory, in particular on chaotic systems. Some of her greatest mathematical moments have come from teaching students.
</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Carola_icon_0.jpg" alt="" width="100" height="100" /> </div> <p><strong><a href="/content/women-mathematics-carola-bibiane-sch-nlieb">Carola-Bibiane Schönlieb</a> </strong> — Carola-Bibiane Schönlieb uses mathematics to process and analyse images. She loves the collaborative nature of maths.
</p></div>
<br clear="all"/>
<p>The Cambridge part of the Women of Mathematics exhibition is the
result of the joint efforts of the photographer <a href="http://henrykenyonphotography.com">Henry Kenyon</a> and
<a href="http://www.damtp.cam.ac.uk/user/cbs31/Home.html">Carola-Bibiane Schönlieb</a>,
<a href="http://www.lucy-cav.cam.ac.uk/about-us/our-people/college-fellows/dr-orsola-rath-spivack/">Orsola Rath-Spivack</a>,
<a href="https://www.dpmms.cam.ac.uk/~hk439/">Holly Krieger</a>,
<a href="http://www.damtp.cam.ac.uk/people/a.c.davis/">Anne-Christine Davis</a>, <a href="/content/people/index.html#marianne">Marianne Freiberger</a>, <a href="/content/people/index.html#rachel">Rachel Thomas</a>,
<a href="http://www.damtp.cam.ac.uk/people/raf59/">Rachel Furner</a>,
<a href="https://www.newton.ac.uk/person/inimar05">Christie Marr</a>,
<a href="http://www.bigdata.cam.ac.uk/directory/yvonne-nobis">Yvonne Nobis</a>,
<a href="http://www.damtp.cam.ac.uk/user/ngb23/">Natalia Berloff</a> and
<a href="http://www.damtp.cam.ac.uk/people/j.r.gog/">Julia Gog</a>.</p></div></div></div>Wed, 12 Apr 2017 16:48:47 +0000mf3446815 at https://plus.maths.org/contenthttps://plus.maths.org/content/women#commentsWomen of mathematics: Holly Krieger
https://plus.maths.org/content/women-mathematics-holly-krieger
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Holly_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article and video accompany the <em>Women of
Mathematics</em> photo exhibition. To see more profiles of female mathematicians and to find out more about the exhibition, see <a href="/content/women">here</a>. Photographs by <a href="http://henrykenyonphotography.com">Henry Kenyon</a>.</em></p>
<hr/>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/Holly_cropped.jpg" alt="Holly Krieger" width="350" height="525" />
<p></p>
</div>
<p>Holly Krieger is a lecturer at the <a href="https://www.dpmms.cam.ac.uk">Department of Pure Mathematics and Mathematical Statistics</a>, University of Cambridge, and Fellow and Director of Studies at <a href="https://www.murrayedwards.cam.ac.uk">Murray Edwards College</a>.</p>
<p><strong><em>Plus</em>:</strong> How and when did you choose to do mathematics?</p>
<p><strong>Holly Krieger:</strong> When I got this job. No, that's not quite true but to an extent it is. This is a difficult career, at least up until the point when you have a permanent position, to know whether or not you're going to be able to go all the way and stay in academia.
</p>
<p>But perhaps more of a reasonable answer is there was a professor when I was an undergraduate who I spoke with a lot about the possibility of going to graduate school and what it would be like to be a mathematician and all these things.</p>
<p>One of the most important things that he told me, which no one else ever since has been willing to say, is that mathematics is actually a good career. You can have it as your career and you don't have to be woken in the middle of the night with passion for mathematics, in fact you can be a mathematician and that can be your job and you can be successful even if that's the case.</p>
<p>So that made me a bit more interested because I was never the sort of person who when I was two years old was dreaming of chalkboards.
So once I thought that it was reasonable and could accommodate other things in my life as well, that was when I really became interested.</p>
<p><strong><em>Plus</em>:</strong> What's it like being a female mathematician?</p>
<p><strong>Holly Krieger:</strong> I should have some reasonable answer to this. I assume it's sort of like being a female in most other aspects of life. Most people are quite reasonable and treat you normally. A few people treat you poorly, but they treat everyone poorly. Then there are a few others where it's not so clear.</p>
<p>So leaving that aside, of course people will behave in the way that they want to behave and that's not under your control, but the thing that I guess is notable about being a female mathematician is that you are very often the strong minority in a room.</p>
<p>The other side of that is that you can have a very large impact just by being in a room. So if I go to a seminar and me being there means that some post-doc or graduate student is not the only woman in the room, that makes a big difference for her.</p>
<p>So there's sort of this dual aspect of it which is that I feel like I can have a lot of impact on other mathematicians and other potential and future mathematicians, but then of course there are some negative aspects to it as well.</p>
<p><strong><em>Plus</em>:</strong> Is there any advice you would give to a young woman who wants to be a mathematician?</p>
<p><strong>Holly Krieger:</strong> The main advice I would give, because this career path is so drawn out in some ways, to get through the undergrad and maybe a Master's and a PhD and all that kind of thing and then post-doc and so on, is that if you want to get through it, all you have to do is not quit.</p><p>
This is not unique to women, there will be times when you want to give up and you feel like you need to give up and in fact somehow, everyone else is more likely to succeed than you, and all you have to do is work through those times instead of being paralysed by your own concerns.</p>
<p><strong><em>Plus</em>:</strong> For you, what are the joys of doing maths and what are the challenges?</p>
<p><strong>Holly Krieger:</strong> Well, the joys are that it's the joy of learning. That's how I spend a decent portion of at least my research time, learning these fantastically amazing things that other people have done or re-learning in some cases what I have done in the past. </p>
<div style="float:left; margin-right: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/HrpqMdKLxF0?rel=0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">Watch the interview with Holly Krieger in this video!</p>
</div>
<p>And the fact that most of the conversations I have with my colleagues are so interesting and really, each day has some unexpected aspect to it.</p><p>
The challenges are mostly the practical matters. I mean, there's not that much time in the day and because you're in this tenuous balance between a job that's not really quite as obviously useful as some other science positions, and you're teaching and all of these things have to be balanced, you don't get to spend as much time doing that learning and doing that communicating as you might like.</p>
<p><strong><em>Plus</em>:</strong> Can you explain your area of mathematics?</p>
<p><strong>Holly Krieger:</strong> Yes, of course. I work in dynamics. Dynamics is the study of long-term behaviour of systems.</p>
<p> I think probably the most familiar example to most people is thinking about orbits or the solar system. In school physics you learn that there are rules for how the planets move. Say, just to simplify things, if two gravitational bodies [are at particular positions] at time one and they're [at other positions] at time two, then what happens at time one million? </p>
<p>This is what the question of dynamical systems is asking; if I know how to go from step A to step B then can I, when I repeat that process, say something about the long-term behaviour?</p>
<p> I particularly study places where dynamics is chaotic, meaning that any small change in the starting configuration might lead to unpredictable behaviour in the long-term. <em>(You can read more about the kind of mathematics involved <a href="/content/unveiling-mandelbrot-set">here</a>.)</em></p>
<p><strong><em>Plus</em>:</strong> Could you describe one of your favourite mathematical moments?</p>
<p><strong>Holly Krieger:</strong> I think probably most of these come from teaching for me. Of course, there's this wonderful thing when you solve a problem and you didn't understand it one moment and then suddenly it was also brilliantly clear to you the next moment, but I think this is the funniest when it relates to teaching.</p><p>
There have been so many things that I thought I understood when I learned them and then I realised when I was preparing teaching that I didn't actually understand them.</p><p>
To use an analogy that most people would be familiar with is when you first learn the Pythagorean theorem, which says that a right triangle has this relationship between the squares of the lengths of the edges. Maybe it's hard to remember okay, if this one is A and this one is B and this one is C, is it A squared plus B squared on one side or is it A squared plus C squared? Which one is it?</p><p>
But then of course if you can draw the picture and put it into the context and you really know the proof, then of course you know where which one goes because one has got to be the longest side, right? So the true understanding of the problem sort of takes care of all of the unpleasant notation and memorisation.</p><p>
So moments like that, when I thought I knew something but really there was something there that was not quite snapped into place in my head, and then I prepare the lecture for it or I discuss it with a student and realise, "Oh, of course. I didn't actually understand this the first time and now I do," — I find that really interesting.</p>
<p><strong><em>Plus</em>:</strong> Thank you very much!</p></div></div></div>Wed, 12 Apr 2017 16:41:26 +0000mf3446811 at https://plus.maths.org/contenthttps://plus.maths.org/content/women-mathematics-holly-krieger#commentsWomen of mathematics: Julia Gog
https://plus.maths.org/content/womem-mathematics-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" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Julia_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article and video accompany the <em>Women of
Mathematics</em> photo exhibition. To see more profiles of female mathematicians and to find out more about the exhibition, see <a href="/content/women">here</a>. Photographs by <a href="http://henrykenyonphotography.com">Henry Kenyon</a>.</em></p>
<hr/>
<div class="rightimage" style="max-width: 450px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/Julia_small.jpg" alt="Julia Gog" width="450" height="525" />
<p></p>
</div>
<p>Julia Gog is a Reader in mathematical biology at the <a href="http://www.damtp.cam.ac.uk">Department of Applied Maths and Theoretical Physics</a>, University of Cambridge, and <a href="http://www.queens.cam.ac.uk/life-at-queens/news-and-events/the-david-n-moore-fellowship-in-mathematics">David N. Moore Fellow of Mathematics</a> at <a href="http://www.queens.cam.ac.uk">Queens' College</a>.</p>
<p><strong><em>Plus</em>:</strong> How and when did you choose to do mathematics?</p>
<p><strong>Julia Gog:</strong> Did I choose? How and when? Well, I'm not sure there was a particular moment when I chose. In fact, for a long time, I wanted to do various other things as a child. As a young child, I actually wanted to be an archaeologist, and then I had a phase where I wanted to do civil engineering.</p>
<p>It was actually during sixth form, when I got involved in the Mathematical Olympiad, and could see that mathematics was something far bigger, and that sucked me in. I chose then to do a maths degree. I can't remember when I chose to be a mathematician. I think it just happened. Have I still chosen? Have I grown up? Do I have a job yet? </p>
<p><strong><em>Plus</em>:</strong> Did your friends and family encourage you?</p>
<p><strong>Julia Gog:</strong> Well, certainly, like probably most mathematicians, I had a crucial teacher in my sixth form, who very much encouraged me, and pushed me to aim a little higher, and think a little harder about things.</p><p<
My family never pushed me particularly towards maths or indeed, academe at all. Actually, both my parents are wonderful people. They're nurses, and they always had the attitude that I was allowed to do absolutely anything, but there were two important things. One, it had to be something I wanted I to do, and two, I had to try and do good with my life. Those were the things.</p>
<p><strong><em>Plus</em>:</strong> What's it like being a female mathematician?</p>
<p><strong>Julia Gog:</strong> Being a female mathematician? Well, I don't have many other points of reference here. I've never been a male mathematician or a female geologist, not sure. Maybe interesting though is last year, I went to one of the <a href="https://www.lms.ac.uk">London Mathematical Society's</a> women in maths conferences in Oxford, where the majority of researchers there were female.</p>
<p>I thought that wouldn't make much difference to me, because I'm kind of used to this, and actually, it did. It felt really different, and then suddenly, it brought it into stark contrast. When I was back at other conferences, the women were in a minority. I was suddenly very aware of it. Yes, it's odd. We're in a minority, but we're quite a vocal minority, and it is changing.</p>
<p><strong><em>Plus</em>:</strong> In what way was it different?</p>
<div style="float:left; margin-right: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/dQaz02HBqks?rel=0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">Watch the interview with Julia Gog in this video!</p>
</div>
<p><strong>Julia Gog:</strong> Just the dynamics. There were more people who I felt were like me, and it was easier to associate with other people quickly. The whole conference just had a whole festival feel about it. We were all clearly enjoying it.
</p>
<p><strong><em>Plus</em>:</strong> For you, what are the joys of doing mathematics, and what are the challenges?</p>
<p><strong>Julia Gog:</strong> The joys, the joys are many. I still love maths. I still love learning new maths. I still go and sit in on undergraduate courses on areas I think, "I don't know enough about this." I enjoy learning about maths, and the maths research, well, that's the real buzz, that's the massive buzz.</p>
<p>I've only had a few real eureka moments in my research career, and they're fantastic, and I can picture them. You don't know they're those moments until afterwards, and you realise what's happened.</p>
<p><strong><em>Plus</em>:</strong> Can you describe one of those mathematical moments?</p>
<p><strong>Julia Gog:</strong> Yes. Actually, the first one was during my PhD. I'd been working in this problem to try and make a system tractable [that is, manageable], and I was getting nowhere. I'd been doing this for weeks, months probably. Actually, I have to confess, I wasn't in the office at all, and I wasn't actually working. I was at home playing computer games.</p>
<p>Suddenly, a system came into my head, I don't know where from, and I didn't think, "Ah, I have the answer. I'm done. I'm wonderful." I thought, "Why is this system not the right one?" I started with, "Is this wrong? Is there any reason why this one shouldn't work? Why is it not right? It can't be right. I can't have just thought of the right one." Then, investigating over a period of days subsequently, yes, it worked.</p>
<p><strong><em>Plus</em>:</strong> Could you describe what your area of mathematics is about?</p>
<p><strong>Julia Gog:</strong> Yes. I do mathematics of infectious disease, so that's very applied maths in the sense that we're really focused on the applications and the outcomes. It's actually quite hard to pin down what area of maths it is. It's not just this type of differential equation or this type of statistics. We'll use whatever we get to hand.</p>
<p>My PhD was fairly mathematical, and it was based on reducing a large system to something that was tractable, so that we can go and investigate some more particular things. Since then, I've been lucky enough to be able to work with some large data sets.</p>
<p>I've had to learn data handling, and programming, and statistics, and I do a little bit of bioinformatics work, which means you use genetic data to design algorithms to find various things, and that suits me. I like this idea of moving around between different areas.</p>
<p>There is a theme that runs through it all, and that big picture drives me, but I like that I don't just do one thing. I can move around between these areas, and that suits my way of working. <em>(You can find out more about some of Julia's mathematics in <a href="/content/list-by-author/Julia%20Gog">these</a> Plus articles.)</em></p>
<p><strong><em>Plus</em>:</strong> Is there any advice that you'd give to a young woman wanting to become a mathematician?</p>
<p><strong>Julia Gog:</strong> A young woman wanting to become a mathematician? As distinct from a young person wanting to become a mathematician? Well, I guess everything I would say would apply to both groups, one being a subset of the other.</p><p>
The most important thing is to pursue areas and things that you enjoy and find interesting. Don't think about, "Is this a good tactical career move? Will this area lead to this or that?" No, follow your interests and they'll be the right ones.</p><p>
Read the books you enjoy, go to the lectures that you enjoy. Pursue a PhD in an area that sparks you, not the one that people tell you, "This area is the right one to go into." The one that makes you think, "Oh, yes. I want to go and do that." You'll instinctively know what that area is, and that would be the same for any young person.
</p>
<p><strong><em>Plus</em>:</strong> Thank you very much!</p></div></div></div>Wed, 12 Apr 2017 15:50:51 +0000mf3446810 at https://plus.maths.org/contenthttps://plus.maths.org/content/womem-mathematics-julia-gog#commentsWomen of mathematics: Nilanjana Datta
https://plus.maths.org/content/nilanjana-datta
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Nilanjana_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article and video accompany the <em>Women of
Mathematics</em> photo exhibition. To see more profiles of female mathematicians and to find out more about the exhibition, see <a href="/content/women">here</a>. Photographs by <a href="http://henrykenyonphotography.com">Henry Kenyon</a>.</em></p>
<hr/>
<div class="rightimage" style="max-width: 450px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/Nilanjana_small.jpg" alt="Nilanjana Datta" width="450" height="300" />
<p></p>
</div>
<p>Nilanjana Datta is a lecturer in quantum information theory at the <a href="https://www.damtp.cam.ac.uk">Department of Applied Mathematics and Theoretical Physics</a>, University of Cambridge, and a fellow of <a href="http://www.pem.cam.ac.uk">Pembroke College</a>.</p>
<p><strong><em>Plus</em>:</strong> How and when did you choose to do mathematics?</p>
<p><strong>Nilanjana Datta:</strong> As far back as I can remember, from my early school days, physics and mathematics were my two favourite subjects. Well, I had two other passions, I also liked dance and creative writing. I still remember when I competed the Indian equivalent of GCSE, my English teacher who was a very enthusiastic Irish nun was rather upset that I chose to do science after that.
<p>But once I started sixth form, it was absolutely clear to me that mathematics and physics were the subjects I wanted to study.</p>
<p><strong><em>Plus</em>:</strong> What's it like being a female mathematician?</p>
<p> <strong>Nilanjana Datta:</strong> Well, it is quite challenging because, as you very well know, we are the minority in this field. Surprisingly to me at least, [because] as far as university students are concerned, we are more of a minority in the Western world than back home where I come from in India, where we were half and half.</p>
<p>So numerous times, for example even during my research, I have been asked by various people, "Why do you do mathematics, why do you do physics?" Well, the answer was clear, because I love the subjects. But once I still remember when I expressed surprise that I was asked this question, I was told, "Don't you know? Mathematics is not feminine."</p>
<p>So there are these prejudices one has to fight against and often I've been mistaken for a secretary in the department. But, well, the challenge makes it all the more worthwhile.</p>
<p><strong><em>Plus</em>:</strong> What advice would you give to a young woman who is thinking about doing mathematics?</p>
<p> <strong>Nilanjana Datta:</strong> Yes, I would have a word of advice for them. If you like mathematics then forget the prejudices, ignore what others think. Just be passionate about it.</p>
<div style="float:left; margin-right: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/NVx90P3SY5k?rel=0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">Watch the interview with Nilanjana Datta in this video!</p>
</div>
<p>I have been asked many a time by young, female students whether it's possible to have a life, a family, children, as an academic doing mathematics. I would say yes, of course. It's completely possible. I have a family, I have a son.</p>
<p>I remember when my son was born some of my older colleagues, male colleagues, told me, "Oh, surely you're going to quit mathematics now." But of course I didn't. You have less time but that doesn't matter. We are women, we can multitask. You have shorter time but you can focus your thoughts and concentrate much better. It's even more satisfying.</p><p>
So I would say that if this is what you want to do, then don't let anything stop you.</p>
<p><strong><em>Plus</em>:</strong> For you, what are the joys of doing mathematics and what are the challenges?</p>
<p> <strong>Nilanjana Datta:</strong> There are a couple of things that appeal to me about mathematics. One of them is that it's about pure reason, it's about absolute truth. There is no room for compromise or vagueness or varying points of view. When you prove a theorem then the statement of the theorem is a fact which nobody can refute, and that's wonderful.</p>
<p>Another thing is the simplicity and elegance of maths. We know that, using mathematics, one can often explain the complexities of nature by elegant and simple mathematical equations. This is very appealing to me.</p>
<p>Also another fact is about doing mathematics, the simplicity of it. We don't need a laboratory, expensive equipment or anything of the sort. All we need is a paper and a pencil and sometimes not even that.</p>
<p>I remember many a time sitting at the dinner table with my family secretly puzzling over some proof that I'd been stuck at and my husband saying, "You're not really with us, are you?"</p><p>
The joy of mathematics is when you prove something. It is wonderful! It might be a very short-lived joy because then you're stuck at the next step, but it's still something I would not give up for anything else.
</p>
<p><strong><em>Plus</em>:</strong> Could you explain your area of mathematics to somebody who doesn't know anything about maths?</p>
<p> <strong>Nilanjana Datta:</strong> Yes, I could try. The field I work on is called quantum information theory. Information theory by itself, which is often called classical information theory, is the mathematical theory of storage, processing and transmission of information.</p>
<p>The importance and relevance of this in our daily lives is obvious to everyone. All of us spend a considerable amount of time every day acquiring information, sending messages, processing information and there are more and more platforms or methods to do it using emails and texts and social media, laptops and mobile phones, etc.</p>
<p>What I work on is quantum information theory. So what's the quantum bit? That's the theory of these information processing tasks when you use quantum mechanical particles, like electrons and photons and atoms, as information carriers. Quantum mechanics is the fundamental theory of microscopic particles, particles on the atomic or subatomic level, where it replaces Newtonian mechanics. Because of the underlying quantum mechanics governing the dynamics of these information carriers, one often gets totally new effects which are not there in classical information theory. These new features can be exploited to perform certain information processing tasks much more efficiently, much more quickly and in an improved manner compared to classical information theory.</p>
<p>So that's my field of research. It's a highly interdisciplinary field. You'll find pure and applied mathematicians, physicists and engineers working in this field. It's very exciting, though my research is more focused on the mathematical aspects of it. <em>(You can read more about quantum information in <a href="/content/quantumcomputing">these</a> Plus articles.)</em></p>
<p><strong><em>Plus</em>:</strong> Could you describe one of your favourite mathematical moments?</p>
<p><p> <strong>Nilanjana Datta:</strong> Oh, yes. Well, I remember one day I was invited to an international conference called TQC which was held in Tokyo in Japan. Unfortunately the time of the conference was during Cambridge term time and I was lecturing a course, so I did something really crazy. I went all the way there just for three days, and that was the stupidest thing to do. So I reached there totally jet-lagged, completely exhausted, stressed about my talk.</p>
<p>
Then I remember attending one talk by a Japanese scientist and suddenly just at the end of the talk, the haze seemed to have cleared. I went out for my coffee break and I had one idea, just one little idea.
Strangely enough, that little idea led me to the main field of my current research. In fact, we now have an annual conference which is based on this small field and my introduction to that field was through that little eureka moment way back in Japan, all those years back.</p>
<p><strong><em>Plus</em>:</strong> Thank you very much!</strong></div></div></div>Wed, 12 Apr 2017 15:48:49 +0000mf3446809 at https://plus.maths.org/contenthttps://plus.maths.org/content/nilanjana-datta#commentsWomen of mathematics: Anne-Christine Davis
https://plus.maths.org/content/women-mathematics-anne-christine-davis
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Anne_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article and video accompany the <em>Women of
Mathematics</em> photo exhibition. To see more profiles of female mathematicians and to find out more about the exhibition, see <a href="/content/women">here</a>. Photographs by <a href="http://henrykenyonphotography.com">Henry Kenyon</a>.</em></p>
<hr/>
<div class="rightimage" style="max-width: 450px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/Anne_small.jpg" alt="Anne Davis" width="450" height="300" />
<p></p>
</div>
<p>Anne-Christine Davis is Professor of Mathematical Physics (1967) at the <a href="http://www.damtp.cam.ac.uk">Department of Applied Mathematics and Theoretical Physics</a>, University of Cambridge.</p>
<p><strong><em>Plus</em>:</strong> How and when you chose to do mathematics?</p>
<p><strong>Anne-Christine Davis:</strong> I'm really a theoretical physicist, so I've always been fascinated by science, how things work, by nature – you know, the stars in the sky. From a very, very young age I became interested in science. A little story there. When I first started school, the teacher put flash cards up: A is for apple, B is for banana, C is for I can't remember what, to teach us all the alphabet. Fine, okay. Then she did it again. And I thought, "That's strange. She's done that once. We all know this now, why's she doing it again?" And the teacher picked this up, that I had learnt my alphabet first time round, and gave me a bucket of water and a pipette to play with. And I learnt a little bit about specific gravity. And I thought, "Oh, I want to be a scientist. This is fun." I was five then.</p>
<p>I realised that to understand physics, one has to understand mathematics; that the two are intertwined, and you can't do physics without mathematics. Since I'm not a natural experimentalist, despite that little story, I became a theoretical physicist, and of course that brought me into mathematics, and some physics ... a lot of physics is underpinned by absolutely beautiful mathematics, like <a href="/content/what-general-relativity">Einstein's equations</a> for example.</p>
<p><strong><em>Plus</em>:</strong> What's it like being a female mathematician/theoretical physicist?</p>
<p><strong>Anne-Christine Davis:</strong> Oh, it's a lot easier now than it used to be. When I was young I didn't really think about it. At school I was the only girl doing A-level physics, A-level mathematics, and chemistry. It was quite isolating because I would work by myself and the boys would work together, until a lad in the year above me failed his A-levels and retook, and then he and I worked together. He liked working with me because I was bright, and I quite liked working with him because I had someone to work with.</p>
<p>When I went to university I went to Royal Holloway to do physics, and it was a bit more of a revelation, because having come from an inner-city comprehensive school being the only girl doing physics, when I got to university there were other women doing a degree in physics. In fact there were about five of us in our lecture room. When I did my PhD I was again in a group with all guys. It was a bit isolating, because you don't really have a close friend. I used to have friends in different research groups to mine, but not in my own research group. And I think it really struck me when I came out of my PhD viva [examination] and one of the lecturers said, "Congratulations ... so when are you going to get married?" I looked slightly surprised. And he said, "Well, you've got your PhD now, so what's there left for you to do but marry Tony?" My then-boyfriend.</p>
<p>And since I had no intentions of marrying Tony, I was really quite shocked. And a senior member in my department had a post-doctoral position that he was offering, and I said, "Are there any women applying?" And he turned around and said, "I wouldn't give a job to a woman when there was a man there, because a man has a family to look after, whereas a woman has a husband to look after her." And I kind of realised that the attitudes that I thought were outdated were still there. This has sort of gone on, I'm afraid, almost throughout my career, that I've come up against being the woman.</p>
<div style="float:left; margin-right: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/6yWoEB_UYRw?rel=0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">Watch the interview with Anne-Christine Davis in this video!</p>
</div>
<p>Well, I survived that experience, I did do a post-doc. My first post-doc was quite hard. My second post-doc was fantastic. I was at Imperial College, I was in the group of <a href="https://en.wikipedia.org/wiki/Tom_Kibble">Tom Kibble</a> [a well-known theoretical physicist]. Tom was definitely a mentor. He had two daughters and a son; he was desperately supportive of women and women in physics. And from that point on, until his recent death, Tom was very supportive of me.</p><p>
<p>So I think my life really changed as a post-doc in Imperial. I went to <a href="https://home.cern">CERN</a> [the European Organisation for Nuclear Research] for my next post-doc. There were other women at CERN; I had friends in the experimental group. There were senior women theorists hanging around, but they were in the University of Annecy or somewhere else, they weren't actually appointed in CERN, and I didn't realise at the time. In fact I've only realised a few years ago that I was the first woman to have a job in the theory division at CERN. I'm pleased I didn't know that when I was there, because I don't know if I could have coped. You know, being the first to go to university in my family, and being the first woman here, and the first woman there, you sort of think, "Hang on a minute, I'm not sure about this."</p>
<p>When I first came to Cambridge on a five-year research fellowship, it was actually quite a hard place to be a woman. Ruth Williams was here. She was a college teaching officer in Girton College, and not really regarded as part of the mainstream. One thing is that her research wasn't central to what the research groups were doing. She was treated a bit like a second-class citizen when I had first arrived. I came with this prestigious five-year fellowship, and I was treated like a second-class citizen. You certainly realised that Cambridge was quite a hard place to be a woman. Slowly things have improved, and now it's changed completely. It's changed out of all recognition. Quite a lot of that was down to an inspirational head of department it had, years ago now, David Crighton, who set the ball rolling, recognised the achievements of some of the women, and essentially found ways of getting them appointed into university positions. His former research student, who is now our head of department, seems to be following in that line.</p>
<p><strong><em>Plus</em>:</strong> What advice would you give to a young woman now who is thinking about going into mathematics?</p>
<p><strong>Anne-Christine Davis:</strong> I think that if they want to go into mathematics, or science in general, if they're interested they need to persevere and follow their dream. If someone says, "You're not good enough," or someone says, "Women don't do maths," just ignore them, or tell them they're wrong, and prove they're wrong. If you really want to do it, just persevere so you can do it.</p>
<p><strong><em>Plus</em>:</strong> For you, what are the joys of doing mathematics and theoretical physics, and what are the challenges?</p>
<p><strong>Anne-Christine Davis:</strong> My biggest joy now is being a PhD supervisor and advisor to younger people, and watching them flower. They just finished their degrees, and they blossom over the three or four years into being great researchers, and this is fantastic: I now have several research students, former research students, who are in faculty positions – in fact, several women. I've got two very successful women, one of them is in Nottingham, another in Imperial, both of them thriving, leaders in the field. And it's fantastic. This is my biggest joy now, watching these people develop.</p>
<p>But also I still get a kick. I've just come back from delivering the Ehrenfest colloquium in Leiden, and as far as I can tell, I'm the second woman in the history of this colloquium to do it. It's quite amusing, because you sign the wall afterwards, and so I signed the wall. There's quite a lot of signatures, but there's also Einstein there! So rather overwhelming company. When I was first thinking about this, I was a bit overwhelmed. Well, I know that they hadn't had women for years, but .... "No, you can do it." You know? And in a way I still get the kick out of realising that my work is appreciated, as well as developing the younger generation.</p>
<p><strong><em>Plus</em>:</strong> Could you tell us what your work is about?</p>
<p><strong>Anne-Christine Davis:</strong> Well, my work recently, over many years now, has been in theoretical cosmology. We discovered that the Universe is undergoing a late period of accelerated expansion. We knew that there was a steady expansion of the Universe, this is what we get from looking at Einstein's equations. But it seems to have undergone this accelerated expansion that, when we found it out, it was completely surprising. And I've been trying to understand that. <em>(You can read more about the expanding Universe and the discovery of its acceleration <a href="/content/exploding-stars-reveal-dark-energy">here</a>.)</em></p>
<p>The approach I've taken is, rather than saying, "Oh, it's a cosmological constant [see <a href="/content/exploding-stars-reveal-dark-energy">here</a>] that Einstein put in," and then try to explain why it's a funny scale, I've been thinking that it's dynamical, and it's coming from the dynamics of a field [fields are related to forces: you might be familiar with the idea of a gravitational field, related to gravity]. So I've added an extra field to Einstein's gravity. It's called a <em>chameleon</em>. Well, that's the one that is easiest to describe. And this should give you an extra force, a fifth force. But the way the chameleon mechanism works, the behaviour of the chameleon depends on the environment. So when the environment's very dense, like in the solar system [there is a lot of mass in the solar system] , the extra force is screened [weakened] because the chameleon becomes quite massive. But cosmologically, it's massless and unscreened, so it gives you extra effects in the cosmos.</p>
<p><strong><em>Plus</em>:</strong> Could you tell us about one of your favourite scientific experiences? </p>
<p><strong>Anne-Christine Davis:</strong> There've been quite a few eureka moments. One of the eureka moments was actually writing down this chameleon theory, and suddenly realising that, " Actually, it works like this: we have this screening around the fifth force when matter is dense. This is how it works." And suddenly realising that on the piece of paper you've got there's the correct idea and thinking, "Wow, that's incredible."</p>
<p><strong><em>Plus</em>:</strong> Thank you very much!</p></div></div></div>Wed, 12 Apr 2017 13:00:08 +0000mf3446814 at https://plus.maths.org/contenthttps://plus.maths.org/content/women-mathematics-anne-christine-davis#commentsWomen of mathematics: Carola-Bibiane Schönlieb
https://plus.maths.org/content/women-mathematics-carola-bibiane-sch-nlieb
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Carola_icon_0.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article and video accompany the <em>Women of
Mathematics</em> photo exhibition. To see more profiles of female mathematicians and to find out more about the exhibition, see <a href="/content/women">here</a>. Photographs by <a href="http://henrykenyonphotography.com">Henry Kenyon</a>.</em></p>
<hr/>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/carola.jpg" alt="Holly Krieger" width="350" height="525" />
<p></p>
</div>
<p>Carola-Bibiane Schönlieb is a Reader in Applied and Computational Analysis and Head of the <a href="http://www.damtp.cam.ac.uk/research/cia/people/">Image Analysis Group</a> at the <a href="http://www.damtp.cam.ac.uk">Department of Applied Mathematics and Theoretical Physics</a>, University of Cambridge.</p>
<p><strong><em>Plus</em>:</strong> How and when did you chose to do mathematics?</p>
<p><strong>Carola-Bibiane Schönlieb:</strong> I went to school in Austria, in Salzburg, to a <em>gymnasium</em> which is like doing A-levels here. Actually the gymnasium I was at was specialised in languages, which it turned out I really don't have a talent for.</p><p>
So I was always quite interested in mathematics and I guess I was just reasonably good at it during school, so it was a natural choice for me to do it at university afterwards.</p><p>
When I graduated in school, my headteacher gave me a book as a present, <em>Fermat's last theorem</em> by Simon Singh, which I read over the summer. I thought wow, this person just spent years of his life proving one very simple statement and then it turns out that pages of proof are involved in verifying it.
That left a deep impression on me and then I thought, "Okay, this is really the right choice, let's just go for it." I'm not a pure mathematician myself now, but this was really something that I thought this must be very fascinating if someone spends years on one question. <em>(You can read more about Fermat's last theorem <a href="/content/very-old-problem-turns-20">here</a>.)</em></p>
<p><strong><em>Plus</em>:</strong> You say you're not a pure mathematician. Can you explain what your area of mathematics is about?</p>
<p><strong>Carola-Bibiane Schönlieb:</strong> My background is in partial differential equations. I use partial differential equations and so-called <em>variational methods</em> in optimisation problems.</p>
<p>I use these mathematical tools for inverse imaging problems, meaning that you have either a measurement of an image or you have an image itself and you would like to extract some information from these measurements.</p>
<p>For example, think about image denoising: you're measuring a noisy image and you want to compute a denoised image. Or it could be you that you're not interested in the whole image, but you want to segment [isolate] certain parts in this image.</p>
<p>For example, very often in medical imaging, you're not measuring an image directly but you measure some indirect <em>transform</em> of this image. In magnetic resonance tomography for instance, that's connected to the <em>Fourier transform</em>: you get Fourier measurements and then you want to reconstruct an image from these measurements. <em>(You can find out more about Fourier transforms of images in <a href="/content/fourier-transforms-images">this article</a>, about some of the maths behind medical imaging in <a href="/content/saving-lives-mathematics-tomography">this article</a>, and about some of Carola's work in <a href="/content/what-eye-cant-see">this article</a>.)</em></p>
<p><strong><em>Plus</em>:</strong> Could you tell us one of your favourite mathematical moments?</p>
<p><strong>Carola-Bibiane Schönlieb:</strong> I think there are many and very different moments that you experience during your research. I think really the best mathematical moments appeared during my research work.</p>
<p>I don't know if there is really a best one but what I appreciate a lot is this process when you start thinking about something and it takes you a long time until you understand it. This is often a kind of torturous process — so this is one of the challenges, I guess, of being a mathematician, that you are always faced with these really difficult problems. But then, this moment when all of a sudden you get an idea in this process of trying to understand the problem and finding out how to solve it, you get an idea and then follow this idea through and see that it actually works, this is really cool.</p>
<p><strong><em>Plus</em>:</strong> A related question: what are the joys of doing mathematics and what are the challenges?</p>
<div style="float:left; margin-right: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/8UOGfztElfI?rel=0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">Watch the interview with Carola-Bibiane Schönlieb in this video!</p>
</div>
<p><strong>Carola-Bibiane Schönlieb:</strong> I think the joys are when you succeed. When you succeed and when you see that what you have done is interesting, not just for you, but for other people.</p><p>
So you go to conferences and you present what you have done and people are excited about that, people read your papers and you get emails saying, "Ah, this is really interesting. Have you thought about this or this?" This is something that I enjoy quite a lot.</p><p>
I also enjoy a quite a lot to work with students, in particular doing research with students.</p>
<p>Actually, collaborating in general is a wonderful thing for me in mathematics, especially if you meet people that are on the same wavelength as you are and that are complementing you, so you can discuss mathematics very freely.</p><p>
Discussing mathematics I think is something very personal and you really need to trust the other person to openly reveal that you have no idea about something. I think only this way can you make progress and actually solve problems. But this collaboration and this personal level about mathematics I like a lot.</p>
<p>The challenges I think are that we really have to work on very hard problems and you need a lot of patience to do that, and you need a lot of confidence, which is often not so easy.</p><p>
So you have to deal with the fact that sometimes you might not be able to answer a question just like that. Maybe you will answer a question a year after you have started thinking about it, maybe you have still not solved it now.<p>
I think this is something that is challenging on a very personal level because your research is your research and it's your question and it's you who has to solve it. If you can't solve it, you still should keep believing in yourself.</p><p>
So I think this thing with confidence is a very important challenge to overcome and to get to know your own strengths and your own weaknesses and to really exploit your strengths and maybe work a bit on your weaknesses.</p>
<p><strong><em>Plus</em>:</strong> What's it like being a female mathematician?</p>
<p><strong>Carola-Bibiane Schönlieb:</strong> When I was doing my undergraduate studies I didn't realise at all that gender makes a difference and I really never thought about it.
Then when I started to do my PhD I for the first time was sharing an office with another female PhD student. Then I thought, "Well, this really makes a difference. It's really nice to not always be the only one."
So this is one thing I think that we should pay attention to and this is one of the key reasons why it's important to increase the number of women in mathematics, because it changes the environment. It's just nice not to be a singularity all the time.</p>
<p><strong><em>Plus</em>:</strong> What advice would you give to a young woman who wants to become a mathematician?</p>
<p><strong>Carola-Bibiane Schönlieb:</strong> An important thing, as I said, is to get to know your strengths and to believe in your strengths, and to accept your weaknesses and don't pull yourself down for your weaknesses.</p>
<p>
I meet so many female mathematicians; students and also women who have already achieved a lot, who still think and still say that they might not be good enough. This really kills you if you don't get rid of it, and there is no reason to believe that actually. Everyone has their strong points and their weak points and I think this is just something that we have to overcome to focus on the strong points, and then just go for it.</p><p>
One other piece of advice I would give is planning your career. Again, I only know this of women, that there is this bad conscience about following your career and leaving family behind — you don't leave family behind, but there is this bad conscience if you want a career.</p>
<p>Just yesterday I met a colleague of mine who is an assistant professor and a woman and her husband has been taking care of their child for the last two years, so he was the one who stayed at home. She said she still has a bad conscience because of that. Just get rid of the bad conscience.</p>
<p><strong><em>Plus</em>:</strong> Thank you very much!</p></div></div></div>Wed, 12 Apr 2017 10:07:08 +0000mf3446813 at https://plus.maths.org/contenthttps://plus.maths.org/content/women-mathematics-carola-bibiane-sch-nlieb#commentsWomen of mathematics: Natalia Berloff
https://plus.maths.org/content/women-mathematics-natalia-berloff
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/Natasha_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article and the video accompany the <em>Women of
Mathematics</em> photo exhibition. To see more profiles of female mathematicians and to find out more about the exhibition, see <a href="/content/women">here</a>. Photographs by <a href="http://henrykenyonphotography.com">Henry Kenyon</a>.</em></p>
<hr/>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/Natalia2.jpg" alt="Natalia Berloff" width="350" height="525" />
<p></p>
</div>
<p>Natalia Berloff is a Professor of Applied Mathematics at the <a href="http://www.damtp.cam.ac.uk">Department of Applied Mathematics and Theoretical Physics</a>, University of Cambridge, and a Fellow of <a href="https://www.jesus.cam.ac.uk">Jesus College</a>.</p>
<p><strong><em>Plus</em>:</strong> How and when you chose to do mathematics?</p>
<p><strong>Natalia Berloff:</strong> Aged about 10, the usual age for this kind of thing. It just happened. I was reading tons of books, shelves after shelves. On one shelf, there were a lot of books about mathematics, some were for children, for example one explaining graph theory. I was suddenly, completely taken by the theory, by how logic works to actually solve unsolvable problems on graphs, so that was the beginning.</p>
<p> I grew up in Russia, so we had a very well developed Olympiad movement in mathematics. I started winning on various levels, and once you get to the country level, then big players like Moscow State University and Landau School start taking notice of people, so that's how I got lured into this wild world.</p>
<p><strong><em>Plus</em>:</strong> What's it like being a female mathematician?</p>
<p><strong>Natalia Berloff:</strong> Being a mathematician, — female or male. I don't know if it makes such a big difference. We're defined by our profession.</p>
<p><strong><em>Plus</em>:</strong> Is there any advice that you would give to a young woman wanting to enter mathematics, or do you think the advice is equal for both genders?</p>
<p><strong>Natalia Berloff:</strong> I think it's equal. I have a boy and a girl, a son and a daughter, so I give them the same advice. The advice is that being good in maths opens a lot of possibilities. You don't have to be a mathematician, but knowing mathematics opens the road for any science; physics, chemistry, biology. You can do whatever you want.</p>
<div style="float:left; margin-right: 1em;"><iframe width="400" height="225" src="https://www.youtube.com/embed/1Ym91MGa1LE?rel=0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; font-size: small; color: purple;">Watch the interview with Natalia Berloff in this video!</p>
</div>
<p><strong><em>Plus</em>:</strong> For you, what are the joys of doing mathematics, and what are the challenges?</p>
<p><strong>Natalia Berloff:</strong> It' a treasure hunt. That's exactly what we do. We're trying to find our jewel, to find the treasure, in a sense we that have some leads. We're solving the problem, and there are hints, leads, but you need to find this holy grail, and then you start using different tools. You start learning different tools, different machinery that probably will help you to discover.</p>
<p>Also, I think at first, when you start being in mathematics, or in physics, you start finding these little jewels, these little things, and you're very happy. The more you go, the more you are really looking for a big thing, and that's what the excitement is about, just expecting that you'll have this big breakthrough.</p>
<p><strong><em>Plus</em>:</strong> Could you explain your area of mathematics?</p>
<p><strong>Natalia Berloff:</strong> It's applied mathematics, so I'm writing equations and trying to understand a physical system. In my case, <a href="https://en.wikipedia.org/wiki/Quantum_fluid">quantum fluids</a>, <a href="https://en.wikipedia.org/wiki/Superfluidity">superfluidity</a>, <a href="https://en.wikipedia.org/wiki/Bose–Einstein_condensate">Bose-Einstein condensate</a>, and for a given physical system, it could be liquid, it could be gas, it could even be a solid state, I'm trying to write down the equations that describe the motions, that describe the phenomena observed.</p>
<p>I'm in close contact with experimentalists, trying to see if my theory works, and what else they present that still doesn't fit into the model, so I can extend it and build a new one.</p>
<p><strong><em>Plus</em>:</strong> Could you describe one of your favourite mathematical moments?</p>
<p><strong>Natalia Berloff:</strong> There were many. The first one, again, was in this book that I mentioned, that I read when I was 10. There was this problem of a given set of points, a graph as a set of points with edges in between — a network. Is it possible to go along every edge just once, without taking your pen off the paper? A very simple problem. The answer is brilliant, as it seemed to me when I was 10. The answer is you count every node, and how many passes are going out of this node.</p>
<div class="rightimage" style="max-width: 194px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/bridges_0.jpg" alt="Graph" width="194" height="158" />
<p>In this graph two nodes have an odd number of edges and two have an even number of edges. This means that you can find a path that crosses every edge exactly once. Image: <a href="http://en.wikipedia.org/wiki/File:Konigsberg_bridges.png">Bogdan Giuşcă</a>.</p>
</div>
<!-- Image by author -->
<p>If the number is odd, it means that you will be able to start, go around, come, and leave. If it's even, then you can't go, come and go, which means that only the graphs where you have two nodes with odd numbers will actually satisfy the property that you can go across every edge exactly once. You have to start at the first odd and finish at the other one. That was a very simple answer. Before knowing the answer you try various graphs, and you see that sometimes you don't know why it works, or why it doesn't work. Maybe you just didn't find the right way. At the end, when you understand, it makes perfect sense. Then, you think, "Oh, wow. Now I can solve any problem because I know this trick." For me being 10, that was a memorable moment. <em>(You can find out more about this problem in <a href="/content/maths-minute-bridges-konigsberg">this</a> article.)</em></p>
<p><strong><em>Plus</em>:</strong> Thank you very much!</p></div></div></div>Wed, 12 Apr 2017 09:56:48 +0000mf3446812 at https://plus.maths.org/contenthttps://plus.maths.org/content/women-mathematics-natalia-berloff#commentsFourier transforms of images
https://plus.maths.org/content/fourier-transforms-images
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_FT.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The sounds we hear – whether music, speech, or background noise – are the
result of vibrations of our ear drum, stimulated by sound waves
travelling through the air, created by our headphones,
musical instruments, people's voice boxes, or that annoying person behind you in the cinema
opening their sweets. These vibrations can be plotted (the intensity, or pressure, of the wave plotted over time) giving us a
visual representation of the sound.
</p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/tuningfork.png"/ alt="Soundwave of a tuning fork">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/speech.png" alt="Soundwave of human speech"/>
<p style="max-width:600px">The sound wave from a tuning fork (top), compared with that of human speech (bottom).</p></div>
<!-- tuning fork from http://www.arborsci.com/wordpress/wp-content/uploads/2013/08/3.png - permission granted by email to RT 27/2/2017 -->
<!-- speech is screenshot of plus interview recording in audacity, created by RT for Plus -->
<p>
<br/>
The sound wave of the middle A on a tuning fork, is a perfect example
of a <em>sine wave</em>, written mathematically as <em>sin(x)</em>. The sound wave of speech is more complicated. But
any sound wave, indeed any repeating function, can be broken up into a
number of sine waves of various frequencies and amplitudes
(intensities). This is the result of work that started with the French mathematician, Joseph Fourier, who lived through the French
revolution in the eighteenth century. The expression of a sound wave, or any signal varying
over time, as the sum of
its constituent sine waves, is known as the <em>Fourier
transform</em> of that signal. (You can read a more detailed
explanation of the maths involved <a
href="https://plus.maths.org/content/os/issue47/features/budd/maths">here</a>
– the maths is quite complicated but the mathematical ideas involved
are lovely!)
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/carola/Fourier_transform.gif"/><p style="max-width:300px">
The function <em>f</em> varies in time –
representing a sound wave. The Fourier transform process takes <em>f</em> and decomposes it
into its constituent sine waves, with particular frequencies and
amplitudes. The Fourier transform is represented as spikes in the frequency domain, the height of the spike showing the amplitude of the wave of that frequency.</p></div>
<!-- movie in public domain - -->
<!-- https://en.wikipedia.org/wiki/Fourier_transform#/media/File:Fourier_transform_time_and_frequency_domains_(small).gif -->
<p>
<br/>
You can also think of an image as a varying function, however, rather than
varying in time it varies across the two-dimensional space of the image. In a grey
scale digital image the pixels each have a value between 0 and 254
representing the darkness of that pixel. So the darkness, or
intensity, of that pixel, is a function of the horizontal and vertical coordinates
giving the location of that pixel. You can think of the image as an
undulating landscape, with the height of the landscape given by the
value of the pixel.
</p>
<div class="centreimage" style="max-width:706px"><img src="/content/sites/plus.maths.org/files/articles/2017/carola/R%2BM-surf.jpg" alt="The plus editors, in person and as a surface"/><p>A digital photograph of the <em>Plus</em> editors, each pixel has a value between 0 and 256, representing the greyness of that pixel. And on the right, the image function of the same digital photograph is shown, with the grey value <i>u(x, y)</i> plotted as the
height of the surface over the <i>(x, y)</i>-plane.</p></div>
<!-- <div class="centreimage"><img src="/issue50/features/schoenlieb/figure2.jpg" alt="A digital image and its numerical representation" width="729" height="216" />
<p>Digital image versus image function: The image on the left shows a zoom into a digital photograph where the image pixels (small squares) are clearly visible; in the middle the grey values of the red selection in the digital photograph are displayed in a matrix form; on the right the image function of the digital photograph is shown, with the grey value <i>u(x, y)</i> plotted as the
height over the <i>(x, y)</i>-plane.</p>
</div> -->
<!-- image provided by Carola Scholieb for https://plus.maths.org/content/restoring-profanity -->
<p>
Images can also be expressed as a sum of sine waves, but this time,
instead of one-dimensional waves, they are waves that vary in
two-dimensions, like ripples on a sheet.
</P><p>
Two-dimensional sine waves are written as
</P><p>
<em>z = a sin(hx+ky)</em>
</P><p>
where <em>x</em> and <em>y</em> give the coordinates for points on the "sheet", <em>z</em> is the height, or intensity, of the wave at that point, <em>a</em> gives the amplitude (the maximum height) of the wave, and <em>h</em> and <em>k</em> give
the number of times the wave repeats in the <em>x</em> and <em>y</em> directions respectively
(they are the <em>x</em> and <em>y</em> frequencies).
</P>
<div class="centreimage" style="max-width:800">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/sinx-sin2y-sinx%2By.jpg" alt="sin(x), sin(2y), sin(x+y)"/>
<p>The waves <em>sin(x), sin(2y)</em> and <em>sin(x+y)</em>. </p>
</div>
<!-- images created by RT for Plus using Matlab -->
<p>
When <em>k=0</em>, the sine wave only fluctuates along the <em>x</em>-axis. When <em>h=0</em>, it
only fluctuates along the <em>y</em>-axis. But if both <em>k</em>
and <em>h</em> are nonzero, the sine wave moves diagonally across the sheet,
with the waves travelling in direction (perpendicular to wave fronts) angled with the slope
<em>h/k</em>.
</P><p>
Adding these waves together just involves adding the respective values, or
heights, of the waves at each pixel. The waves can constuctively
interfere creating a final wave with a higher value at that point. And
the waves can destructively interfere and cancel out. If the
amplitude of one of the constituent waves is much larger than the
others, it will dominate.
</P>
<div class="centreimage" style="max-width:700">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/sinx%2By-5sinx-5siny.jpg" alt="Adding two-dimensional sine waves"/>
<p>The waves <em>sin(x)+sin(y), 5sin(x)+sin(y)</em> and <em>sin(x)+5sin(y)</em>. You can see how the larger amplitude wave – <em>5sin(x)</em> in the middle image and <em>5sin(y)</em> in the image on the right – dominate the resulting wave.</p>
</div>
<!-- image made by RT for Plus using Matlab -->
<p>
The Fourier transform of an image breaks down the image function (the undulating landscape) into a sum of
constituent sine waves. Just as for a sound wave, the Fourier transform is plotted
against frequency. But unlike that
situation, the frequency space has two dimensions, for the frequencies
<em>h</em> and <em>k</em>
of the waves in the <em>x</em> and <em>y</em> dimensions. So it is plotted not as a
series of spikes, but as an image with (roughly) the same dimensions
in pixels as the original image.
</p>
<p>
Each pixel in the Fourier transform has a coordinate (<em>h</em>,<em>k</em>)
representing the contribution of the sine wave with <em>x</em>-frequency <em>h</em>, and
<em>y</em>-frequency <em>k</em> in the Fourier transform. The centre point represents the
(0,0) wave – a flat plane with no ripples – and its intensity (its brightness in
colour in the
grey scale) is the average value of the pixels in the image. The
points to the left and right of the centre, represent the sine waves
that vary along the <em>x</em>-axis, (ie <em>k=0</em>). The brightness of these points
represent the intensity of the sine wave with that frequency in the
Fourier transform (the intensity is the amplitude of the sine wave, squared). Those vertically above and below
the centre point represent those sine waves that vary in <em>y</em>, but remain
constant in <em>x</em> (ie <em>h=0</em>). And the other points in the Fourier transform
represent the contributions of the diagonal waves.
</P>
<div class="centreimage" style="max-width:800">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/sinx-ft-zoom.jpg" alt="sin(x) and its Fourier transform"/><p>The wave <em>sin(x)</em> represented as a grayscale image, and the Fourier transform of that image.</p>
</div>
<!-- images created by RT for Plus using Matlab -->
<p>For example, consider the image above, on the left. This is the two-dimensional wave <em>sin(x)</em> (which we saw earlier) viewed as a grayscale image. Next to it is the Fourier transform of this grayscale image. It has the same dimensions in pixels as the original, and is entirely black except for a few bright pixels at the very centre. If we zoom into the centre of the Fourier transform (which you can see above, on the right) you can see there are exactly three pixels which are not black. One is the bright centre point, with coordinates (0,0), representing the contribution of the (0,0) wave to the image. The bright pixel on either side, with coordinates (1,0) and its reflection (-1,0), represents the contribution of the (1,0) wave (the sine wave in our original image). All the rest of the pixels in the Fourier transform are black, as the original image is exactly described using just the original (1,0) wave. </p>
<div class="centreimage" style="max-width:800">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/Eg-simpleFTs.jpg" alt="Some simple Fourier transforms"/><p>Top: The wave <em>sin(20x)+sin(10y)</em> and its Fourier transform, showing two pairs of bright pixels (at the coordinates (20,0) and (0,10) and their reflections) representing these contributions of these two waves. <br/>
Bottom: The wave <em>sin(100x+50y)</em> and its Fourier transform, showing just the pair of bright pixels at the coordinates (100,50) and its reflection.</p>
</div>
<!-- images created by RT for Plus using Matlab -->
<p>
The Fourier transforms of simple combinations of waves have only a
few bright spots. But for more complex images, such as digital photos,
there are many many bright spots in its Fourier transform, as it takes
many waves to express the image.
</P>
<p>
In the Fourier transform of many digital photos we'd normally take,
there is often a strong intensity along the <em>x</em> and <em>y</em> axis of the Fourier
transform, showing that the sine waves that only vary along these
axes play a big part in the final image. This is because there are
many horizontal or vertical features and symmetries in the world around us – walls,
table tops, even bodies are symmetrical around the vertical axes.
You can see this by rotating an image a little (say by 45%). Then its Fourier
transform will have a strong intensity along a pair of perpendicular
lines that are rotated by the same amount.
</p>
<div class="centreimage" style="max-width:600px">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/Plus-FT.jpg" alt="Fourier transform of Plus">
<p><em>Plus</em> editors and their Fourier transform, showing a series of contributions of vertical wave represented by bright points along the vertical axis. </p></div>
<div class="centreimage" style="max-width:600px">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/Plus45-FT.jpg" alt="Fourier transform of angled Plus">
<p>The <em>Plus</em> editors rotated by 45 degrees, and their Fourier transform.</p></div>
<p>
Fourier transforms are incredibly useful tools for the analysis and manipulation of sounds and images. In particular for images, it's the mathematical machinery behind image compression (such as the JPEG format), filtering images and reducing blurring and noise.
</p>
<p><em>
The images of 2D sine waves, surfaces and Fourier transforms were made in <a href="https://uk.mathworks.com/products/matlab.html">MATLAB</a> – in case you'd like to try it yourself you can see the commands we used <a href="/content/sites/plus.maths.org/files/articles/2017/carola/MATLAB.TXT">here</a>.
</em></p>
<h3>About this article</h3>
<p>
<a href="/content/people/index.html#rachel">Rachel Thomas</a> is Editor of <em>Plus</em>.</p></div></div></div>Fri, 07 Apr 2017 14:54:15 +0000Rachel6783 at https://plus.maths.org/contenthttps://plus.maths.org/content/fourier-transforms-images#commentsThe hidden beauty of multiplication tables
https://plus.maths.org/content/hidden-beauty-multiplication-tables
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/multi_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Zoheir Barka</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In this article we explore some of the symmetries that hide within the multiplication table of positive whole numbers. </p>
<p>Let us start with the standard multiplication table. The table below contains the numbers 1 to 10 in the first row and the first column. Any other square contains the product of the first number in its row and the first number in its column.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table0.png" alt="Table 0" width="486" height="489" />
<p style="max-width: 486px;"></p>
</div>
<!-- Image produced by author -->
We will add a row of <img src="/MI/533a11057dd9f84c2725ee773f14c76f/images/img-0001.png" alt="$0s$" style="vertical-align:0px;
width:15px;
height:12px" class="math gen" /> at the top and a column of <img src="/MI/533a11057dd9f84c2725ee773f14c76f/images/img-0001.png" alt="$0s$" style="vertical-align:0px;
width:15px;
height:12px" class="math gen" /> on the left. This still gives a consistent table — the first row and column contain multiples of <img src="/MI/533a11057dd9f84c2725ee773f14c76f/images/img-0002.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, the second row and column contain multiples of <img src="/MI/533a11057dd9f84c2725ee773f14c76f/images/img-0003.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" />, the third row and column contain multiples of <img src="/MI/533a11057dd9f84c2725ee773f14c76f/images/img-0004.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, etc — and it will provide a nice frame for our patterns.
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table1.png" alt="Table 1" width="544" height="542" />
<p style="max-width: 544px;"></p>
</div>
<!-- Image produced by author -->
<p>In the following, we will colour the squares of the multiplication table that correspond to multiples of a number <img src="/MI/cd1706804a9b91d0b6556328e5eafeba/images/img-0001.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> for various values of <img src="/MI/cd1706804a9b91d0b6556328e5eafeba/images/img-0001.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" />. And we’ll discover some beautiful symmetries. </p>
<h3>Single multiples</h3>
We begin with <img src="/MI/e7e1c2363475ec2578df491eb8d0ab4c/images/img-0001.png" alt="$k=2$" style="vertical-align:0px;
width:38px;
height:12px" class="math gen" />: we assign the colour blue to every square in the multiplication table that is a multiple of <img src="/MI/e7e1c2363475ec2578df491eb8d0ab4c/images/img-0002.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />. (The number <img src="/MI/e7e1c2363475ec2578df491eb8d0ab4c/images/img-0003.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> is a multiple of <img src="/MI/e7e1c2363475ec2578df491eb8d0ab4c/images/img-0002.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, so all the <img src="/MI/e7e1c2363475ec2578df491eb8d0ab4c/images/img-0003.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> squares are blue.)
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table2.png" alt="Table 2" width="606" height="397" />
<p style="max-width: 606px;"></p>
</div>
<!-- Image produced by author -->
<p>Here we have extended the table a bit so that it runs until the number 15 in the horizontal direction. Indeed, since the complete multiplication table on positive integers is infinite on two sides, we will continue
to tweak the dimensions of the tables in what follows to display the emerging
patterns more clearly.</p>
<p>Note that the whole pattern above can be pieced together using the fundamental building block:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/block.png" alt="Fundamental building block" width="74" height="73" />
<p style="max-width: 74px;"></p>
</div>
<!-- Image produced by author -->
<p>The fundamental building block contains <img src="/MI/db5d24bd1132ea957603690494c23003/images/img-0001.png" alt="$k \times k = 2 \times 2 =4$" style="vertical-align:0px;
width:126px;
height:12px" class="math gen" /> cells of the multiplication table. The squares defined by the white cells in the pattern consist of </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/db5d24bd1132ea957603690494c23003/images/img-0002.png" alt="\[ (k-1)^2=(2-1)^2=1 \]" style="width:166px;
height:19px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> cells. </p>
<p>Below are two more images in which the multiples of a number <img src="/MI/d7365982ae156698b6d0676ef66cedd8/images/img-0001.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> have been coloured blue. Can you tell what the value of <img src="/MI/d7365982ae156698b6d0676ef66cedd8/images/img-0001.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> is in each case? Can you tell what the fundamental building blocks are, how many cells they contain, and how many cells make up the squares defined by the white cells? You can post your answers in the comment field below — in case you can’t work them out, we’ll publish the answers in a few weeks’ time. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table3.png" alt="Table 3" width="639" height="640" />
<p style="max-width: 639px;"></p>
</div>
<!-- Image produced by author -->
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table4.png" alt="Table 4" width="523" height="398" />
<p style="max-width: 523px;"></p>
</div>
<!-- Image produced by author -->
<h3>Multiple multiples of consecutive numbers</h3>
A more interesting pattern emerges if we use multiple multiples, and corresponding to them, multiple colours. In the following figure, the numbers that are multiples of <img src="/MI/b3bf7cecf54cc14ba9232641d4dffa66/images/img-0001.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> are coloured red, and those that are multiples of <img src="/MI/b3bf7cecf54cc14ba9232641d4dffa66/images/img-0002.png" alt="$3$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> are coloured orange (with the orange taking precedence over the red in the case of multiples of both <img src="/MI/b3bf7cecf54cc14ba9232641d4dffa66/images/img-0001.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> and <img src="/MI/b3bf7cecf54cc14ba9232641d4dffa66/images/img-0002.png" alt="$3$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, that is, multiples of <img src="/MI/b3bf7cecf54cc14ba9232641d4dffa66/images/img-0003.png" alt="$6$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />).
<p>This gives the following pattern.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table5.png" alt="Table 5" width="698" height="536" />
<p style="max-width: 698px;"></p>
</div>
<!-- Image produced by author -->
<p>Note that this time our fundamental building blocks consist of <img src="/MI/5f85d04f7ee0b987a12b6330d786d46e/images/img-0001.png" alt="$ 6 \times 6 = 36$" style="vertical-align:0px;
width:75px;
height:12px" class="math gen" /> little squares, which makes sense, because <img src="/MI/5f85d04f7ee0b987a12b6330d786d46e/images/img-0002.png" alt="$6$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> is the least common multiple of <img src="/MI/5f85d04f7ee0b987a12b6330d786d46e/images/img-0003.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> and <img src="/MI/5f85d04f7ee0b987a12b6330d786d46e/images/img-0004.png" alt="$3$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />. The symmetry emerges from repeated copies of a <img src="/MI/5f85d04f7ee0b987a12b6330d786d46e/images/img-0005.png" alt="$5 \times 5$" style="vertical-align:0px;
width:36px;
height:13px" class="math gen" /> square with a nice four-fold symmetry. </p>
The next figure takes this a step further, assigning red to numbers that are multiples of <img src="/MI/2cbaec196b22591693e1f5bf9788e6f9/images/img-0001.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, orange to numbers that are multiples of <img src="/MI/2cbaec196b22591693e1f5bf9788e6f9/images/img-0002.png" alt="$3$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, and yellow to numbers that are multiples of <img src="/MI/2cbaec196b22591693e1f5bf9788e6f9/images/img-0003.png" alt="$4$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />. If a cell is a multiple of two of these numbers (eg <img src="/MI/2cbaec196b22591693e1f5bf9788e6f9/images/img-0004.png" alt="$6=2\times 3$" style="vertical-align:0px;
width:67px;
height:12px" class="math gen" />) then it will be assigned the colour of the larger of these two numbers (orange in the example). We will stick to this convention for the rest of this article.
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table6.png" alt="Table 6" width="696" height="694" />
<p style="max-width: 696px;"></p>
</div>
<!-- Image produced by author -->
<p>This time, our fundamental building blocks contain <img src="/MI/efc9343c3e79305050d4859633f7b9bf/images/img-0001.png" alt="$12 \times 12$" style="vertical-align:0px;
width:52px;
height:12px" class="math gen" /> cells, which again makes sense, given that <img src="/MI/efc9343c3e79305050d4859633f7b9bf/images/img-0002.png" alt="$12 $" style="vertical-align:0px;
width:15px;
height:12px" class="math gen" /> is the least common multiple of <img src="/MI/efc9343c3e79305050d4859633f7b9bf/images/img-0003.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, <img src="/MI/efc9343c3e79305050d4859633f7b9bf/images/img-0004.png" alt="$3$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> and <img src="/MI/efc9343c3e79305050d4859633f7b9bf/images/img-0005.png" alt="$4$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />. The symmetry emerges from repeated copies of an <img src="/MI/efc9343c3e79305050d4859633f7b9bf/images/img-0006.png" alt="$11\times 11$" style="vertical-align:0px;
width:51px;
height:12px" class="math gen" /> square, which contains nine small <img src="/MI/efc9343c3e79305050d4859633f7b9bf/images/img-0007.png" alt="$3 \times 3$" style="vertical-align:0px;
width:36px;
height:12px" class="math gen" /> squares which together create a nice four-fold symmetry. </p>
<p>We can carry on playing this game indefinitely. The next four figures use multiples of four, five, six and seven consecutive numbers respectively, and four, five, six and seven colours respectively. What patterns can you discern? Can you find any axes of (reflectional) symmetry? What should be the size of the fundamental (repeating) building blocks of symmetry in each case? Remember that you can post your answers in the comment field below, and in case you can't work them out, we'll publish the answers in a few weeks' time.
</p>
<p>(Click on the images to see a larger version.)</p>
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table7_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table7.jpg" alt="Table 7" width="600" height="509" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table8_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table8.jpg" alt="Table 8" width="600" height="452" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table9_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table9.jpg" alt="Table 9" width="600" height="482" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table10_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table10.jpg" alt="Table 10" width="600" height="402" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<h3>Multiples of non-consecutive numbers</h3>
<p>Next we use some non-consecutive values of <img src="/MI/86dec8be33d9611a6db07180ed89de9f/images/img-0001.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" />. The following figure uses blue for numbers that are multiples of <img src="/MI/86dec8be33d9611a6db07180ed89de9f/images/img-0002.png" alt="$6$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, and green for numbers that are multiples of <img src="/MI/86dec8be33d9611a6db07180ed89de9f/images/img-0003.png" alt="$9$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />. </p>
<p>(Click on the image to see a larger version.)</p>
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table11_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table11.jpg" alt="Table 11" width="600" height="363" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<p>The fundamental building blocks will now consist of <img src="/MI/2d58ae61245436d7a3d9af677fbe8df5/images/img-0001.png" alt="$18 \times 18 = 324$" style="vertical-align:0px;
width:98px;
height:12px" class="math gen" /> little squares, as <img src="/MI/2d58ae61245436d7a3d9af677fbe8df5/images/img-0002.png" alt="$18$" style="vertical-align:0px;
width:15px;
height:12px" class="math gen" /> is the least common multiple of <img src="/MI/2d58ae61245436d7a3d9af677fbe8df5/images/img-0003.png" alt="$6$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> and <img src="/MI/2d58ae61245436d7a3d9af677fbe8df5/images/img-0004.png" alt="$9$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />. Still the additional symmetries within the nine <img src="/MI/2d58ae61245436d7a3d9af677fbe8df5/images/img-0005.png" alt="$5 \times 5$" style="vertical-align:0px;
width:36px;
height:13px" class="math gen" /> squares that make up the repeated <img src="/MI/2d58ae61245436d7a3d9af677fbe8df5/images/img-0006.png" alt="$17 \times 17$" style="vertical-align:0px;
width:52px;
height:13px" class="math gen" /> squares may come as pleasant surprises. Can you find mathematical explanations for these? </p>
<p>Here are a few more patterns for you to admire. In each case we colour the multiples of non-consecutive numbers. Can you tell what numbers these are and describe the patterns that emerge? Remember that you can post your answers in the comment field below, and in case you can't work them out, we'll publish the answers in a few weeks' time.</p>
<p>(Click on the images to see a larger version.)</p>
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table12_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table12.jpg" alt="Table 12" width="600" height="386" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table13_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table13.jpg" alt="Table 13" width="600" height="431" />
<p style="max-width: 724px;"></p>
</div></a>
<!-- Image produced by author -->
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table14_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table14.jpg" alt="Table 14" width="600" height="353" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table15_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table15.jpg" alt="Table 15" width="600" height="403" />
<p style="max-width: 686px;"></p>
</div></a>
<!-- Image produced by author -->
<a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table16_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table16.jpg" alt="Table 16" width="600" height="345" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<h3>Remainders</h3>
<p>Finally, if we fix a number <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0001.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> and assign colours to cells depending on their remainder with respect to <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0001.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" />, then all the squares can be filled in. For example, let multiples of <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0002.png" alt="$5$" style="vertical-align:0px;
width:8px;
height:13px" class="math gen" /> be black, numbers with a remainder <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0003.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" /> with respect to <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0002.png" alt="$5$" style="vertical-align:0px;
width:8px;
height:13px" class="math gen" /> be green, numbers with a remainder <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0004.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> with respect to <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0002.png" alt="$5$" style="vertical-align:0px;
width:8px;
height:13px" class="math gen" /> be red, numbers with a remainder <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0005.png" alt="$3$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> with respect to <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0002.png" alt="$5$" style="vertical-align:0px;
width:8px;
height:13px" class="math gen" /> be purple, and numbers with a remainder <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0006.png" alt="$4$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> with respect to <img src="/MI/68a094433dc0749433229fc143f141ce/images/img-0002.png" alt="$5$" style="vertical-align:0px;
width:8px;
height:13px" class="math gen" /> be yellow; the following figure is obtained.: </p><p>(Click on the image to see a larger version.)</p><a href="/content/sites/plus.maths.org/files/articles/2017/multitables/table17_large.jpg">
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2017/multitables/table17.jpg" alt="Table 17" width="600" height="403" />
<p style="max-width: 600px;"></p>
</div></a>
<!-- Image produced by author -->
<h3>Conclusion</h3>
<p>
We've discovered some of the symmetries that hide within the multiplication table of positive integers.
It's easy to create these patterns (for example, using Excel) and they can all be explained without much difficulty using the arithmetic of whole numbers and divisibility criteria. Displaying these symmetries using colours introduces a new facet to the maths. These images and others created in similar ways may appeal to students of mathematics and the arts, and may lead to new collaborations. At the very least such images may, we hope, intrigue, amaze, and inspire.</p>
<hr/>
<h3>About this article</h3>
<p>
Zoheir Barka , from Laghouat in Algeria, is an amateur and self-educated mathematician. He has a Masters degree in French language from Laghouat University and is currently a French teacher in elementary school.</p>
<p>A version of this article first appeared as <em><a href="http://scholarship.claremont.edu/jhm/vol7/iss1/15">The hidden symmetries of the multiplication table</a></em> in the Journal of Humanistic Mathematics, Volume 7, Issue 1 (January 2017), pages 189-203. </p></div></div></div>Fri, 07 Apr 2017 13:36:02 +0000mf3446821 at https://plus.maths.org/contenthttps://plus.maths.org/content/hidden-beauty-multiplication-tables#comments