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Packing spheres: Maryna Viazovska
https://plus.maths.org/content/node/7114
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/maryna_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><div class="rightimage" style="maxwidth: 304px;"><img src="/content/sites/plus.maths.org/files/articles/2018/sphere_packings/maryna_vazovska_mfo_2013_crop.jpg" alt="Maryna Viazovska" width="304" height="243" /><p>Maryna Viazovska. Photo: Petra Lein, <a href="https://creativecommons.org/licenses/bysa/2.0/de/deed.en">CC BYSA 2.0 DE</a>.</p></div>
<! image in public domain ><p>This month we met <a
href="https://en.wikipedia.org/wiki/Maryna_Viazovska">Maryna
Viazovska</a> who in 2016 made a breakthrough in the theory
of sphere packings. She
explained her results at the Royal Society's <a href="https://royalsociety.org/scienceeventsandlectures/2018/10/srinivasaramanujan/">celebration</a> of the
legendary mathematician <a href="/content/happybirthdayramanujan0">Srinivasa Ramanujan</a>, who was elected a fellow
of the Society 100 years ago. During a coffee break she gave us
a condensed description of sphere packing problems and her work. </p>
<p><em>You can also read <a href="/content/packingspheres">an article</a> based on our interview with Viazovska.</p></div></div></div><div class="field fieldnamefieldremoteencl fieldtypefile fieldlabelinline clearfix clearfix"><div class="fieldlabel">Podcast download link: </div><div class="fielditems"><div class="fielditem even"><span class="file"><img class="fileicon" alt="Audio icon" title="audio/mpeg" src="/content/modules/file/icons/audioxgeneric.png" /> <a href="https://plus.maths.org/content/sites/plus.maths.org/files/podcast/sphere_packings.mp3" type="audio/mpeg; length=11126941">sphere_packings.mp3</a></span></div></div></div>
Tue, 13 Nov 2018 16:48:11 +0000
Marianne
7114 at https://plus.maths.org/content
https://plus.maths.org/content/node/7114#comments

Packing spheres
https://plus.maths.org/content/packingspheres
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/oranges_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Marianne Freiberger</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>When it comes to transportation fruit are awkward. Not only are they
easily squashed, they also have shapes that don't lend
themselves to being packed in boxes. Even the simplest possible
fruit shape, the sphere as seen in oranges and apples, causes a problem, because no matter how you pack spheres,
there'll always be gaps between them. This raises a geometrical
question: how should you pack spheres into a box to make sure the volume
of the gaps is as small as possible?</p>
<p>This month we met an expert in the field: the mathematician <a
href="https://en.wikipedia.org/wiki/Maryna_Viazovska">Maryna
Viazovska</a> who in 2016 made a breakthrough in the theory
of sphere packings. She
explained her results at the Royal Society's <a href="https://royalsociety.org/scienceeventsandlectures/2018/10/srinivasaramanujan/">celebration</a> of the
legendary mathematician <a href="/content/happybirthdayramanujan0">Srinivasa Ramanujan</a>, who was elected a fellow
of the Society 100 years ago. During a coffee break she gave us
a condensed description of sphere packing problems and her work. (You can also listen to the interview in a <a href="/content/node/7114">podcast</a>.) </p>
<h3>The problem</h3>
<p>"Suppose you have a very big box and a supply of spheres," Viazovska explains. "To make the problem easier suppose the spheres are of equal size and also hard, so we cannot squeeze them. We put as many spheres as we can into the box." The question is, what's the largest number of spheres you can fit in? </p><p>
<div class="leftimage" style="maxwidth: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2018/sphere_packings/hexagonal_circle.png" alt="Hexagonal circle packing" width="250" height="250" /><p>The hexagonal circle packing.</p></div>
<! image in public domain >
<p>
If the box is small, then the answer depends on the shape of the box. But if the box is very large, the effect of the shape is negligible, and the answer depends only on the volume of the box. "It's intuitively clear, though mathematically one has to work a little bit to see this, that there is a maximal possible [proportion of the volume you can fill with spheres]." The sphere packing problem is to find this highest proportion, also called the <em>sphere packing constant</em>.</p>
<p>For an easier example, let’s drop down a dimension: instead of packing spheres into 3D space let’s pack discs into 2D space. "In dimension two the best packing [comes from the] honeycomb," explains Viazovska. In a traditional honeycomb every cell is a hexagon and the hexagons fit neatly together leaving no space between them. If you arrange discs in the same pattern you do get gaps, but the packing turns out to have the highest density possible. "This way we will cover slightly over 90% of the area with these equallysized discs." The precise value of the sphere packing constant in two dimension is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/244a0b2ff313e2c4ba367a33122e12cb/images/img0001.png" alt="\[ \frac{\pi \sqrt {3}}{6} \approx 0.9069. \]" style="width:106px;
height:37px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>"In dimension three [the problem] was known as <em>Kepler’s conjecture</em> and remained open for [nearly 400] years," says Viazovska. "[Here] we don’t have only one best packing, we have many equally good packings. One of them you can see on the market, where oranges are stacked in pyramids (see the figure below, which illustrates this with balls rather than oranges). The density [of this packing] is about 74%." The precise value of the sphere packing constant in three dimensions is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/ec7bdc1feb3664fd93423ec715eec246/images/img0001.png" alt="\[ \pi /(3\sqrt {2}) \approx 0.7405. \]" style="width:133px;
height:20px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>The proof that you really can't do any better than this only arrived in
1998 when the mathematician
<a href="http://www.mathematics.pitt.edu/person/thomashales">Thomas
Hales</a> produced 250 pages of traditional
mathematical proof supplemented by over
3 gigabytes of computer code and data for making necessary calculations. This was a controversial approach.
Since no human could possibly check the computer
calculations in their life time, mathematicians weren't sure if Hales' work really counted as a proof. A panel of
experts eventually decided they were 99% sure that it did, and since then the proof has been verified using formal computer logic. </p>
<h3>Higher dimensions</h3>
<div class="rightimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/sphere_packings/cubic.jpg" alt="Face centred cubic packing" width="350" height="337" /><p>This is the facecentred cubic packing, one of the packings with maximal density in dimension three. Image: <a href="https://commons.wikimedia.org/wiki/File:Closepacked_spheres,_with_umbrella_light_%26_camerea.jpg">Greg A L</a>, <a href="https://creativecommons.org/licenses/bysa/3.0/deed.en">CC BYSA 3.0</a>.</p></div>
<p>Involving no computers and filling just a few
pages, Viazovska's proofs are as solid as proofs can
get. They involve packing higherdimensional spheres into higherdimensional spaces, namely spaces of dimensions 8 and 24. The endeavour might seem both useless and impossible to get your head around, but it's neither. Higherdimensional sphere packings are important in communications technology, where they ensure that the messages we send via the internet, a satellite, or a telephone can be understood even if they have been scrambled in transit (find out more <a href="/content/communicationandballpacking">here</a>). </p>
<! <div style="maxwidth: 340px; float: left; border: thin solid grey; background: #CCC CFF; padding: 0.5em; marginright: 0.5em; fontsize: 75%">
<h3>Measuring distance</h3>
<p>Suppose you have two points in 3D space given by the coordinates <img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0001.png" alt="$(a_1, a_2,a_3)$" style="verticalalign:4px;
width:74px;
height:18px" class="math gen" /> and <img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0002.png" alt="$(b_1,b_2,b_3).$" style="verticalalign:4px;
width:74px;
height:18px" class="math gen" /> The distance between them is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0003.png" alt="\[ \sqrt {(a_1b_1)^2+ (a_2b_2)^2+(a_3b_3)^2}. \]" style="width:273px;
height:21px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> (This fact follows from two applications of Pythagoras’ theorem.) </p><p>By analogy, the distance between two points with coordinates <img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0004.png" alt="$(a_1, a_2,...,a_8)$" style="verticalalign:4px;
width:95px;
height:18px" class="math gen" /> and <img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0005.png" alt="$(b_1, b_2,...,b_8)$" style="verticalalign:4px;
width:90px;
height:18px" class="math gen" /> in 8D space is </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0006.png" alt="\[ \sqrt {(a_1b_1)^2+ (a_2b_2)^2+...+(a_8b_8)^2}. \]" style="width:308px;
height:21px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>And the distance between two points with coordinates <img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0007.png" alt="$(a_1, a_2,...,a_ n)$" style="verticalalign:4px;
width:96px;
height:18px" class="math gen" /> and <img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0008.png" alt="$(b_1, b_2,...,b_ n)$" style="verticalalign:4px;
width:91px;
height:18px" class="math gen" /> in <img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0009.png" alt="$n$" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" />D space, for any other natural number <img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0009.png" alt="$n$" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" />, is </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/8e7f468f0a5648afbe9152dc2ebe8bdb/images/img0010.png" alt="\[ \sqrt {(a_1b_1)^2+ (a_2b_2)^2+...+(a_ nb_ n)^2}. \]" style="width:311px;
height:21px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
</div> >
<p>To get a grip on higher dimensions, cast your mind back to school when you moved from doing geometry in two dimensions to doing it in three. If you were one of the many people who struggle visualising things in 3D, you'll have appreciated the help of algebra. Points in three dimensions are given by three coordinates, and shapes such as lines, planes or spheres by equations involving the coordinates. If you can't visualise how shapes relate to each other, you use the equations to help you.</p>
<p>In higher dimensions the same principle applies. Points in <img src="/MI/be232e8bc29ada5b8c7b2fb2e3ddb4bd/images/img0001.png" alt="$n$" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" />dimensional space are given by <img src="/MI/be232e8bc29ada5b8c7b2fb2e3ddb4bd/images/img0001.png" alt="$n$" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" /> coordinates. You can come up with notions for distance and volume in analogy to two and three dimensions, and you can define shapes, such as spheres, using equations. Although you can’t visualise these shapes, the algebra allows you to deal with them, so you can also define what you mean by a sphere packing and its density. </p>
<p>Going back to the familiar dimensions two and three for a
moment, you can see how a 3D packing can be made from a 2D one by
stacking: start with spheres arranged in the honeycomb pattern that
gives the best density for circles in 2D and stack another layer of
spheres above it, so that the spheres of the second layer sit in the
dips between spheres in the first layer. Then add a third layer, and a
fourth layer, etc. This does generate a 3D packing
with optimal density, so it's tempting to think that stacking
might also work in higher dimensions.</p>
<p>Alas, it doesn't. Knowing
the densest packing in one dimension doesn't give you
a clue of what it should be in the next. The graph below shows the
density of the
densest packings we know for dimensions 4 to
26, but these might not be the densest overall. The graph suggests, and this turns out to be true, that the sphere packing density decreases exponentially as the dimension increases.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2018/sphere_packings/best_known.jpg" alt="The density of the bestknown sphere packings in dimensions 4 to 26" width="642" height="343" /><p style="maxwidth: 642px;">The density of the bestknown sphere packings in dimensions 4 to 26.</p></div>
<! image made by MF >
<h3>Finding bounds</h3>
<p>When you're trying to find a number that attains some sort of a
maximum, like being the highest packing density, but haven't got much
luck, one approach is to lower your bar and look only for an <em>upper
bound</em>: in our case a number you can prove the packing constant can't exceed. </p>
<p>Various upper bounds for packing densities have been known for some time, but in 2003 <a href="http://math.mit.edu/~cohn/">Henry
Cohn</a> and <a href="http://www.math.harvard.edu/~elkies/">Noam Elkies</a> came up with a particularly interesting recipe for finding them in any dimension. The recipe is hard to put into practice, so Cohn
an Elkies were only able to approximate the associated upper bounds for dimensions up to 32. The
result is shown in the graphs below, together with the bestknown values for
the packing density, for dimensions 4 to 12 and 20 to 28.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2018/sphere_packings/8_24.jpg" alt="The density of the bestknown sphere packings in dimensions 4 to 26" width="700" height="254" /><p style="maxwidth: 642px;">The CohnElkies upper bound (blue) and the density of the bestknown packing (green) for dimensions 4 to 12 and 20 to 28.</p></div>
<! image made by MF >
<p>What's striking here is that in dimensions 8 and
24 the two graphs appear to coincide. If they really do coincide, then the densest packings we know in these dimensions are also the densest overall, and their density will be the sphere packing constant. But Cohn and Elkies weren't
able to prove this: there remained the annoying possibility that denser packings exist, whose densities fall right into
invisibly tiny gaps between the two graphs.
"That's not a plausible scenario for anyone with faith in the beauty
of mathematics, but faith does not amount to a proof," Cohn
wrote in a beautiful (though more advanced) <a href="http://www.ams.org/journals/notices/201702/rnotip102.pdf">article</a> in the <em>Notices of the American Mathematical Society</em>. </p>
<h3>Closing the gap</h3>
<p>Viazovska's work, which closed the gap for dimension 8 and was
later extended with the help of Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko to dimension 24, builds on the
centre piece of Cohn and Elkies' work. If you forget about the actual spheres
in a sphere packing and only consider their centres, you're left with a configuration of points. Rather than giving the coordinates of all the points, you can also characterise the configuration by the statistics of the distances between points: what is the smallest distance that occurs, how often does it occur, etc.</p>
<div style="maxwidth: 340px; float: left; border: thin solid grey; background: #CCC CFF; padding: 0.5em; marginright: 1em; fontsize: 75%">
<h3>The E<sub>8</sub> lattice sphere packing</h3>
<p>The spheres in this eightdimensional packing are centred on points whose coordinates are either all integers or all lie half way between two integers, and whose coordinates sum to an even number. The radius of the spheres is <img src="/MI/a298f246da6672b2cbebe6b567ef7d33/images/img0001.png" alt="$1/\sqrt {2}$" style="verticalalign:4px;
width:38px;
height:19px" class="math gen" />.</p>
<p>The E<sub>8</sub> lattice is related to the <a href="/content/beautifulsymmetryprovidesglimpsequantumworld">exceptional Lie group E<sub>8</sub></a>. As the name suggests the group is an exceptional object in mathematics, so it's perhaps not surprising that it is connected to an exceptional sphere packing. </p>
</div>
<p>It's a fruitful approach that is often used in physics. "Astronomers also do this," explains Viazovska. "They look at the stars and compute the
distances between different stars and then forget
about the geometry of space and remember only the statistics of
pairwise distances. It turns out that these statistics have
to satisfy certain restrictions. If you want to have so many stars at
this distance, and so many stars at that distance, and so many stars
at another distance, it might not be possible to realise this in
space."</p>
<p>In a similar spirit, Cohn and Elkies had shown that the distance statistics for a sphere packing should also satisfy certain restrictions, which in turn give an upper bound for a packing's density. To be sure the restrictions
are really satisfied, though, you have to find a mathematical function with particular properties, and that's not easy to do. Cohn and Elkies were only able to approximate these functions, which is why they could only approximate the upper bounds you can theoretically get from their recipe. </p>
<p>To find the sphere packing constant in dimension 8 (and later 24) Viazovska had to take a step further. She had to find a "magic function" which not only gives you an upper bound, but also shows that this upper bound is <em>sharp</em>, in other words that it equals to the density of the best packing we know in dimension 8. Cohn and Elkies thought that such a function must exist, but had no idea how to find it — "The magic functions seemed to come out of nowhere," Cohn writes in the <a href="http://www.ams.org/journals/notices/201702/rnotip102.pdf">article</a> quoted above.</p><p> This is the trick Viazovska managed to pull off: using a "bold construction" nobody had thought of before, she came up with a function that behaved just as required. The construction uses mathematical objects called <em>modular forms</em>, which were studied extensively by Ramanujan. This is why we met Viazovska at a conference celebrating the Indian genius.</p>
<p>With her result Viazovska proved that the highest possible density of a sphere packing in dimension 8 is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/d01b633867faede4b306a0cf62d00b99/images/img0001.png" alt="\[ \frac{\pi ^4}{384}\approx 0.25367, \]" style="width:106px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>which means that around 25% of 8D space can be filled with nonoverlapping spheres of the same size. </p>
<p>The packing which gives this density (and is marked as the bestknown packing in the graph above) is called the <em>E<sub>8</sub> lattice sphere packing</em>. We can't visualise it because it lives in eight dimensions, but we can describe it quite easily via the
coordinates of the centre points of all the spheres — see the box. The packing that gives the highest density in 24 dimension is called the <em>Leech lattice sphere packing</em>. It's similar to, but more complex than, the E<sub>8</sub> lattice packing, and its density is</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/ea883995b3e5de7756c9479e337b1aec/images/img0001.png" alt="\[ \frac{\pi ^{12}}{12!} \approx 0.001929. \]" style="width:114px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<h3>Where next?</h3>
<div class="rightimage" style="maxwidth: 304px;"><img src="/content/sites/plus.maths.org/files/articles/2018/sphere_packings/maryna_vazovska_mfo_2013_crop.jpg" alt="Maryna Viazovska" width="304" height="243" /><p>Maryna Viazovska. Photo: Petra Lein, <a href="https://creativecommons.org/licenses/bysa/2.0/de/deed.en">CC BYSA 2.0 DE</a>.</p></div>
<p>The obvious next question is whether similar techniques can prove sphere packing constants in other dimensions. The answer, sadly is, no. "For other dimensions the method of Cohn and Elkies gives some bound, but the bound is not optimal," says Viazovska.
"Everybody asks what is special about dimensions 8 and 24 — I don't know, it's a mystery. In these dimensions we have these two extremely great configurations, which we don't have in other dimensions. They are so good that methods which fail in all other dimensions in these dimensions give a sharp estimate. If you ask me why, I don't know."</p>
<p>Not all is lost, however. Cohn and Elkies approach hinges on a deep result (known as the <em>Poisson summation formula</em>), but doesn't make use of all the information the result could theoretically give you. "There are many inequalities the statistics [of pairwise distances] have to satisfy," explains Viazovska. "Theoretically we know all these inequalities, but describing them is quite difficult." For dimensions 8 and 24 it's possible to get away with only two of these inequalities, but for other dimensions we might need more. "One way to go would be to use those additional inequalities. I'm working on this, but it's not all that easy."</p>
<p>Another approach would be to make better use of the geometry of sphere packings, which was ignored in Viazovska's proofs in favour of statistics. "Maybe in some dimensions [forgetting the geometry] is not such a fruitful idea," she says. "Maybe we should come back to geometry, or use a combination of methods."
</p>
<p>"In other dimensions we know that there are [packings] that are good and people keep records of them. In small dimensions, up to ten, people are quite confident that the [packings] we think are the best. But of course as the dimension increased, our confidence decreases — it decreases exponentially."</p>
<! <p>But even only sorting out the sphere packing densities in dimensions 8 and 24 counts as a major breakthrough. Viazovska's results have already earned her some prestigious awards, including the <a href="https://en.wikipedia.org/wiki/Breakthrough_Prize_in_Mathematics">Breakthrough Prize in Mathematics</a> and the <a href="https://en.wikipedia.org/wiki/SASTRA_Ramanujan_Prize">SASTRA Ramanujan Prize</a>. At 34 years of age, she'll still be eligible for one of the highest accolades in mathematics, the <a href="https://www.mathunion.org/imuawards/fieldsmedal">Fields medal</a>, when it is next awarded in 2022. We will be watching her!</p> >
<hr/>
<h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. She interviewed Maryna Viazovska at the Royal Society's <a href="https://royalsociety.org/scienceeventsandlectures/2018/10/srinivasaramanujan/">celebration</a> of the centenary of the election as a Fellow of Srinivasa Ramanujan. She would like to thank Alison Kiddle for her take on higher dimensions.</p>
<p>You can also listen to the interview in a <a href="/content/node/7114">podcast</a>.</p></div></div></div>
Tue, 13 Nov 2018 14:56:58 +0000
Marianne
7113 at https://plus.maths.org/content
https://plus.maths.org/content/packingspheres#comments

From an ecologist's nightmare to a mathematician's dream
https://plus.maths.org/content/ecologistsnightmaremathematiciansdream
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/possum_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Wim Hordijk</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>"Since you are a computer scientist, I have an optimisation problem for you!", my colleague said, half jokingly. As an ecologist, one of the things my colleague studies is invasive plant species. The question he was facing is how to reconstruct the most likely routes along which these species travel when they invade new territory, based on historical records on when and where they first appeared. As it turns out, this question is an instance of a known mathematical optimisation problem called the <em>minimum cost arborescence problem</em>.</p>
<h3>Invasive species: Spotted knapweed</h3>
<div class="rightimage" style="maxwidth: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2018/arborescence/knapweed.jpg" alt="Spotted knapweed" width="250" height="333" /><p><em>Centaurea stoebe</em> (spotted knapweed), a member of the daisy family. Photo: <a href="https://commons.wikimedia.org/wiki/File:Centaurea_maculosa_Bozeman.jpg">Matt Lavin</a>, <a href="https://creativecommons.org/licenses/bysa/2.0/deed.en">CC BYSA 2.0</a>.</p></div>
<p>Due to increased human activity and mobility, many invasive species have been introduced to various parts of the world. Examples are possums in New Zealand, rabbits in Australia, and spotted knapweed in North America. Spotted knapweed (scientific name: <em>centaurea stoebe</em>) is considered a pest in North America, outcompeting many native species. It was most likely brought over from Europe on ships in the late 19th century. In fact, there were at least two introduction sites: one on the east coast near Boston, and one on the west coast near Vancouver.</p>
<p>My ecologist colleague was interested in reconstructing the most likely dispersal routes of spotted knapweed after its (dual) introduction in North America. He already had data on when this species was first observed in many different locations across the continent, but did not know how to analyse this data mathematically to get the result he wanted. The key to the problem is a mathematical object known as a <em>minimum cost arborescence</em>.</p>
<h3>Minimum cost arborescence</h3>
<p>In mathematics a <em>directed graph</em> is a collection of nodes with arrows between them, which point from one node to another. An <em>arborescence</em> is a directed graph in which there is one unique node <img src="/MI/76310672f2a9d932f5ffddb4e4446b44/images/img0001.png" alt="$r$" style="verticalalign:0px;
width:8px;
height:7px" class="math gen" /> (called the root) that has only outgoing arrows. Every other node has exactly one incoming arrow (and any number of outgoing ones). As a consequence, there is always a unique directed path from the root <img src="/MI/76310672f2a9d932f5ffddb4e4446b44/images/img0001.png" alt="$r$" style="verticalalign:0px;
width:8px;
height:7px" class="math gen" /> to any of the other nodes. Technically, an arborescence forms a <em>directed rooted tree</em>. </p>
<p>Now suppose you are given a directed graph <img src="/MI/202e25550cbde48313587f9918d3dde9/images/img0001.png" alt="$G$" style="verticalalign:0px;
width:13px;
height:11px" class="math gen" /> with a root node <img src="/MI/202e25550cbde48313587f9918d3dde9/images/img0002.png" alt="$r$" style="verticalalign:0px;
width:8px;
height:7px" class="math gen" /> that has only outgoing arrows. Imagine that each arrow in this graph has a number attached to it — you could interpret this number as the cost which something that travels along that arrow has to pay. </p><p>A <em>minimal cost arborescence</em> <img src="/MI/202e25550cbde48313587f9918d3dde9/images/img0003.png" alt="$A$" style="verticalalign:0px;
width:12px;
height:11px" class="math gen" /> is a subgraph of <img src="/MI/202e25550cbde48313587f9918d3dde9/images/img0001.png" alt="$G$" style="verticalalign:0px;
width:13px;
height:11px" class="math gen" />, consisting of all the nodes of <img src="/MI/202e25550cbde48313587f9918d3dde9/images/img0001.png" alt="$G$" style="verticalalign:0px;
width:13px;
height:11px" class="math gen" /> (including the root node <img src="/MI/202e25550cbde48313587f9918d3dde9/images/img0002.png" alt="$r$" style="verticalalign:0px;
width:8px;
height:7px" class="math gen" />) and some of its arrows, which is an arborescence, and for which the sum of the costs associated to arrows is as small as possible. Here is an example: </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2018/arborescence/arborescence.jpg" alt="A directed graph <em>G</em> with root node
<em>r</em>
and a cost assigned to each edge" width="407" height="372" /><p style="maxwidth: 407px;">A directed graph <em>G</em> with root node
<em>r</em>
and a cost assigned to each arrow. The minimum cost arborescence for this graph is indicated in bold. For example, the path {<em>r</em>, <em>v</em><sub>2</sub>, <em>v</em><sub>1</sub>, <em>v</em><sub>4</sub>}
from the root node <em>r</em> to node <em>v</em><sub>4</sub>
has a lower cost (1+2+2=5) than the two shorter paths {<em>r</em>, <em>v</em><sub>1</sub>, <em>v</em><sub>4</sub>}
(7+2=9) and {<em>r</em>, <em>v</em><sub>2</sub>, <em>v</em><sub>4</sub>}
(1+6=7).
</p></div>
<! Image provided by author >
Given a directed graph <img src="/MI/688012969506bc2e74e48dc7e25b52b3/images/img0001.png" alt="$G$" style="verticalalign:0px;
width:13px;
height:11px" class="math gen" /> with a root and a cost function, there are algorithms that find you a minimal cost arborescence in a reasonable amount of time (for experts, the problem is in the <a href="/content/whatsyourproblem">complexity class P</a>). Such algorithms were proposed independently first in 1965 by YoengJin Chu and TsengHong Liu (who called it a <em>shortest arborescence</em>), then in 1967 by <ahref="https://en.wikipedia.org/wiki/Jack_Edmonds">Jack Edmonds</a> (who called it an <em>optimum branching</em>), and finally in 1971 by F. Bock (who called it a <em>minimum spanning tree in a directed network</em>). These algorithms are all based on wellknown and efficient <a href="https://en.wikipedia.org/wiki/Minimum_spanning_tree">minimum cost spanning tree</a> algorithms for undirected graphs, with a recursive extension to check for, and resolve, potential cycles. Later on, an improvement was made by <a href="https://en.wikipedia.org/wiki/Robert_Tarjan">Robert Tarjan</a>, resulting in an even more efficient algorithm.
<h3>Reconstructing dispersal routes</h3>
<p>When you are trying to reconstruct the route along which a species, such as the spotted knapweed, has dispersed you usually have the following pieces of information:</p>
<ul><li>A number <img src="/MI/4ab3ab6f4bea4adfd9809ec9edd43c18/images/img0001.png" alt="$n$" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" /> of geographic locations where the species has been observed. The locations are points on the map, so we can give them coordinates <img src="/MI/4ab3ab6f4bea4adfd9809ec9edd43c18/images/img0002.png" alt="$(x_1,y_1)$" style="verticalalign:4px;
width:50px;
height:18px" class="math gen" />, <img src="/MI/4ab3ab6f4bea4adfd9809ec9edd43c18/images/img0003.png" alt="$(x_2,y_2)$" style="verticalalign:4px;
width:50px;
height:18px" class="math gen" />, etc, up to <img src="/MI/4ab3ab6f4bea4adfd9809ec9edd43c18/images/img0004.png" alt="$(x_ n,y_ n)$" style="verticalalign:4px;
width:53px;
height:18px" class="math gen" />. </i>
<li>For each location there is a time at which the particular species of interest was first observed at that location. We write <img src="/MI/bf1bd894f8ef6b851e780e1a3f9babc3/images/img0001.png" alt="$t_1$" style="verticalalign:2px;
width:12px;
height:12px" class="math gen" /> for the time corresponding to the location <img src="/MI/bf1bd894f8ef6b851e780e1a3f9babc3/images/img0002.png" alt="$(x_1,y_1)$" style="verticalalign:4px;
width:50px;
height:18px" class="math gen" />, etc, up to <img src="/MI/bf1bd894f8ef6b851e780e1a3f9babc3/images/img0003.png" alt="$t_ n$" style="verticalalign:2px;
width:14px;
height:12px" class="math gen" /> for the time corresponding to the location <img src="/MI/bf1bd894f8ef6b851e780e1a3f9babc3/images/img0004.png" alt="$(x_ n,y_ n)$" style="verticalalign:4px;
width:53px;
height:18px" class="math gen" />.</li></ul>
<p>Using this information we can construct a directed graph <img src="/MI/1827a5211f4b749d66acbacf29611e5a/images/img0001.png" alt="$G$" style="verticalalign:0px;
width:13px;
height:11px" class="math gen" /> with a root. Each node in the graph corresponds to one of the locations <img src="/MI/1827a5211f4b749d66acbacf29611e5a/images/img0002.png" alt="$(x_ i,y_ i)$" style="verticalalign:4px;
width:46px;
height:18px" class="math gen" /> (where <img src="/MI/1827a5211f4b749d66acbacf29611e5a/images/img0003.png" alt="$i$" style="verticalalign:0px;
width:5px;
height:11px" class="math gen" /> runs from <img src="/MI/1827a5211f4b749d66acbacf29611e5a/images/img0004.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> up to <img src="/MI/1827a5211f4b749d66acbacf29611e5a/images/img0005.png" alt="$n$" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" />). We connect one node to another by an arrow pointing from the first to the second node if the time associated to the first node is earlier than the time associated to the second node. The root node is the one with the earliest associated time. The cost associated to an arrow is the (Euclidean) distance between the two locations represented by the nodes it connects.</p>
<p>Assuming that a location is always invaded from the nearest earlier occupied location, it doesn't take much thought to see that finding the minimum cost arborescence for the graph <img src="/MI/90e80a14c7534f030ff9124ac36a8ffa/images/img0001.png" alt="$G$" style="verticalalign:0px;
width:13px;
height:11px" class="math gen" /> immediately provides the most likely dispersal routes of the given species. The figure below shows the most likely dispersal routes for spotted knapweed in North America, based on historical data of first occurrences in many different locations, using the arborescence method as described above.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2018/arborescence/solution.jpg" alt="The most likely disperal routes of spotted knapweed in North America" width="600" height="600" /><p style="maxwidth: 600px;">The most likely dispersal routes of spotted knapweed in North America, as reconstructed using the minimum cost arborescence method. The two red dots indicate the two (independent) introduction sites.
</p></div>
<! Image provided by author >
<p>My colleague Olivier Broennimann and I published this formal correspondence between the dispersal routes reconstruction problem in ecology and the minimum cost arborescence problem in mathematics in an <a href="https://www.sciencedirect.com/science/article/pii/S0022519312002846">article in the <em>Journal of Theoretical Biology</em></a>.</p>
<h3>Generalisations and other applications
</h3>
<p>Some of the assumptions underlying the method above are rather simplistic. For example, an invasion event may not always happen from the nearest already occupied location. There may be a barrier between two locations, such as a river or a mountain, that prevents the dispersal from one location to another. Or dispersal may depend on predominant wind direction (such as with plant seeds), which may preclude dispersal in certain directions. Such cases can be dealt with by setting the cost value of the arrow connecting the two locations to a very large value or even infinity. In other words, the cost function
can be easily adjusted according to "real world" situations, to take geographical, climatic, or biological constraints into account.</p>
<div class="rightimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/arborescence/possum.jpg" alt="Common bushtail possum" width="350" height="234" /><p>Cute or not, the common bushtail possum is an invader.</p></div>
<! Image from fotolia.com >
<p>There is also likely to be some noise in the data in terms of the first occurrence times. In the data set, the times <img src="/MI/7e3d2a8d2690bc325276df599e15e902/images/img0001.png" alt="$t_ i$" style="verticalalign:2px;
width:10px;
height:12px" class="math gen" /> indicate the first time a given species was observed in a given location. However, this means that the actual time of invasion may have been some time before <img src="/MI/7e3d2a8d2690bc325276df599e15e902/images/img0001.png" alt="$t_ i$" style="verticalalign:2px;
width:10px;
height:12px" class="math gen" />. To take this into account we can, for example, subtract some random value from each <img src="/MI/7e3d2a8d2690bc325276df599e15e902/images/img0001.png" alt="$t_ i$" style="verticalalign:2px;
width:10px;
height:12px" class="math gen" /> and run the minimum cost arborescence algorithm on multiple "randomised" instances. This way, a confidence score can be assigned to each arrow in the most likely dispersal routes, by counting in how many of these random instances that same arrow is included in the resulting minimum cost arborescence.</p>
<h3>Why do we do it?</h3>
<p>Being able to reconstruct the most likely dispersal routes of invasive species has important practical applications. For example, it can help in the design and implementation of quarantine strategies, or of conservation actions such as preventing the establishment of new "seed" populations, rather than focusing on established invasion fronts. Furthermore, the method as described here can be used to study various "what if?" scenarios. Both the data and the cost function can be adjusted to reflect hypothetical (possibly even future) scenarios, for which it can then be studied how that affects the most likely dispersal of invasive species.</p>
<p>In conclusion, the ecological dispersal routes reconstruction problem can be stated in terms of a known mathematical problem, for which there exists an efficient algorithm to find the optimal solution. This provides a mathematically sound and versatile tool to study the dynamics of invasive species, thus turning an ecologist's nightmare into a mathematician's dream.</p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2016/altruism/wim.jpg" alt="Wim Hordijk" width="200" height="200" /></div>
<p>Wim Hordijk is an independent scientist currently visiting the <a href="http://www.santafe.edu">Santa Fe Institute</a> in the USA. He has worked on many research and computing projects all over the world, mostly focusing on questions related to evolution and the origin of life. More information about his research can be found on his <a href="http://www.worldwidewanderings.net">website</a>.</p>
</div></div></div>
Tue, 06 Nov 2018 16:15:44 +0000
Marianne
7121 at https://plus.maths.org/content
https://plus.maths.org/content/ecologistsnightmaremathematiciansdream#comments

Maths around the clock
https://plus.maths.org/content/mathsaroundclock
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/clock_icon_1.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Antonella Perucca</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p><p>On clock dials you find the numbers from 1 to 12 — but for a bit of a change, could you write those numbers in a different way? </p></p>
<div class="rightimage" style="maxwidth:350px">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/clockexample.jpg" width="350" height="350" alt="Pi clock"/>
<p>A π clock. See <a href="/content/mathsaroundclock#examples">below</a> for more examples.</p>
</div>
<p><p>The answer is yes. For example, you can write <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0001.png" alt="$4$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> as <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0002.png" alt="$2+2$" style="verticalalign:1px;
width:36px;
height:13px" class="math gen" /> or <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0003.png" alt="$2\times 2$" style="verticalalign:0px;
width:36px;
height:12px" class="math gen" /> or <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0004.png" alt="$2^2$" style="verticalalign:0px;
width:14px;
height:14px" class="math gen" /> by using the sum, the product, or by raising to a power. If you like the number <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0005.png" alt="$\pi $" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" /> you could also write <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0001.png" alt="$4$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> as <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0006.png" alt="$\lceil \pi \rceil $" style="verticalalign:4px;
width:19px;
height:17px" class="math gen" /> by using the <em>ceiling function</em>, which approximates <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0005.png" alt="$\pi $" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" /> from above with the closest integer. There are infinitely many possibilities for expressing the number <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0001.png" alt="$4$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" />, and which one is the best is a matter of personal preference. In general, you can produce your own exclusive mathematical clock by writing <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0007.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/7940b03138b5d9fafd7da34102714e40/images/img0008.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> in your favourite way. </p></p>
<p>If you are looking for something extremely fancy, you can use <a href="/content/mathsminuteeulersidentity">Euler's identity</a> to express <img src="/MI/f704e723443594298bfadee4103f7b4d/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> using the numbers <img src="/MI/f704e723443594298bfadee4103f7b4d/images/img0002.png" alt="$e$" style="verticalalign:0px;
width:7px;
height:7px" class="math gen" />, <img src="/MI/f704e723443594298bfadee4103f7b4d/images/img0003.png" alt="$\pi $" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" />, and the imaginary number <img src="/MI/f704e723443594298bfadee4103f7b4d/images/img0004.png" alt="$i$" style="verticalalign:0px;
width:5px;
height:11px" class="math gen" />:
<p><img src="/MI/890c51c8fee4d9c8a874104a730bdde2/images/img0001.png" alt="$1=e^{\pi i}\, .$" style="verticalalign:0px;
width:70px;
height:14px" class="math gen" /> </p></p>
<p>Or, if you like the <a href="/content/baselproblem">Basel problem</a> you could display <img src="/MI/de402f5f1fbcfcaa316e516666b0d2f9/images/img0001.png" alt="$6$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> as the sum of all reciprocals of the squares of the positive multiples of <img src="/MI/de402f5f1fbcfcaa316e516666b0d2f9/images/img0002.png" alt="$\pi :$" style="verticalalign:0px;
width:17px;
height:8px" class="math gen" />
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7f4b66a293e62ff8d3670a3ca9461c0f/images/img0001.png" alt="\[ 6=\frac{1}{(\pi )^2}+\frac{1}{(2\pi )^2}+\frac{1}{(3\pi )^2}+\frac{1}{(4\pi )^2}+\frac{1}{(5\pi )^2}+\ldots \]" style="width:356px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>For something more concrete, you can pick any digit from <img src="/MI/dee0308b43decb74e73f39cc54a68ec9/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/dee0308b43decb74e73f39cc54a68ec9/images/img0002.png" alt="$9$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> and express the numbers on the clock dial only with this digit and mathematical symbols (some examples are provided in the images <a href="/content/mathsaroundclock#examples">below</a>).</p>
<p>Much more generally, it is possible to write the integers from <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> by using only any given real number and mathematical symbols. The reason is that we can always find a suitable expression for the number <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" />. Indeed, for a positive real number which is at most <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" />, it suffices to take the ceiling function (ie taking the integer approximation from above) to produce <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" />. For any real number <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0003.png" alt="$x$" style="verticalalign:0px;
width:9px;
height:7px" class="math gen" /> greater than <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> we can write <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> as the floor function of <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0004.png" alt="$\sqrt [x]{x}$" style="verticalalign:4px;
width:22px;
height:17px" class="math gen" /> (ie taking the integer approximation from below of the <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0003.png" alt="$x$" style="verticalalign:0px;
width:9px;
height:7px" class="math gen" />th root of the number <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0003.png" alt="$x$" style="verticalalign:0px;
width:9px;
height:7px" class="math gen" />), as can be checked with the logarithmic identity </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0005.png" alt="\[ \sqrt [x]{x}=e^{\frac{\log x}{x}}\, . \]" style="width:87px;
height:22px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Finally, negative numbers may be turned positive with the absolute value, and for <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0006.png" alt="$0$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> one may use the factorial identity <img src="/MI/9f98e83c9b6cd7d90506d9abdfeb3b44/images/img0007.png" alt="$0!=1$" style="verticalalign:0px;
width:42px;
height:12px" class="math gen" />. </p>
<p> There are plenty of mathematical clocks in circulation with various expressions on the clock dial. Occasionally, rather than having the numbers from <img src="/MI/5a8bc6bad83ddbea1d76e9077e8b431c/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/5a8bc6bad83ddbea1d76e9077e8b431c/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" />, you find the first twelve terms of some sequence. For example, you may have the first twelve <a href="/content/lifeandnumbersfibonacci">Fibonacci numbers</a>:</p>
<p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0001.png" alt="\[ 1,1,2,3,5,8,13,21,34,55,89,144\, . \]" style="width:245px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> In the very same way, you could use the letters from <img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0002.png" alt="$A$" style="verticalalign:0px;
width:12px;
height:11px" class="math gen" /> to <img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0003.png" alt="$L$" style="verticalalign:0px;
width:11px;
height:11px" class="math gen" />. </p><p>Another option is writing down an equation such that the desired number is the only solution, for example conveying <img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0004.png" alt="$5$" style="verticalalign:0px;
width:8px;
height:13px" class="math gen" /> with </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0005.png" alt="\[ x^2+7=10x18\, . \]" style="width:137px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> Some mathematical clocks show equations with more than one solution, but with exactly one solution among the integers from <img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0006.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0007.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" />. In general, beware of mathematical inaccuracies in mathematical clocks (for example <img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0008.png" alt="$3$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> is not really <img src="/MI/99cd59ab767a573d9244d5d35c790d46/images/img0009.png" alt="$\pi 0.14$" style="verticalalign:0px;
width:59px;
height:12px" class="math gen" />)! </p><p>Nevertheless, everything is allowed to have fun playing with numbers. We challenge you to produce your own mathematical clock, but if you need some inspiration here are some examples. </p>
<a name="examples"></a>
<h3>Mathematical Clock Themes</h3>
<p>In the image gallery below you find various mathematical clocks, with alternative ways of writing the numbers from <img src="/MI/b35fdb714e57869d7ca2f545b84ca171/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/b35fdb714e57869d7ca2f545b84ca171/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" />. You can print out one of those images and use it as a clock dial (e.g. you buy a clock with a custom dial), or you can simply place selected numbers' expressions above or around your clock. You can also buy clocks that you can write on, which allow you to change your mathematical clock at leisure. </p>
<p>Here are some examples:</p>
<h4>1 to 9 clocks
</h4>
<p> Fix any decimal digit from <img src="/MI/820b5ecf9438385cfbb16c6adf04ea36/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/820b5ecf9438385cfbb16c6adf04ea36/images/img0002.png" alt="$9$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> and then display all integers <img src="/MI/820b5ecf9438385cfbb16c6adf04ea36/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/820b5ecf9438385cfbb16c6adf04ea36/images/img0003.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> with short expressions involving only that digit and some arithmetic operations. We use the basic arithmetic operations, taking powers, and taking the squareroot.
</p>
<p>Here is an example of a 7 clock. Examples for 1, 2, 3, 4, 5, 6, 8, and 9 are given <a href="/content/mathsaroundclock#numbers">below</a> (notice that with the digits 5, 6, and 7 we only use the basic arithmetic operations).</p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/7clock.jpg" width="500" height="500" alt="7 clock"/>
<p style="maxwidth:500px">A 7 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/7clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<h4>The 123 theme</h4>
<p>It is possible to write all integers from <img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> using only the digits <img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" />,<img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0003.png" alt="$2$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" />,<img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0004.png" alt="$3$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> exactly once and in this order. We use the basic arithmetic operations, taking powers, taking the squareroot, taking the factorial (the factorial of a natural number <img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0005.png" alt="$n$" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" />, denoted by <img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0006.png" alt="$n!$" style="verticalalign:0px;
width:14px;
height:12px" class="math gen" />, is defined as the product of all natural numbers from <img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0005.png" alt="$n$" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" />), and applying the floor function <img src="/MI/c96c4b050f41fc4a7acf0b6944b5c807/images/img0007.png" alt="$\lfloor \, \rfloor $" style="verticalalign:4px;
width:12px;
height:17px" class="math gen" /> (this function approximates a real number with the closest integer from below).</p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/123clock.jpg" width="500" height="500" alt="123 clock"/>
<p style="maxwidth:500px">A 123 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/123clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<h4>The π clock</h4>
<p>It is possible to write all integers from <img src="/MI/db718f24f4cbd34fdfb1067fe9d98aa0/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/db718f24f4cbd34fdfb1067fe9d98aa0/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> using only the number <img src="/MI/db718f24f4cbd34fdfb1067fe9d98aa0/images/img0003.png" alt="$\pi $" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" />, the basic arithmetic operations, and the floor/ceiling functions <img src="/MI/db718f24f4cbd34fdfb1067fe9d98aa0/images/img0004.png" alt="$\lfloor \, \rfloor $" style="verticalalign:4px;
width:12px;
height:17px" class="math gen" /> and <img src="/MI/db718f24f4cbd34fdfb1067fe9d98aa0/images/img0005.png" alt="$\lceil \, \rceil $" style="verticalalign:4px;
width:12px;
height:17px" class="math gen" /> (these functions approximate a real number with the closest integer from below and above, respectively).</p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/piclock_large.jpg" width="500" height="500" alt="Pi clock"/>
<p style="maxwidth:500px">A pi clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/piclock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<h4>The <em>e</em> clock</h4>
<p>It is possible to write all integers from <img src="/MI/2976d426c1037b03918b4e4b3828ba42/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/2976d426c1037b03918b4e4b3828ba42/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> using only Euler's number <img src="/MI/2976d426c1037b03918b4e4b3828ba42/images/img0003.png" alt="$e$" style="verticalalign:0px;
width:7px;
height:7px" class="math gen" />, the basic arithmetic operations, taking powers, taking the squareroot, and applying the floor/ceiling functions <img src="/MI/2976d426c1037b03918b4e4b3828ba42/images/img0004.png" alt="$\lfloor \, \rfloor $" style="verticalalign:4px;
width:12px;
height:17px" class="math gen" /> and <img src="/MI/2976d426c1037b03918b4e4b3828ba42/images/img0005.png" alt="$\lceil \, \rceil $" style="verticalalign:4px;
width:12px;
height:17px" class="math gen" />.</p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/eclock.jpg" width="500" height="500" alt="e clock"/>
<p style="maxwidth:500px">An <em>e</em> clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/eclock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<h4>The binary clock</h3>
<p>Here we write all numbers from <img src="/MI/bc95d7b1c4ba9903c248f9af4e8045d9/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/bc95d7b1c4ba9903c248f9af4e8045d9/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> in the <a href="/content/mathsminutebinarynumbers">binary system</a>, which only uses the digits <img src="/MI/9a220fe90adbcc1d4d222e36caff3471/images/img0001.png" alt="$0$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> and <img src="/MI/9a220fe90adbcc1d4d222e36caff3471/images/img0002.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" />.</p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/binaryclock.jpg" width="500" height="500" alt="Binary clock"/>
<p style="maxwidth:500px">An binary clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/binaryclock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<h4>The prime numbers clock</h4>
<p> Here we only write those numbers from <img src="/MI/cb3ca129440e135925c3b1e9b7cc4caa/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/cb3ca129440e135925c3b1e9b7cc4caa/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> which are prime numbers.</p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/primenumbersclock.jpg" width="500" height="500" alt="Prime number clock"/>
<p style="maxwidth:500px">A prime number clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/primenumbersclock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<h4>The Chinese numerals clock</h4>
<p> Here we write all numbers from <img src="/MI/0fc10ad6c5973f42165550897043e2dd/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/0fc10ad6c5973f42165550897043e2dd/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> using Chinese numerals: you may notice how the numbers <img src="/MI/0fc10ad6c5973f42165550897043e2dd/images/img0003.png" alt="$11$" style="verticalalign:0px;
width:14px;
height:12px" class="math gen" /> and <img src="/MI/0fc10ad6c5973f42165550897043e2dd/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> are composed from the symbol for <img src="/MI/0fc10ad6c5973f42165550897043e2dd/images/img0004.png" alt="$10$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> and the numbers <img src="/MI/0fc10ad6c5973f42165550897043e2dd/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> and <img src="/MI/0fc10ad6c5973f42165550897043e2dd/images/img0005.png" alt="$2$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> respectively. <p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/chinesenumeralsclock.jpg" width="500" height="500" alt="Chinese numerals clock"/>
<p style="maxwidth:500px">A Chinese numerals clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/chinesenumeralsclock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<h4>The Maya numerals clock</h4>
<p> Here we write all numbers from <img src="/MI/c0e71baac932a89f439f7d0698c1fa5e/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/c0e71baac932a89f439f7d0698c1fa5e/images/img0002.png" alt="$12$" style="verticalalign:0px;
width:15px;
height:12px" class="math gen" /> using Maya numerals: a dot stands for the number <img src="/MI/c0e71baac932a89f439f7d0698c1fa5e/images/img0001.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> and a bar stands for <img src="/MI/c0e71baac932a89f439f7d0698c1fa5e/images/img0003.png" alt="$5$" style="verticalalign:0px;
width:8px;
height:13px" class="math gen" />.</p>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/mayanumeralsclock.jpg" width="500" height="500" alt="Maya numerals clock"/>
<p style="maxwidth:500px">A Maya numerals clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/mayanumeralsclock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<p>The <img src="/MI/5961c7f9be82973b03c8807a58e9f97f/images/img0001.png" alt="$\pi $" style="verticalalign:0px;
width:10px;
height:7px" class="math gen" /> clock, the <img src="/MI/5961c7f9be82973b03c8807a58e9f97f/images/img0002.png" alt="$e$" style="verticalalign:0px;
width:7px;
height:7px" class="math gen" /> clock, the <img src="/MI/5961c7f9be82973b03c8807a58e9f97f/images/img0003.png" alt="$1$" style="verticalalign:0px;
width:6px;
height:12px" class="math gen" /> to <img src="/MI/5961c7f9be82973b03c8807a58e9f97f/images/img0004.png" alt="$9$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> clocks and the <img src="/MI/5961c7f9be82973b03c8807a58e9f97f/images/img0005.png" alt="$123$" style="verticalalign:0px;
width:23px;
height:12px" class="math gen" /> clock were developed by the author (for the <img src="/MI/5961c7f9be82973b03c8807a58e9f97f/images/img0005.png" alt="$123$" style="verticalalign:0px;
width:23px;
height:12px" class="math gen" /> theme, we got some inspiration from <em><a href="http://www.sbcrafts.net/clocks/">Math clocks</a></em>).</p>
<a name="numbers"></a><hr/>
<h3>More examples...</h3>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/1clock.jpg" width="500" height="500" alt="1 clock"/>
<p style="maxwidth:500px">A 1 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/1clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/2clock.jpg" width="500" height="500" alt="2 clock"/>
<p style="maxwidth:500px">A 2 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/2clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/3clock.jpg" width="500" height="500" alt="3 clock"/>
<p style="maxwidth:500px">A 3 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/3clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/4clock.jpg" width="500" height="500" alt="4 clock"/>
<p style="maxwidth:500px">A 4 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/4clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/5clock.jpg" width="500" height="500" alt="5 clock"/>
<p style="maxwidth:500px">A 5 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/5clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/6clock.jpg" width="500" height="500" alt="6 clock"/>
<p style="maxwidth:500px">A 6 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/6clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/8clock.jpg" width="500" height="500" alt="8 clock"/>
<p style="maxwidth:500px">A 8 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/8clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<div class="centreimage">
<img src="/content/sites/plus.maths.org/files/articles/2018/clocks/9clock.jpg" width="500" height="500" alt="9 clock"/>
<p style="maxwidth:500px">A 9 clock. Click <a href="/content/sites/plus.maths.org/files/articles/2018/clocks/9clock.pdf">here</a> for a printable pdf of this clock face.</p>
</div>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2018/clocks/peruccapic.jpg" alt="Antonella Perucca" width="200" height="280" />
<p></p>
</div>
<p>Antonella Perucca is Professor for Mathematics and its Didactics at the University of Luxembourg. She is a researcher in number theory and invents mathematical exhibits (for example the "Chinese Remainder Clock"). To find out more, explore her <a href="http://www.antonellaperucca.net/">webpage</a>.</p>
<br clear="all" />
</div></div></div>
Tue, 06 Nov 2018 14:10:30 +0000
Marianne
7122 at https://plus.maths.org/content
https://plus.maths.org/content/mathsaroundclock#comments

Mathscon: Reshaping perceptions of mathematics
https://plus.maths.org/content/mathsconreshapingperceptionsmathematics
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/icon_9.png" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Farheen Zehra</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p><em>Plus is looking forward to taking part in the next Mathscon in February 2019, after thoroughly enjoying the last event. In this article Farheen Zehra, part of the Mathscon team, explains what you can expect at the next event.</em></p>
<p>Anyone who is passionate about mathematics knows that its beauty lies in concepts, not calculations. Yet somewhere along the line we have ended up with a reputation for just number crunching.
</p><p>
Enter Mathscon: an organisation with a mission to reshape the world's perception of mathematics. Directed at undergraduates, young researchers and science lovers in general, each event we hold aims to expose the public to a broad range of applications that showcase the beauty of mathematics. The launch event took place in February 2017 at Imperial College London. Our delegates included some of the brightest minds and mathematics enthusiasts from universities all over the UK.
</p><p>
Speakers at last year's event included Nira Chamberlain – one of the VicePresidents of the Institute of Mathematics and its Applications and listed by the Science Council as "one of UK's top 100 Scientists"; Katie Steckles – an influential female mathematics communicator who works with schools, festivals and on BBC Radio and YouTube; and the multiple award holding science author and science journalist Simon Singh. (<em>You can see interviews with <a href="/content/mathematicalmomentsnirachamberlain">Nira</a> and <a href="content/mathematicalmomentskatiesteckles">Katie</a> on Plus</em>.)
</p><p>
The conference also included two panel discussions on thought provoking topics within the science community: "Is God a Mathematician?" and "Do Parallel Universes exist?". Each panel had some of the most eminent speakers related to the topics, such as James Baggot, author of <em>Higgs: The Invention and Discovery of the 'God Particle'</em>, and Carlo Contaldi, who studies the distribution of galaxies, dark matter, and the cosmic microwave background.
</p>
<div class="rightimage" style="maxwidth:400px"><img src="/content/sites/plus.maths.org/files/news/2018/mathscon/chess.jpg"></div>
<p>
Maths con also offers delegates the chance to see some of the coolest applications of mathematics through workshops, covering topics such as augmented reality, mathematics and medicine, and a workshop on chess led by an international chess competitor, Dagnė Čiukšytė, and chess grandmaster and threetime world champion John Nunn.
</p><p>
Mathscon wants to support people who not only share an appreciation for mathematics, but also use the subject to be innovative and make a difference. At the 2018 conference Mathscon introduced the Mathscon Grant. The winner of the grant receives £1000 from Mathscon to support them in their mathematical studies. The grant is available to any mathematics enthusiast who can show how they are contributing to the mathematical community in an exciting or original way. Think this could be you? Keep an eye on the <a href="www.mathscon.com">Mathscon website</a>, details for how to apply should appear there soon.
</p><p>
Mathscon is the largest student led mathematics conference in the UK and we intend to grow even bigger and organise many Mathscon events around the world. We are already beginning to build partnerships with educational institutions, create relations with pioneer corporations and receive support from great names in mathematics.
</p><p>
The upcoming edition of Mathscon will have some exciting additions. Mathscon has expanded and will be hosting multiple smaller scale events throughout November in different universities across the UK along with our main event on Saturday 23 February 2019 at Imperial College London. Our main event will also have an additional section for student talks in which selected students will be invited to present their research work in ten minutes. You can apply at any time to be involved in these talks, see the <a href="http://www.mathscon.com">website</a> for more information.
</p><p>
You can stay updated with information, including dates for the November events and early bird ticket sales for the February 2019 conference by checking out the <a href="http://www.mathscon.com">Mathscon website</a>, and following us on <a href="https://www.facebook.com/themathscon/">Facebook</a>, <a href="https://twitter.com/themathscon">Twitter</a>, or <a href="https://www.instagram.com/themathscon/?hl=en">Instagram</a>.
</p>
<div class="centreimage" style="maxwidth:600px"><img src="/content/sites/plus.maths.org/files/news/2018/mathscon/nira.png"><p>Nira Chamberlain and the Mathscon team</p></div></div></div></div>
Tue, 23 Oct 2018 15:51:09 +0000
Rachel
7112 at https://plus.maths.org/content
https://plus.maths.org/content/mathsconreshapingperceptionsmathematics#comments

Fantastic fractals
https://plus.maths.org/content/fantasticfractals
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/icon_32.jpg" width="100" height="97" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Marianne Freiberger</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>The Austrian painter and architect <a href="http://en.wikipedia.org/wiki/Friedensreich_Hundertwasser">Friedensreich Hundertwasser</a> once bemoaned what he called the "tyranny" of the straight line:
"It is the line which does not exist in nature," he said and condemned
any design based on straight lines as "stillborn".</p>
<div class="rightimage" style="width: 200px;"><img src="/sites/plus.maths.org/files/news/2010/mandelbrot/koch.png" alt="" width="200" height="200" />
<p>The first four steps in creating the Von Koch snowflake. Image from <a href="http://en.wikipedia.org/wiki/File:KochFlake.svg">Wikipedia</a>. <a href="/sites/plus.maths.org/files/news/2010/mandelbrot/Von_Koch_curve.gif" target="blank">Click here</a> to see the snowflake emerge from an infinite number of steps.</p>
</div>
<p> A bit dramatic perhaps, but
he had a point that wasn't just relevant to art and design. The
straight lines and perfect circles of classical geometry have given
rise to millennia worth of beautiful mathematics, but there are
natural forms whose intricate complexity they just cannot capture. The
outlines of clouds, shorelines and the intricacies of the humble
broccoli are examples. And it turns out that, far from being beyond the reach of mathematics, these complex forms can emerge from simple mathematical rules — though mathematicians didn't fully realise this until computers arrived on the scene.</p>
<p>Fractals are geometric objects that exhibit complex structure at
every scale. No matter how closely you zoom in on a fractal, its
complexity doesn't diminish and you often see the same structures
appearing again and again.</p>
<div class="leftimage" style="width: 300px;"><img src="/sites/plus.maths.org/files/news/2010/mandelbrot/mandelbrot.jpg" alt="" width="300" height="225" />
<p>The famous Mandelbrot set is a fractal. Image <a href="http://en.wikipedia.org/wiki/File:Mandel_zoom_00_mandelbrot_set.jpg">Wolfgang Beyer</a>. </p>
</div>
<p> A famous example is the Von Koch snowflake. Start with an equilateral triangle and replace the middle third of each side by a "spike" consisting of two sides of a smaller equilateral triangle. Now do the same for each of the twelve straightline segments of the resulting shape and repeat, ad infinitum. The shape you get in the end, after an infinite number of steps, exhibits the same spiky structure at every level of magnification: you'll never see a piece of straight line in its outline, because every straightline piece that was once there has been broken up and adorned with a spike.</p>
<p>Mathematically, fractals live in a strange world in between dimensions. You couldn't call the Von Koch snowflake "onedimensional" because it contains no straight lines or smooth curves at all. No amount of zooming in will reveal such onedimensional components. On the other hand, the snowflake isn't twodimensional either, because it occupies no area. In fact, it takes a new definition of "dimension" to sort out the snowflake's place in the dimensional hierarchy. According to this definition, the snowflake has a dimension of around 1.26. Having a <em>fractional</em> dimension, one that's not a whole number, is what characterises the fractals.</p>
<div class="rightimage" style="width: 300px;"><img src="/sites/plus.maths.org/files/news/2010/mandelbrot/mandel_zoom.jpg" alt="" width="300" height="225" />
<p>Zooming in on the Mandelbrot set reveals its rich structure and a tiny baby Mandelbrot set hidden inside. Image <a href="http://en.wikipedia.org/wiki/File:Mandel_zoom_03_seehorse.jpg">Wolfgang Beyer</a>. </p>
</div>
<p>Examples of fractals have been known to mathematicians for some
time, the snowflake was first published in 1904 by the Swedish
mathematician <a
href="http://wwwgroups.dcs.stand.ac.uk/~history/Biographies/Koch.html">Helge
von Koch</a>, but they were regarded as mathematical oddities; strange
artificial constructs. The fact that fractals can also emerge from
everyday mathematical objects, such as quadratic functions, only
became apparent with the advent of computers. Making the fractals
visible can involve hundreds of thousands calculations that are
impossible to perform if you're just working with pencil and paper.</p>
<p>Fractal structures appear all over the place in real life, too.
The fluctuations of financial markets look similar whether you look at
a graph representing a whole year, or just a week, in other words,
they exhibit selfsimilarity. There is an abundance of fractal
patterns in nature: coast lines, galaxies, turbulent waters, and
more. Using mathematics it is possible to program a computer to draw fantastical, yet lifelike, fractal landscapes. </p>
<div class="leftimage" style="width: 320px;"><img src="/issue6/turner2/mountain.jpg" alt="A fractal landscape created by Professor Ken Musgrave." maxwidth="320" height="213" /><p>A fractal landscape created by Ken Musgrave (Copyright: <a href="http://www.kenmusgrave.com/">Ken Musgrave</a>).</p>
</div>
<p>Like the Von Koch snowflake, natural fractal structures can
often be modelled by repeatedly applying a set of relatively simple
mathematical rules. This doesn't only make them perfect candidates
for computer exploration, but also illustrates how astonishing
complexity can emerge in apparently simple mathematical or physical
systems. Indeed, fractals are intimately linked to
<em>chaos theory</em>, which explores processes that, while not being
random, are still unpredictable.</p>
<p>The theory of fractals has many practical applications. It is used to understand the <a href="/content/howbigmilkyway">fate of our Universe</a>, of financial markets and of <a href="/content/fatchancechaos">populations of animals and humans</a>. In medical research it's used to shed light on the <a href="/content/brain">structure of our brains</a> and our <a href="/content/eatdrinkandbemerry0">digestive system</a>, and in developing medical imaging techniques. Fractal geometry has applications in image and video compression and even to spot <a href="/content/fractalexpressionism">art forgeries</a>. And that's quite aside from the new world fractal geometry has opened up for artists and <a href="/content/os/issue55/features/kormann/index">musicians</a>.</p>
<hr/>
<p><em>If you would like to explore dynamic geometry then take a look at <a href="https://nrich.maths.org/13899">dynamics explorations feature</a> on our sister site NRICH.</em></p>
<hr/>
<h3>Further reading</h3>
<ul><li><a href="/content/whatmandelbrotset">What is the Mandelbrot set?</a></li>
<li><a href="/content/computingmandelbrotset">Computing the Mandelbrot set</a></li>
<li><a href="/content/fatchancechaos">A fat chance of chaos</a></li>
<li><a href="/content/extractingbeautychaos">Extracting beauty from chaos</a></li>
<li><a href="/content/modellingnaturefractals">Modelling nature with fractals</a></li>
<li><a href="/content/noneuclideangeometryandindraspearls">Noneuclidean geometry and Indra's pearls</a></li></ul>
<p><em>If you would like to explore dynamic geometry then take a look at <a href="https://nrich.maths.org/13899">dynamics explorations feature</a> on our sister site NRICH.</em></p>
<hr/>
<h3>About this article</h3>
<p>This article is based on our <a href="/content/benoitmandelbrothasdied">report of the death of Benoît Mandelbrot</a> in 2010. <a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>.</p>
</div></div></div>
Tue, 23 Oct 2018 15:01:35 +0000
Marianne
7111 at https://plus.maths.org/content
https://plus.maths.org/content/fantasticfractals#comments

Dodging Doppler: Atomic clocks and laser cooling
https://plus.maths.org/content/atomicclocksandlasercooling
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/molasses_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Marianne Freiberger</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>
To measure the time you need a ticker: something that "clicks" at regular intervals, isn't easily upset, and works the same everywhere. Pendulums are good tickers. The time it takes them to complete a swing back and forth depends on their length and the force of gravity, and only very slightly on the width of the swing — nothing else. The problem is that their length can vary over time and the strength of gravity over the surface of the Earth, so pendulum clocks aren't as accurate as we'd like.</p>
<div style="width: 330px; float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; marginright: 1em; marginleft: 1em; fontsize: 75%">
<h3>The period of a pendulum</h3>
<p>For pendulum with a sufficiently small swing the period <img src="/MI/4cf4806adc561d270890ccb484e7066a/images/img0001.png" alt="$T$" style="verticalalign:0px;
width:12px;
height:11px" class="math gen" /> is approximately <table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/4cf4806adc561d270890ccb484e7066a/images/img0002.png" alt="\[ T \approx 2\pi \sqrt {\frac{L}{g}}, \]" style="width:88px;
height:50px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table> where <img src="/MI/4cf4806adc561d270890ccb484e7066a/images/img0001.png" alt="$T$" style="verticalalign:0px;
width:12px;
height:11px" class="math gen" /> is time, <img src="/MI/4cf4806adc561d270890ccb484e7066a/images/img0003.png" alt="$L$" style="verticalalign:0px;
width:11px;
height:11px" class="math gen" /> is the length of the pendulum and <img src="/MI/4cf4806adc561d270890ccb484e7066a/images/img0004.png" alt="$g$" style="verticalalign:3px;
width:8px;
height:10px" class="math gen" /> is acceleration due to gravity. You can find out more about pendulum maths <a href="https://en.wikipedia.org/wiki/Pendulum_(mathematics)">here</a>.</p></div>
<p>Better tickers are quartz crystals which, using electricity, can be made to vibrate like tuning forks. The frequency of their vibration is more reliable
than that of pendulums, but since no two quartz crystals are exactly the same, even when they have been machined, and since the crystals are also sensitive to temperature changes, quartz clocks also aren't as accurate as scientists and engineers would like for their most sophisticated applications.</p>
<p>The best tickers are atoms. Two atoms of the same kind behave in exactly the same way, always and everywhere. This isn't an obvious fact, but our best physical theories predict it and no experiment has ever proved it false. Clocks that use atoms are accurate to one second in 300 million years. </p>
<p>But they are not easy to maintain. To function, the atoms need to be kept at an incredibly low temperature, as near to absolute zero as possible. Strangely, the cooling is achieved by shining laser light on the atoms — not what you'd expect since usually things you a shine a light on get hot. The physicist <a href="https://www.nist.gov/people/williamdphillips">Bill Phillips</a> received the <a href="https://www.nobelprize.org/prizes/physics/1997/summary/">1997 Nobel prize in physics</a> for his work on cooling, and trapping, atoms using lasers. We met him at the <a href="https://www.heidelberglaureateforum.org/event_2018/">Heidelberg Laureate Forum</a> (HLF) 2018, where he explained
the basics to a riveted audience of young researchers and gave us interview (which you can watch in the <a href="/content/atomicclocksandlasercooling#video">video</a> below).</p>
<h3>What makes atoms tick?</h3>
<p>"Every atom [comes with] certain energy levels," explains Phillips. "[These] energy levels are very specific and they are the same for every atom of the same kind. One of the ways of [making an atom] go from one energy level to another is to put in light, or microwaves, or radio waves, at exactly the right frequency. That's the key."</p>
<div class="leftimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phillips/atom.jpg" alt="Solar system model of the atom" width="350" height="234" /><p>An illustration of the solar system model of the atom.</p></div>
<! image from fotolia.com >
<p>To understand what Phillips means, think of the solar system model of the atom, which has electrons orbiting the nucleus of an atom, just like planets orbit a sun.
Naively you would think that an electron can orbit the nucleus at any distance, but quantum mechanics tells us that it can't: only certain distances are allowed. </p><p>The distances are related to the energy of the electron. When an electron loses energy it moves closer to the nucleus and when it gains energy it moves further away. And because not all distances are allowed, the electron doesn't gradually inch closer when it loses energy, but performs a leap to the next allowed distance. The lost energy is emitted in the form of electromagnetic radiation, that is, light waves or microwaves. As all waves, the radiation released has a certain frequency, which can be measured. So rather than measuring time in terms of the frequency of a pendulum's swing or a quartz crystal's vibration, we can measure it in terms of the frequency of the radiation emitted by an electron changing energy levels inside an atom.</p>
<p> For various reasons, <a href="https://en.wikipedia.org/wiki/Caesium">caesium</a> atoms lend themselves as tickers particularly well. This is why a second is officially defined as</p>
<p><em>The duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.</em></p>
<p>To build a clock that uses this definition of a second, you need to measure the frequency associated to the caesium atoms. This is done by bombarding the atoms with microwaves. As Phillips explained above, the microwaves can cause the electrons to change energy level, but only if the frequency of the microwaves is just right. By seeing how many electrons change energy level, the frequency resulting from the electrons' quantum leaps can be established.</p>
<h3>Why do atomic clocks need to be cold?</h3>
<p>At the microscopic level, temperature is related to the speed at which atoms move: fastmoving atoms result in a higher temperature than slowmoving ones.</p>
<div class="rightimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phillips/phillips.jpg" alt="Bill Phillips at HLF." width="350" height="246" /><p>Bill Phillips playing with liquid nitrogen in his lecture at the <a href="https://www.heidelberglaureateforum.org/event_2018/">HLF 2018</a>. Liquid nitrogen can instantly freeze flowers, but it's not cold enough for atomic clocks. Don't try it at home! Photo: Bernhard Kreutzer for HLF, © Bernhard Kreutzer.</p></div>
<p>"The faster the atoms are going, the harder it is to measure them," says Phillips. "For one thing [they] don't stay very long in your apparatus, and it's not easy to measure something when you don't have very long to measure it. The other thing is that there are shifts in the frequency."</p>
<p>Phillips is referring to the familiar <em>Doppler shift</em>. Imagine you're standing on the beach with
gentle waves lapping at your feet. The waves will be coming at you with a particular frequency, say one every 10 seconds. If you start running into the sea, the waves will hit you in a more rapid succession, so their frequency will seem increased to you. If you run back to the beach, the waves will hit you less often, appearing to have a lower frequency. The Doppler effect comes into play whenever the observer of a wave and the source emitting a wave move at different speeds (find out more <a href="/content/dopplerdetectives">here</a>). Atoms experience the effect too. The frequency they "detect" in microwaves depends on how the atoms are moving with respect to the waves.</p>
<p>Effects from more advanced physics also impact on the frequency measurements an atomic clock needs to make. One is <em>time dilation</em>, as predicted by Einstein's theory of relativity, and another is Heisenberg's <em>uncertainty principle</em> from quantum mechanics. We won't get into the details here; you can find out more about time dilation in <a href="/content/whatssospecialaboutspecialrelativity">this article</a> and about the uncertainty principle in <a href="/content/heisenbergsuncertaintyprinciple">this article</a>.</p>
<p>To minimise these unwanted effects, it helps to slow the atoms down, in other words, to cool them. The temperature at which the motion of atoms comes to a minimum is called <em>absolute zero</em>: it's 0 degrees on the Kelvin scale for measuring temperature, and 273.15 degrees on the Celsius scale. Atoms don't come to a complete stop at absolute zero — quantum effects mean that they are never totally still — but the motion is minimal. Ideally, we'd like an atomic clock to be cooled to as near to absolute zero as possible. Simply sticking the atoms into a superpowerful refrigerator won't work, however. The gas containing the atoms would condense on the walls of the refrigerating container, and for atomic clocks, that's no good. What's needed is a way of cooling the atoms without anything touching them. This is where laser cooling comes in.</p>
<h3>What is laser cooling?</h3>
<p>"We cool the atoms with lasers, which seems like a crazy idea because typically when we shine light at something it gets hot, rather than cold," says Phillips. "But the thing to remember is that the idea of temperature has to with the motion of the atoms and molecules that make up whatever it is we're measuring the temperature of. If we have a gas, which is what we are dealing with in my laboratory, it means you have atoms going every which way. You can shine light on the atoms, and the light pushes on the atoms in such a way that they slow down."</p>
<p>Simply shining light from one (or even more) directions at a bunch of atoms moving all over the place won't do the trick. If an atoms happens to meet a light beam headon, you might hope it'll be slowed by the beam, but an atom moving in the opposite direction might get a push and speed up.
</p>
<div class="leftimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phillips/molasses.png" alt="Optical molasses" width="350" height="281" /><p>Atoms being trapped and slowed in the "optical molasses" created by six laser beams.</p></div>
<! image made by MF>
<p>The key fact is that light can only exert a force on an atom if it is absorbed — and caesium atoms preferentially absorb light from the violet part of the spectrum, which has a frequency of about 10<sup>15</sup> Hz. "What we do is tune the frequency of the light to be a little bit lower than the exact frequency that the atoms would like to absorb. If the atoms were at rest, then that would mean that they wouldn't absorb very much. But when the atom is moving [towards the light], there's a Doppler shift which makes it appear to the atom as if the light that is shining on it has an higher frequency; which is to say closer to the frequency that it wants to absorb." </p>
<p>The idea is that those atoms that meet the laser beam headon will slow down, while others will hardly be affected.
"This is the clever trick that came from [<a href="https://en.wikipedia.org/wiki/David_J._Wineland">David J. Wineland</a>, <a href="https://en.wikipedia.org/wiki/Hans_Georg_Dehmelt">Hans Georg Dehmelt</a>, <a href="https://en.wikipedia.org/wiki/Theodor_W._H%C3%A4nsch">Theodor W. Hänsch</a> and <a href="https://en.wikipedia.org/wiki/Arthur_Leonard_Schawlow">Arthur Leonard Schawlow</a>], who came up with the idea of laser cooling back in 1975, and we have been using that trick ever since. One laser cooling method using this trick has been dubbed "<a href="https://en.wikipedia.org/wiki/Optical_molasses">optical molasses</a>" because atoms find it really hard to through the light created by six lasers.</p>
<! <p>(If you are wondering how this works when we think of a beam of light as a stream of photons, here is how it goes: atoms that meet the stream headon will absorb a photon, and this will slow them down. The photon will excite the atom and move an electron to a higher energy state. The atom will then spit the photon out again, a process that comes with a small recoil effect towards some random direction. However, the atom will immediately absorb a new photon and the process will repeat. Since the many recoil effects go in random directions, their net effect on the atom is zero. What remains is the slowingdown effect.)</p> >
<p>Phillips and others have used this idea as a basis for methods that can cool caesium atoms down to about one billionth of a degree above absolute zero. The best caesium atomic clocks are accurate to one part in 10<sup>16</sup>, which means that after 300 million years have elapsed they will still not have slipped by more than a second. "We have clocks that are even better than that, but they're not caesium," says Phillips. "So eventually we may change the definition of the second to take advantage of those better clocks."</p>
<h3>Why do we need to keep time so accurately?</h3>
<p>One application that does need very accurate time keeping is the global positioning system (GPS). The system pinpoints your location by working out your distance to several GPS satellites. Each satellite sends out signals encoding the current time. When a GPS device picks up such a signal, it compares the broadcast time with the current time, which tells it how long the signal took to get to it, and therefore how far the satellite in question is away (you can find out more <a href="/issue51/package/MMP_navigation_toolkit.pdf">here</a>).</p>
<p>"The clocks that are in the satellites of the GPS are not laser cooled clocks, or at least not yet," says Phillips. "But they are controlled from the ground by laser cooled clocks. It's really convenient to have clocks on the ground that you don't have to worry about; they're just so good."</p>
<p>That's a practical application of atomic clocks, but there are also scientific ones. "We use them to study really fundamental questions, like 'is Einstein's theory of general relativity correct?' or 'Are the constants of nature really constant?' These [questions] we can explore using very good atomic clocks."</p>
<p>For everyday use at home, a beautiful grandfather clock is still more than sufficient. But to use your satnav and all sorts of other devices the future may bring and to probe the frontiers of science atomic clocks are the way to go.</p>
<hr/>
<a name="video"></a>
<iframe width="560" height="315" src="https://www.youtube.com/embed/oVgIjnm1iuc?rel=0" frameborder="0" allow="autoplay; encryptedmedia" allowfullscreen></iframe>
<hr/>
<h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. She interviewed Bill Phillips at the <a href="https://www.heidelberglaureateforum.org/event_2018/">Heidelberg Laureate Forum</a> in September 2018.</p>
</div></div></div>
Thu, 11 Oct 2018 09:48:10 +0000
Marianne
7107 at https://plus.maths.org/content
https://plus.maths.org/content/atomicclocksandlasercooling#comments

Tools made of light win 2018 Nobel prize in physics
https://plus.maths.org/content/toolsmadelightwin2018nobelprizephysics
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/icon_8.png" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Rachel Thomas</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><div class="rightimage" style="maxwidth:250px">
<img src="/content/sites/plus.maths.org/files/news/2018/nobel/spectrum.png" alt="Spectrum of visible light"/><p>The spectrum of electromagnetic waves, including the visible light range. (Image <a href="https://commons.wikimedia.org/wiki/File:ElectromagneticSpectrum.svg">Victor Blacus</a> – <a href="https://creativecommons.org/licenses/bysa/3.0">CC BYSA 3.0</a>)</p></div>
<p>
The <A href="https://www.nobelprize.org/prizes/physics/2018/summary/">2018 Nobel prize in physics</a> has been awarded to <A href="https://history.aip.org/phn/11409018.html">Arthur Ashkin</a> "for the optical tweezers and their application to biological systems" and to <a href="https://www.polytechnique.edu/annuaire/en/users/gerard.mourou">Gérard Mourou</a> and <a href="https://uwaterloo.ca/physicsastronomy/peopleprofiles/donnastrickland">Donna Strickland</a> "for their method of generating highintensity, ultrashort optical pulses". These inventions have revolutionised laser physics and had wide ranging applications in science, technology, industry and medicine.
</p>
<p> Light consists of electromagnetic waves. The wavelength <img src="/MI/d6cf271c9a961693ec98cb1fe7e67935/images/img0001.png" alt="$\lambda $" style="verticalalign:0px;
width:8px;
height:11px" class="math gen" /> is related to the frequency <img src="/MI/d6cf271c9a961693ec98cb1fe7e67935/images/img0002.png" alt="$f$" style="verticalalign:3px;
width:8px;
height:14px" class="math gen" /> of a light wave by the formula that works for all waves, <img src="/MI/d6cf271c9a961693ec98cb1fe7e67935/images/img0003.png" alt="$\lambda = c/f$" style="verticalalign:4px;
width:54px;
height:16px" class="math gen" />, where <img src="/MI/d6cf271c9a961693ec98cb1fe7e67935/images/img0004.png" alt="$c$" style="verticalalign:0px;
width:7px;
height:7px" class="math gen" /> is the speed of the light wave, around <img src="/MI/d6cf271c9a961693ec98cb1fe7e67935/images/img0005.png" alt="$3\times 10^8 ms^{1}$" style="verticalalign:0px;
width:91px;
height:14px" class="math gen" />. The wavelength and corresponding frequency of the wave determine the colour of the light we see in the visible spectrum. </p>
<p>
Naturally occurring light consists of light waves of many different wavelengths scattering in every direction. The light waves in a laser are different. They are <em>coherent</em>: they have the same wavelength, they travel in the same direction, and they are in phase – the peaks and troughs of the waves line up perfectly.
</p>
<br class="brclear"/>
<h3>Cloning light</h3>
<p>
The identical waves can be created thanks to the dual nature of light that was first proposed by Albert Einstein in the beginnings of quantum physics. Light can simultaneously act like a wave and act like a particle called a photon. (You can read more in <a href="/content/lightsidentitycrisis"><em>Light's identity crisis</em></a>.)
</p>
<div class="leftimage" style="width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2016/quantum/bohr_atom.png" alt="Bohr model" width="250" height="218" />
<p>The Bohr model of the atom. Electrons are only permitted to be at certain distances from the centre. (Image: <a href="https://commons.wikimedia.org/wiki/File:BohratomPAR.svg">JabberWok</a> – <a href="http://creativecommons.org/licenses/bysa/3.0/">CC BYSA 3.0</a>)</p>
</div>
<p>
Photons can be generated by the quantum behaviour inside atoms. Most of us have in our head a solar system model of an atom held together by electrical rather than gravitational forces. This picture was refined by <a href="https://en.wikipedia.org/wiki/Niels_Bohr">Niels Bohr</a> in 1913 when he proposed that electrons could not orbit the nucleus of an atom at any distance: only certain distances are allowed. The distances are related to the energy of the electron, with lower energies corresponding to the electron being nearer the nucleus. When an electron loses energy, it doesn't gradually inch inwards towards the nucleus, but performs a leap — a quantum jump — to the next allowed distance. And when an electron performs such a leap the energy it loses is released as a photon. (You can read more in <a href="/content/quantumuncertainty"><em>Quantum uncertainty</em></a> and <a href="/content/whyquantummechanics"><em>Why quantum mechanics?</em></a>.)
</p>
<p> Due to the dual nature of light this photon can also be seen as a wave with a frequency and corresponding wavelength, and these are determined by the amount of energy lost by the electron that created the photon: <table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/9b0cf3237ee61695ac6c11f0236f8203/images/img0001.png" alt="\[ E=hf=\frac{hc}{\lambda } \]" style="width:96px;
height:33px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table> where <img src="/MI/9b0cf3237ee61695ac6c11f0236f8203/images/img0002.png" alt="$h$" style="verticalalign:0px;
width:8px;
height:11px" class="math gen" /> is <a href="https://en.wikipedia.org/wiki/Planck_constant">Planck's constant</a>.
</p><p>
Lasers start with a material made up of the same type of atoms that is primed by some external energy source so that enough electrons are in specific high energy orbits around the nuclei. Then, if the material is in a particular state, something called <em>stimulated emission</em> occurs. A photon generated by the drop of one electron between orbits moves through the material. As it passes through another atom with an electron in the specified high energy orbit, it causes that electron to drop and another photon is released, with the same energy (and hence frequency and wavelength) as the first photon. Now we have two photons moving through the material which not only have the same wavelength, they are also moving in the same direction and their waves are in phase.
</p><p>
This process repeats over and over as the growing number of cloned photons move through the material. Mirrors are placed on either side of the material to amplify this effect. And the stream of cloned photons is released as a beam of laser light through a small transparent patch in one of the mirrors.
</p>
<h3>Light trap</h3>
<p>
The coherence of these cloned photons means that it is possible to control the beam of light from a laser very precisely. For example the beam can be kept very tight, rather than dispersing in all directions like the beam of light from a torch. Lasers can travel huge distances – they have been used to accurately measure the distance between the Moon and the Earth. And as well as enabling scientific advances, lasers are now part of our everyday lives. For example, they are used to store and read the information stored as microscopic pits in the surface of a CD or DVD and are used in barcode scanners at the supermarket. This year's Nobel prize in physics recognises two developments that have revolutionised the use of lasers.
</p>
<! <div class="rightimage" style="maxwidth:300px">
<img src="/content/sites/plus.maths.org/files/news/2018/nobel/tweezers_edit.png"><p>The particle was pushed to the centre of the laser beam (top image). The particle is trapped where the lens focuses the beam (bottom image). (Images ©Johan Jarnestad/The Royal Swedish Academy of Sciences)</p></div> >
<p>
When the first lasers were built in the 1960s, Ashkin began to experiment with using them to move small particles. He was successful, but to his surprise he also discovered an unexpected behaviour – the particles were drawn to move along the centre of the laser beam where the light was most intense. This was explained by the pressure the light exerted on the particle varying with the intensity of the beam (which was greater at the centre, reducing to the weakest at the edge of the beam): the sum of all these forces pushes the particle to the centre.
</p><p>
When Ashkin added a strong lens to focus the beam, the particles were drawn to and trapped in the spot where the light was most intense. Over several years Ashkin was able to develop the technique until these optical tweezers could capture atoms. This tool has had many applications but Ashkin’s own application has also been recognised by the Nobel committee. He was able to use optical tweezers to capture living bacteria without harming them, creating a new way to study biological systems now widely used in medical and biological research.
</p>
<div class="centreimage" style="maxwidth:600px">
<img src="/content/sites/plus.maths.org/files/news/2018/nobel/tweezers.jpg"></div>
<h3>Boosting light</h3>
<p>
Strickland and Mourou's breakthrough appeared in Stickland's first scientific publication in 1985 when Mourou was her supervisor. Lasers can be used to emit short bursts of light and researchers were keen to find ways to find ways to increase the intensity of these pulses. After an initial period of success in increasing the power of laser pulses since lasers were first developed, progress plateaued. By the mid 1980s researchers had reached the limit of how much they could amplify a laser pulse without damaging the amplifiers they were using.
</p><p>
Strickland and Mourou came up with an ingenious technique that is now known as <em>chirped pulse amplification</em> (CPA). They first stretched out a short laser pulse, so that its peak power would be lower. They then passed this lower power, longer pulse through the amplifier. And finally, they squeezed the pulse back to its original length, dramatically increasing the power of the final pulse.
</p>
<div class="centreimage" style="maxwidth:600px">
<img src="/content/sites/plus.maths.org/files/news/2018/nobel/cpa.jpg">
</div>
<p>
Strickland and Mourou's CPA technique is now used throughout science and industry to produce highintensity lasers. It has had many applications and enabled new areas of physics, chemistry and medicine, include laser eye surgery, which is now routine and enhancing the vision of many people every day.
</p><p>
The work of Ashkin, Mourou and Strickland really does fulfil Alfred Nobel's wish that the prizes recognise work that has been of the "greatest benefit to mankind". The Nobel prize in physics this year is also very welcome as it is the first time it has been awarded to a woman for 55 years, and only the third to be awarded to a woman. When this was pointed out to Strickland during the press conference she responded: "Is that all, really? I thought there might have been more." Perhaps the Nobel prize will amplify her positive experience as a physicist and encourage the recognition of others. "We need to celebrate women physicists because we’re out there. Hopefully, in time, it will start to move forward at a faster rate."
</p></div></div></div>
Wed, 03 Oct 2018 20:31:54 +0000
Rachel
7106 at https://plus.maths.org/content
https://plus.maths.org/content/toolsmadelightwin2018nobelprizephysics#comments

Blockchain: Spreading trust
https://plus.maths.org/content/blockchainspreadingtrust
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/bitcoin_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Marianne Freiberger</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p><em><a href="https://bitcoin.org/en/faq#howdoesbitcoinwork">Bitcoin</a></em> made the headlines last year, when it
shot up in value from $998 at the beginning of 2017 to a phenomenal
$19,666 in December 2017. It then fell again, partly due to a ban by
China and several hacks, and is currently trading at around
<a href="https://www.xe.com/currency/xbtbitcoin">$6600</a>. Nevertheless, bitcoin has entered the public consciousness as a potential
alternative currency.</p>
<h3>What is bitcoin?</h3>
<div class="rightimage" style="maxwidth: 379px;"><img src="/content/sites/plus.maths.org/files/articles/2018/bitcoin/bitcoin_rate.png" alt="Bitcoins" width="379" height="257" /><p>Bitcoin experienced a massive surge in value at the end of 2017. Chart: <a href="https://commons.wikimedia.org/wiki/File:Bitcoin_usd_price.png>Ster3oPro</a>, <a href="https://creativecommons.org/licenses/bysa/4.0/deed.en">CC BYSA 4.0</a>.</p></div>
<! image from fotolia.com >
<p>Bitcoin is a digital currency: it is not tied to any
tangible asset, such as banknotes or gold. Bitcoin users
trade in digital tokens, that is, encrypted files that live on
computers. What's more, bitcoin operates without any kind of central bank
that checks and records transactions.</p>
<p>At first, the idea is puzzling. If there isn't a central
bank that issues the currency, then where do new bitcoins come from? And
since it's presumable easy to copy a digital token, after all it's
just a file, how do you stop
users from duplicating their tokens and spending the same bitcoin more than once?</p>
<p>Bitcoin has an elegant answer to both these problems. The services of
a central bank, such as validating and recording transactions, are
performed by users of the currency. As a reward
these users receive
new bitcoins that are generated according to a fixed mathematical
formula. Performing banking services to receive bitcoins is a bit like
mining the ground for gold, which is why users who volunteer to
do this are known as
<em>miners</em>. </p>
<p>The first bitcoins where mined in January 2009 by the mysterious person, or
group of people, who developed bitcoin, known as <a
href="https://en.wikipedia.org/wiki/Satoshi_Nakamoto">Sakoshi
Nakamoto</a>. The first person to receive bitcoins (ten of them) was the <a href="https://en.wikipedia.org/wiki/Cypherpunk">cypherpunk</a> <a href="https://en.wikipedia.org/wiki/Hal_Finney_(computer_scientist)">Hal Finney</a> on January 12, 2009. In 2010 programmer Laszlo Hanyecz bought two pizzas for 10,000 bitcoin in what is believed to be the first commercial bitcoin transaction. </p>
<p>Once an amount of bitcoins is in circulation and there are users to check transactions and mine for new coins, the system can roll on of its own accord — at least that's the idea. Underneath it, stopping the system from
descending into chaos and fraud, is a structure for storing data called
a <em>blockchain</em>. Because blockchains can potentially be used, not
just in cryptocurrencies such as bitcoin, but in all sorts of other
contexts, they are the focus of a lot of
attention in the tech world. Blockchain was also a hot topic at this year's
<a href="https://www.heidelberglaureateforum.org/event_2018/">Heidelberg Laureate Forum</a> (HLF), which allows young researchers to meet the
best minds of mathematics and computer science to pave the way for
future research. At the HLF we met <a href="https://people.csail.mit.edu/silvio/">Silvio Micali</a>, Ford Professor of Engineering and Computer Science at MIT, who
took upon him the daunting task of explaining how blockchain works (you can also watch a video of some of the interview at the end of this article). </p>
<h3>What is blockchain?</h3>
<p> "In cryptocurrency the idea is not to trust [a central authority], but
to spread the trust over many, many entities, ideally everybody who is
part of the system," says Micali. To achieve this, the record of
valid bitcoin transactions, which would normally be held privately by a central
authority, is public and maintained by all the users of the bitcoin
currency. Anybody who wants to make a transaction broadcasts their
intention to the network of bitcoin users. Other users (miners) will
check the transaction is valid, for example making sure the bitcoins
in question haven't been spent before, and then append the valid
transaction to the record. </p>
<div class="leftimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/bitcoin/micali.jpg" alt="Silvio Micali" width="350" height="233" /><p>Silvio Micali talking at the Heidelberg Laureate Forum 2018. Photo ©HLFF/Mueck</p></div>
<p>If the record were just a very long list, then it could
of course be tampered with: if you wish you hadn't spent all those
bitcoins on a new dress, you simply go back and erase the
payment.
This is why the transactions are combined into
blocks. Once a block is complete it is encoded using a mathematical
function called a <a
href="https://en.wikipedia.org/wiki/Cryptographic_hash_function">cryptographic
hash</a>, which turns the block into a string of 0s and 1s according to a
mathematical recipe. This code, the block's "fingerprint", is then included in the next block that
comes along, which, when complete, will also be encoded and have its
fingerprint included in the following block, and so on.</p>
<p> In this way, every block
carries the fingerprints of all the blocks that came before it in the
chain. "The data base is secure in the sense that a
change in one block causes changes in the next block, and the one
after that, etc, so you can see if someone has tampered with it," says
Micali. The only way to alter a block without detection would be to
change it in a way that leaves the block's fingerprint unchanged. And this, by the design of the
cryptographic hash function, is practically impossible. The order of
blocks also can't be changed without detection. "[Blockchain] is a way
to freeze, at every point in time, a data base that keeps on going and
going," says Micali.</p>
<h3>Who makes the blocks?</h3>
<p>An obvious question, however, is this: without a central authority
to orchestrate the block chain, who decides which transactions go into
the next block? Bitcoin uses a competitive system called <em>proof of
work</em>. If you're a miner, you are constantly listening out for new
transactions waiting to be validated. Once you have checked
sufficiently many, you come up with a candidate block of new
transactions. However, the system will immediately generate a
hard maths problem uniquely connected to your block. It'll do the same
for every other user proposing a block, and although all the problems are
different, they are of comparable difficulty. The user whose computer
cracks its problem first gets to propose the next block along, and earn
Bitcoins as a reward. And since all previous transactions are common knowledge, a block containing fraudulent transactions can
be detected and rejected by the other users. At the moment bitcoin generates a new block roughly every
ten minutes. </p>
<h3>Expensive blocks</h3>
<p>As it stands, blockchain as used by bitcoin isn't free of
problems. Some people simply reject the idea that a decentralised currency devoid of context could actually work. Micali points to a more practical problem: the fact that the system for generating new blocks through a problem
solving competition costs a lot of computing power. "You can think of as having an
equation that is uniquely tied to a block," says Micali. "I want to solve it.
so I plug in random values for the variables and see if the equation
is solved. It is a very expensive process, so if I have [many]
computers I have a big edge over everybody else in solving the [problem
associated to a block]. So I need to have a lot of capital to have a
voice in the system."</p>
<p>"Right now in systems [using proof of work] blocks can only be
proposed by relatively few individuals. These individuals have
consolidated into pools of miners. We are now at a point where, in
bitcoin, block proposing is concentrated in the hands of three
pools."</p>
<p>This doesn't only contravene the philosophy of decentralisation,
but also opens the door to fraud. If a single entity controls more
than 50% of the computing power used to propose blocks, then it could
in theory include fraudulent transactions in the blocks, or spend
bitcoins twice. </p>
<div class="rightimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/bitcoin/bitcoin.jpg" alt="Bitcoins" width="350" height="234" /><p></p></div>
<! image from fotolia.com >
<p>Another problem with proof of work is that
it's expensive in terms of energy. "The farms of computers that are
trying to solve the [problems associated to blocks] now consume more
electricity than the country of Switzerland by a margin of over 20%,"
says Micali. "As people join the race to solve [these problems] it's
going to be more expensive." This cost, so Micali, will eventually come back to the users, in the shape of inflation or transaction fees. Current bitcoin transaction fees are less
than $0.1, but in the past they have peaked at $34 — not what
you want to pay if you're only buying a meal in a restaurant.</p>
<p>The race is currently on to find other systems for proposing blocks
in the blockchain. "The idea is to look at the money that all
users have," he says. "And to orchestrate the system so
that if the majority of the money is in honest hands then the system
works and there is no need to punish anybody because you make it
impossible to cheat." Micali has developed his own system for
choosing blocks, called <a
href="https://www.algorand.com/">Algorand</a>. It works by randomly
selecting a single user to propose the new block, with the probability
of a user being selected proportional to the amount of money they have
in the system, and then randomly selecting a collection of verifiers,
with probabilities also in proportion to the amount of money those
users own, to agree on the block the first user has chosen (find out
more <a href="https://www.algorand.com/howitworks/whyalgorandscales/">here</a>).</p>
<p>What method is best remains to be seen. It also remains to be seen whether the blockchain itself, whatever context it is used in will turn out to be a good idea —
some people query the fact that blockchain and cryptocurrencies, despite having been around fora good few years, haven't really taken off as yet. But Micali urges patience. "For thousands of years we either walked or went on horse
back," says Micali. "Then we had some puffing machines you risked
your life getting into, then we had locomotives and eventually
planes. I think that with blockchains and cryptocurrencies that's
what's happening. We started with simple technology that vaguely
addressed the goal of decentralisation, but for poor design ended up
being centralised. But we're going to see better and better
technologies so that we are going to realise the dreams people have
about cryptocurrencies."</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/EYn7hC0aPw?rel=0" frameborder="0" allow="autoplay; encryptedmedia" allowfullscreen></iframe>
<hr/>
<h3>About this article</h3>
<p><a href="">Marianne Freiberger</a> is editor of <em>Plus</em>. She interviewed Silvio Micali in September 2018 at the <a href="https://www.heidelberglaureateforum.org/event_2018/">Heidelberg Laureate Forum 2018</a>. </p>
</div></div></div>
Fri, 28 Sep 2018 12:38:46 +0000
Marianne
7105 at https://plus.maths.org/content
https://plus.maths.org/content/blockchainspreadingtrust#comments

What is pharmaceutical statistics?
https://plus.maths.org/content/pharmaceuticalstatistics
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/pills_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Marianne Freiberger</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>In this datadriven age, studying statistics at Alevel or beyond opens up a wide range of career options. One interesting area for statisticians to work in, which we recently had the chance to find out more about, is the pharmaceutical industry.</p>
<h3>How are new medical treatments tested?</h3>
<p>Before a new drug comes on the market it needs to be tested on people who have the condition the drug has been developed for. Medical scientists will already have an idea that the drug might work from tests on biological cells and on animals, but they still need to see if it also works in humans, measure just how effective it is, and what kind of side effects it might have. </p>
<div class="rightimage" style="maxwidth: 300px;"><img src="/latestnews/janapr10/rct/pills.jpg" alt="Pill bottles" width="300" height="253" /><p>In a medical trial one group of people is given the new treatment and another a placebo or an existing treatment.</p></div>
<p>At first sight, conducting such a test seems easy: give the treatment to a group of people with the condition and see what happens. In reality though, things aren't that simple. One problem is the famous placebo effect, also known as the sugar pill effect: if you have a medical condition, then simply thinking that you’re being treated for it might make you feel better, even if the treatment doesn’t actually work.</p>
<p>Another problem is that even if your patients do appear to get better, this might just be a fluke, or down to some other factors you’re not even aware of. For example, if the patients all live in the same area, and have recently been getting out and about a lot because the weather in that area has been nice, then they might have got better because of all the exercise and fresh air. The treatment may have nothing to do with it.</p>
<p>To get around these problems medical treatments are tested on two groups of people: one group will receive the treatment and the other will receive either a placebo (a "sugar pill" that doesn't work) or an existing treatment you want to compare the new treatment to. People are allocated to these groups randomly. That's the best way of making sure they don’t all share some particular characteristic (such as living in the same area) that can influence the result. A trial which works in this way is called a randomised controlled trial.</p>
<h3>What do medical statisticians do?</h3>
<div class="rightimage" style="maxwidth: 157px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phamastats/2017_05_15_mci_psi_conference_day_2_168.jpg" alt="" width="157" height="254" />
<p>Mary ElliottDavey.</p>
</div>
<p>Statisticians are a critical part of every stage of the trial process: from the initial design of a randomised controlled trial to analysing the results and communicating them to colleagues. "My job begins before the trial starts," explains biostatistics manager Mary ElliottDavey. "I will work with colleagues from other departments to design the trial. For example, how many patients do we need in each group? How long are we going to observe the patients for? How are we going to measure the disease outcome to see if the new medication works better?"</p>
<! <div class="rightimage" style="maxwidth: 300px;"><img src="/latestnews/janapr10/rct/blood.jpg" alt="blood pressure measurement" width="300" /><p>How do we test whether a new drug for reducing blood pressure really works?</p></div>
>
<p>Even only deciding how many people should be part of a study requires many of the statistical tools you learn about at school. The number of people needed depends on how confident you would like to be that you draw the correct conclusion from the outcome of your trial: the more confident you'd like to be, the more people you need. Measuring this confidence involves significance levels and confidence intervals. The number of people you need also depends on how variable the quantity (or quantities) that will be measured in the trial (for example blood pressure) is in the population you are interested in: the more it varies naturally from person to person, the more people you need in your trial. Working out this variation requires you to understand probability distributions and their measures of variability, such as standard deviation. There are standard formulae that tell you how many people you need depending on these, and other, factors. (You can find out more about this in <a href="/content/evaluatingmedicaltreatmenthowdoyouknowitworks">this</a> <em>Plus</em> article, or explore the idea yourself with <a href="https://nrich.maths.org/13762">this activity</a> on our sister site NRICH.) </p>
<p>"Once the study has finished I will then analyse the data that was collected," explains ElliottDavey. "There is a possibility that the results observed were just down to chance, and maybe the new medicine isn't any better. My job is to assess this chance and help the team determine if our new medicine does work better. I also explore the characteristics of the patients that were in the study. For example, what was the average age of patients in the study? How severe was their disease? Did they have any other [diseases]?"</p>
<div class="leftimage" style="maxwidth: 157px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phamastats/pic2.png" alt="Rhian Jacob" width="157" height="254" />
<p>Rhian Jacob.</p>
</div>
<p>To assess whether the results of a study were down to chance or indicate that a new treatment really is effective, you need to be clear about what questions you are asking of the data. You need to test a clearly formulated hypothesis (the drug reduces blood pressure by x amount) against the possibility that the new treatment is not effective, or no more effective than others. A statistical technique called hypothesis testing delivers the framework to do just that It also involves probability distributions and significance levels. (You can learn more in <a href="/content/evaluatingmedicaltreatmenthowdoyouknowitworks">this</a> article on <em>Plus</em> or <a href="https://nrich.maths.org/13722">this</a> article on our sister site NRICH, and try out the statistical techniques yourself in <a href="https://nrich.maths.org/13764">this</a> collection of NRICH resources.)</p>
<p>Applying sophisticated mathematics in reallife contexts is never just about the maths, however. It also involves a lot of teamwork. "A medical statistician partners with other nonstatistical experts, such as doctors, pharmacologists, and logistics/operations personnel," says statistical scientist Rhian Jacob. "They all have different expertise and generally don't understand statistics. Being able to communicate complex, statistical concepts in simple terms is essential." The ability to visualise complex data sets that potentially involve many variables is another essential tool in this context. </p>
<p>Teamwork is one of the things Jacob likes most about her job: "Being a single member of a very large, global team where everyone has different opinions and expertise but ultimately have the same common goal: to deliver safe and effective medicines to patients."</p>
<h3>Changing lives</h3>
<p>A pharmaceutical statistician usually works on several projects at once. They might investigate the design of a new trial one day, and then analyse the results from the latest study, or explain them to external doctors, the next. They might also travel to attend training sessions, meetings or conferences. "The work is varied, challenging, has large scope for progression and can require international travel," says Jacob. "Most companies are open to new ideas and welcome curiosity; it's nice that contribution from you personally is recognised and valued early on."</p>
<div class="rightimage" style="maxwidth: 157px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phamastats/jlf_plus_photo.jpg" alt="James LayFlurrie." width="157" height="254" />
<p>James LayFlurrie.</p>
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<p>Statistics leader James LayFlurrie points to the flexibility of career paths in the industry and the potential for exciting discovery. "Many companies will encourage you to take career decisions that play to your strengths. For example, if you have a real interest in being on the cutting edge of new statistical methods then it's possible to go down the statistical scientist route, acting in a consultant role across many projects."</p>
<p>"The great thing about the industry is that statisticians are always striving to develop cutting edge methods and trial designs to ensure medicines can be approved more quickly and at less expense, helping these medicines reach patients as quickly and cheaply as possible," says LayFlurrie. "Despite often working for rival companies there is a great statistical community within the industry and willingness to share and develop new ideas." LayFlurrie, Jacob and ElliottDavey are all members of <a href="https://www.psiweb.org/">PSI</a>, an organisation comprising members from many different companies in all areas of drug development, dedicated to leading and promoting best practice and industry initiatives for statisticians.</p>
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But what is perhaps the most satisfying aspect of the job is how the results are put to use. "I love how statistics can be used to improve people's lives," says ElliottDavey. "It is exciting to know that the study I am working on may be for a medicine that could treat cancer or prevent Alzheimer's."</p>
<p>"Every person will be affected by serious diseases, either directly or indirectly, at some point in their lives. To be able to say that my work might help even just a little bit is what gets me out of bed in the morning."</p>
<p><em>You can find out more about statistics careers in the pharmaceutical industry on the <a href="https://www.psiweb.org/careershomepage">PSI careers page</a>.</em></p>
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<h3>About this article</h3>
<p>We are grateful to <a href="https://www.psiweb.org/">PSI</a> for their support in producing this article.</p>
<p>You can try out some of the statistical techniques mentioned here in <a href="https://nrich.maths.org/13764">this</a> collection of resources on our sister site NRICH.</p>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>.</p>
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Thu, 20 Sep 2018 10:40:36 +0000
Marianne
7102 at https://plus.maths.org/content
https://plus.maths.org/content/pharmaceuticalstatistics#comments