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Plus advent calendar door #24: The sagacity of bees
https://plus.maths.org/content/plusadventcalendardoor24sagacitybees
<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_24_5.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>Honeycombs are made out of hexagonal cells. This makes them pretty to look at, but it begs the question of why. Why hexagons and not, for example, squares, or triangles?</p>
<div class="rightimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2019/advent/honey_comb.jpg" alt="Honeycomb" width="350" height="263"/><p>A honeycomb. Image: <a href="https://commons.wikimedia.org/wiki/File:Honey_comb.jpg">Merdal</a> at Turkish Wikipedia, <a href="https://creativecommons.org/licenses/bysa/3.0/deed.en">CC BYSA 3.0</a>.</p></div>
<p>The answer is to do with the fact that the dividing walls between the cells are made out of wax, and that wax is expensive to produce for the bees. To make one pound of it, they would have to collectively consume about six pounds of honey and they would need to fly the equivalent of nine times around the world to collect enough pollen to make that much honey.</p>
<p>This is why it makes sense for bees to choose a grid in which the total length of the dividing walls between cells is as small as possible. And this is the case for the honeycomb grid: out of all the grids that divide the plane into cells of (a given) equal area, the honeycomb has the smallest perimeter, that is, the smallest total length of dividing lines.</p>
<p>People have known this since at least 300 AD when the Greek mathematician <a href="http://mathshistory.standrews.ac.uk/Biographies/Pappus.html">Pappus of Alexandria</a> wrote about it in <em>On the sagacity of bees</em>. Proving the statement mathematically, however, was another story. It took over 2000 years, until 1999, when a <a href="https://arxiv.org/abs/math/9906042">general proof</a> was finally provided by the mathematician <a href="https://en.wikipedia.org/wiki/Thomas_Callister_Hales">Thomas Hales</a>. Mathematicians were overjoyed that the millennia old problem had finally been solved. What the bees thought of the proof we do not know. </p>
<p>Happy Christmas!</p>
<p><em>Return to the <a href="/content/plusadventcalendar2019">Plus advent calendar 2019</a>.</em></p></div></div></div>
Tue, 24 Dec 2019 16:50:51 +0000
Marianne
7234 at https://plus.maths.org/content
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Plus Advent Calendar Door #23: Flying home with quantum physics
https://plus.maths.org/content/plusadventcalendardoor23flyinghomequantumphysics
<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_23_2.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" width="250px"><img src="/sites/plus.maths.org/files/news/2010/birds/bird.jpg" alt="bird" width="250px" height="249" /><p></p></div> <p>Quantum mechanics is usually associated with weird and counterintuitive phenomena we can't observe in real life. But it turns out that quantum processes can occur in living organisms, too, and with very concrete consequences. Some species of birds, for example, use quantum mechanics to navigate. And as <em>Plus</em> found out at a conference in 2010, studying these little creatures' quantum compass may help us achieve the holy grail of computer science: building a quantum computer.</p>
<p>To find out more, read <a href="/content/learningquantumphysicsbirds"><em>Flying home with quantum physics</em></a>
<p><em>Return to the <a href="/content/plusadventcalendar2019">Plus advent calendar 2019</a>.</em></p></div></div></div>
Mon, 23 Dec 2019 13:25:56 +0000
Marianne
7244 at https://plus.maths.org/content
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Plus Advent Calendar Door #22: It pays to be nice!
https://plus.maths.org/content/plusadventcalendardoor22itpaysbenice
<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_22_2.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"><img src="/content/sites/plus.maths.org/files/articles/2012/nowak/sticklebackfish.jpg" width="250" height="208" alt="Stickleback fish" /><p style="width:250px">Stickleback fish rely on titfortat when approaching a potential predator to determine how dangerous it is. A pair of stickleback approach in short spurts, each spurt can be thought of as a round of the prisoner's dilemma. (Image by <a href="http://en.wikipedia.org/wiki/File:Gasterosteus_aculeatus.jpg">Ron Offermans</a>)</p></div>
<! image from http://en.wikipedia.org/wiki/File:Gasterosteus_aculeatus.jpg CC BYSA 3.0 ><p>We humans pride ourselves on our ability to be altruistic and charitable, but it turns out that we're not the only ones. Many animals also perform apparently selfless acts, including those we wouldn't normally credit with much sophistication, such as stickleback fish.</p>
<p>This raises a baffling question: since evolution is based on survival of the fittest, and helping someone else doesn't immediately increase your own fitness, how did altruism first evolve? Mathematics has an answer, based on a famous mathematical game called the <em>prisoner's dilemma</em>. To find out more, read <a href="/content/doesitpaybenicemathsaltruismparti"><em>Does it pay to be nice?</e></a></p>
<p><em>Return to the <a href="/content/plusadventcalendar2019">Plus advent calendar 2019</a>.</em></p></div></div></div>
Sun, 22 Dec 2019 13:22:13 +0000
Marianne
7243 at https://plus.maths.org/content
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Plus Advent Calendar Door #21: Can cats do logic?
https://plus.maths.org/content/plusadventcalendardoor21cancatsdologic
<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_21_2.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>Of course they can! Any cat knows that pawing the box of
cat biscuits will make it fall over and the biscuits
pour out. That's a firm grip on the law of cause and
effect. But what if the box is empty? Can cats deduce this fact from the
sound or feel of the box, or are they simply taking their chances?
How much can cats infer from incomplete information?</p>
<div class="rightimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/news/2016/cats/cat.jpg" alt="Cat" width="350" height="234" />
<p></p>
</div>
<! Image from fotolia,com >
<p>It's a question
that has received a partial answer in 2016, in a (refreshingly simple)
<a href="http://link.springer.com/article/10.1007/s1007101610016">experiment conducted by scientists in Japan</a>.
The study showed that cats know what it means when a container
rattles when shaken, and that they expect something to fall out when
the container is turned over. This may not appear hugely impressive,
but our relatives, the great apes, have failed
similar tests. Cats themselves haven't fared well in other tasks
testing their causal understanding (for example <a href="http://www.ncbi.nlm.nih.gov/pubmed/19449193">tests involving the
pulling of strings</a>) and were therefore thought a little
unsophisticated in that respect. The 2016 study appears to vindicate
them.</p>
<p>Thirty cats took part in the study, and here is how the researchers tested them. They presented each cat with a container that
did or did not rattle when shaken. The container was then turned
over and an object did or did not fall out — but not always in line
with what you'd expect. In some cases a rattling box
did reveal an object when turned over, but in others it
didn't. Conversely, a nonrattling box could also reveal an object, or not. Once the container had been turned over, the cat was allowed to go and explore. If cats do understand the
connection between rattling and the existence of objects and
if they understand gravity, then, so the reasoning goes, they should act surprised when
the unphysical situation occurs.</p>
<p>This, the researchers say, is exactly what happened. Cats spent
more time looking at the containers that didn't confirm with the laws of
physics, than they did at those which did. This suggests they do
understand the causal connection between sound and object and have a
grasp on gravity. </p>
<p>In some ways the result isn't surprising. Cats often hunt in dark
places, which means they need to rely on their ears as much as their eyes to detect their prey. For primates this isn't the case —
and rather than indicating food, sound for primates often indicates
danger and something to turn away from.</p>
<p>
More generally the study sheds light on how ecological factors
influence animals' ability to make inferences from sound. It doesn't indicate that cats
are likely to take up mathematical logic any time soon.</p>
<p><em>Return to the <a href="/content/plusadventcalendar2019">Plus advent calendar 2019</a>.</em></p></div></div></div>
Sat, 21 Dec 2019 13:05:43 +0000
Marianne
7242 at https://plus.maths.org/content
https://plus.maths.org/content/plusadventcalendardoor21cancatsdologic#comments

Plus Advent Calendar Door #20: The tree of life
https://plus.maths.org/content/plusadventcalendardoor20treelife
<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_20_3.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="width: 400px;"><img src="/issue46/features/phylogenetics/Haeckel.png" alt="The tree of life as drawn by Ernst Haeckel.." width="400" height="612" /><p>Ernst Haeckel's <em>Monophyletic tree of organisms</em>, 1866. </p>
</div>
<! Image is in public domain since copyright has expired. >
<p>At the heart of the theory of evolution lies a beautifully simple mathematical object: the evolutionary tree, which depicts the evolution of species, like ourselves, from other species that came before them. The big question in <em>phylogenetics</em>, the study of genetic relationships, is how to infer the evolutionary tree from the information we have about existing species and fossil records. </p>
<p>In Darwin's time this would have involved looking for outward resemblances like the ones between humans and apes. These days scientists look at molecular information and gene frequency data to infer similarities. Whichever method you use, one way of dealing with your information is to try and come up with a single number that quantifies the difference between any two of the species in question.</p>
<p>Once you have such a number for each pair of species you are faced with a mathematical question: given the distance information, is there a tree that realises it? And if yes, how many different trees are there?</p>
<p>To find out the answer, and to learn more about the role of maths in phylogenetics, read <a href="/content/reconstructingtreelife"><em>Reconstructing the tree of life</em></a>.</p>
<p><em>Return to the <a href="/content/plusadventcalendar2019">Plus advent calendar 2019</a>.</em></p></div></div></div>
Fri, 20 Dec 2019 13:02:29 +0000
Marianne
7241 at https://plus.maths.org/content
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Spencer BeckerKahn: Minimal surfaces
https://plus.maths.org/content/spencerbeckerkahnminimalsurfaces
<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/spencer_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>Spencer BeckerKahn is a mathematician who has joined the quest for some of the most elusive objects in mathematics: <em>minimal surfaces</em>. In this video he explains what they are and why he likes them.</p>
<p>You can also read <a href="/content/lessmorequestminimalsurfaces">an article</a> based on this interview with BeckerKahn.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/HaGe0w5EFto" frameborder="0" allow="accelerometer; autoplay; encryptedmedia; gyroscope; pictureinpicture" allowfullscreen></iframe></div></div></div>
Fri, 20 Dec 2019 10:26:57 +0000
Marianne
7246 at https://plus.maths.org/content
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Less is more: The quest for minimal surfaces
https://plus.maths.org/content/lessmorequestminimalsurfaces
<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/helicoid_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"><div class="rightshoutout">You can watch a video interview with BeckerKahn <a href="/content/lessmorequestminimalsurfaces#video">below</a>.</div>
<p>For over 250 years <em>minimal surfaces</em> have been playing hide and seek with mathematicians, turning themselves into an irresistible object of study in the process. But what are they and why are they interesting? We spoke to mathematician <a href="https://spencerbeckerkahn.wordpress.com/">Spencer BeckerKahn</a> of the University of Cambridge to find out more.</p>
<h3>Blowing bubbles</h3>
<p>The notion of a minimal surface comes from something we're all familiar with. When blowing soap bubbles you first dip a circular plastic frame into soapy water. The film which then forms within the plastic loop, before you actually blow the bubble, takes the simplest shape possible: it's perfectly flat without any bumps or bulges. This flat shape minimises surface tension (nature likes to be frugal) and it also minimises area. If the soap film had bulges and bumps, then its surface area would be bigger.</p>
<div class="rightimage" style="maxwidth: 400px;"><img src="/content/sites/plus.maths.org/files/news/2019/Abel/640pxgirl_blowing_bubbles.jpg" alt="soap bubbles" width="400" height="267" /><p>Before we blow the bubbles the soap forms a film spanning the plastic frame (or in this case each of two frames!). Image: <a href="https://commons.wikimedia.org/wiki/File:Girl_blowing_bubbles.jpg">fir0002</a>, CC.</p></div>
<p>The discshaped soap film (to be precise the mathematical idealisation of the film) is an example of an <em>area minimising surface</em>. The shape of the frame (a circle, a triangle, or something more twisted) dictates the shape of the resulting area minimising surface, so each such surface is defined by the shape of its boundary.</p>
<h3>Minimal surfaces</h3>
<p>A <em>minimal surface</em> is a slightly more general beast. If you draw a (sufficiently small) loop of any shape on the surface then the bit of surface inside the loop needs to be the area minimising surface defined by that loop. But the entire minimal surface itself doesn't need to minimise any area.</p>
<p>"One analogy I sometimes use to explain minimal surfaces is to imagine walking in a straight line," explains BeckerKahn. "In theory, if you do this for a very long time, you go all around the surface of the Earth and you come back to a point which is just a few metres behind you. Clearly you haven't taken the shortest path between your start point and your end point — to do that you should have walked backwards a few steps. But at any given time during your journey you were taking the shortest path between where you were and the point right in front of you."</p>
<p>"So this big circle around the surface of the Earth captures the property that if you take a little small portion of it, what you're looking at is the shortest path between its two end points. But if you stand back and look at the whole object it's obviously not the shortest path to take." Analogously, a minimal surface is made up of lots of area minimising surfaces without itself needing to be one.</p>
<p>The simplest example of a minimal surface is the twodimensional plane. If you draw a closed loop on the plane and ask for the surface of minimal area that takes the loop as its boundary, you'll find that this surface is exactly the bit of plane that lies inside the loop. Any other shape will have bulges or bumps, which will always lead to a bigger area. As a whole, though, the plane can't be considered an area minimising surface, not least because it doesn't come with a boundary to define the area that needs to be minimised.</p>
<h3>Finding minimal surfaces</h3>
<p>But what other minimal surfaces are there apart from the plane? That's the interesting question. At the end of the 18th century, even after mathematicians such as Leonhard Euler and JosephLouis Lagrange had turned their attention on the subject, only two other minimal surfaces were known. One was the <em>catenoid</em>, an illustration of which can be formed with soap using two pieces of frame, and the other was the <em>helicoid</em>, which looks a bit like an infinite spiral staircase with a ramp instead of steps. The 19th and 20th centuries saw minimal surface theory go in and out of fashion, with 'golden ages' followed by stagnation. Today the quest for finding more, and understanding them properly, is far from complete.</p>
<div class="centreimage"><img alt="catenoid" src="https://www.maths.cam.ac.uk/features/files/catenoid.png" style="maxwidth: 400px; height: 330px;" />
<p style="maxwidth: 400px;">This soap film represents a piece of the catenoid. The actual catenoid has no boundary and continues indefinitely towards the top and bottom.</p>
</div>
<div class="centreimage"><img alt="helicoid" src="https://www.maths.cam.ac.uk/features/files/helicoid%20%282%29.png" style="maxwidth: 400px; height: 201px;" />
<p style="maxwidth: 400px;">This soap film represents a piece of the helicoid. The actual helicoid is unbounded and continues indefinitely in all directions. Photo © <a href="http://www.exploratorium.edu" target="blank">The Exploratorium</a>. </p>
</div>
<p>Yet, despite periods of slow progress, the quest for minimal surfaces opened up fertile mathematical ground. "One thing that isn't often conveyed to people outside of academic mathematics is how easy it is to ask questions that are completely impossible for any human mathematician to solve," says BeckerKahn. "It's also very common to ask a question which turns out to be completely trivial, so a week later you wonder [why you were even thinking about the question at all]."</p>
<p>"The rarer situation is to have a plentiful supply of questions which are just right. That's when a field starts to gain steam, and that just seemed to happen with minimal surfaces. When people first coined the term they asked some very innocent questions which turned out to be much deeper mathematical themes than they originally knew."</p>
<! <p>Some of these themes eventually spawned questions in other areas of mathematics where they developed a life of their own. BeckerKahn's PhD thesis focussed on questions of this nature in the area of analysis, though he wasn't even aware of this at the start. As he gradually found out that minimal surfaces had provided the motivation for his PhD work he became enthralled by the beauty of the subject.</p> >
<h3>Pinched and wrinkled</h3>
<p>In his own work BeckerKahn is particularly interested in minimal surfaces that are a little more awkward than you would like them to be. "Sometimes you can't quite find a minimal surface that is a nice smooth sheet and is pretty and elegant to look at," he explains. You might be able to prove that a given method for constructing a surface produces a surface that's minimal, but not be able to rule out that this surface intersects itself, or is pinched, or badly contorted in some way.</p>
<p>"Ideally you'd like a more beautiful theorem at the end which rules out the possibility of your surface looking crumpled, wrinkled or pinched. But sometimes we don't know how to do that, so we are very interested in studying these regions of minimal surfaces where you get these [socalled <em>singularities</em>]. These 'bad' areas retain some of the properties of area minimisation, but they are not a smooth surface. We can use lots of the ideas and techniques that come from studying the classical setting of minimal surfaces, but often with a lot more analytical machinery to understand what the surface looks like close to the [singularities]."</p>
<p>Studying singularities is one aspect of minimal surface theory, but the field has many more strings to its bow. For example, some researchers look for minimal surfaces in spaces other than the ordinary threedimensional space we are used to. The idea that infinitely many minimal surfaces should exist in certain such settings has only recently been confirmed by a proof. "And obviously, as soon as you close off one question you realise there's another one coming," says Spencer. "It's very exciting!"</p>
<p>After over 250 years of history, minimal surfaces still provide fruitful ground for mathematical research. This was recognised earlier this year when mathematician Karen Uhlenbeck was awarded the Abel Prize, one of the highest honours in mathematics, for a body of work that includes major advances in minimal surface theory. You can find out more about this work <a href="/content/abelprize2019">here</a>. </p>
<a name="video"></a>
<iframe width="560" height="315" src="https://www.youtube.com/embed/HaGe0w5EFto" frameborder="0" allow="accelerometer; autoplay; encryptedmedia; gyroscope; pictureinpicture" allowfullscreen></iframe>
<hr/>
<h3>About the author</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>.</p>
</div></div></div>
Thu, 19 Dec 2019 13:32:18 +0000
Marianne
7245 at https://plus.maths.org/content
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Plus Advent Calendar Door #19: The mathematical horse
https://plus.maths.org/content/plusadventcalendardoor19mathematicalhorse
<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_19_2.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>Research has shown that various animals have an ability to count, but in the nineteenth century a horse amazed audiences by completing far more advanced mathematical tasks. Hans Von Osten was an Arabian stallion bought by Wilhelm Von Osten in 1888. Before Hans, Wilhelm had tutored a cat and a bear in addition. When both attempts failed, he tried a horse. Someone would ask a simple arithmetic question — for example <em>What's 7 times 4?</em> — and they would then start counting aloud. When they reached 28 Hans would start stamping his hooves.</p>
<div class="centreimage"><img src="/issue41/features/daniel/clever_hans.jpg" alt="Clever Hans performing." width="500" height="336" /><p>Clever Hans performing.</p>
</div>
<! image from wikipedia who says it's in the public domain >
<p>A panel of scientists — The Hans Commission — led by Carl Stumpf, was asked to conduct research into the horse's ability. When they struggled to find a reason, the responsibility was passed to Oskar Pfungst, another psychologist. After a series of tests Pfungst concluded that the horse was neither doing mathematics, nor involved in fraud: Hans was reading subconscious messages! When the
questioner was counting, he would emit subtle subconscious signals as the answer was reached, Hans recognised these, and responded accordingly. Hans's case study is now infamous, with the generalised case of his ability now named after him: the <em>Clever Hans Effect</em>.</p>
<em>This is an excerpt from the article <a href="/content/deathlightningcalculator">The death of the lightning calculator</a>, which looks at how, before the advent of technology, humans (and a horse) performed amazing calculational feats.</em></p>
<p><em>Return to the <a href="/content/plusadventcalendar2019">Plus advent calendar 2019</a>.</em></p></div></div></div>
Thu, 19 Dec 2019 12:47:20 +0000
Marianne
7240 at https://plus.maths.org/content
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Plus Advent Calendar Door #18: Infinite monkey business
https://plus.maths.org/content/plusadventcalendardoor18infinitemonkeybusiness
<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_18_1.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="width: 244px;"><img src="/issue54/risk/Monkeytyping_web.jpg" alt="Monkey typing" width="244" height="175" />
<p>It's just a matter of time...</p>
</div>
<p>The idea that an infinite number of monkeys typing at random on an infinite number of typewriters will eventually produce the complete works of Shakespeare apparently <a href="http://en.wikipedia.org/wiki/Infinite_monkey_theorem">dates from 1913</a>, and has appeared <a href="http://en.wikipedia.org/wiki/Infinite_monkey_theorem_in_popular_culture">repeatedly in popular culture</a> ever since.</p>
<p>But how can we be so certain that they really will, and how long do we have to wait until they do? David Spiegelhalter worked out the probabilities to help the BBC with a Horizon programme about infinity. You can see the results in <a href="/content/infinitemonkeybusinesst">this article</a>.</p>
<p><em>Return to the <a href="/content/plusadventcalendar2019">Plus advent calendar 2019</a>.</em></p></div></div></div>
Tue, 17 Dec 2019 12:39:10 +0000
Marianne
7239 at https://plus.maths.org/content
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Plus Advent Calendar Door #17: Ants and vectors
https://plus.maths.org/content/plusadventcalendardoor17antsandvectors
<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_17_2.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="width: 280px"><img src="/content/sites/plus.maths.org/files/articles/2011/ants/ants_coloured.jpg" alt="" width="280" height="200" /><p>Ants used in research are painted so that researchers can keep track of individuals. This doesn't harm them! Photo: Stuart Robinson.</p> </div>
<p>The life of a foraging ant
involves many repeated trips between food sources and the nest. These trips are arduous and long. A single foraging trip of an ant, one of many in a day, might be hundreds of metres. We can put this in human terms by comparing this foraging distance to the bodylength of an ant. A 200m journey for an ant represents a distance of over 26,000 body lengths. For a human of average height that would equate to a trip of 30 miles. An ant forager will repeat this journey until she drops dead from exhaustion.</p>
<p>The foraging trips aren't just long, they also follow complex zigzag paths. So how do ants manage to find their way back home? And how do they manage to do so along a straight line? Their secret lies in ... vector geometry!</p>
<p>To find out more, read <a href="/content/findingwayhome"><em>Finding your way home without knowing where you are</em></a>.</p>
<p><em>Return to the <a href="/content/plusadventcalendar2019">Plus advent calendar 2019</a>.</em></p></div></div></div>
Tue, 17 Dec 2019 09:40:21 +0000
Marianne
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https://plus.maths.org/content/plusadventcalendardoor17antsandvectors#comments