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enThe festival of the spoken nerd
https://plus.maths.org/content/node/6770
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In our ongoing research into the hypothesis that maths and science is the new rock'n'roll we ventured to the seedy streets of Soho, to see <em><a href="http://festivalofthespokennerd.com">Festival of the spoken nerd</a></em> at Soho Theatre. Otherwise known as <a href="http://stevemould.com">Steve Mould</a>, <a href="http://helenarney.com">Helen Arney</a> and <a href="http://standupmaths.com">Matt Parker</a>, the FotSN have been touring their combination of comedy, science and maths for a while and are now filling the Soho Theatre for a run of five nights. The show was a selection of their favourite bits of the last two shows toured, with each of the three taking their turn to do their thing.</p>
<p>The first thing to strike us was the similarity of the crowd, both in type and behaviour, to the crowds attending any of the other comedy gigs here. Nerdery is no longer a niche market — we're all nerds now. Once ushered into the theatre we were presented with a
slick show, with (slightly irritating) sound effects and power point slides, and a number of health-and-safety-bending experiments. A fantastically well-rehearsed team worked together to facilitate each others' performances.</p>
<div style="float:right; margin-left: 1em;">
<iframe width="400" height="225" src="https://www.youtube.com/embed/_dQJBBklpQQ" frameborder="0" allowfullscreen></iframe></div>
<p>Mould started with his speciality — the <em>Mould effect</em> — where a chain of beads pours out of a jar, seemingly defying gravity by rising higher and higher in the air before clattering back down to the stage. With effortless humour Mould explained the journey from discovering this phenomenon (in possibly dubious circumstances), to becoming a YouTube sensation, finding a scientific explanation and finally immortality in the effect being named after him. A final demonstration showed that there is always room for dub step to dial up the appreciation of any scientific phenomenon to 11. </p>
<p>Parker represented the mathsy part of the show, relentlessly exploiting mathematics' role as the queen of nerdery, but not without surprises. Parker proved we are all obsessed with spreadsheets, even if we don't realise it, by brilliantly showing our addiction to all things screens is really just an ongoing worship of Excel spreadsheets. And we particularly liked his great demonstration of parabolic mirrors as he sent heat across the stage causing a remote fire —
all in honour of one of our own favourite mathematical shapes, the parabola. </p>
<p>Arney's electric ukulele and heavenly voice explored love scientifically and mathematically with a suitable helping of dark humour. The final number showed how well these three performers work together: Arney playing a maths based love song, supported by competing visual gags from Parker on an OHP and Mould on Power Point slides. It's always impressive when something is that slick but so relaxed that you can't tell which bits are improvised and which are rehearsed. Definitely worth seeing!</p></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">Review by </div><div class="field-items"><div class="field-item even">Rachel Thomas with Marianne Freiberger</div></div></div>Tue, 17 Jan 2017 13:31:22 +0000mf3446770 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/6770#comments'A farewell to ice'
https://plus.maths.org/content/node/6769
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="leftimage" style="width: 150px; border: 1px;"><img src="/content/sites/plus.maths.org/files/reviews/2016/wadhams_cover.png" alt="cover" width="150" height="229" /></div><h2>A farewell to ice: A report from the Arctic</h2>
<h3>by Peter Wadhams</h3>
<p><a href="http://www.damtp.cam.ac.uk/user/pw11/">Peter Wadhams</a> has been to the Arctic over 40 times, nearly every year since 1970. He is Professor of Ocean Physics at the Centre for Mathematical Sciences in Cambridge. In a scientific career spanning five decades Wadhams has seen the object of his study undergo dramatic changes. In <em>A farewell to ice</em> he shares with readers in a largely non-technical style, yet with many scientific explanations, references, and data, the view of a sea-ice expert on the current status and the future of Arctic sea ice. The fate of ice is inevitably intertwined with that of life on Earth. </p>
<p>And the future does not look good. As Wadhams states in the closing chapter, "If we destroy our planet, we destroy ourselves. There is nowhere else for us to go. There is no Planet B. It will not just be farewell to ice, but farewell to life."</p>
<p>The book starts gently. In chapter 2 we are introduced to ice, "the magic crystal". Wadhams neatly describes some of the myriad ways in which the water-ice pair differs from most other common solids and liquids. He talks of different mechanisms of sea ice formation, that lead to rather different kinds of sea ice, ranging from what is called pancake ice, to the most important kind, multi-year ice. The tender affection the scientist has for sea ice is palpable even through his simple and lucid style of writing. </p>
<p>After explaining the mechanisms by which Arctic sea ice effects global climate, the main theme of the book unfolds in full in chapter 5, starting with the greenhouse effect. Chapters 6 and 7 discuss respectively how yearly sea ice data points to an undeniable "Arctic death spiral", and the accelerating effects of positive feedbacks within the Arctic system. This forms the core of the book, elaborating why we are nearing a stage where we must bid farewell to sea ice. The most terrifying feature of this "Arctic amplification" is aptly stated by Wadhams: "We are not far from the moment when the feedbacks will themselves be driving the change — that is, we will not need to add more CO<sub>2</sub> to the atmosphere at all, but we will get the warming anyway."</p>
<p>Allowing myself one exaggeration to drive home the urgency, in view of what might be in store for climate in the future, what we're seeing right now is just a warning, not a warming!
</p>
<p>The book goes further than just reporting on the state of the Arctic. Wadhams enters into impassioned descriptions of the political inaction and lack of collective will in tackling the problem of climate change induced by human activity. He laments at one point, "But the unfortunate reality is that the global population, especially in the West, is extraordinarily reluctant to give up the conveniences of living in a fossil fuel world." This is food for thought for all readers: how much longer can we live thinking, "just one more Ryanair flight, and isn't that SUV a good way of getting the children to school?"</p>
<p>Another alarming reality we are told about is the growth of climate change denial organisations. Good science needs critics and sceptics. But these are not scientists disagreeing over mechanisms for climate change. Instead they are parties with enormous wealth and vested interests, especially in the oil industry, usually having nothing to do with science. They seek to sow doubt and coax the world into inaction, misleading people for their own short term monetary gain. </p>
<p>In the last two chapters, Wadhams rallies all the foregoing discussion of sea ice, climatic patterns, and the mechanisms that might link them, to discuss the global climatic effects we can expect in the future, and the economic and human spinoffs of these potentially catastrophic changes. He proposes an <em>integrated arctic science</em> to bring together scientists, policymakers and economists. </p>
<p>Coming at a time when Arctic sea ice has thinned by over 75% and its extent in area diminished to record lows, this book offers a first-hand insight into the world of Arctic science. Mathematics is, of course, an essential component of this science, and the book points towards its applications in this area. The wealth of scientific data and references add significantly to the genre of climate science books. Wadhams describes in an accessible way things that are not normally presented to the lay person, be it the discussion of multi-year ice, the thermohaline circulation, or hot research topics like chimneys and methane plumes.</p>
<p>The target audience for this book is broad. The content is primed for the lay reader, but with enough detail to keep those acquainted with science alert as well. The material is well organised, and the theme of melting ice is constantly reiterated in every chapter, each time viewed through the lens of a different phenomenon. The chapter titles offer crisp glimpses of the content. </p>
<p>Wadhams has previously been labelled an "alarmist" by climate change deniers: a blanket term they use for outspoken scientists and environmentalists alike. However, in presenting the story through empirical observations and discussing different hypotheses that try to explain these, he succeeds in making a detailed and convincing argument that climate change is very real, and a very urgent problem facing humanity.</p>
<p>There is reasonable evidence to believe that a molecule of CO<sub>2</sub> stays active in the climate system for almost 100 years. Wadhams' view that <em>carbon drawdown</em> is the only way to rein in climate change certainly makes sense, seeing as precaution is better than cure. Stressing the need for action at the level of nations and governments, he quotes the late David MacKay, former chief scientific adviser to UK Govt. on Energy and Climate, who exclaimed, "If everyone does a little, we will achieve only a little".</p>
<p>Happily, the book has very few noticeable shortcomings. There could have been a few more sentences on the ecological havoc being wrought by the loss of Arctic ice, especially the habitat loss for such representative species as the polar bear. A curious unconscious bias of the author might be inferred at some places : he frequently implicates Russia and "developing countries" (China and India for their fossil fuel usage, reference to a predicted quadrupling of population in Africa) against the stark backdrop of not once explicitly referring to the role of the industrialised countries and their colonial rule where the problem originally began.</p>
<p>In sum, <em>A farewell to ice</em> is bound to touch the reader. This is a book that I would recommend one and all to read. For an awareness of the issues it discusses and the voice of concern that it raises, a voice of concern that is not represented enough objectively in the media, are potentially crucial for the future of humankind. </p>
<dl> <dt><strong>Book details:</strong></dt>
<dd><em>A farewell to ice</em></dd> <dd>Peter Wadhams</dd> <dd>hardback — 256 pages</dd>
<dd>Allen Lane (2016)</dd>
<dd> ISBN 978-0241009413</dd>
</dl>
<hr/><h3>About the author</h3>
<p>Sathyawageeswar Subramanian is a PhD student in Quantum Computing at the <a href="http://www.damtp.cam.ac.uk/">Department of Applied Mathematics and Theoretical Physics</a> at the University of Cambridge.</p>
</div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">Review by </div><div class="field-items"><div class="field-item even">Sathyawageeswar Subramanian</div></div></div>Mon, 09 Jan 2017 15:14:58 +0000mf3446769 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/6769#comments'Can you solve my problems?'
https://plus.maths.org/content/node/6768
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="leftimage" style="width: 150px; border: 1px;"><img src="/content/sites/plus.maths.org/files/reviews/2016/Alex_cover.png" alt="cover" width="150" height="193" /></div><h2>Can you solve my problems?: A casebook of ingenious, perplexing and totally satisfying puzzles</h2>
<h3>by Alex Bellos</h3>
<p>In <em>Can you solve my problems?</em> Alex Bellos offers yet another light-hearted exploration into the surprising world of mathematics. Following the acclaimed <em><a href="/content/alexs-adventures-numberland">Alex's adventures in numberland</a></em> and <em><a href="/content/alex-through-looking-glass">Alex through the looking-glass</a></em>, Bellos uses a curated set of 125 thought puzzles to underscore the approachability of mathematics, without hiding any of its complexity. The puzzles are organised into five broad conceptual categories, and interspersed with historical commentary. Throughout, Bellos is intrigued more by the process of puzzle creation than by their solutions (though fear not, the book concludes with an extensive <em>answers</em> section). By weaving together the minds, cultures, and concepts that produced centuries of playful wonder, Bellos successfully brings an obscure corner of mathematics to life. </p>
<p>One of the key merits of puzzles, according to Bellos, is that they call upon pure mathematical reasoning, rather than on previous training. Puzzles require virtually no prior knowledge, and yet can confound even the most brilliant minds. This irony makes puzzles inherently humorous, a quality that can be amplified by how a problem is posed. These traits – accessibility and humour – are why puzzles are so entertaining, and apparently what led Bellos to author this book. They characterise the entire book's tone, which is readable and playful throughout, while never shying away from a fiendishly difficult problem. </p>
<p>Refreshingly, Bellos focuses not just on puzzles, but on their writers and their contexts. The first chapter opens in Charlemagne's court, where the scholar Alcuin offered the king "some arithmetical curiosities" to "amuse" him. Many of the puzzles found in that document remain popular today. By consistently focusing his narrative on puzzle writers, Bellos reminds us that making good puzzles is far trickier than solving them. </p>
<p>Bellos also shows how a puzzle can take on a life of its own, providing a window into the cultures that produced it and passed it down. He describes how the wolf, goat, and cabbage in the famous <a href="http://britton.disted.camosun.bc.ca/jbwolfgoat.htm">stream-crossing problem</a> were replaced in different countries with the local predators, herbivores, and vegetables, and how three men and their sisters in a puzzle based on dishonourable conduct have been replaced by various social archetypes – husbands and wives, masters and valets – as the tastes of early modernity gave way to later sensibilities and prejudices.</p>
<p>The book tries to appeal to both amateur mathematical historians and puzzle aficionados – and risks leaving both dissatisfied. The historical narratives are short and, with few exceptions, unrelated to one another. Those that do have recurring characters or themes are scattered throughout the text with little warning or clear pattern. On the other hand, Bellos speaks directly to the reader throughout the text, commiserating with and often teasing them when puzzles are particularly challenging. This is usually entertaining, but comes off in moments as trite, drawing attention away from the puzzles themselves. The book also lacks a conclusion, leaving a feeling of irresolution that proves unsettling for a self-proclaimed puzzle lover — though perhaps that's precisely the effect he intends! </p>
<p>Overall, this is a book worth reading. It is best taken in pieces, a few pages and a few puzzles at a time. Bellos reiterates throughout the text that puzzles are, and have always been, intended to entertain and to sharpen. With this book, Bellos fixes himself into a millennia-long tradition of using mathematics to inspire and entertain. </p>
<dl> <dt><strong>Book details:</strong></dt>
<dd><em>Can you solve my problems?</em></dd> <dd>Alex Bellos</dd> <dd>hardback — 352 pages</dd>
<dd>Guardian Faber Publishing (2016)</dd>
<dd> ISBN 978-1783351145</dd>
</dl>
<hr/><h3>About the author</h3>
<p>Stephen Kissler is a PhD candidate in applied mathematics at the University of Cambridge. He studies mathematical biology, currently focusing on how infectious diseases (primarily influenza) spread.</p>
</div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">Review by </div><div class="field-items"><div class="field-item even">Stephen Kissler</div></div></div>Mon, 09 Jan 2017 11:42:20 +0000mf3446768 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/6768#comments'Calculating the cosmos'
https://plus.maths.org/content/node/6760
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="leftimage" style="width: 150px; border: 1px;"><img src="/content/sites/plus.maths.org/files/reviews/2016/cover.png" alt="cover" width="150" height="226" /></div><h2>Calculating the cosmos</h2>
<h3>by Ian Stewart</h3>
<p>There is a quote attributed to Albert Einstein: "Everything should be made as simple as possible, but not simpler". This is the approach that Ian Stewart takes in his most recent book <em>Calculating the cosmos</em>, while taking his readers through a journey that starts at the beginning of the Universe and continues on through space and time to the present day. The subtitle <em>How mathematics unveils the universe</em> is aptly chosen: this is a book that chronicles the history of the Universe through people's attempts to understand it using mathematics. And yet there are very few formulas and technical definitions in the book. Stewart focuses instead on explaining the intuition behind mathematical models, making this book a smooth and friendly read to everyone irrespective of their scientific background.</p>
<p>
The book is divided into nineteen relatively short chapters, each dedicated to answering a single central question, such as "how did our solar system form?" or "are there intelligent alien life forms?". Most chapters follow a similar structure: Stewart offers some motivation for the question at hand and then examines in chronological order the various ways people have approached it, from when it was first posed up to the modern day. He transitions from explaining one theory to the next very organically by pointing out the problems of the original theory, which in turn motivated and necessitated the development of new ones. Newton's laws might describe the effects of gravity as a force between two bodies very elegantly, but they cannot explain how exactly that force works. Many scientists, including Newton, tried to address that problem, but it remained unsolved until Einstein formulated his theory of general relativity centuries later. These transitions turn the multitude of scientific theories that Stewart examines into a coherent, easily-followed narrative, despite their apparent disconnectedness in the eyes of someone who might not have a good foundation in cosmology and physics.</p>
<p>The first few chapters concern the more traditional questions and approaches in cosmology, while the latter ones start delving into more modern theories such as dark matter and dark energy. Stewart makes a point of emphasising that all models are to some extent a simplification of reality, which often leaves room for interpretation. As a result, even questions that seem to have had definitive answers for hundreds of years can come back to the spotlight with new scientific discoveries and technological progress. </p>
<p>An example comes from a widely-accepted theory about the formation of the : that something large collided with the Earth head-on and part of the resulting debris became the Moon. It was only a few years ago that scientists were able to perform accurate simulations to challenge the two main assumptions of that theory, namely the size of the impactor and the direction of the collision. This led to two new theories: maybe the impactor was much larger than originally thought but collided with the Earth sideways instead of head-on, or maybe it was much smaller but the Earth was spinning faster than it is today. So even something as seemingly resolved as the origin or the Moon still remains somewhat of a mystery. Therefore if there is a single most important message that Stewart is trying to convey through this guided voyage into the Universe, it's that we should always remain inquisitive and humble and ready to revise, because despite the incredible amount of knowledge we have accumulated over the years, there are still so many stones left unturned.</p>
<p>Stewart has a long history of writing popular science books and knows how to make the science accessible to everyone interested in finding out more about it. Whenever a theory might sound very convoluted or weakly motivated, he incorporates graphs, diagrams and illustrations as a visual aid. One of the very few instances where he includes mathematical formulas is when he examines the Titius-Bode law, which states that in our solar system the distances of the planets from the Sun follow a (nearly) geometric sequence. In the discussion that follows he presents a power-law formula for the distance of the <em>n</em>th planet from the Sun and solves it using logarithms. To illustrate why such a mathematical manipulation is useful, he then plots the actual log distances of the planets to show that they indeed follow a straight line, exactly as the power law formula would predict. To satisfy and help readers from a wide range of (non)scientific backgrounds, he also includes a few pages listing the most common scientific terminology that he needs to use, as well as references to the literature for the more mathematically inclined. The result is a book that will satisfy anyone who wishes to learn more about the Universe independently of how good their mathematical knowledge is.</p>
<p>Perhaps the best feature of <em>Calculating the cosmos</em> is Stewart's passion for mathematics and physics. His enthusiasm is contagious: in the end, we too are left in awe of the Universe and admiring the effort that science has put into understanding the final frontier.</p>
<dl> <dt><strong>Book details:</strong></dt>
<dd><em>Calculating the cosmos</em></dd> <dd>Ian Stewart</dd> <dd>hardback — 360 pages</dd>
<dd>Basic Books (2016)</dd>
<dd> ISBN 978-0465096107</dd>
</dl>
</div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">Review by </div><div class="field-items"><div class="field-item even">Venetia Karamitsou</div></div></div>Mon, 09 Jan 2017 11:25:17 +0000mf3446760 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/6760#commentsPlus Advent Calendar Door #24: Twelve minutes of Christmas
https://plus.maths.org/content/plus-advent-calendar-door-24-merry-christmas-1
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_24.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/blog/122016/dogs.jpg" alt="" width="350" height="235" /><p>Merry Christmas!</p></div>
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<p>Finally the wait is over! Just one more night and Christmas is here. We hope that you'll get a well-deserved break over the holidays — some time to cuddle up in front of the fire with something nice to read.</p><p> In case you're short of reading material, or would like to try out that brand new e-reader you got for Christmas, we've produced a new ebook comprising twelve favourite topics from our <em>Maths in a minute</em> library. Your very own twelve minutes of Christmas! Enjoy!</p>
<p><a href="/content/sites/plus.maths.org/files/ebooks/Twelve Minutes of Christmas.pdf">Download as pdf</a></br>
<a href="/content/sites/plus.maths.org/files/ebooks/Twelve Minutes of Christmas.mobi">Download as mobi</a> (for kindle)</br>
<a href="/content/sites/plus.maths.org/files/ebooks/Twelve Minutes of Christmas.epub">Download as epub</a></br> </p>
<p>As this is the first ebook we have produced in a while, we'd really appreciate if you could answer the following quick questions. Thank you!</p>
<iframe src="https://docs.google.com/forms/d/e/1FAIpQLSfHYQdP7RYXdj6E82XLnByeiSkZ7nPEzyfV_2CSqQ-BGfRgdQ/viewform?embedded=true" width="600" height="500" frameborder="0" marginheight="0" marginwidth="0">Loading...</iframe></div></div></div>Sat, 24 Dec 2016 11:03:00 +0000mf3446766 at https://plus.maths.org/contenthttps://plus.maths.org/content/plus-advent-calendar-door-24-merry-christmas-1#commentsPlus Advent Calendar Door #23: Navigation
https://plus.maths.org/content/plus-advent-calendar-door-23-navigation
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_23.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Chris Budd</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightshoutout">This article is adapted from one of Budd's Gresham College lectures. See <a href="http://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world">here</a> to find out more about this free, public lecture series.</div>
<p>Perhaps the leading scientific question of the 17th and 18th centuries was how to find out exactly where you are when you are at sea. It had almost a mythical status and appeared in <em>Gulliver's Travels</em> as an example of an impossible problem. This difficult question not only stimulated a lot of mathematics, but also led, directly, to the modern world in which mathematics and machines work together.</p>
<p>The first breakthrough in navigation came when it was realised that the position of the Sun and the stars in the sky depend upon where you were on the (round) Earth. By seeing how the angle of the Sun changed the Greek mathematician <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Eratosthenes.html">Eratosthenes</a> was able to calculate the radius of the Earth to surprising precision. Having worked out a coordinate system for the Earth (latitude and longitude) it was apparent that the latitude could be determined by measuring the angle of the noonday Sun above the horizon. Doing this required a good knowledge of angles and <a href="https://en.wikipedia.org/wiki/Trigonometry">trigonometry</a>. The angle itself could be measured using a <em>sextant</em>, again using ideas from trigonometry.</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2016/navigation/HMS_Association_%281697%29.jpg" alt="Boxing gloves" width="350" height="254" /><p>The <a href="https://en.wikipedia.org/wiki/Scilly_naval_disaster_of_1707">Scilly naval disaster</a> of 1707 was a direct result of sailors' inability to pin-point their longitude. 1550 men lost their lives as their ships struck rocks.</p></div>
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<p>At that point the mathematics became much harder. In order to find the longitude it was necessary to determine the time at the location with reference to some absolute standard (for example, you would need to find out the local time with reference to the time in London). Mathematicians such as Newton struggled with this problem, and in a sense solved it: they found that the longitude could be determined from very accurate measurements of the location of the Moon, combined with a fearsome amount of calculation. Unfortunately, none of this was possible in the conditions at sea (and before the invention of the pocket calculator). </p>
<p>An accurate method of finding longitude had to wait until the development of a clock called <em><a href="https://en.wikipedia.org/wiki/John_Harrison#H4">H4</a></em> by <a href="https://en.wikipedia.org/wiki/John_Harrison">John Harrison</a> as a means of finding the time at Greenwich. See Dava Sobel's book <a href="/content/longitude"><em>Longitude</em></a> for an excellent account of Harrison's struggles to build H4. </p><p>However, even with H4 a large amount of calculation was needed and here mathematics came into its own. In particular the development of <a href="https://en.wikipedia.org/wiki/Spherical_trigonometry">spherical trigonometry</a>, which was needed to solve the triangles on the surface of the Earth that were the results of the navigational measurements. Tables were constructed which solved triangles with a vast range of different angles. These were used in parallel with <em>ephemerides</em>, which were tables of the location of the Sun, planets and many stars for frequent time intervals in every day throughout the year. Looking at nautical tables from the 18th century I am overwhelmingly impressed by the amount of calculation needed to produce them, most of which would have been done by human computers.</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2016/navigation/Harrison_H4_clockwork_1.jpg" alt="H4" width="350" height="213" /><p>The clockwork in Harrison's H4 clock. Photograph by <a href="https://commons.wikimedia.org/wiki/User:Mike_Peel">Mike Peel</a>, <a href="https://creativecommons.org/licenses/by-sa/4.0/">CC-BY-SA-4.0</a>.</p></div>
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<p>The result of all of this mathematics, combined with the mechanical brilliance of Harrison, was completely transformative. It utterly transformed navigation by sea, making it much safer and cheaper to transport goods around the world. It revolutionised both the economy and also the process of exploration, leading to the modern world.</p>
There was a nice side product of all of this effort. The process of producing the navigational tables was essentially routine and open for being mechanised. Driven by this idea, in the 19th Century, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Babbage.html">Charles Babbage</a> was inspired to design the <em>difference engine</em>, arguably the ancestor of all modern computers. Sadly, Babbage did not live to see his machines actually working (although a working model of the difference engine was built by the <a href="http://www.sciencemuseum.org.uk">Science Museum</a> in London), but his ideas have certainly led to the modern computer, and thus the modern world. </p>
<hr/>
<h3>Further reading</h3>
<p>You can find out more in the following articles:</p>
<ul><li><a href="/content/finding-your-place-world"><em>Finding your place in the world</em></a> explains the global coordinates of latitude and longitude,</li>
<li><em><a href="/content/latitude-stars">Latitude by the stars</a></em> explains how latitude can be determined by measuring the angle of the Sun above the horizon,</li>
<li><em><a href="/content/longitude-problem">The longitude problem</a></em> explains just that,</li>
<li><a href="/content/ada-lovelace-visions-today"><em>Ada Lovelace: visions of today</em></a> explores Babbage's attempt at building the first ever computer, and the contributions of his friend Ada Lovelace.</li></ul>
<p><a href="/content/2016-plus-advent-calendar">Return to the <em>Plus</em> advent calendar 2016</a>.</p>
<p>This article comes from our <em><a href="/content/maths-in-a-minute">Maths in a minute library</a></em>.</p>
<hr><h3>About this article</h3>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/blog/092016/budd.jpg" alt="Chris Budd" width="250" height="250" /><p>Chris Budd.</p></div>
<p>This article is adapted from one of Chris Budd's Gresham College lectures, part of a series called <em><a href="http://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world">Mathematics and the making of the modern and future world</a></em>. The lectures take place in London, are aimed at a general audience and free to attend. </p>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org/">Royal Institution</a>, <a href="http://www.gresham.ac.uk/professors-and-speakers/professor-chris-budd/">Gresham Professor of Geometry</a>, and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p>
</div></div></div>Fri, 23 Dec 2016 15:37:21 +0000mf3446764 at https://plus.maths.org/contenthttps://plus.maths.org/content/plus-advent-calendar-door-23-navigation#commentsPlus Advent Calendar Door #22: What's the problem with quantum gravity?
https://plus.maths.org/content/plus-advent-calendar-door-22-whats-problem-quantum-gravity
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_22.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>At the heart of modern physics lurks a terrible puzzle: the
two main theories that describe the world we live in just won't fit
together.</p>
<div class="rightimage" style="width: 300px;"><img src="/latestnews/sep-dec06/rixmas/Presents.jpg" alt="Dice with question marks" width="300" height="276" /><p></p>
</div>
<p> The force of gravity is
described by Einstein's general theory of relativity (which celebrates
its 100th birthday this year). General relativity says that space and
time can be curved by massive objects.</p>
<p> The other two fundamental forces of nature (the
electroweak force and the strong nuclear force), as well as the
fundamental particles, are described by quantum physics. A main result
in this context is Heisenberg's uncertainty principle, which implies
that you can never determine the location and motion of particles at the same time
with complete precision (more accurately, the classical concepts of position and momentum cannot coexist with perfect sharpness). </p><p>But if those particles have mass, then
according to general relativity, their motion effects the shape of space and
time. Putting both theories together implies that you can't
determine the space and time in which particles exist and move. That's
clearly a problem, and displays a significant incompatibility between relativity and quantum theory. Physicists are hard at work developing
a unifying <em>theory of quantum gravity</em>, but it's proving to be very, very tricky. One contender is <em>string theory</em>, which has taken the radical step of giving up the fundamental notion of a "point" in space and time.</p>
<p>To find out more read:</p>
<ul><li><a href="/content/einstein-and-relativity-part-ii">Einstein
and relativity</a></li>
<li><a href="/content/string-theory-newton-einstein-and-beyond">String
theory: From Newton to Einstein and beyond</a></li></ul>
<p><a href="/content/2016-plus-advent-calendar">Return to the <em>Plus</em> advent calendar 2016</a>.</p>
<p>This article comes from our <em><a href="/content/maths-in-a-minute">Maths in a minute library</a></em>.</p></div></div></div>Thu, 22 Dec 2016 15:34:13 +0000mf3446763 at https://plus.maths.org/contenthttps://plus.maths.org/content/plus-advent-calendar-door-22-whats-problem-quantum-gravity#commentsSpontaneous spirals
https://plus.maths.org/content/spontaneous-spirals
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/spiral_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Wim Hordijk</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Spirals are common in nature. We've all admired the beautiful spirals that occur in sea shells, we can find spirals in plants, and even in the arms of galaxies or weather patterns. There are also situations in which spirals aren't a result of slow growth, but
occur spontaneously in biological or chemical systems. A famous example from chemistry is the <a href="https://en.wikipedia.org/wiki/Belousov–Zhabotinsky_reaction#History"><em>Belousov-Zhabotinsky (BZ) reaction</em></a>: when several chemicals are mixed together in a petri dish, the resulting solution forms changing spiral patterns, as seen in this video:</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/PnOy1fSxBdI" frameborder="0" allowfullscreen></iframe>
<p> In biology a particular <a href="http://dictybase.org/tutorial/">slime mould</a>, called <a href="http://dictybase.org/tutorial/">dictyostelium discoideum</a>, gives rise to similar patterns. The organism normally lives as individual <a href="https://en.wikipedia.org/wiki/Amoeba">amoebae</a> that consist of a single cell. The spirals arise when these single-celled amoebae aggregate into a body made up of many cells, which then produces spores that develop into individual amoebae again. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/spirals/mould.png" alt="Spiral waves in dictyosteliumn" width="325" height="327" />
<p style="max-width: 283px;">Spiral waves in <em>dictyostelium</em>. Image from <em><a href="http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1004367">Regulation of Spatiotemporal Patterns by Biological Variability</a></em> by Miriam Grace and
Marc-Thorsten Hütt. Published in PLOS Computational Biology, reproduced under Creative Commons.</p>
</div>
<p>What's remarkable about both these examples is that there is no single unit, or even a small group of units, that controls the behaviour of the system as a whole and "dictates" the formation of the spiral patterns. Instead, the individual units interact only locally with whatever other units happen to be nearby. Yet, despite this lack of a central control and the limitation to local interactions only, these systems produce <em>global</em> patterns, such as the spiral waves. They also produce coordinated behaviours, such as aggregating into one multi-cellular body, that are at the scale of the system as a whole, well beyond that of a single individual. </p>
<p>These are examples of <em>decentralised spatial systems</em>. But how do their patterns form?</p>
<h3>Cellular automata</h3>
<p>Spontaneous spiral wave formation in decentralised spatial systems can be reproduced and studied with simple mathematical models known as <em>cellular automata</em>. A cellular automaton (CA) consists of a large number of similar individual units ("cells") arranged in a particular way. For example, the cells could be squares that line up in a row (a one-dimensional arrangement) or they could form a two-dimensional grid, like graph paper.</p>
<p> Each cell in this grid contains a variable that has one of a number of possible values, for example either 0 or 1. The values of these variables can be visualised by simply colouring each cell accordingly, such as white and black for 0 and 1. The variables could also take as values numbers between 0 and 1, in which case we can use or a colour gradient to represent them.</p>
<p>At regular time steps, each cell updates the value of its variable according to a given mathematical formula that takes account of the current values of the cell itself and those directly surrounding it. For example, the formula might decree that a cell that currently has the value 0, and whose neighbours also all have the value 0, will change its value to 1. Thus, at each time step, the colours of the cells in the grid change accordingly, which gives rise to a constantly changing visual pattern in the system as a whole. However, there is no central control in the system, and each cell only interacts with its closest neighbours.</p>
<p>The simplest version of a CA consists of a row of cells whose values can only be 0 or 1. The new value (or colour) of a cell depends only on the cell itself and one neighbour on either side, that is, a three-cell local neighbourhood. This is known as an <em>elementary cellular automaton</em> (ECA). In this case, the update formula can simply be represented by a lookup table that lists each of the possible triples of values that represent a local neighbourhood, together with the new value of the centre cell for each of these configurations. Since each cell can takes one of two values, there are 2<sup>3</sup>=8 possible triples. An example is shown in the table below. </p>
<table class="data-table">
<tr><td>Neighbourhood</td><td>000</td><td>001</td><td>010</td><td>011</td><td>100</td><td>101</td><td>110</td><td>111</id></tr>
<tr><td>New value of middle cell</td><td>0</td><td>1</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td></tr></table>
<p>In this example (which is known as ECA 18) a string of cells that consists of a single 1 and the rest 0s , that is,</p>
<table class="data-table">
<tr><td>...</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>...</td></tr></table>
<p>at the next time step changes into the following string: </p>
<table class="data-table"><tr><td>...</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>...</td></tr></table>
<p>Which at the next time step changes into:</p>
<table class="data-table"><tr><td>...</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>...</td></tr></table>
<p>This process of updating the cells according to the rule can be continued indefinitely.</p>
<p>Replacing the numbers by colours (black for 1 and white for 0) and placing the strings, as they evolve at each time step, underneath each other gives the <em>space-time diagram</em> of the ECA.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/spirals/ECA18.png" alt="ECA 18" width="539" height="308" />
<p style="max-width: 539px;">ECA 18 starting from a row with only one back cell (top). The picture shows the rows you get from 18 time steps. </p>
</div>
<!-- Image produced by MF -->
<p>A different starting string will give a slightly different picture, but retain some of the overall patterns. Here is the diagram for a starting string in which the colour of each cell has been chosen at random.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/spirals/ECA18-2.gif" alt="ECA 18" width="539" height="539" />
<p style="max-width: 539px;">ECA 18 starting from a random string (top). </p>
</div>
<!-- image provided by author -->
<p>The global patterns, even though generated by a simple and local update formula, look very natural. These patterns resemble, for example, ones that occur on sea shells.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/spirals/shell.jpg" alt="Sea shell" width="500" height="375" />
<p style="max-width: 500px;">Image: <a href="https://www.flickr.com/photos/tehsma/3722970830/in/gallery-clicksnappy-72157625545604414/">tehsma</a>, <a href="https://creativecommons.org/licenses/by-nc/2.0/">CC BY-NC 2.0</a>.
</p>
</div>
<!-- image provided by author -->
<p>
Note that the new cell values in an ECA, that is the 0s and 1s in the bottom row of the lookup table above, could have been assigned in any way. There are two possible values for each of the eight triples, giving rise to 2<sup>8</sup>=256 different possible lookup tables for ECAs. The particular patterns that show up in a ECA's global dynamics depend on which of these lookup tables is used. You can see a picture of each ECA on <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Wolfram Mathworld</a>.</p>
<h3>Spiral waves</h3>
<div style="max-width: 340px; float: right; border: thin solid grey;
background: #CCC CFF; padding: 0.5em; margin-left: 1em; font-size:
75%">
<h3>The ISCAM model for spiral waves</h3>
<p>For the ideal storage model imagine a container that contains a certain amount of a substance. At regular time intervals a proportion <img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0001.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> of the substance is released from the container, and a fixed amount <img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0002.png" alt="$C_ s$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> is replenished. If the amount of the substance in the container is <img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0003.png" alt="$S$" style="vertical-align:0px;
width:10px;
height:11px" class="math gen" /> at the current time step, then it will be </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0004.png" alt="\[ S^{\prime } = S - RS+C_ S \]" style="width:134px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>at the next time step. The release proportion is itself dynamic, depending on the amount released at the previous step. If the proportion was <img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0001.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> at one time step, then the release proportion <img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0005.png" alt="$R^{\prime }$" style="vertical-align:0px;
width:16px;
height:13px" class="math gen" /> for the next time step is </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0006.png" alt="\[ R^{\prime } = \frac{1}{1+e^{-5RS+C_ r}}, \]" style="width:147px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>where <img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0007.png" alt="$C_ r$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> is a fixed constant. The two parameters, <img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0002.png" alt="$C_ s$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> and <img src="/MI/399da7830b87417dce9332882bbc0601/images/img-0007.png" alt="$C_ r$" style="vertical-align:-2px;
width:18px;
height:13px" class="math gen" /> can be tuned to change the dynamical behaviour of the model. </p>
<p>Now imagine this dynamical system running separately in each individual cell of a 2D CA. In other words, each cell in the CA now contains two variables: <img src="/MI/6ac202163f722fd408afafde0cfbc331/images/img-0001.png" alt="$S$" style="vertical-align:0px;
width:10px;
height:11px" class="math gen" /> and <img src="/MI/6ac202163f722fd408afafde0cfbc331/images/img-0002.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" />. However, instead of each cell updating its variables in isolation, there is a coupling between neighbouring cells, as there needs to be for the system to form a CA. </p>
<p>Write <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0001.png" alt="$R_ i$" style="vertical-align:-2px;
width:17px;
height:13px" class="math gen" /> for the release proportion of a given cell at the current time step. First, calculate the average of the <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0002.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> values of the eight neighbours of the given cell. Let’s write <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0003.png" alt="$R_ A$" style="vertical-align:-2px;
width:23px;
height:13px" class="math gen" /> for this average. Next, for a fixed value <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0004.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> between 0 and 1 calculate </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0005.png" alt="\[ \bar{R_ i} = (1-p)R_ i+pR_ A. \]" style="width:164px;
height:20px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Now, each time the cell calculates its new values <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0006.png" alt="$S^{\prime }$" style="vertical-align:0px;
width:14px;
height:13px" class="math gen" /> and <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0007.png" alt="$R^{\prime }$" style="vertical-align:0px;
width:16px;
height:13px" class="math gen" />, using the ISM formulas given above, its current <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0008.png" alt="$\bar{R_ i}$" style="vertical-align:-2px;
width:17px;
height:18px" class="math gen" /> value is used instead of <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0001.png" alt="$R_ i$" style="vertical-align:-2px;
width:17px;
height:13px" class="math gen" />. This way, the dynamics of each cell is coupled to that of its direct neighbours. </p><p>The additional parameter <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0004.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> (between 0 and 1) determines the strength of this coupling. If <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0009.png" alt="$p=0$" style="vertical-align:-3px;
width:39px;
height:15px" class="math gen" /> there is no influence of neighbouring cells on the new values <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0006.png" alt="$S^{\prime }$" style="vertical-align:0px;
width:14px;
height:13px" class="math gen" /> and <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0007.png" alt="$R^{\prime }$" style="vertical-align:0px;
width:16px;
height:13px" class="math gen" /> in a given cell. When <img src="/MI/ba0bdac2daa09655464bb0333842db66/images/img-0010.png" alt="$p=1$" style="vertical-align:-3px;
width:38px;
height:15px" class="math gen" /> there is complete influence. </p>
</div>
<p>To generate spiral waves with cellular automata, we need to use a somewhat more elaborate version of a CA. First, a two-dimensional space is needed. This means that the CA now consists of a 2D regular grid, where the new value of each cell is determined by the cell itself and its eight directly surrounding neighbours. </p>
<p>Next, instead of just using the two possible values 0 and 1, a continuous range of values is needed, for example the real numbers between 0 and 2. Finally, a more sophisticated update formula is needed.</p>
<p>The particular update formula we will use here is based on the so-called <em>Ideal Storage Model</em> (ISM). Imagine that each cell in our 2D grid is a container that contains a certain amount of a substance. At regular time intervals a certain proportion of the substance disappears from the cell, and a fixed amount is replenished. The proportion that disappears depends on the amount of substance that disappeared from the cell at the previous time step, but it also depends on the proportion that disappeared from the eight neighbouring cells at the previous time step (see the box for the precise rule). In this way, the amount of the substance in each cell varies over time in a way that depends on its eight neighbours and the past. The amount is represented by a number between 0 and 2, which in turn can be represented by a colour from black (for a zero amount) to bright red (for an amount value equal to 2). This embedding of the ISM in a 2D CA
is known as the <em>Ideal Storage Cellular Automaton Model</em> (ISCAM).</p>
<p>The ISCAM is capable of generating spontaneous spiral waves very similar to the ones observed in our two examples at the beginning of this article, the BZ reaction and dictyostelium. The figure below shows several selected time steps from one particular run of the ISCAM on a 100x100 grid, starting from a purely random initial configuration, and with parameter values <img src="/MI/e7154e259faf3c4bd4ed66f5b525e7d9/images/img-0001.png" alt="$C_ s=0.5$" style="vertical-align:-2px;
width:62px;
height:15px" class="math gen" />, <img src="/MI/e7154e259faf3c4bd4ed66f5b525e7d9/images/img-0002.png" alt="$C_ r=2.5$" style="vertical-align:-2px;
width:62px;
height:15px" class="math gen" />, and <img src="/MI/e7154e259faf3c4bd4ed66f5b525e7d9/images/img-0003.png" alt="$p=0.2$" style="vertical-align:-3px;
width:52px;
height:15px" class="math gen" /> (see the box above to see how these parameters influence the ISCAM). The different shades of red represent the amount <img src="/MI/e7154e259faf3c4bd4ed66f5b525e7d9/images/img-0004.png" alt="$S$" style="vertical-align:0px;
width:10px;
height:11px" class="math gen" /> of the substance in each cell, with black denoting <img src="/MI/e7154e259faf3c4bd4ed66f5b525e7d9/images/img-0005.png" alt="$S=0$" style="vertical-align:0px;
width:40px;
height:12px" class="math gen" />, bright red <img src="/MI/e7154e259faf3c4bd4ed66f5b525e7d9/images/img-0006.png" alt="$S=2$" style="vertical-align:0px;
width:40px;
height:12px" class="math gen" />. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/spirals/sequence-1.jpg" alt="An ISCAM" width="539" height="420" />
<p style="max-width: 539px;">One particular sequence of global patterns generated by the ISCAM in a regular grid of 100x100 cells. The system starts (at <em>t</em>=0) with completely random values in the cells. These values are colour-coded with a gradient from black (<em>S</em>=0) to bright red (<em>S</em>=2). After 300 time steps (<em>t</em>=300), several "nucleation sites" start to appear, which then grow into real spirals (<em>t</em>=1000) that start to interact (<em>t</em>=2000) and compete (<em>t</em>=3500) with each other, until eventually one spiral dominates the system (<em>t</em>=6000).
</p>
</div>
<!-- image provided by author -->
<p>There is a small problem when working out the behaviour of the ISCAM on a finite grid: in order to work out what is happening in the cells on the edge of the grid, we need to know what is happening in the cells that are their neighbours, but which aren't on the grid and for which we haven't got any values. To get around this problem, we simply wrap the grid around so that the cells in the leftmost column and the rightmost column are considered each other's neighbours, and similarly for the top and bottom rows. In other words, the CA grid actually exists on a doughnut shape (or <em>torus</em>) which is what you get when you glue together opposing sides of a square.
</p>
<p>The one-minute movie below shows a similar run of the ISCAM in real time, using the same parameter values as in the figure above. Click the "play" button to start the movie.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/-WGB2RaGVfA" frameborder="0" allowfullscreen></iframe>
<p>It turns out that these spirals are remarkably robust. For example, when a perturbation ("noise") is introduced in a spiral by resetting the <img src="/MI/b58c1322994bda064d79b9950b0f3465/images/img-0001.png" alt="$S$" style="vertical-align:0px;
width:10px;
height:11px" class="math gen" /> values in a 3x3 block of cells to zero (black), and the system is then allowed to proceed, the spiral will simply continue to go around, quickly removing the perturbation, seemingly as if nothing ever happened. This process is shown in the figure below on a 50x50 grid. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/spirals/noise-1.jpg" alt="A perturbation" width="539" height="155" />
<p style="max-width: 539px;">Noise (the black square) is introduced into the system at a given time <em>t</em>. After nine time steps (<em>t</em>+9) the spiral has gone around once, and the noise is reduced to a vague smudge. After 90 time steps (<em>t</em>+90) everything looks just as before.</p>
</div>
<!-- image provided by author -->
<p>Natural systems tend to produce many beautiful and often complicated patterns. However, as the above example shows, the underlying mechanisms and mathematical principles can be surprisingly simple. Using a basic mathematical model (a cellular automaton with an update formula based on the ISM), spontaneous spirals in spatial systems cannot only be accurately reproduced, but also studied and understood in more detail. This certainly makes for some interesting psychedelic science!</p>
<hr/>
<h3>Acknowledgements</h3>
<p>The author would like to thank his former colleagues Andreas Dress, Lin Wei, and Peter Serocka (at the Partner Institute for Computational Biology, Chinese Academy of Sciences, and the Centre for Computational Systems Biology, Fudan University, Shanghai), with whom the ISCAM was originally developed.</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 a computer scientist currently on a fellowship at the Konrad Lorenz Institute in Klosterneuburg, Austria. 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>Thu, 22 Dec 2016 11:04:26 +0000mf3446758 at https://plus.maths.org/contenthttps://plus.maths.org/content/spontaneous-spirals#commentsHappy birthday Ramanujan!
https://plus.maths.org/content/happy-birthday-ramanujan-0
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="width: 267px;"><img src="/content/sites/plus.maths.org/files/blog/122014/ramanujan.png" alt="Ramanujan" width="267" height="325"/>
<p>Srinivasa Ramanujan (1887 - 1920).</p>
</div>
<p>December 22nd would have been the 129th birthday of the legendary Indian mathematician <a
href="http://www-history.mcs.st-and.ac.uk/Biographies/Ramanujan.html">Srinivasa
Ramanujan</a>, who recently achieved wider fame through the film <em><a href="https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity_(film)">The man who knew infinity</a></em>. His story really is remarkable. Born in 1887 in a small village around 400km
from Madras (now Chennai), Ramanujan developed a passion for maths
very early on. By age 15 he routinely solved maths problems
that went way beyond what his classmates were dealing with. He worked out his own method for solving quartic equations, for example, and even had a go at quintic ones (and failed of course, since <a href="/content/stubborn-equations">the general quintic is unsolvable</a>). But since he neglected all other
subjects apart from maths, Ramanujan never got into university, and was forced to continue
studying maths alone and in poverty. Only after a plea to an eminent mathematician, who described Ramanujan as "A short uncouth figure, stout, unshaven, not over clean," did Ramanujan eventually get a job as a clerk at the Madras Port Trust.</p>
<p>It was during his time at the Port Trust that Ramanujan decided to write a letter that was to change his
life. It was addressed to the famous Cambridge number theorist G. H. Hardy who, accustomed to this early-twentieth-century form of spam, was irritated at first: a letter from an unknown Indian containing crazy-looking theorems and no proofs at all. But as he went about his day, Hardy couldn't quite forget about the script:</p>
<div style="color: 80a096; margin-left: 20px; margin-right: 20px;">
<p><em> At the back of his mind [...] the Indian manuscript nagged away. Wild theorems. Theorems such as he had never seen before, nor imagined. A fraud of genius? A question was forming itself in his mind. As it was Hardy's mind,
the question was forming itself with epigrammatic clarity: is a fraud of genius more probable than an unknown mathematician of genius? Clearly the answer was no. Back in his rooms in Trinity, he had another look at the script. He sent word to Littlewood that they must have a discussion after hall...</em></p>
<p><em>Apparently it did not take them long. Before midnight they knew, and knew for certain. The writer of these manuscripts was a man of genius.</em></p>
</div>
<div style="colour: 80a096;width: 50%; float: right; font-size: small;">
<p>From the foreword by C. P. Snow to Hardy's <em>A Mathematician's Apology</em></p>
</div>
<br clear="all" />
<p>Hardy invited Ramanujan to
Cambridge, and on March 17, 1914 Ramanujan set sail for England to start one of the most fascinating
collaborations in the history of maths. Right from the start the pair
produced important results and Ramanujan made up for the gaps in his
formal maths education by taking a degree in Cambridge. Perhaps the most famous story to emerge from this period has Hardy visiting Ramanujan as he lay ill in bed. Hardy complained that the number of the taxi he had arrived in, 1729, was a boring number, and that he worried this was a bad omen. "No," Ramanujan replied, apparently without hesitation. "It is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways":</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/df5e19b189000dcd20e0cd15b2df5661/images/img-0001.png" alt="\[ 1729 = 1^3 + 12^3 = 9^3 + 10^3. \]" style="width:201px;
height:17px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>Unfortunately, Ramanujan's sickness wasn't a one-off. His health had always been feeble, and the cold weather and unaccustomed English food didn't help. Ramanujan decided to return to India in
1919 and died the following year, aged only 33. He is still celebrated as one of India's greatest mathematicians.</p>
<p>You can find out more about Ramanujan's mathematics in these <em>Plus</em> articles:</p>
<ul><li><a href="/content/ramanujan">Ramanujan surprises again</a></li>
<li><a href="/content/disappearing-number">A disappearing number</a> (accompanied by a <a href="/podcasts/PlusPodcastFeb09.mp3">podcast</a>)</li>
<li><a href="/content/conversation-manjul-bhargava">Numbers, toys and music: A conversation with Manjul Bhargava</a></li>
<li><a href="/content/chaos-numberland-secret-life-continued-fractions">Chaos in numberland</a></li>
</ul>
<p>We have also recently <a href="/content/node/6725">reviewed the famous essay <em>A mathematician's apology</em></a> by Ramanujan's collaborator G.H. Hardy.</p></div></div></div><div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon-5.jpeg" width="100" height="100" alt="" /></div></div></div>Thu, 22 Dec 2016 09:24:08 +0000mf3446767 at https://plus.maths.org/contenthttps://plus.maths.org/content/happy-birthday-ramanujan-0#commentsPlus Advent Calendar Door #21: Combinatorics
https://plus.maths.org/content/plus-advent-calendar-door-21-combinatorics
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_21.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>How likely are you to win the lottery?</p>
<p>In the UK lottery you have to choose 6 numbers out of 49, and for a chance at the jackpot you need all of your 6 numbers to come up in the main draw. So the question is really how many possible combinations of 6 numbers can be drawn out of 49? There are 49 possibilities for the first number, 48 for the second, and so on to 44 possibilities for the sixth number, so there are 49 x 48 x 47 x 46 x 45 x 44 = 10068347520 ways of choosing those six numbers... in that order. But we don't care which order our numbers are picked, and the number of different ways of picking 6 numbers are 6 x 5 x 4 x 3 x 2 x 1 = 720</i>. Therefore our six numbers are one of 49 x 48 x 47 x 46 x 45 x 44 / 720 = 13983816 so we have about a one in 14 million chance of hitting the jackpot.
Hmmm...</p>
<p>But on a brighter note, we have just discovered a very useful mathematical fact: the number of combinations of size k (sets of objects in which order doesn't matter) from a larger set of size n is <em>n! / (n-k)! / k!</em>,
where <em>n!</em> stands for <em>n</em> x (<em>n</em> - 1) x (<em>n</em> - 2) x ... x 2 x 1. </p>
<p>This sort of argument lies at the heart of <em>combinatorics</em>, the mathematics of counting. It might not help you win lotto, but it might keep you healthy. It is used to understand how viruses such as influenza reproduce and mutate, by assessing the chances of creating viable viruses from random recombination of genetic segments.</p>
<p>You can read more on combinatorics, including money (lotto), love (well kissing frogs) and fun (juggling and rubiks cubes) on <a href="https://plus.maths.org/content/tags/combinatorics"><em>Plus</em></a>.</p>
<p><a href="/content/2016-plus-advent-calendar">Return to the <em>Plus</em> advent calendar 2016</a>.</p>
<p>This article comes from our <em><a href="/content/maths-in-a-minute">Maths in a minute library</a></em>.</p>
</div></div></div>Wed, 21 Dec 2016 14:42:12 +0000mf3446762 at https://plus.maths.org/contenthttps://plus.maths.org/content/plus-advent-calendar-door-21-combinatorics#comments