Plus Blog

March 17, 2015

Were you stuck in traffic on your way to work or school this morning? Then you might take consolation in the thought of yourself as a tiny experimental subject in the giant petri dish of the UK's motorways, contributing your data points to a greater understanding of traffic flow.

Traffic heat map

Our image of the week shows a heat map of the traffic on a clockwise section of the M25 (junction 9 to junction 14) showing the speed of cars as a colour, from pale yellow for around 80 mph to red for stationary, as they pass along this stretch of road on a typical Monday. The image was created by Richard Gibbons, from the University of Cambridge's Computer Laboratory. It is based on data collected by the Highways Agency from MIDAS (the Motorway Incident Detection and Automatic Signalling) system, archived minute by minute since 1997 at loop detectors every few hundred metres under many UK motorways.

The diagonal red stripes streaking down the image from the left to right between 7am and 11am are driver's commuting pain manifested in data. A driver's path through the data will start at the bottom and move upwards as they passed along the motorway and to the right as time passed. And if they passed junction 9 at, say, 7.30 in the morning, they would have hit red stripe after red stripe, like striking the incoming wave fronts as you try to swim out from a beach.

These stripes are the wave fronts of a stop and go wave, the infuriating pattern of repeated stopping (red) and starting (yellow) that is well known to both motorists and traffic modellers. This is a kind of shock wave that moves back through the traffic, opposite to the direction the cars are travelling. This can be easily seen in the heat map: one starts around junction 12 and moving backwards down the motorway as time goes on.

Click here to see previous images of the week.

March 14, 2015

Today, written as 3/14 the American way, is $\pi $ day! This special day happens every year, but today is extra special! It’s 3/14/15, giving us the first five digits of that lovely number, rather than just three! But why should we care about all these digits?

If you had a sloppy maths teacher at school you might have grown up with the idea that the number $\pi $ is equal to $22/7.$ Now that is completely wrong. Writing those numbers out in decimal gives $22/7 = 3.142...$ while $\pi = 3.141...$. There’s a difference in the third decimal place after the decimal point!


How accurately do we need to know the value of π?

Surely this small inaccuracy doesn't matter? Well, as the following extract from a longer article by Chris Budd shows, it really does.

The point is that $\pi $ is not any number. It lies at the heart of any technology that involves rotation or waves, and that is much of mechanical and electrical engineering. If rotating parts in, say, a typical jet engine are not manufactured to high tolerance, then the parts simply won’t rotate. This typically involves measurements correct to one part in 10,000 and, as these measurements involve $\pi $, we require a value of $\pi $ to at least this order of accuracy to prevent errors. In medical imaging using CAT or MRI scanners, the scanning devices move on a ring which has to be manufactured to a tolerance of one part in 1,000,000, requiring an even more precise value of $\pi $.

However, even this level of accuracy pales into insignificance when we look at modern electrical devices. In high frequency electronics, with frequencies in the order of 1GHz (typical for mobile phones or GPS applications), electrical engineers require a precision in the value used for $\pi $ in the order of one part in 1,000,000,000,000,000.

So, the modern world needs $\pi $ and it needs it accurately!

March 10, 2015

Maryam Mirzkhani, the first female Fields medallist

We may be a little late in celebrating International Women's Day but it's not for a shortage of excellent female mathematicians to celebrate! Here are some of the most inspiring women we've worked with or learnt about in the last year...

Maryam Mirzakhani is one of the best mathematicians in the world. Last August she received the Fields medal (one of the highest accolades in mathematics), recognised for her "rare combination of superb technical ability, bold ambition, far-reaching vision, and deep curiosity". She was the first woman to be awarded this prize and we were lucky enough to meet Mirzakhani and learn about her fascinating work in topology at the International Congress of Mathematicians (ICM) in Seoul last Summer.

Ingrid Daubechies

Ingrid Daubechies at the opening ceremony of the International Congress of Mathematicians in 2014. Image copyright ⓒ 2010-2014 International Congress of Mathematicians 2014, all rights reserved.

At the ICM we were also fortunate to meet Ingrid Daubechies, President of the International Mathematical Union. As well as talking to her about the importance of a mathematical community, we heard her speak about her recent work virtually restoring paintings, just one application of wavelet theory. Famously it has also been used by the FBI to digitise finger prints, and it is widely used in medical imaging. Debauchies made fundamental discoveries in wavelet theory that opened up the field and has had an important role in making this mathematics into a practical tool for analysis in other areas outside mathematics. (You can read more about wavelets and how Debauchies work opened up this field here.)

We also found out about a fantastic mathematician, Margharita Piazolla Beloch, who discovered a new type of origami fold (the sixth axioms of origami) in 1936 and proved that this fold solves cubic equations. We'll write more about Beloch in upcoming articles, but it's thanks to her discovery that we can now solve the unsolvable and trisect an angle.

And finally, one of the people we've really enjoyed working with this last year (and years before that!) is Vicky Neale, our former colleague and now Whitehead Lecturer at the Mathematical Institute in Oxford. As well as learning many fascinating things working with Vicky on a project producing new and exciting resources for sixth form students, we've also really enjoyed listening to Vicky's contributions to many radio shows and her popular lectures and workshops. You can listen to many of her recordings online, the details are here.

February 26, 2015

If you feel in need of some love this cold, dark February, then our images of the week might be just what you need. These hearts, created by Hamid Naderi Yeganeh, aren't drawn by hand, but defined by mathematical equations.

Image by Hamid Naderi Yeganeh.

Image by Hamid Naderi Yeganeh.

The first curve consists of points in the plane whose coordinates $(x,y)$ satisfy

  \[ x=\frac{48}{25}\sin (5t)+\frac{40}{25}\sin (6t) \]    
  \[ y=\frac{48}{25}\cos (4t)+\frac{40}{25}\cos (5t) \]    

for $\Theta \leq t \leq 2\pi -\Theta ,$ where $\Theta \approx 2.32.$ ($\Theta $ is a zero of $\frac{48}{25}\sin (5t)+\frac{40}{25}\sin (6t)$.)

The second curve consists of points in the plane whose coordinates $(x,y)$ satisfy

  \[ x=-(\sin (3t))^{3} \]    
  \[ y=\sin \left(4t+\frac{7\pi }{36}\right)-\frac{\sin \left(\frac{7\pi }{36}\right)-\sin \left(\frac{103\pi }{36}\right)}{2}(\cos (\frac{3t}{2}))^{5} \]    

for $0 \leq t \leq \frac{2\pi }{3}$.

You can see more of Hamid's images on this website, in The Guardian and on the American Mathematical Society website.

Click here to see previous images of the week.

February 25, 2015

Science festival

The Cambridge Science Festival will kick off on March 9, and there is some great maths in the programme. Here are some maths-related events we are particularly excited about (in chronological order). Click on the links for price, booking and location information. For the full programme visit the festival website. And if you can't make it to Cambridge, then read the Plus articles by some of the speakers, linked to from the individual entries.

  • Colour, new dimensions and the geometry of physics (March 10, 5pm - 6pm) — Nobel laureate Frank Wilczek talks about his work on finding a theory that explains all the fundamental forces and particles we see, and his popular book The lightness of being. To find out more about Wilczek's fascinating research, read the Plus article Strong but free.
  • Order out of chaos (March 11, 6pm - 7pm) — Imre Leader explores how order can be found in chaos, as long as there is enough of it. Leader is a great speaker and the topic's very accessible. Find out more in this video or this Plus article.
  • Let Newton be! (March 11, 7:30pm - 9:30pm public performance, 2:30pm - 4:30pm schools performance) — This play brings the complex and controversial character of Sir Isaac Newton to life: a devout, difficult, obsessive man who sought and found god in universal laws of light and motion. Theatrical, entertaining and informative, the play explores the life and thought of a genius whose scientific theories still provide the foundations for our understanding of the Universe today.
  • Some essential links between maths and the arts (March 12, 5:30pm - 6:30pm) — John D. Barrow explores a range of links between mathematics and some of the arts; including Dali's use of 4D geometry, how fractals are used to distinguish abstract art works, the plan of the subterranean Tunnel of Eupalinos in 520BC, and how smooth curves inform Henry Moore's stringed sculptures. For a taster, read Barrow's Plus article Where to stand to look at sculptures.
  • Enigma and the secret world of code breaking (March 14, 10:30am - 11:30am) — James Grime looks at the fascinating history and mathematics of codes and code breaking - from ancient Greece to the present day - including a demonstration of an original WWII Enigma Machine! For a brief introduction, see the Plus article Exploring the Enigma.
  • Don't believe this talk: maths that can't be true! (March 14, 2pm - 3pm) — Steve Mould looks at the maths that confounds our expectations and laughs in the face of our intuition. Find out how many numbers in the universe start with a 1, how to cheat on your homework and other mathematical surprises. (This event is fully booked but limited tickets are available on the day.)
  • Happy birthday, Fermat's last theorem (March 14, 3:30pm - 4:30pm) — Simon Singh, author of a book and director of a BBC documentary about Fermat's Last Theorem, celebrates the fact that it has been exactly twenty years since a proof was published. He will discuss the origin of the problem, describe the people who tried and failed to prove it, and tell the story of Professor Andrew Wiles, who conquered Fermat's challenge after working in secret for seven years. You can prepare yourself by reading about Fermat's last theorem on Plus. (This event is fully booked but limited tickets are available on the day.)
  • Illuminating statistics (March 14, 10am - 4pm) — The modern world is full of numbers. It is more important than ever that we know how to ask, answer and understand questions about public health in this sea of numbers and data. Come join the MRC Biostatistics Unit and take part in activities that will show how and what we can learn from numbers. For a brief intro, listen to our interview with Sheila Bird from the Biostatistics unit.
  • Seeing further than others: Isaac Newton's world of light and colour (March 14, 11am - 12:30pm) — Rob Iliffe, Director of the Isaac Newton Papers Project in the University of Sussex, talks about Newton's work on light and his great publication Opticks. The talk will be accompanied by a display of Newton's manuscripts from the University Library's treasured collections.
  • Stamping through mathematics (March 16, 6pm - 7pm) — Robin Wilson explores the entire history of mathematics in one hour, as illustrated by around 300 postage stamps featuring mathematics and mathematicians from across the world. From Euclid to Euler, from Pythagoras to Poincaré, and from Fibonacci to the Fields Medals, all are featured in attractive, charming and sometimes bizarre stamps. No particular knowledge of mathematics or philately required.
  • Sex by numbers: statistics of our intimate lives (March 18, 7:30pm - 8:30pm) — The latest survey of British sexual behaviour suggests that we are becoming more experimental in our sex lives, but there is less of it going on. David Spiegelhalter will look at sex statistics going back to 1580, show that more boys are born at the end of wars, and argue that this means sexual activity was at a historical minimum in 1898. (You can read David Spiegelhalter's column on Plus.)
  • El Niño: what on earth will happen next? (March 19, 5:50pm - 6:30pm) — El Niño events are the largest causes of year-to-year climate variability on a global scale, bringing floods to some regions and droughts to others. Join Michael Davey as he explains the phenomenon, explores the impacts, and describes how maths helps us understand how they occur and evolve.
  • Einstein's legacy: 100 years of general relativity (March 19, 7:45pm - 9pm) — 2015 is the 100th anniversary of Albert Einstein's full formulation of the theory of general relativity. This research fundamentally changed our concepts of space and time. John D Barrow and Ulrich Sperhake from the University of Cambridge and Michael Kramer from Max Planck Institute for Radio Astronomy and University of Manchester will discuss Einstein's legacy. You can read more about Einstein and his theories on Plus. (This event is fully booked but limited tickets are available on the day.)
  • Maths public open day (March 21, 12noon - 4pm) — Come and visit the home of Plus! From Isaac Newton onwards, Cambridge has been associated with some of the most famous mathematicians in history. Modern mathematicians and theoretical physicists work on everything from the Big Bang to prime numbers, fluid dynamics or investigating the spread of disease. Join the Plus editors, as well as students and staff from the Faculty of Mathematics, to explore the excitement of mathematics and theoretical physics through hands on activities, demonstrations and displays.
  • Thinking mathematically (March 21, 12:15pm - 1:15pm) — Join Charlie Gilderdale to work on some of his favourite mathematical problems from our sister site NRICH, and discover that everyone can think mathematically. Come prepared to explore, discuss, conjecture, question, explain and generalise!
  • The Large Hadron Collider and the dark matter mystery (March 21, 2pm - 3pm) — The Large Hadron Collider will start operation again at a higher energy at the beginning of 2015. Join Ben Allanach for an introduction to the machine, particle physics, the discovery of the Higgs boson, and what its observations tell us about the mysterious dark matter. For a brief intro, read Ben Allanach's Plus article Particle hunting at the LHC.
  • Why are stringed instruments so hard to play? (March 21, 6:30pm - 9:45pm) — The motion of a bowed violin string has been studied since the 19th century, and shows some quite unexpected behaviour. This talk by Jim Woodhouse, University of Cambridge Department of Engineering, will show some examples of how computer modelling combined with acoustic measurements can shed light on playability of bowed-string instruments. This talk is followed by a concert of works by Mussorgsky, Elgar and Tchaikovsky including the cello concerto by Elgar with soloist Olivia da Costa. To prepare, read Jim Woodhouse's Plus article Why is the violin so hard to play?.
February 9, 2015

This short video features one of our favourite mathematicians, Corinna Ulcigrai from the University of Bristol, talking about mathematical billiards, its connection to chaos theory, and why mathematicians study it. You can also read our article Chaos on the billiard table, based on an interview with Ulcigrai.

This video was originally produced for the Cambridge Mathematics Education Project.

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