How do you judge the risks and benefits of new medical treatments, or of lifestyle choices? With a finite health care budget, how do you decide which treatments should be made freely available on the NHS? Historically, decisions like these have been made on the basis of doctors' individual experiences with how these treatments perform, but over recent decades the approach to answering these questions has become increasingly rational. Statistics and maths are used not just to test new treatments, but also to measure such fuzzy terms as quality of life, and to figure out which treatments provide most "health for money".
While the decisions of health authorities affect all our lives, the underlying calculations are rarely discussed in the media. To explore these difficult decisions and the role of maths in evidence-based medicine, we have put together a package of six articles, three podcasts, a career interview and a classroom activity.
Everyone has the chance to create mathematical beauty as part of a competition during the Cambridge Science Festival. As part of the Imaginary exhibition of beautiful mathematical images and artwork taken from algebraic
geometry and differential geometry, visitors (both real and virtual) can create their own mathematical art.
By downloading the SURFER program, anyone can create images of algebraic surfaces by simple equations using the three spatial coordinates of x, y and z. For example, the equation x2 + y2 + z2 = 1 results in a sphere.
The competition requires creativity, intuition and mathematical skill in order to create equations yourself or to change given equations to produce beautiful images. The images are easily generated with the SURFER programme, and you can then upload your artwork to the competition gallery by 20 March. Everybody is invited to take part, including group entries from classes and families. The
entries will be judged by a distinguished panel including Sir Christopher Frayling (Royal College of Art and Arts Council England) and Conrad Shawcross (sculptor and artist-in-residence at the Science Museum, London).
So good luck to all aspiring artists, and if you need some inspiration why not browse through the Plus articles on maths and art.
If you have ever wondered what it feels like to do mathematics, take a look at the series of beautiful short films produced by the mathematics department at the University of Bristol. Chrystal Cherniwchan, Azita Ghassemi and Jon Keating interviewed over 60 mathematicians, asking them to describe the emotional aspects of maths research.
The discussions range from the role of creativity and beauty in maths, to what it feels like to pursue the wrong research path, and the eureka moment of discovering mathematical truth. You can view them all on the Mathematical Ethnographies site.
Marcus du Sautoy is a mathematician and Charles Simonyi Professor for the Public Understanding of Science. In this TED talk he explores how the world turns on symmetry — from the spin of subatomic particles to the dizzying beauty of an arabesque — complete with an introduction to groups.
Marcus du Sautoy has also written several articles for Plus:
Achilles and a tortoise are competing in a 100m sprint. Confident in his victory, Archilles lets the tortoise start 10m ahead of him. The race starts, Achilles zooms off and the tortoise starts bumbling along. When Achilles has reached the point A from where the tortoise started, it has crawled along by a small distance to point B. In a flash Achilles reaches B, but the tortoise is already at
point C. When he reaches C, the tortoise is at D. When he's at D, the tortoise is at E. And so on. He's never going to catch up with the tortoise, so he has no chance of winning the race.
Something's wrong here, but what? Let's assume that Achilles is ten times faster than the tortoise and that both are moving at constant speed. In the times it takes Achilles to travel the first 10m to point A, the tortoise, being ten times slower, has only moved by 1m to point B. By the time Achilles has travelled 1m to point B, the tortoise has crawled along by 0.1m to point C. And so on.
After n such steps the tortoise has travelled
1+1/10+1/100+1/1000+ .... +1/10(n-1) metres.
And this is where the flaw of the argument lies. The tortoise will never cover the 90m it has to run using steps like these, no matter how many of them it takes. In fact, the distance covered in this way will never exceed 10/9=1.111... metres. This is because the geometric progression
converges to 10/9. Since the tortoise is travelling at constant speed, it covers this distance in a finite time, and it's precisely when it's done that that Achilles overtakes it.
This problem is known as one of Zeno's paradoxes, after the ancient Greek philosopher Zeno, who used paradoxes like this one to argue that motion is just an illusion.
A central prediction of Albert Einstein's general theory of relativity is that gravity makes clocks tick more slowly — time passes slower when you're close to a massive body like the Earth, compared to when you're further away from it where its gravitational pull is weaker. This prediction has already been confirmed in experiments using airplanes and rockets, but a new experiment in an atom
interferometer measures the slowdown 10,000 times more accurately than before — and finds it to be exactly what Einstein predicted.
The longest interval of time for some process (eg a heart beat or a human lifetime) is that measured by a clock moving with the observer. It is
called proper time. The length of that interval measured by some other clock in relative motion to that observer is always less than the proper time and as the relative speed approaches that of light, it tends to zero.
The twin paradox (see http://plus.maths.org/issue36/features/aiden) is related. If a twin stays at home and lives for 10 years on his watch while his identical twin goes off on a spacetrip at near light speed, then the travelling twin will return to find that he is
younger than his stay-at-home twin when he is reunited with him on earth.
So the maximal time is given by the proper time measured by a clock moving with you, and the minimum time can be arbitrarily small as the relative speed approaches that of light, or the gravitational field approaches the value needed to
make a black hole.
There is no such thing as the General Theory Of Relativity. Einstein was successful in developing the Special Theory Of Relativity which seems to describe much of what we know about our own universe in accord with conditions that he defined in the theory. He was unsuccessful in deriving the General Theory Of Relativity which, if ever derived, would apply under any set Of conditions. The
Special Theory Of Relativity is to the Generla Theory Of Relativity as a Square is to a Polygon.