Image produced by R.R. Hogan, University of Cambridge.
This pretty picture looks like something you'd see through a kaleidoscope, but it's more than a bit of fun. It's an image of a two-dimensional crystal — but not as you would see it if you looked at it. Instead, it shows the Brillouin zones of the crystal, which give important information about how waves, such as light or X-rays, pass through it.
Alex Bellos, one of our favourite maths authors, recently conducted a survey to find the world's favourite number. After polling more than 30,000 people from around the world, he found that the winner was ... 7!
We rather like 7 ourselves and decided that this is because 7 is the first prime number that really "feels" like a prime number, and because it's the most common result you get from throwing a couple of dice. Alex has come up with a different reason — see the video below. To find out more about the poll, see favouritenumber.net or read Alex's new book Alex through the looking glass (reviewed in Plus).
This image appears to be a procession of elephants but is, in fact, a much-magnified small detail of one of the Mandelbrot set. The Mandelbrot set is a famous example of a fractal – mathematical objects whose structure is infinitely complex. Whether you're viewing them from afar or zooming in on them with a mathematical microscope, the same complexity is always visible. This self-similarity even extends to the some of the same structures repeating at all scales.
This unending complexity means that fractals live between dimensions. For example, there are shapes that are so crinkly, they are "more" than a one-dimensional curve, but not extensive enough to give a two-dimensional shape like a disc or square. Instead, they have a fractional, or fractal, dimension between 1 and 2, which is why Benoît Mandelbrot, the father of fractals, named them so.
Fractals famously appear in nature, from snowflakes to coastlines, and also have revolutionised mathematics by inspiring the field of chaos theory which is used in weather prediction and stock market analysis.
Five mathematicians turned into millionaires this week when they were awarded the Breakthrough Prize set up by Facebook founder Mark Zuckerberg and internet entrepreneur Yuri Milner. Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terry Tao and Richard Taylor will each receive £1.8m. That's more than the £930,000 that's awarded for the Nobel prize and should make up for the fact neither of the five is likely to ever to receive a Nobel: there isn't one for mathematics.
It's the first time this prize has been awarded to mathematicians. It joins a string of other prizes Milner has set up to reward physicists and life scientists. The idea is to give mathematicians (and other scientists) the "rock star status" they deserve. "We think scientists should be much better appreciated. They should be modern celebrities, alongside athletes and entertainers," Milner told the Guardian. "We want young people to get more excited. Maybe they will think of choosing a scientific path as opposed to other endeavours if we collectively celebrate them more."
The five mathematicians that have been chosen certainly made big splashes in the world of mathematics. You may have come across Richard Taylor and Terence Tao on Plus before: Taylor (among other things) helped Andrew Wiles solve a famous 350-year-old problem called Fermat's last theorem. Tao was one of the youngest people to ever receive the Fields medal (another prestigious maths prize) in 2006 and has delivered a range of ground breaking work, including on prime numbers.
The other three are equally eminent. Simon Donaldson (with whom we had the pleasure of working on a forthcoming series of articles) has provided deep insight into geometrical structures inspired by theoretical physics. Lurie, at 36 the youngest of the five, has also been honoured for work on the mathematical end of physics, as well as results that lie on the interface between geometry and topology on the one hand and algebra on the other. Kontsevich is also a theoretical physicist, but has been awarded the prize for an astounding range of contributions to pure maths.
Million dollar prizes are always a contentious issue. The money spent on a few individuals could be used to fund maths education at home or in developing countries, help support young researchers or those disadvantaged in some way, or fund important research projects. Milner and Zuckerberg's reply will no doubt be that a hefty prize tag, even if slightly unseemly,
raises awareness and sends a clear message. Maths is essential, even the sort of pure maths that some of the laureates are being honoured for, which may at first sight seem useless. In a world of short attention spans money talks.
UK mathematicians seem to have responded to the prize with mixed feelings. One unnamed mathematician cast some doubt on the prize's aim when he told the Guardian, "I find it interesting that they think it's possible to make rock stars out of people who do something that 99% of the population have no hope of understanding, and I include most professional mathematicians in that." This is something we don't agree with. While there is not much hope of explaining a mathematician's work to a general audience in great detail, it is certainly possible to give the gist of an area, and communicate its excitement, attraction and motivation. You won't convince everyone, just as you won't convince everyone of the worth of a piece of art, but you will always enthuse a few and interest many. If this is something mathematicians find hard to do themselves, then perhaps a little chunk of those millions should go towards funding maths communication.
Monday 23 June 2014 would have been Alan Turing's 102nd birthday. One of the 20th century's great mathematicians, Turing made profound contributions to a wide range of fields including computer science, artificial intelligence and mathematical biology. He also played a crucial role in the Allied codebreaking work at Bletchley Park during World War II, credited by some historians with shortening the conflict by two years.
More recently, much public attention has focused on the controversy surrounding his 1952 conviction for gross indecency under the punitive anti-homosexuality laws in force in Britain at the time: an official pardon was issued by the British government in December 2013. Turing's life and work has also inspired several representations on stage and screen, including the acclaimed play and BBC drama Breaking the Code and a forthcoming film, The Imitation Game, starring Benedict Cumberbatch and Keira Knightley.
To celebrate Turing's birthday, learn more about his contributions to mathematics with these Plus articles, and watch the video below, produced by the University of Cambridge to mark Turing's centenary in 2012, in which James Grime (who works alongside Plus as part of the Millennium Mathematics Project) gives an overview of Turing's story.
Recently we had to learn about tensors for an upcoming article. What are those, you ask? We didn't know at first either. Like some other concepts in maths, they seem confusing at first but actually are just a way of capturing information we are all used to.
First of all, let's start with scalars. Scalars are just your ordinary, everyday, real numbers. A scalar field is used to describe something that has a particular value at every point in the space you are considering. For example, a map of temperatures across the UK, or indeed the world, is a scalar field; with a value for the temperature at each point on the map. You can read more about how scalar fields describe dark energy and the Higgs boson.
Then we get to the more dynamic concept of vector fields. A vector field is something that associates a vector (a magnitude and direction) to every point in space. Again, thanks to meteorology we have the familiar example of a wind map as a vector field. Vector fields are incredibly important in maths and physics and, like in our example of a wind map, usually describe how things move. You find out how fluid dynamics uses vector fields to model the movement of tears, wind and waves in Births and deaths in fluid chaos
A tensor field
A tensor just extends this definition to one where the value of some property depends on the direction in which you measure it. So where a vector is a magnitude and a particular direction from some point, a tensor gives a magnitude for every direction from that point. A tensor field is something that assigns a tensor to every point in space. Naturally it's harder to picture a tensor field but if you've ever played with a piece of chewing gum you've actually seen one in action. As you pull a piece of gum (or some other rubbery substance) between your fingers, it stretches in tension along one direction but compresses in the other perpendicular directions. So for each point in the gum the stress is a function of direction: in each direction the stress will take a certain value that is a combination of the contributions from these tension and compression stresses.
Tensors are incredibly useful tools, particularly when describing things in higher dimensions. The curvature of multidimensional surfaces (called manifolds) is described with tensors and Einstein used tensors to describe both the curvature and distribution of matter of four-dimensional space-time. You can read more about Einstein and the role of curvature on Plus.
So there you go. Tensors are nothing to get tense about!