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!
Congratulations to Plus contributors David Spiegelhalter and Helen Mason, who have been awarded a knighthood and OBE respectively in the 2014 Queen's Birthday Honours list! Joining famous faces including Dame Maggie Smith, Angelina Jolie and Daniel Day-Lewis, Helen Mason and David Spiegelhalter's honours have been awarded to recognise their outstanding work in mathematics, science and public communication.
Professor — now Sir — David Spiegelhalter is Winton Professor of the Public Understanding of Risk at the University of Cambridge, and has been knighted for "services to statistics". A medical statistician, he has played a leading role in developing simulation technologies and clinical trials on drug safety, and has supported the UK health service through many inquiries, including the public inquiry into children’s heart surgery at Bristol Royal Infirmary.
As the official government press release notes, and as regular readers of Plus will already know, Spiegelhalter has also 'made a significant difference to how to communicate with patients and the public about risk'. He started the brilliant project Understanding Uncertainty, writes public articles, lectures to public and school audiences and has appeared on TV and radio (ranging from being interviewed on the BBC's heavy-hitting Newsnight to appearing as a contestant in Winter Wipeout). Watch a lighthearted example of David Spiegelhalter's public understanding work below, in a video produced by the University of Cambridge, and read his articles on Plus.
Helen Mason leads the Atomic Astrophysics group in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. Helen Mason's extensive outreach and education work has included writing for Plus and setting up the Sun|Trek website for schools, as well as giving frequent talks and appearing on radio and TV.
Helen Mason's OBE has been awarded "for services to Higher Education and to Women in Science, Engineering and Technology". Learn more about her work in this video from the Royal Institution:
Today's Google doodle!
(Google and the Google logo are registered trademarks of Google Inc., used with permission.)
Oh Paul the psychic octopus… How we miss you and your uncannily accurate predictions of World Cup glory! And we aren't the only ones – Google is celebrating Paul in today's Google Doodle!
Paul, an English octopus then living in Germany, became world famous during the last World Cup when he picked the winner of all of Germany's seven matches, including their two defeats, and got the 2010 World Cup final correct too. Paul provided a welcome watery break between the matches, even though, as David Spiegelhalter's statistical analysis showed, he wasn't actually psychic after all.
Sadly, Paul can't help us predicting this year's World Cup – psychic or not. He died in October 2010 after living a full and happy octopus life. But you can read more about Paul's contribution to the 2010 World Cup in our article Understanding uncertainty: how psychic was Paul?.
We are the outcome of a process which took nearly 14 billion years during which atoms, stars, planets and biospheres emerged from a hot and dense big bang. The details of this process are sensitive to a few important numbers — the so-called constants of physics.
Martin Rees, the Astronomer Royal, discussed the key stages in this process in the lecture below, given on 17 March 2014 as a public event during the Cambridge Science Festival, and linked to a conference on the philosophy of cosmology. The talk also addressed two questions: What would our cosmos be like if the key numbers were different? And could a huge variety of other universes exist as part of physical reality, each the aftermath of a different big bang?
Physicists are hard at work on these questions. In fact, just a couple of hours before Lord Rees gave this talk, US researchers announced a result that could signal a major breakthrough in our understanding of how our Universe evolved (see thisPlus article). Watch out for the references to this in the lecture — good timing!
"It's failure to prepare mentally and failure to take practicing penalties really seriously." This is Ken Bray's explanation for England's dismal performance in penalty shootouts. England are successful in only 17% of their encounters, compared to Germany's impressive 80%. Bray is an expert in the science of football, and he has studied the physics as well as the psychology of penalties and analysed the statistics. The result are three steps to ensure a perfect penalty, which he explains in this video. You can find out more about the science and maths of football in Bray's Plus articles on football.
I was lucky enough to see the beautiful Matisse exhibition at the Tate Modern in London last week. A few days later I was asked, by a TV researcher, how do you make maths interesting and understandable to people when so many people, in her experience, had hated and avoided it at school? In response I found myself telling her about the Matisse exhibiton. Not to change the subject or avoid discussing the bad reputation maths has in many people's minds, but to explain what we at Plus, along with many other maths communicators, try to do.
Now I loved maths at school. It made sense, I could express what was in my brain and how I saw the world through maths, and I found it fun. What I found a challenge was art class, I didn't feel I had any artistic ability at all. People who could accurately draw or realistically paint people (or bowls of fruit or jugs of water...) amazed me - it was like they had a magic power. And for a long time I didn't think much of modern art. For example Picasso's pictures seemed ugly and didn't look like the things they apparently depicted, and Matisse's bright art seemed simplistic.
My opinion on art, particularly modern art, has changed, thanks particularly to two fantastic exhibitions. The first was over a decade ago at a light-filled gallery in Spain that showed many of Picasso's preliminary sketches and studies leading up to the Guernica, the full work of which filled the final room of the gallery. Following the journey that Picasso took as he built towards this huge masterpiece, along with the commentary of the social and artistic context of the piece, gave me a new appreciation of the skill, effort and genius of this work. It was awesome. I began to see all his work differently.
The Matisse show at the Tate contains many of what are called the Cut-Outs. Very late in his life, after a serious illness, he initially wasn't able to paint the huge canvases he previously produced. Instead, sitting in a chair or up in bed, he began to cut out coloured paper which he would instruct assistants to pin to the walls of his room and studio, moving it a little to the left or right, a little up or down, experimenting with these coloured shapes to build collages that had enormous power. He was so taken by this new genre he had invented that even when he recovered enough to paint he preferred this new approach.
The exhibition contains video of Matisse at work, photographs of his studio and commentary that explain the personal, artistic and social context for his work. Walking through the chronologically arranged rooms it gave me a sense of his motivation for working this way, and how it changed his perspective, and the significance of this work both artistically but also its wider cultural context. The exhibition included different types of content: recordings of the artist himself (or someone speaking his words) about his motivation, clear descriptions of the works and how they were made, explanations of the significance of this work in art, and how they interpreted Matisse's world. And most importantly, of course, was seeing the works themselves, up close, smooth curves, jagged edges, bright colours, pinned together.
It was the brilliant curation of this exhibition that reminded me and inspired me about the work we do here at Plus. We want to allow anyone a chance to see some of the wonders of mathematics up close. We hope that hearing the words of researchers will give a sense of their motivations in doing their mathematics, and we aim to show the significance of this work both in mathematics, and in a wider, cultural setting. We hope to give people a glimpse into how mathematicians perceive the world and how they use mathematics to express their perception of the world. And we hope this gives our readers a new appreciation of maths, of its power and beauty, that they might not have noticed or enjoyed before.
Now I'll never be an artist. But thanks to clever and passionate curation I have over time developed an appreciation of many different types of art, that makes seeing new art less confronting and more exciting. And sometimes, when the urge arises, I might have a go at capturing something on pencil and paper or in cut-outs of coloured paper, just for my own pleasure. I hope that Plus plays a small part in helping everyone have a similar appreciation of mathematics. We might not all be mathematicians, but I hope we can all enjoy and engage with mathematical ideas, appreciate their beauty and power. And that the next time maths pops up in our lives, it's something to be excited about, rather than avoided.