Plus Blog

March 14, 2014

We love every number here at Plus equally (ok, not really, 3 and i are the best numbers), but there is no denying the fame, ubiquity and usefulness of the number $\pi $. Today, written as 3.14 in US date format, is known as Pi Day, and we are celebrating with this lovely image created by Mick Joyce.

pi

Graphic representation of $\pi $, created by Mick Joyce

Joyce wrote a computer programme to create the image, with each of the digits in the decimal expansion of $\pi $ represented by a different coloured pixels, demonstrating that the expansions is essentially without repeats or structure. Joyce's page is one of our favourites from the upcoming book, 50 Visions of Mathematics. The book (which we were lucky enough to help edit) will be released in May to coincide with the 50th anniversary of the Institute of Mathematics and its Applications, but you can preorder it now!

Our friend Alex Bellos is also celebrating Pi Day with beautiful pictures in his Guardian blog, as well as translating the classics into Pilish. And we loved the, perhaps photoshopped, $\pi $ in the sky from @inFinnityPi.

You can read all about $\pi $ on Plus:

But, of course, if you are anti-pi, you are in good company….

March 7, 2014
Marianne and Rachel with Cedric Villani

The Plus Team, Marianne and Rachel, with one of the Fields Medallists, Cédric Villani, at the 2010 ICM

Tomorrow is International Women's Day, and it got us wondering… could this be the year we finally have a female winner of a Fields Medal?

The four Fields medallists will be announced in August at the International Congress of Mathematicians (ICM) in Seoul, Korea. We asked this question back at the 2010 ICM to the then, president of the International Union of Mathematicians (IMU), László Lovász, and Ragni Piene, the chair of the Abel Prize committee.

This year, we hope to discuss this again with the current IMU president, Ingrid Daubechies, the first woman to hold this position. One of our favourite lectures at the 2010 conference was by Irit Dinur, will she be in line for the prize this year? We are looking forward to interviewing all the winners this year, but we must admit, as two female mathematicians ourselves, we will be incredibly excited if one or more of them are women!

Read more about the Fields Medal and the 2010 ICM, and meet some of our favourite women in mathematics.

2 comments
February 27, 2014
Snowboarder

Snowboarders are vulnerable to gravity. Image: Picswiss.ch.

How do you test the effects of gravity? One way is to tip yourself over the edge of your snowboard as you are elegantly gliding along to see how long it takes until you hit the ground. We tried that last week, but it didn't work (the Plus scientist contracted concussion). Another is to win a £4.2 million grant to develop sensitive equipment to detect elusive gravitational waves. This is what the University of Glasgow has just done, having applied for the funding from the Science & Technology Funding Council.

The Glasgow experiment will significantly extend our own using the snowboard. Had it been successful, our experiment would probably have confirmed Newton's universal law of gravitation, which says that the gravitational force between two point masses is

  \[ F = G \frac{m_1 m_2}{r^2}, \]    

where $m_1$ and $m_2$ are the two bodies’ respective masses, $r$ is the distance between them and $G$ is the gravitational constant, approximately equal to $6.67×10^{−11} N(m/kg)^2$. From this you can work out how long it should take a falling snowboarder to meet the snow face-on. The Glasgow experiment, however, will test a more sophisticated theory.

Newton came up with his law in 1687 and it remained unchallenged until 1905, when Einstein published his special theory of relativity. The theory says that there is a universal speed limit in the Universe: nothing can travel faster than light, that is, nothing can travel faster than roughly 300,000 metres per second. According to Newton, however, the effect of gravity is instantaneous. Take away the Sun, and the effect will be felt on Earth immediately. Einstein himself later remedied this problem by proposing that gravity isn't a force that wafts across the ether in some mysterious way, but a result of the curvature of space. An analogy that is often given is that of a bowling ball sitting on a trampoline. The ball creates a dip in the trampoline, curving its surface, so a marble placed nearby will roll into the dip towards the ball. According to Einstein, massive bodies warp space in a similar way, causing less massive bodies to be attracted to them.

One of the consequences of Einstein's theory of gravity is that when gravitational monsters such as black holes shunt their weight around, they should create ripples that can be felt across space and time. "Near black holes the curvature of spacetime is extremely high," explains Bangalore Sathyaprakash, a gravity expert. "Now imagine two black holes moving around each other: the curvature is large but also changing. It's a bit like taking a stick and moving it around in a pond. That's going to generate ripples in the water. Only in the case of black holes, we're talking about ripples in the very fabric of spacetime." These ripples are the gravitational waves researchers at the University of Glasgow will be looking for. They will develop instrumentation for gravitational wave detectors with a sensitivity of around 1/1000th of the diameter of a proton (10-18m).

Sheila Rowan, Professor of Physics and Astronomy at the University of Glasgow, said: "We are entering a very exciting time in the search for gravitational waves. Experiments aimed at detecting gravitational waves have been in development for several decades and we are now reaching sensitivity levels where detection is expected in the next few years." We hope they won't come away with concussion!

You can find out more about gravitational waves and gravity in general in Catching waves with Kip Thorne and How does gravity work?

February 27, 2014

We've read the book. We've bought the T-shirt. And now, finally, here it is: the movie of one of our favourite maths problems, the bridges of Königsberg. Though admittedly, we made it ourselves. We learnt several interesting lessons in the process. For example that a bin doesn't make a good supporting character and that people who shouldn't be in the frame should get out of it. But other than that, we're well on course for an Oscar this weekend!


You can read more about the bridges of Königsberg here.

February 25, 2014

There was a brief pause in research at the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge (DAMTP, also the home of Plus) this afternoon to celebrate the newly established Chair in Cosmology. The Chair, funded by a $US 6 million donation from Avery-Tsui Foundation, is named after Stephen Hawking and he will be the first to hold the Professorship. Paul Shellard, Director of the Centre for Theoretical Cosmology, said that the honour recognised Hawking's contributions to changing our understanding of the Universe.

Hawking

Stephen Hawking experiencing zero gravity (Image: NASA)

"When I arrived at DAMTP in 1962 cosmology was a speculative science and we didn't know if the Universe had a beginning or had existed forever in a steady state," Hawking said. He went on to say that the new Professorship recognised the role of the department in taking cosmology from this speculative start to the remarkably successful field it is today.

We'd like to congratulate Hawking on his new post (and thank him for the cake and champagne!) and look forward to the next exciting discovery from our cosmologist neighbours.

You can read more about Hawking's life and work in his articles 60 years in a nutshell and A brief history of mine, and in our coverage of his 60th and 70th birthday symposia. And of course, there's much more about cosmology on Plus.

February 25, 2014
Cantor

Georg Cantor

Are there more irrational numbers than rational numbers, or more rational numbers than irrational numbers? Well, there are infinitely many of both, so the question doesn't make sense. It turns out, however, that the set of rational numbers is infinite in a very different way from the set of irrational numbers.

As we saw here, the rational numbers (those that can be written as fractions) can be lined up one by one and labelled 1, 2, 3, 4, etc. They form what mathematicians call a countable infinity. The same isn't true of the irrational numbers (those that cannot be written as fractions): they form an uncountably infinite set. In 1873 the mathematician Georg Cantor came up with a beautiful and elegant proof of this fact. First notice that when we put the rational numbers and the irrational numbers together we get all the real numbers: each number on the line is either rational or irrational. If the irrational numbers were countable, just as the rationals are, then the real numbers would be countable too — it's not too hard to convince yourself of that.

So let’s suppose the real numbers are countable, so that we can make a list of them, for example

1. $0.1234567\dots $

2. $1.4367892\dots $

3. $2.3987851\dots $

4. $3.7891234\dots $

5. $4.1415695\dots $

and so on, with every real number occurring somewhere in the infinite list. Now take the first digit after the decimal point of the first number, the second digit after the decimal point of the second number, the third digit after the decimal point of the third number, and so on, to get a new number $0.13816\dots $.

Now change each digit of this new number, for example by adding $1$. This gives the new number $0.24927\dots $. This new number is not the same as the first number on the list, because their first decimal digits are different. Neither is it the same as the second number on the list, because their second decimal digits are different. Carrying on like this shows that the new number is different from every single number on the list, and so it cannot appear anywhere in the list.

But we started with the assumption that every real number was on the list! The only way to avoid this contradiction is to admit that the assumption that the real numbers are countable is false. And this then also implies that the irrational numbers are uncountable.

It's easy to see that an uncountable infinity is "bigger" than a countable one. An uncountable infinity can form a continuum, such as the number line, in a way that a countable infinity can't. Cantor went on to define all sorts of other infinities too, one bigger than the other, with the countable infinity at the bottom of the hierarchy. When he first published these ideas, Cantor faced strong opposition from some of his colleagues. One of them, Henri Poincaré, described Cantor's ideas as a "grave disease" and another, Leopold Kronecker, went so far as to denounce Cantor as a "scientific charlatan" and "corrupter of youth". Cantor suffered severe mental health problems which may have resulted in part from the rejection his work had met with. But we now know that his work had simply come too soon: 150 years on, Cantor's ideas form a central pillar of mathematics and many of his results can be found in standard textbooks.

See our infinity page to find out more about this and other things to do with infinity.

Syndicate content