In the eighteenth century the city we now know as Kaliningrad was
called Königsberg and it was part of Prussia. Like many other great
cities Königsberg was divided by a river, called the Pregel. It contained two
islands and there were seven bridges linking the various land masses. A
puzzle at the time was to find a walk through
the city that crossed every bridge exactly once. Many people claimed
they had found such a walk but when asked to reproduce it no one was able to. In
1735 the mathematician Leonhard Euler explained why: he showed that such a walk didn't exist.
Euler's solution is surprisingly simple — once you look at the
problem in the right way. The trick is to get rid of all
unnecessary information. It doesn't matter what path the walk takes on
the various land masses. It doesn't matter what shape the
land masses are, or what shape the river is, or what shape the bridges are. So you might as well
represent each land mass by a dot and a bridge by a line. You don't have
to be geographically accurate at all: as long as you don't disturb the
connectivity of the dots, which is connected to which, you can distort
your picture in any way you like without changing the problem.
Once you have represented the problem in this way, its features are
much easier to see. After playing around with it for a while you might
notice the following: when you arrive at a dot via a line (enter a
land mass via the bridge), then unless it is the final dot at which your
walk ends, you need to leave it again, by a different line as those are
the rules of the game. That is, any dot that is not
the starting and end-point of your walk needs to have an even number of
lines coming out of it: for every line along which you enter there has
to be one to leave.
For a walk that crosses every line exactly once to be possible, at most two dots can have an odd number of lines coming out of them. In fact there have to be either two odd dots or none at all. In the former case the two correspond to the starting and end points of the walk and in the latter, the starting and end points are the same. In the Königsberg problem, however, all dots have an odd
number of lines coming out of them, so a walk that crosses every
bridge is impossible.
Euler's result marked the beginning of graph theory, the
study of networks made of dots connected by lines. He was also able to
show that if a graph satisfies the condition above, that the number of
dots with an odd number of lines is either zero or two, then there
will always be a path through it that crosses every line exactly
The result also marked the beginning of topology, which
studies shapes only in terms of their connectivity, without taking
note of distances and angles. The London tube map is a great example
of the topological triumph. By distorting distances and angles it
turns what would otherwise be an unintelligible mess into a map that
every tourist can read effortlessly. You can find out more here.
"That's when I thought, 'this could be the moment we all die'. The de-pressurisation alarm went off and this meant that there really was a hole in the station. I felt the pressure fall in my ears. And then it got cold. Really, really cold."
No, it's not a quote from Gravity, but the words of astronaut Michael Foale recalling an incident in 1997 that nearly cost him his life. Foale's story has all the ingredients for a movie: a damaged space station, no ground communications, no power, and only one chance to get it right. The crew used the soyuz's thrusters — their only means of getting back home — to control the station's chaotic spin. Their main tools were their brains and the basic maths of motion. As Foale put it, "You cannot be an effective mission designer, astronaut, flight controller or engineer if you don't know maths."
Read our interview with Foale for some real-life adventure, space and maths, or watch the Gravity trailer below.
How do you balance a cardboard cut-out of a triangle on a pencil? Trial and error is one way, but maths can save you lots of bending down and picking it up. Take the pencil and a ruler and connect the mid-point of each side to the opposite corner. You'll find that the three lines intersect in a single point, which lies exactly a third of the way from the midpoint of each side to the opposite vertex. That point, called the centroid, is the centre of mass of the triangle. If the triangle is made from uniform material, so it's not more lumpy in some places than in others, then the centroid is the unique point on which you can balance it without it tipping over. Amazingly, the centroid would also be the centre of mass of the triangle if its mass was concentrated only at its corners, and evenly divided between them.
Instead of drawing a line from the mid-point of a given side to the opposite corner, you could also draw the line which passes through the mid-point but forms a right angle with the side the mid-point is on. If you do this for each side you again get three lines, and again these meet at a single point, called the circumcentre of the triangle. If you now draw a circle with the circumcentre as its center passing through one of the triangle's corners, you will find that the other two corners of the triangle lie on the circle too! The circumcentre of the triangle is also the centre of the unique circle that contains the three corners of the triangle. But it doesn't need to lie inside the triangle — in fact, it only does if all the triangle's angles are less than 90 degrees (so the triangle is acute). If one angle is greater than 90 degrees (the triangle is obtuse) then the circumcenter lies outside the triangle, and if one angle is exactly equal to 90 degrees then it is the mid-point of the hypothenuse.
But there's another point that qualifies as a centre of a triangle. You find it by drawing aline
from each corner that is perpendicular to the opposite side. Amazingly, the three lines again meet in a single point, called the orthocentre. As for the circumcentre, the orthocentre lies inside the triangle if the triangle is acute and outside it if it is obtuse. If one of the angles is exactly equal to 90 degrees then the orthocenter will be one of the corners.
And what links all these points together? A straight line! The beautiful fact that the centroid, circumcentre and orthocentre of a triangle all lie on a straight line was first noticed in the 18th century by Leonhard Euler, one of the most prolific mathematicians of all time. That line now carries his name: it's called the Euler line of a triangle. You can play around with the three different centres and the Euler line on Math Open Reference which has beautiful interactive demonstrations, from which we made the images illustrating these articles.
Last year we asked Plus readers to send in their own popular maths articles, with winning entries to be published in a forthcoming book. The winners were chosen a while back and we're now proud to announce that 50: Visions of mathematics is with the printers (well, nearly) and will be out for the world to read in spring 2014. The book celebrates the 50th anniversary of the Institute of Mathematics and its Applications and will be published by Oxford University Press.
Besides the winning contributions from Plus readers it contains articles from our favourite popular maths authors, including Ian Stewart, Marcus du Sautoy, Simon Singh, Chris Budd and John D. Barrow. There will be a foreword by Dara Ó Briain and also fifty images showcasing the beauty and diversity of maths. A big thank you for taking part in the competition and sending in image suggestions.
The book was edited by Sam Parc with the help of a devoted editorial team including our humble selves. To keep up to date with its progress follow Sam on Twitter!
It's Ada Lovelace day, celebrating the work of women in mathematics, science, technology and engineering. To celebrate we bring you a selection of articles by female authors that were published on Plus this year (including those written by the entirely female Plus editorial staff). To find our more about the pioneering work of Ada Lovelace herself read our article Ada Lovelace - visions of today. You can also check out our Ada Lovelace day blogs from 2012 and 2011 for more by and about women mathematicians.
Do infinites exist in nature?
— Marianne Freiberger and Rachel Thomas talk to physicists, philosophers and cosmologists in search of the unbounded.
Next week will be an exciting one for a handful of scientists with the announcement of the 2013 Nobel prizes. You'll be able to watch live at the Nobel site as the awards for physiology or medicine are announced on Monday, physics on Tuesday, chemistry on Wednesday and the peace prize on Friday. Economics will follow on Monday 14 October and the literature prize soon after. (The announcements start about 11-11.30am Central European Time or 10-10.30am British Summer Time. And yes, despite the rubbish weather, we are still on summer time!)
With the recent experimental confirmation of the Higgs boson last year at the Large Hadron Collider, rumours are beginning to swirl that the physics prize might go to some of the physicists who predicted its existence and the mechanism that gives mass to matter in the Universe. Six physicists, Robert Brout, François Englert, Peter Higgs, Gerry Guralnik, Carl Richard Hagen and Tom Kibble, contributed to three revolutionary papers published in 1964 that explained this theory. The Nobel prize can be awarded to up to three people so it will be interesting to see who will be recognised and what the reaction of the physics community will be.
We'll be watching the announcements of all the prizes with great interest. There may not be a Nobel prize specifically for mathematics, but you can be sure that maths will have played a vital role in the research of many of the 2013 Nobel Laureates. You can read more about the discovery of the Higgs boson and the previous Nobel prizes on Plus.
Secret symmetry and the Higgs boson
It's official: the notorious Higgs boson has been discovered at the Large Hadron Collider at CERN. The Higgs is a subatomic particle whose existence was predicted by theoretical physics. Also termed the god particle, the Higgs boson is said to have given other particles their mass. But how did it do that? In this two-part article we explore the so-called Higgs mechanism, starting with the humble bar magnet and ending with a dramatic transformation of the early Universe.
The Higgs boson: a massive discovery
If it looks like the Higgs... and it smells like the Higgs... have we finally found it? In July 2012 most physicists finally agreed it's safe to say we've finally observed the elusive Higgs boson. And perhaps that is not all....
A Nobel Prize for quantum optics
The 2012 Nobel Prize for Physics was awarded to Serge Haroche and David J. Wineland for ground-breaking work in quantum optics. By probing the world at the smallest scales they've shed light on some of the biggest mysteries of physics and paved the way for quantum computers and super accurate clocks.
How to make a marriage stable
How do you best allocate students to universities, doctors to hospitals, or kidneys to transplant patients? The solution to this tough problem was recognised in the 2013 Nobel Prize in Economics.
Shattering crystal symmetries
In 1982 Dan Shechtman discovered a crystal that would revolutionise chemistry. He was awarded the 2011 Nobel Prize in Chemistry for his discovery — but did the Nobel committee miss out a chance to honour a mathematician for his role in this revolution as well?
And the Nobel Prize in Mathematics goes to...
Well, it goes to no-one because there isn't a Nobel Prize for maths. Some have speculated that Alfred Nobel neglected maths because his wife ran off with a mathematician, but the rumour seems to be unfounded. But whatever the reason for its non-appearance in the Nobel list, it's maths that makes the science-based Nobel subjects possible and it usually plays a fundamental role in the some of the laureates' work. Here we'll have a look at two of the 2010 prizes, in physics and economics.
There might not be a Nobel Prize for mathematics, but you can read more about the winners of the highly prestigious Abel prize and Fields medals.