Here's a little game to play. Take the interval from 0 to 1, including the end points. Now delete the middle third of this interval, from 1/3 to 2/3, but without deleting the two end points 1/3 and 2/3. Then delete the middle thirds of the two intervals that are left, [0,1/3] and [2/3,1], again without deleting the end points. Keep going, deleting middle thirds ad infinitum — well, you can't really do this in practice, but you can imagine it. What will you be left with in the end?
Seven steps towards C.
Let's call the set of left-over points C. Clearly, C contains the end points of middle third intervals because these don't get removed: 0, 1, 1/3, 2/3, 1,9, 2/9, 7/9, 8/9, and so on. But other points are left too. For example, 1/4 does not get removed at the first step because it's less than 1/3, it does not get removed at the second step either because it's bigger than 2/9, and so on.
Does C contain any intervals at all? It's pretty easy to convince yourself that it doesn't: any line segment, no matter how tiny, will have been broken up by having its middle third removed. C is what's called totally disconnected because all its connected components are just points.
But these points are not isolated. If you draw a little circle around any point x in C, then no matter how small the circle, it will contain other points of C. For example, take the point 0. Arbitrarily close to it there is a point of the form 1/3k for some k, which is in C because it's the end point of a middle third interval. The fact that every point has other points arbitrarily close to it makes C into a perfect set.
All this means that C inhabits a strange world between dimensions. It's more than a collection of isolated points, which has dimension 0, and less than a line, which has dimension 1. If you calculate its Hausdorff dimension, which is an extension of our usual concept of dimension, the answer is ln(2)/ln(3)=0.63, which is between 0 and 1 as expected.
The set C is also self-similar. For example, if you take the part of C contained in some tiny interval [0,1/3k] and blow it up, what you get looks exactly the same as C itself. Being self-similar and having a dimension that's not a whole number makes C into a fractal.
C has a name too: it's called the middle-thirds Cantor set, after Georg Cantor, who wrote about it in 1883, although it had previously been discovered by Henry John Stephen Smith. Cantor was one of the first people to give a rigorous description of a fractal, though the term "fractal" was not coined until almost 100 years later by Benoit Mandelbrot.
Cantor sets often turn up in mathematics when things are being broken up according to the same recipe ad infinitum. For example, you can create a similar construct in two dimensions by dividing a square up into nine smaller squares and repeatedly removing the middle one. The result is called the Sierpinski carpet, which has a Hausdorff dimension of ln(8)/ln(3)=1.892. The three-dimensional analogue is the Menger sponge.
read more about Cantor sets and fractals in
Will he rocket to the lead? (Most excellent tortoise suit by Katie Bradley.)
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
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 m. This is because the geometric progression
converges to . 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.
Here's a strange fact: if you look up some numbers, for example the
numbers in your tax return, population sizes of Chinese provinces, or
the length of the world's rivers, then most likely around 30% of these
numbers start with the digit 1, around 18% start with the digit 2,
12.5% start with a 3, and so on, all the way to 9 (which only heads up
around 5% of the numbers) - the larger the digit, the fewer numbers in
your list start with it. This fact, known as Benford's law, applies to
so many different kinds of data sets that it's often used to detect
fraud. But why does it work?
Well, if the processes that give rise to your list of numbers do
produce a universal distribution of first digits, then this
distribution should apply no matter what units you use. It should work
no matter if you do your tax return in pounds or in euros, or measure
your rivers in metres or miles - it's universal after all. This means
that the distribution of first digits remains the same when you
multiply your numbers by whatever constant you need to change between
units. And it turns out that the only distribution with this property
of scale invariance is precisely the Benford distribution.
As an example, imagine that your first digits are distributed equally
(roughly the same proportion of numbers begin with the digit 1, 2,
3...) – so NOT according to the Benford distribution. Is this
distribution scale invariant? Let's see what happens when we multiply
by 2. All numbers starting with 5, 6, 7, 8, and 9, when multiplied by
2, give a number starting with 1. By contrast, the only way to end up
with a number beginning with, say, 3, is to start out with a number
starting with 1. In other words, the resulting distribution of first
digits, after multiplying by 2, is skewed towards 1. It's not uniform,
so your original distribution is not scale invariant. It's not too
hard to show that in order to be scale invariant, the first digits
have to be distributed in the way stipulated by the Benford
distribution. It's worth noting though that Benford's law only
applies to data sets that are neither too random, nor too constrained:
alas, it doesn't work for lottery numbers.
To find out more about Benford's law read our article Looking out for number one. This news story explores an application of Benford's law to uncover potentially fraudulent
We were delighted when the BBC approached us back in 2005 to ask if some of the Plus posters we had created for schools could be used as set dressing in an episode of Doctor Who with a storyline revolving around maths classrooms. And accordingly, in April 2006, the Plus posters made their TV debut in the Doctor Who episode School Reunion, alongside tenth Doctor David Tennant. The Doctor has infiltrated a school under cover as a science teacher, and discovers an alien plot to use the enhanced intelligence of the school students to solve the Skasis Paradigm, the key to the fundamental secrets of the universe. Solving the Paradigm requires imagination as well as intelligence - and when you're looking for maths students who are both imaginative and intelligent, of course they'll be reading Plus...
The posters which featured in the episode were launched in 2004, funded by EPSRC, in order to showcase the wide range of careers open to people with a background in maths. As Plus readers know know, the huge range of subjects featured in our articles and career interviews is proof of the amazing range of possibilities open to mathematicians : from avalanche researcher to audio software engineer, almost every field you can think of is represented on the Careers with Maths posters. The posters were intended to spread this message and to be a resource for teachers and career advisors dealing with mathematically-minded students.
All the Plus posters
We are extremely proud of the appearance of the Plus posters on Doctor Who, and very grateful to Charles Trevelyan, Plus's graphic designer. The posters were hugely successful: at one point we were receiving over 1000 requests for them a day! Although the posters are long out of print and no longer being distributed you can still download and print out a copy for yourself. With The Doctor's help, maths fame has now spread throughout the TV-viewing world and beyond, reaching intelligent civilisations of all ages and dimensions. But then, if they're intelligent they probably already got the message. Enjoy the 50th anniversary episode tomorrow night, whichever time and relative dimension in space you find yourself!
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.