post

Heads and Tails

Recently I attended a quiz night at our local club. During the break the attendees played Heads and Tails. The rules are as follows:

                                                                                                      Heads and Tails

 

Two coins are thrown by the organiser. The three possible (but not equally likely) outcomes are 2 heads, 2 tails and a head and a tail. Players place their two hands on their heads, two hands on their bottom or one hand on their head and one hand on their bottom signifying that they are betting on the outcome being 2 heads, 2 tails or a head and a tail respectively. Those players guessing incorrectly are eliminated and the game continues until one person is left who is the winner.

Continue reading

post

Integration: It’s more than the sum of its parts

When you’re taught integration in school, it can look weird and random. What’s that curly line? Why do you have to write “dee ecks” after the function like it’s a magic spell? You’re taught that it finds the area under a curve, and that it’s the opposite of differentiation. But what’s often lost in-between learning tricks like integration by parts is a sense of what integration is and what it means. It turns out that explaining this leads naturally to an explanation of a huge area of integration overlooked in school but vital to science and engineering: numerical integration.

Continue reading

post

Fun phenomena in fluids

One important use of mathematics is to describe things that happen in the natural world around us. For hundreds of years now, mathematicians have worked together with physicists to explain, predict and quantify all manner of natural phenomena; from the solar eclipse and the movement of the planets, to the spread of diseases and infections.

But what happens when mathematics predicts something that seems too strange to be true? Is it a bug in the system, or is the world sometimes stranger than we might expect? This week, we take a look at some weird results from the field of fluid mechanics. Continue reading

post

Modelling blood

From understanding the effect of aneurysms and what causes strokes to simulating and constructing artificial organs, maths has a huge role to play in developing new medical treatments. But one key part of the human physiology is the study of blood. It’s fairly obvious that blood is key to life – if you bleed too much you die. It has been studied by many eminent figures, from Aristotle who believed blood was required to transport heat around the body to Poiseuille who derived derived a simplified model of mathematical flow in a pipe to describe flow through arteries. We now understand that blood carries oxygen and essential nutrients to our cells, and carries waste products such as urea away to be processed.

Continue reading

post

Maths trumps review

On a blustery early spring afternoon, three of the Chalkdust team gathered to test out an intriguing new product: a mathematically-themed version of the classic “my-car-is-faster-than-yours” card game, trumps. If you’ve not played trumps before, the idea is simple. Each card in a set of trumps depicts a member of a certain group and statistics about that member. Players take it in turns to read out a statistic from the top card in their hand, and the one with the highest number wins all of the cards from that round. For example, a set might be all about wild animals, and each card will show a picture of the animal along with its weight, speed, life-span etc. Your aim is to collect all of your opponents cards by choosing which statistic you will do the best in.

So it’s a game involving sets, statistics and probability… seems only natural that mathematicians might want to get involved, right? Right! We recently got our hands on some maths trumps, a new twist on the game with six different sets of cards all themed around mathematics. Read on to hear what we thought about two of the sets, “2D shapes” and the mysteriously titled “Connections”.
Continue reading

post

Our favourite (and not-so-favourite) Euler equations

In previous issues of Chalkdust, we shared with you a selection of our favourite “things” in maths, such as our favourite functions, shapes and sets. On the other hand, there are also some things we find annoying and very much dislike, such as bad notation, certain numbers, etc. For this special occasion (commemorating Euler’s birthday Euler a few weeks ago), we decided to spread some of our favourite, and not-so-favourite, examples of things named after Euler throughout issue 07.

We would also like to hear yours! Send them to us at contact@chalkdustmagazine.com. Continue reading

post

The new Chalkdust T-shirt

If you’ve been to one of our issue 07 launch events or you’ve been keeping an eye on Twitter, you may have spotted the new Chalkdust T-shirt. If you like it, you can order one here!

Whether or not you’d seen it before, you’re probably wondering what the pattern on the T-shirt means… Wonder no more, we’re about to reveal all in this blog post. If you’d like to try to work it out yourself then stop reading now; spoilers ahead.

The pattern on the new T-shirt is a position in John Conway’s Game of Life. Life is a cellular automaton that was invented by John Conway in 1970, and popularised soon afterwards by Martin Gardner.

In Life, cells on a square grid are either alive or dead. In this post and on the T-shirt, we use white for alive cells and black (or the colour of the T-shirt) for dead cells. Life begins at generation 0 with some cells alive and some dead. The aliveness of a cell in the following generations is determined by the following rules:

  • Any live cell with four or more live neighbours dies of overcrowding.
  • Any live cell with one or fewer live neighbours dies of loneliness.
  • Any dead cell with exactly three live neighbours comes to life.

These three simple rules leads to some surprisingly complicated behaviour.

The pattern to the left is called a glider. This is because as the generations progress it glides across the grid. You can see what I mean in the GIF below.

Another fan favourite is Gosper’s glider gun, which is shown below. It is called Gosper’s glider gun as it fires gliders across the grid, and it was discovered by Bill Gosper.

But before we get too distracted by all the things you can make in Life, let’s get back to the T-shirt.

The T-shirt shows a position in Life. But it’s not just any old position: if you go forward one generation, you get the following:

If you like that, you can buy a T-shirt with it printed on here! You can also use this tool to write any word/phrase you like in Life.

Of course, you could continue to look at what happens to the T-shirt’s pattern after more generations. Unfortunately, not much of interest happens:

post

A few of Euler’s masterpieces

Leonard Euler wrote more mathematics than anyone in history. It is said that he was responsible for around a third of all the mathematics, physics and mechanical engineering research published in all of Europe between the years 1726-1800. Much of our modern notation is due to him. He left his mark on every subject he touched. In fact, there is a whole Wikipedia page dedicated to simply listing all the things named after him. Almost everything on the list has its own Wikipedia page. Instead of attempting the impossible by trying to summarise of all of his work, we will present a few personal favourites from the world of pure maths and hope that it encourages others to read further and find their own personal favourites.

Geometry

There’s quite a lot going on in this picture but let’s just focus on the miraculous red line in the middle known as the Euler line.

What’s so miraculous about it? After reading what it is, perhaps you’ll agree that it’s very existence is a miracle. Start with three arbitrary points $A, B, C$ and draw the triangle $ABC$. Next, construct the three perpendicular bisectors of the edges of $ABC$. These are the green lines and they all meet at a single point which we label $O$. The gold lines are the medians of the triangle. They are the lines through the vertices and opposing midpoints and they also meet in a single point, which we label $G$. The blue lines are constructed by dropping a perpendicular from each vertex to its opposite edge of the triangle. Again, they meet in a single point which we call $H$. It turns out, and this is what Euler proved, that no mater how the original points $A,B,C$ are arranged, the points $O,G,H$ always line up in on straight line. What’s more, the distance $GH$ is always exactly twice that of $OG$.

Analysis

Logarithms are introduced in school nowadays as being related to exponentiation by the formulas
$$y= a^x \text{ if and only if } x = \log_a(y).$$
It was Euler who first clearly perceived logarithms in this way. Before Euler, logarithms were used by scientists and engineers to simplify calculations by converting multiplication (which was hard) into addition (which was easier). Euler recognised the significance of logarithms as mathematically interesting functions in their own right, independently of their use in calculations. He observed that $a^\delta$ is very slightly larger than 1 when $\delta$ is very slightly larger than 0. In fact, for $\delta$ positive but very small, $a^\delta \approx 1 + k \delta$ for some proportionality constant $k$ which depends on $a$. He gives the numerical examples, $a=10$, $\delta = 0.000001$ for which $k = 2.3026$ and $a=5$, $\delta = 0.000001$ for which $k = 1.60944$, and found that the number $e = \sum_{n=1}^{\infty} \frac{1}{n!} = 2.7182818284\ldots$ is exactly that number with proportionality constant $k=1$. This number $e$ is appropriately called Euler’s number. The power series
$$e^x = 1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.$$
is also due to him. Many expressions become simpler because $k=1$ and this is why it is natural to take logarithms to the base $e$. He also discovered the power series
$$\log_e(1+x) = x-\frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \frac{x^5}{5} – \cdots.$$

Number theory

“These works are recorded to have been completed in six days $\ldots$ because six is a perfect number – not because God required a protracted time, as if He could not at once create all things,$\ldots$ but because the perfection of the works was signified by the number six. For the number six is the first which is made up of its own parts, i.e., of its sixth, third and half, which are respectively one, two and three, and which make a total is six.”

This is an excerpt from St Augustine’s City of God (Part XI Chapter 30) explaining that God created the world in six days because six is the first perfect number. A perfect number is a positive integer which is equal to the sum of all its proper divisors, so excluding the number itself. For example, 6 is perfect because its proper divisors are 1, 2 and 3, and 1+2+3 = 6. The next smallest perfect numbers are 28, 496 and 8128. Perfect numbers have been entertaining the imaginations of mathematicians and non-mathematicians alike for literally thousands of years. In fact, as far back as c. 300 BC, Euclid proved in book IX of his Elements that if $n = 2^p(2^p-1)$ where $p$ and $2^p-1$ are both prime numbers then $n$ is a perfect number. A prime number of the form $2^p-1$ is known as a Mersenne prime. Although it had been conjectured previously, it wasn’t until Euler worked on the problem (around 2000 years later!) that someone finally succeeded in proving that if $n$ is an even perfect number then $n=2^p(2^p-1)$ where $p$ and $2^p-1$ are both prime numbers. This result, now called the “Euclid-Euler Theorem”, establishes a strikingly curious one-to-one correspondence between even perfect numbers and Mersenne primes.

Analytic number theory

“The remarks I have decided to present here refer generally to that kind of series
which are absolutely different from the ones usually considered till now.”

Euler wrote a lot of his work in Latin. The quotation above is the first sentence from a paper he wrote whose title translates into English as “Several Remarks on Infinite Series”. Theorem 7 of that paper is the following enigmatic identity
$$1 + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6} + \cdots = \frac{2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdots}{1\cdot2\cdot4\cdot6\cdot10\cdot12\cdot16\cdot18\cdots} $$
where the numerator of the right hand side is the product of all the prime numbers and the denominator is the product of all the numbers 1 less than a prime. It is well known that the left hand side of this identity diverges to infinity. In fact, in a different work, Euler proved the stronger statement
$$\lim_{n \rightarrow \infty}\left(-\log n + \sum_{k=1}^{n}\frac{1}{k}\right) = \gamma$$
where $\gamma = 0.57721\ldots$ is the Euler–Mascheroni constant. It follows straight away from the fact that the left hand side diverges that there are infinitely many primes – because the right hand side cannot be a finite product. Euler didn’t stop there though. He used his product formula to prove the much more impressive result that
$$\frac{1}{2} + \frac{1}{3} +\frac{1}{5} + \frac{1}{7} +\frac{1}{11} +\frac{1}{13} + \frac{1}{17} + \cdots = \infty.$$

Combinatorics

This next one is truly astonishing – both the statement and Euler’s proof. It concerns the number of different ways of expressing a positive integer as a sum of other positive integers. For example, there are 15 ways of expressing 7 like this and they are

\begin{align*}
&1+1+1+1+1+1+1+1, \:\: 1+1+1+1+1+2, \:\: 1+1+1+1+3, \\
&1+1+1+2+2, \:\: 1+1+1+4, \:\: 1+1+2+3, \:\: 1+2+2+2, \:\: 1+1+5, \\
&1+2+4, \:\: 1+3+3, \:\: 2+2+3, \:\: 1+6, \:\: 2+5, \:\: 3+4, \: \text{ and } \:7.
\end{align*}

The thing to notice is that there are exactly 5 ways where all the numbers are odd and also exactly 5 ways in which there are no repeats. This is no accident. In fact Euler proved that this always happens.

The number of ways of expressing a given number as a sum of distinct positive integers is the same as the number of ways of expressing it a sum of odd positive integers.

It’s hard to believe this at first since it seems like it has no right to be true, but it is. In order to better appreciate Euler’s ingenious proof, it is worth trying to imagine how helpless you would feel if you were asked to show this in an exam. Euler’s proof is as shocking as the statement.

He starts by noticing that the number of ways of writing $n$ as a sum of distinct positive integers is precisely the coefficient of $x^n$ in the expression $(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots.$ Next, manipulate this infinite product to get
\begin{align*}
(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots &= \frac{(1-x^2)(1-x^4)(1-x^6)(1-x^8)\cdots}{(1-x)(1-x^2)(1-x^3)(1-x^4)\cdots} \\
&=\frac{1}{(1-x)(1-x^3)(1-x^5)(1-x^7)\cdots}
\end{align*}
and expand using the formula for a geometric series to get that this is equal to
$$(1+x+x^2+\cdots )(1+x^3+x^6 + \cdots )(1+x^5+x^{10}+\cdots )(1+x^7+x^{14}+\cdots )\cdots.$$
Now finish by recognising the coefficient of $x^n$ in this last expression as being exactly the number of ways of writing $n$ as a sum of positive odd integers, where now we allow repeats.

Infinite series

Jakob Bernoulli’s 1689 Tractatus de seriebus infinitis was a state-of-the-art account of infinite series, as they were understood at the time. It included results like the fact that the harmonic series $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges and explicitly evaluated a number of convergent series. For example, the geometric series $\sum_{n=1}^{\infty}a^n = \frac{1}{1-a}$ for $|a|<1$, the sum of the reciprocals of the triangular numbers,
$$1 + \frac{1}{3} + \frac{1}{6} +\frac{1}{10} + \frac{1}{15} + \cdots = \sum_{n=1}^{\infty}\frac{1}{n(n+1)} = 2,$$
and others like $\sum_{n=1}^{\infty}\frac{n^2}{2^n} = 6$ and $\sum_{n=1}^{\infty}\frac{n^3}{2^n} = 26$ were all known at the time. At some point Jakob decided to think about $\sum_{n=1}^{\infty}\frac{1}{n^2}.$ He knew that it converged but tried and failed, as did a number of others, to evaluate it explicitly. Concerning this sum, the Tractatus included the line

“If anyone finds and communicates to us that which thus far has eluded our efforts, great will be our gratitude.”

Euler rose to the challenge in spectacular fashion by showing that
$$1+\frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \cdots = \frac{\pi^2}{6}.$$
His original argument, although not entirely justified at the time, is undoubtedly the work of a genius. He reasoned as follows. Just like polynomials can be factored according to their roots, Euler factorised $\frac{\sin x}{x}$ according to it’s (infinitely many!) roots, which are $\pm n \pi$ for $n = 1, 2, 3, \ldots$, as
$$\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi}\right)\left(1+\frac{x}{n\pi}\right) = \prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2\pi^2}\right).$$
The power series expansion
$$\frac{\sin x}{x} = 1 – \frac{x^2}{3!} + \frac{x^4}{5!} – \frac{x^6}{7!} – \cdots$$
was well known to Euler. To evaluate the sum in question, it just remains to expand the infinite product and compare coefficients of $x^2$ in these two representations of $\frac{\sin x}{x}.$

Further reading

To learn more about Euler’s mathematics the following resources are highly recommended.

  • W. Dunham, Euler Master of Us All. This is an excellent book that explains in much more detail some of things written about here. It nicely puts Euler’s contributions into context by outlining the state of knowledge before Euler and explaining what later work it inspired.
  • P. Nahin, Dr Euler’s Fabulous formula: Cures Many Mathematical Ills. This whole book is dedicated to Euler’s formula $e^{i \theta } = \cos \theta + i \sin \theta$. It’s packed full of wonderful identities and important applications.
  • eulerarchive.maa.org is a website that describes itself as “A digital library dedicated to the work and life of Leonard Euler”. Amongst other things, it contains links to a huge number of his original papers, many of which have been translated into English.
post

Let them share cake

There comes a point in every person’s life where they have to learn how to share fairly. Admittedly some people seem to ignore this point, sailing on through life gleefully seizing more than would be justified, but we’re willing to bet that the situation of having to divide up a resource (for example some food or a list of chores) into parts that everybody is happy with is pretty much universal.

If there are just two people who want to split the resource, then there is a simple method to ensure that it is divided fairly. This concept (called the “I cut, you choose” method) is so old that it’s even mentioned in the Bible. As the name suggests, the method involves one person splitting the resource into what they consider to be equal halves, and then the other person picking which (if any) of the pieces they think is worth more. The person who chooses is bound to be happy, and the person who cut can’t complain since they were supposed to divide the resource into equal pieces. The solution for two people, then, is so simple that it doesn’t seem like mathematics at all. However, the problem becomes significantly harder once you start to include more people — so difficult, in fact, that a completely ‘satisfactory’ answer for an arbitrary number of sharers was not found until 2016… Continue reading