# Analogue computing: fun with differential equations

When it comes to differential equations, things start to get pretty complicated—or at least that’s what it looks like. When I studied mathematics, lectures on differential equations were considered to be amongst the hardest and most abstract of all and, to be honest, I feared them because they really were incredibly formalistic and dry. This is a pity as differential equations make nature tick and there are few things more fascinating than them.

When asked about solving differential equations, most people tend to think of a plethora of complex numerical techniques, such as Euler’s algorithm, Runge–Kutta or Heun’s method, but few people think of using physical phenomena to tackle them, representing the equation to be solved by interconnecting various mechanical or electrical components in the right way. Before the arrival of high-performance stored-program digital computers, however, this was the main means of solving highly complicated problems and spawned the development of analogue computers.

# Roots: the legacy of Fibonacci

Most initial thoughts when the name Fibonacci is mentioned centre around sequences, rabbits, nature and spirals. However, the Fibonacci legacy is much more fundamental to modern scientific studies, and without his influence, mathematics—as we know it—would not exist.

The famous Fibonacci spiral

Leonardo, of the family of Bonacci, was born in Pisa, Italy, in around 1170. It wouldn’t be until the French mathematician, Édouard Lucas, wrote extensively about the $1, 1, 2, 3, 5, 8, \ldots$ sequence in 1877 that the “Fibonacci sequence” would become more well-known. Leonardo’s father was a successful merchant and customs officer, travelling around the Mediterranean with his family in tow.

# The Mathematical Games of Martin Gardner

It all began in December 1956, when an article about hexaflexagons was published in Scientific American. A hexaflexagon is a hexagonal paper toy which can be folded and then opened out to reveal hidden faces. If you have never made a hexaflexagon, then you should stop reading and make one right now. Once you’ve done so, you will understand why the article led to a craze in New York; you will probably even create your own mini-craze because you will just need to show it to everyone you know.

The author of the article was, of course, Martin Gardner.

# Spherical Dendrite by Mark J Stock

We are surrounded by complex structures and systems that appear to be lawless and disorderly. Mathematicians try to look for patterns in the seemingly chaotic behaviour and build models that are simple, and yet have the capacity to accurately predict the reality around us. But can a scientific or mathematical model have any artistic value? It seems that the answer is yes. There is a group of digital and algorithmic artists that use science and computational mathematics to create visual art. However, there is an even smaller group of people whose art and science coincide. Meet Mark J Stock. Continue reading

# You can count on Dirichlet

In the days before email, mathematicians relied upon pen, paper and the postman to share ideas and communicate fiendish numerical taunts. An excited Dirichlet wrote to Kronecker in 1858:

… that sum, which I could only describe up to an error of order $\sqrt{x}$ at the time of my last letter, I’ve now managed to home in on significantly.

# A mathematical view of voting systems

Soon after I began my undergraduate degree, Barack Obama won the 56th United States presidential election. The next day, my pure maths tutor asked me if I had followed it closely, saying that elections really interested him. I was surprised to hear this: surely the only maths involved was adding up the number of votes? In reality, voting systems hold considerable interest for mathematicians, and there are several mathematical results and theorems concerning electoral processes. The main thing that I like about the language of mathematics is that it allows us to make extremely precise statements without ambiguity and, as you’ll see, we can make precise mathematical statements about voting systems—with some surprising results.

# Menace: the Machine Educable Noughts And Crosses Engine

The use of machine learning to teach computers to play board games has had a lot of interest lately. Big companies such as Facebook and Google have both made recent breakthroughs in teaching AI the complex board game, Go. However, people have been using machine learning to teach computers board games since the mid-twentieth century. In the early 1960s Donald Michie, a British computer scientist who helped break the German Tunny code during the Second World War, came up  with Menace (the Machine Educable Noughts And Crosses Engine). Menace uses 304 matchboxes all filled with coloured beads in order to learn to play noughts and crosses. Continue reading

# Fractional calculus: the calculus of witchcraft and wizardry

Differentiating a function is usually regarded as a discrete operation: we use the first derivative of a function to determine the slope of the line that is tangent to it, and we differentiate twice if we want to know the curvature. We can even differentiate a function negative times—ie integrate it—and thanks to that we measure the area under a curve. But why stop there? Is calculus limited to discrete operations, or is there a way to define the half derivative of a function? Is there even an interpretation or an application of the half derivative?

# In conversation with Ian Stewart

In an airy office in the Mathematics Institute of the University of Warwick we find Ian Stewart, the prominent maths professor, Fellow of the Royal Society and one of the UK’s most prolific popularisers of mathematics. He has published over 80 books, between 1991 to 2001 took over Martin Gardner’s original Mathematical Games column for the magazine Scientific American, and in 1995 won the Michael Faraday Prize for excellence in communicating science to UK audiences. He greets us with a kind smile, a warm handshake and leads us to his desk.