In the classic arcade game Pac-Man, the player moves the title character through a maze. The aim of the game is to eat all of the pac-dots that are spread throughout the maze while avoiding the ghosts that prowl it.
While playing Pac-Man recently, my concentration drifted from the pac-dots and I began to think about the best route I could take to complete the level.
Manhattan is a Cartesian plane brought to life. The original design plan for the streets of Manhattan, known as the Commissioners’ Plan of 1811, put in place the grid plan that defines Manhattan to this day. Using rectangular grids in urban planning is common practice but Manhattan went as far as naming its streets and avenues with numbers: 1st Street, 42nd Street, Fifth Avenue and so on.
The idea of a system of coordinates was first published in 1637 by René Descartes (hence Cartesian) and revolutionised mathematics by providing a link between geometry and algebra. According to legend, school boy René was lying in bed, sick, when he noticed a fly on the ceiling. He realised that he could describe the position of the fly using two numbers, each measuring the perpendicular distance of the fly to the walls of the room. Voilà! The Cartesian plane was born.
It is only natural then that we interpret the Cartesian plane in terms of spatial coordinates but what happens if we take the same old Cartesian plane and re-invent the 2D space as a space of functions?
Have you ever browsed one of your friend’s playlists and discovered that you both share almost the same music? Maybe you have some additional songs and perhaps he has a band that you have never heard of before, but basically you have the same music. Is there a way to get a proper measure of how similar your music is? If you could compare your music with every person you know, is there a way of ranking playlists according to which is most similar to yours? It is possible that the playlist that is closest to yours contains songs that you have never heard before but that you will really enjoy.
Hannah Fry is a lecturer at the Centre for Advanced Spatial Analysis (CASA) at UCL. In addition to her research on the mathematics of social systems, Hannah also does a lot of public engagement – showing the general public some of the fascinating ways that Maths can be used in the real world. She’s given TED talks, spoken on TV and radio, made YouTube videos, and performed in science stand-up and stage shows. Most recently, she has written a book called The Mathematics of Love, and presented the BBC documentary Climate Change by Numbers.
Maths and the Real World
Would you like to tell our readers a bit about your mathematical background?
I did my undergraduate degree in Maths here at UCL and I much preferred the applied side. I then did my PhD in fluid dynamics with Prof. Frank Smith, doing lots of lovely asymptotic analysis. My postdoc was a bit different. I think fluids is a really great place to train, but it’s hard to find a really good postdoc in fluids, and you can’t pick the subject that you want to work on. So this postdoc came up using mathematics to look at social systems – things like trade, migration and security. And I just thought it was an interesting topic and came over here to CASA, and I’ve been here ever since!
Be proud if you are studying Mathematics at UCL! Looking back, we have numerous famous alumni who later gained significant achievements in their field. One of them is Klaus Roth, who was once a research student at UCL, and later was a lecturer and professor at the university, during which time he won the Fields Medal.
If you haven’t heard of the Fields Medal, it is seen as the equivalent of the ‘Nobel Prize’ in Mathematics (although unfortunately it has a much lower monetary reward) and is awarded every four years by the International Mathematical Union. The award is given to a maximum of four mathematicians each time, all of whom must be under the age of 40 and have made a great contribution to the development of Mathematics. Roth won the Medal in 1958, when he was 33 years old and still a lecturer at UCL (show more respect to your lecturers … you never know!), for having “solved in 1955 the famous Thue-Siegel problem concerning the approximation to algebraic numbers by rational numbers and proved in 1952 that a sequence with no three numbers in arithmetic progression has zero density (a conjecture of Erdös and Turàn of 1935).”
This issue’s cover picture is a creation of Anthony Lee, a young British artist, who has always been fascinated by exploring the possibilities of creating images through light. In Anthony’s eyes, this experimental process is the result of “the idea of an ephemeral substance or state, the idea that the captured moment was never intended to last or be repeated. In my light images neither the light nor the shape can last and yet they stay captured in the image I present.”
It is interesting to notice where both the artistic and scientific processes intersect and interact with each other – and where they do not. The artist, Anthony, is looking for a way to use scientific knowledge to express his personal emotions and inner thrills; and the resultant art is the outcome and purpose that elevates and distinguishes the science. And yet Anthony is bending and filling reality with his own meanings – his “ephemeral” ideas of light and shape – that are changeable and unique to him. Contrast this with the aims of scientists, who look for permanent truths that affect every observer, irrespective of their uniqueness in this space-time continuum.
Wormholes have been fascinating science fiction writers for decades, allowing protagonists to travel instantaneously to remote parts of the cosmos, the distant future or even entirely different universes. Last year Christopher Nolan’s blockbuster film Interstellar featured a wormhole as its key plot device, and recently won an Oscar for its visual depiction of them. The film centres around a crew of astronauts travelling through a rip in space and time in the hope of finding a future home for humanity.
Hold on tight
Although the idea of a wormhole might sound like one of the more farfetched ideas coming from the minds of science fiction writers, in fact there is a considerable amount of active research into the science behind them. The original screenplay for Interstellar was actually developed by a physicist at Caltech named Kip Thorne, who has been studying the mathematical properties of wormholes for nearly thirty years. Thorne collaborated with Double Negative Ltd, a special effects company based in Great Portland Street in central London, to ensure that the wormholes displayed in the film obeyed the correct laws of physics. And amazingly the cutting-edge super-accurate visualization software available to the Hollywood special-effects team actually enabled physicists to see new phenomena that they hadn’t anticipated.
So what are wormholes, and how can we describe them mathematically? The answer to this question requires us to know a little bit about physicists’ currently favoured description of gravity, the general theory of relativity. Kip Thorne has been one of the big names in this field of general relativity for over half a century, and so before we explore the mathematics of wormholes we will take a short detour through Einstein’s crowning achievement.