Would Alien (Non-Euclidean) Geometry Break Our Brains?

The author H. P. Lovecraft often described his fictional alien worlds as having ‘Non-Euclidean Geometry’, but what exactly is this? And would it really break our brains?

 

Produced by Tom Rocks Maths intern Joe Double, with assistance from Tom Crawford. Thanks to the Oxford University Society East Kent Branch for funding the placement.

Tom Crawford, and Rockin’ Maths Matters

Esther Lafferty meets Dr Tom Crawford in the surprisingly large and leafy grounds of St Hugh’s College Oxford as the leaves begin to fall from the trees. It’s a far cry from the northern town of Warrington where he grew up.

Tom is a lecturer in maths at St Hugh’s, where, defying all ‘mathematics lecturer’ stereotypes with his football fanaticism, piercings, tattoos, and wannabe rock musician attitude, he makes maths understandable, relevant and fun.

‘It was always maths that kept me captivated,’ he explains, ‘ever since I was seven or eight. I remember clearly a moment in school where we’d been taught long multiplication and set a series of questions in the textbook: I did them all and then kept going right to the end of the book because I was enjoying it so much! It was a bit of a surprise to my teacher because I could be naughty in class during other subjects, messing around once I’d finished whatever task we’d been set, but I’ve loved numbers for as long as I can remember and I still find the same satisfaction in them now. There’s such a clarity with numbers – there’s a right or else it’s wrong. In English or History you can write an essay packed with opinion and interpretation and however fascinating it might be, there are lots of grey areas, whereas maths is very black and white. I like that.’

‘My parents both left school at sixteen for various reasons but they appreciated the value of education. My mum worked in a bank so she perhaps had an underlying interest in numbers but it wasn’t something I was aware of. I went to the local school and was lucky enough to be one of the clever children but it wasn’t until I got my GCSE results [10 A*s] that the idea of Oxford or Cambridge was suggested to me. I would never have thought to consider it otherwise.

‘I remember coming down for an interview in Oxford, at St John’s, arriving late on a Sunday night and the following morning I took a stroll around the college grounds  – I could feel the history and traditions in the old buildings and it was awesome. I really wanted to be part of everything it represented. I thought it would be so cool to study here so I was very excited when I was offered a place to read maths.

‘Studying in Oxford I found I was most interested in applied maths, the maths that underpins physics and engineering for example. ‘Pure’ maths can be very abstract whereas I prefer to be able to visualise the problems I am trying to solve and then when you work out the answer, there’s a sudden feeling when you just know it’s right.’

In his second year, Tom became interested in outreach work, volunteering to take the excitement of maths into secondary schools under the tutelage of Prof Marcus Du Sautoy OBE as one of Marcus’s Marvellous Mathematicians (or M3), a group who work to increase the public understanding of science.

‘I went to China one summer to teach sixth formers and it was great to have the freedom to talk about so many different topics. I spent another summer in an actuary’s office because I was told that was the way to make real money out of maths – it was a starkly different experience. I realised I was not at all cut out for a suit and a screen!’ Tom smiles. ‘I am a real people-person and get a real buzz from showing everyone and anyone that you can enjoy maths, and that it is interesting and relevant. I love the subject so much and I think numbers get a bad press for being dull and difficult and yet they underpin pretty much everything in the whole universe. They can explain almost everything and you’ll find maths in topics from the weather to the dinosaurs.

Take something like the circus for example – hula-hoops spinning and circles in the ring, and then the trapeze is all about trigonometry: the lengths and angles of the triangle. Those sequinned trapeze artists are working out the distances and directions they need to leap as they traverse between trapezes and its maths that stops them plummeting to the floor!’

Having spent four years in Oxford Tom then spent five years at Cambridge University looking at the flow of river water when it enters the sea, researching the fluid dynamics of air, ice and water, and conducting fieldwork in the Antarctic confined to a boat for six weeks taking various measurements in sub-zero temperatures. You’d never expect a mathematician to be storm-chasing force 11 gales in a furry-hooded parka, but to get the data needed to help to improve our predictions of climate change, that was what had to be done!

Tom also spent a year as part of a production group known as the Naked Scientists, a team of scientists, doctors and communicators whose passion is to help the general public to understand and engage with the worlds of science, technology and medicine. The skills he obtained allowed him to kick-start his own maths communication programme Tom Rocks Maths, where he brings his own enthusiasm and inspiring ideas to a new generation alongside his lectureship in maths at St Hugh’s.

A keen footballer (and a massive Manchester United fan) it’s no surprise Tom has turned his thoughts to football and as part of IF Oxford, the science and ideas festival taking over Oxford city centre in October, Tom is presenting a free interactive talk (recommend for age twelve and over) on Maths versus Sport – covering how do you take the perfect penalty kick? What is the limit of human endurance – can we predict the fastest marathon time that will ever be achieved? And over a 2km race in a rowing eight, does the rotation of the earth really make a difference? Expect to be surprised by the answers.

Esther Lafferty, OX Magazine

The original article can be found here.

Tom Rocks Maths S02 E02

The second episode of season 2 of Tom Rocks Maths on Oxide Radio – Oxford University’s student radio station. Featuring the numbers behind the sub 2-hour marathon world record attempt, P versus NP and the battle for control of the world, and the usual dose of Funbers with my super sweet 16. Plus, music from Blink 182, Billy Talent and Hollywood Undead. This is maths, but not as you know it…

Christmas Stamps

This Christmas themed puzzle was featured on BBC Radio 4’s Today programme as the 375th ‘Puzzle for Today‘. You can listen to the broadcast here at 48:55.

Christmas stamps are sold with the following values 16p, 17p, 23p, 24p, 39p and 40p. You want to send a present which has a postage cost of £1.00. How many stamps do you need to buy to make the exact amount?

Send your answers to @tomrocksmaths on Facebook, Twitter, Instagram, YouTube, or using the contact form on my website. The solution will be posted with the next puzzle.

Merry Christmas!

Not so smooth criminals: how to use maths to catch a serial killer

The year is 1888, and the infamous serial killer Jack the Ripper is haunting the streets of Whitechapel. As a detective in Victorian London, your mission is to track down this notorious criminal – but you have a problem. The only information that you have to go on is the map below, which shows the locations of crimes attributed to Jack. Based on this information alone, where on earth should you start looking?

Picture1

The fact that Jack the Ripper was never caught suggests that the real Victorian detectives didn’t know the answer to this question any more than you do, and modern detectives are faced with the same problem when they are trying to track down serial offenders. Fortunately for us, there is a fascinating way in which we can apply maths to help us to catch these criminals – a technique known as geospatial profiling.

Geospatial profiling is the use of statistics to find patterns in the geographical locations of certain events. If we know the locations of the crimes committed by a serial offender, we can use geospatial profiling to work out their likely base location, or anchor point. This may be their home, place of work, or any other location of importance to them – meaning it’s a good place to start looking for clues!

Perhaps the simplest approach is to find the centre of minimum distance to the crime locations. That is, find the place which gives the overall shortest distance for the criminal to travel to commit their crimes. However, there are a couple of problems with this approach. Firstly, it doesn’t tend to consider criminal psychology and other important factors. For example, it might not be very sensible to assume that a criminal will commit crimes as close to home as they can! In fact, it is often the case that an offender will only commit crimes outside of a buffer zone around their base location. Secondly, this technique will provide us with a single point location, which is highly unlikely to exactly match the true anchor point. We would prefer to end up with a distribution of possible locations which we can use to identify the areas that have the highest probability of containing the anchor point, and are therefore the best places to search.

With this in mind, let’s call the anchor point of the criminal z. Our aim is then to find a probability distribution for z, which takes into account the locations of the crime scenes, so that we can work out where our criminal is most likely to be. In order to do this, we will need two things.

  1. A prior distribution for z. This is just a function which defines our best guess at what z might be, before we have used any of our information about the crime locations. The prior distribution is usually based off data from previous offenders whose location was successfully determined, but it’s usually not hugely important if we’re a bit wrong – this just gives us a place to start.
  2. A probability density function (PDF) for the locations of the crime sites. This is a function which describes how the criminal chooses the crime site, and therefore how the criminal is influenced by z. If we have a number of crimes committed at known locations, then the PDF describes the probability that a criminal with anchor point z commits crimes at these locations. Working out what we should choose for this is a little trickier…

We’ll see why we need these in a minute, but first, how do we choose our PDF? The answer is that it depends on the type of criminal, because different criminals behave in different ways. There are two main categories of offenders – resident offenders and non-resident offenders.

Resident offenders are those who commit crimes near to their anchor point, so their criminal region (the zone in which they commit crimes) and anchor region (a zone around their anchor point where they are often likely to be) largely overlap, as shown in the diagram:

Picture2

If we think that we may have this type of criminal, then we can use the famous normal distribution for our density function. Because we’re working in two dimensions, it looks like a little hill, with the peak at the anchor point:

Picture3

Alternatively, if we think the criminal has a buffer zone, meaning that they only commit crimes at least a certain distance from home, then we can adjust our distribution slightly to reflect this. In this case, we use something that looks like a hollowed-out hill, where the most likely region is in a ring around the centre as shown below:

Picture4

The second type of offenders are non-resident offenders. They commit crimes relatively far from their anchor point, so that their criminal region and anchor region do not overlap, as shown in the diagram:

Picture5

If we think that we have this type of criminal, then for our PDF we can pick something that looks a little like the normal distribution used above, but shifted away from the centre:

Picture6

Now, the million-dollar question is which model should we pick? Determining between resident and non-resident offenders in advance is often difficult. Some information can be made deduced from the geography of the region, but often assumptions are made based on the crime itself – for example more complex/clever crimes have a higher likelihood of being committed by non-residents.

Once we’ve decided on our type of offender, selected the prior distribution (1) and the PDF (2), how do we actually use the model to help us to find our criminal? This is where the mathematical magic happens in the form of Bayesian statistics (named after statistician and philosopher Thomas Bayes).

Bayes’ theorem tells us that if we multiply together our prior distribution and our PDF, then we’ll end up with a new probability distribution for the anchor point z, which now takes into account the locations of the crime scenes! We call this the posterior distribution, and it tells us the most likely locations for the criminal’s anchor point given the locations of the crime scenes, and therefore the best places to begin our search.

This fascinating technique is actually used today by police detectives when trying to locate serial offenders. They implement the same steps described above using an extremely sophisticated computer algorithm called Rigel, which has a very high accuracy of correctly locating criminals.

So, what about Jack?

If we apply this geospatial profiling technique to the locations of the crimes attributed to Jack the Ripper, then we can predict that it is most likely that his base location was in a road called Flower and Deane Street. This is marked on the map below, along with the five crime locations used to work it out.

Picture7

Unfortunately, we’re a little too late to know whether this prediction is accurate, because Flower and Deane street no longer exists, so any evidence is certainly long gone! However, if the detectives in Victorian London had known about geospatial profiling and the mathematics behind catching criminals, then it’s possible that the most infamous serial killer in British history might never have become quite so famous…

Francesca Lovell-Read

Growing human hearts

Growing a human heart from a single cell may seem like science fiction, but scientists at the Gladstone Institute at the University of California San Francisco, have taken a huge step forward, by producing the first three-dimensional, beating, human heart chamber. Previously, it had been possible to produce a two dimensional sheet of beating heart cells, but to really gain an understanding of heart formation in a developing foetus and perhaps more importantly, how drugs given to women during pregnancy may affect this development, a three dimensional structure was needed. By treating stem cells with drugs and then confining them to a very small spherical geometry, Bruce Conklin and his team have managed to grow their very own three dimensional model of a human heart, as he explains…

Bruce – the cells around the edge became fibroblasts – a particular type of cell that you use to heal wounds and then only in the very centre were cardiac cells that beat. What this is forming is more of a little organoid is what we call it, where there’s beating cells but there’s also multiple other cell types and that’s what makes it so interesting is that these cell types are somehow talking to each other and somehow collaborating in some way so that they can actually make this structure that we didn’t expect.

Tom – It’s almost like they’re trying to form a heart…

Bruce – That certainly is the impression. They’re heart cells, they’re forming cavities so it could be a model of how parts of human development occurs, but it certainly is not a real human heart in the sense that there’s probably many things that we’re missing. We just have a simplified version with just one chamber, but having it in a controlled way where it happens the same way over and over again we can start asking questions about ‘how do these cells talk to each other?’ So once you have a system which is reproducible you can do experiments to break it in some way or to enhance it in another way.

Tom – What are the applications of this work then?

Bruce – The most obvious application of the work is to study human development. How do cells actually form a heart is something of basic interest. And also, the most common form of birth defects is actually cardiac defects. But the other application is that we can expose these developing human micro-chambers to drugs which are thought to cause developmental defects, specifically of the heart, and in fact one of the key experiments in this study was to use the drug thalidomide which is notorious for causing birth defects. When we expose these cells to the thalidomide they had a dramatic change in the morphology so that you could see that it was altering the developmental process in this micro-chamber. Thalidomide was tested in rodents before it was tried in people and there were no cardiac defects in the rodents. I think that more and more we’re thinking how do we get tests which use real human cells so that we can actually make safer drugs. And in this case say you turn back the hands of time and you had this sort of test perhaps you would have discovered that thalidomide was dangerous before it had gone on to be given to people.

You can listen to the full interview with the Naked Scientists here.

Killer robots a step too far say AI researchers

War has always happened throughout human history and, chances are, it will continue to do so in the future. With this in mind, it’s important to ensure that if it does occur it’s carried out as humanely as possible, which is why treaties such as the Geneva Convention exists. Violating certain parts of this treaty, such as the use of chemical and biological weapons, for example, constitutes a war crime. With recent developments in artificial intelligence, a new version of the convention may be required. There have been two major revolutions in warfare so far: gunpowder and nuclear weapons, and the use of artificial intelligence is seen by many as the third such revolution. In an open letter to the United Nations, more than 100 leading robotics experts, including Elon Musk, Stephen Hawking, and the founder of Google’s Deepmind have called for a ban on the use of AI in managing weapons systems. I spoke to Peter Clark, founder of Resurgo Genetics and an expert in machine learning…

  • The letter aims to trigger a debate about having international legislation for AI weapons systems, much in the same way that we have for nuclear or chemical weapons.
  • Current drones require a pilot (even if thousands of miles away) and therefore still maintain an element of human morals and ethics, which means they are very different to a fully autonomous weapons system.
  • One possible example of this technology could be a swarm of mini drones carrying small packets of explosives that could target individuals in a population.
  • Techniques that are currently used to profile people’s online behaviour could be easily applied to such weapons systems to identify and eliminate people that opposed a particular ideology.
  • The technologies being discussed are all available, and could be put together now into a system that could be catastrophic for the globe, which is why this letter is so important.

You can listen to the full interview for the Naked Scientists here.

SJC Inspire: how to design a successful video game

Very excited to announce the launch of the SJC Inspire digital magazine this week – a project I’ve been working on for the past few months in my role as Access and Outreach Associate for STEM at St John’s College, Oxford.

The first issues is ‘how to design a successful video game’ and features articles by researchers at St John’s, video interviews with students at the college, and practice puzzles set (and solved) by real Oxford tutors (myself included). I’ve highlighted some of my favourites below, but be sure to check out the full contents of the issue on the website here.

Maths in video games

My former tutorial partner, James Hyde, now works for Creative Assembly developing hit titles such as Halo Wars and Halo Wars 2. Here he explains how maths has helped him to land his dream job…

halowars1

Fun and games at the circus

Try out this maths puzzle set by St John’s maths tutor Dr David Seifert. If you send your answers in to inspire@sjc.ox.ac.uk you might even win a goodie bag!

circus-828680_1920

How to earn billions by giving something away for free

St John’s Economics tutor Dr Kate Doornik explains the pricing strategy behind the incredibly successful ‘Fortnite: Battle Royale’. Originally given away for free, it is expected to make over $3 billion in sales in 2018…

gamer-3725586_1920

Goldbach’s Conjecture: easy but hard

Often in Mathematics problems that are easy to state turn out to be extremely difficult to solve. Two hundred and seventy-five years ago, Goldbach wrote a letter to the famous Swiss mathematician Leonhard Euler in which he wrote the simple statement:

“Every even integer greater than 2 can be expressed as the sum of two primes.”

Just in case you are not up to speed with your maths (and let’s face it why would you be if you’re not a mathematician), let’s break this statement down. The even integers are the numbers divisible by two: 2, 4, 6, 8, …, 256, … and so on. The prime numbers are the ones that can only be obtained by multiplying one by themselves. For example, 3 and 5 are prime numbers because 3=1×3 and 5=1×5 and they have no other representations as a product of two numbers. However, 6 for instance is not prime because 6=1×6=2×3. In fact, all even integers, greater than 2 that were mentioned above, are not primes because they are all divisible by 2 and therefore can be represented as a product of two numbers in at least two ways: 4=1×4=2×2, 6=1×6=2×3, 8=1×8=2×4 etc.

And so, to Goldbach’s conjecture. It says that all even numbers: 4, 6, 8, 10, … can be written as a sum of two primes. Let’s see a couple of examples:

4=2+2

6=3+3

8=3+5

10=3+7

12=5+7

….

A nice way to represent the conjecture visually is through a “pyramid” and because we all love pretty pictures let’s see how this magic happens.

First, we write all of the prime numbers on two of the sides of a triangle as below: 2, 3, 5, 7 etc. We then draw a line leaving each prime number which is parallel to the opposite side of the triangle (stick with me), and finally at the points of intersection of these lines, we write the sum of the numbers. It sounds more complicated than it is as you’ll see with the following example. In the picture below, take the blue line coming out of the number 7 on the left and the red line coming out from the number 11 on the right. They intersect at 18 because 11+7=18. This means that the even integer 18 can be represented as a sum of the two prime numbers 11 and 7. If you look at the intersections of all of the red and blue lines in the pyramid, you’ll see that we actually get all of the even numbers. In other words, any even integer can be written as the sum of two prime numbers, and we can see what those two numbers are by finding the corresponding intersection on our diagram. This is Goldbach’s Conjecture.

goldbach

It is not very difficult to show that a small even number greater than 2 is the sum of two prime numbers – either by finding the corresponding point on the picture or by trying all of the possibilities. Let’s take 96. We start by checking the smallest prime number 3. 96=3+93, but 93 is not a prime, because 93=1×93=3×31. We continue with the next prime – 5. 96=5+91, which again doesn’t work because 91=1×91=7×13. Next, we try with 7: 96=7+89. Since 89 is a prime, we have obtained a representation of the number 96 as a sum of two primes.

We were able to quickly check whether 96 satisfies Goldbach’s conjecture because the number is relatively small. It becomes much harder to make these checks for larger numbers. It’s been verified with the use of a computer that the conjecture is true for numbers as big as 4×1018 and this is why the conjecture is believed to be true, but we do not yet have a formal mathematical proof. And being mathematicians, we cannot say something is true until we can prove it.

There have of course been many efforts over the last 275 years to try to prove the conjecture, most of which followed one of two routes. Either by proving that all even integers can be represented as a sum of some number of primes – as a sum of 6 primes (1995, Ramare) and as a sum of 4 primes (Herald Helfgott) – or by proving that almost all even integers can be written as a sum of 2 primes. But, as of yet, the secret formula required to unlock the proof of Goldbach’s Conjecture remains elusive.

You may be wondering why on earth mathematicians are spending their time and effort to prove this seemingly random result about prime numbers? Is it really that important? Whilst you may have a valid point about the applications of this particular conjecture, the value in proving such a result is not in the statement itself, but rather in the new methods, theories and techniques that will need to be developed to solve the problem. So, in 20, 10 or even 2 years from now when you hear that Goldbach’s conjecture has been proved, you should be happy not because we now know for sure that it’s true, but rather because some incredible new area of mathematics has been developed in the process. And who knows, this new area of maths may even pose a new, even more complicated conjecture that will occupy mathematicians for the next 275 years…

Mariya Delyakova

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