Science issue: Mathematical Bridges

5B Vicky Neale (2)NewsMy day job combines teaching maths undergraduates, public communication of mathematics, and working with school students. When I’m not doing these things, I enjoy various forms of craft. I’ve recently been teaching myself to make beaded beads. So far, I’ve been concentrating on making the five Platonic solids, the regular polyhedra.

These are highly symmetrical objects, and part of my teaching last year for the first-year Oxford maths undergraduates was about understanding the structure of the symmetries of these polyhedra (part of an area of maths called Group Theory).

This summer I realised that my beading was closely connected to my teaching in another way too. I spent six weeks this summer helping to run a summer school, PROMYS Europe, in Oxford for exceptionally keen school-aged mathematicians. I chose to teach a course on one of my favourite areas of maths, namely Graph Theory.

In this context, a graph isn’t something like a bar chart or a pie chart. Here, a graph is a collection of points, called vertices, some of which are connected by lines, called edges.

5B Graph_examples

Graph Theory is a fascinating subject in its own right within pure mathematics, and also turns to have many real-world applications, such as to the delivery driver seeking to find the shortest route, or understanding network connectivity on the internet.

The subject begain with Euler’s solution of the problem of the bridges of Königsberg in the 1730s. The city of Königsberg had seven bridges spanning the rivers, and the problem was to determine whether it was possible for a resident of the city to go for a Sunday afternoon stroll that crossed every bridge just once (no doubling back).

5B Konigsberg_bridges5B KonigsbergEuler recognised that the bridges and regions of land are not in themselves important, all that matters is the corresponding graph of vertices and edges. He was able to prove that the proposed walk crossing each bridge exactly once was impossible, by considering the number of edges out of each vertex. Every time our hypothetical rambler crosses a bridge to a different piece of land, she must then use a different bridge to leave, and so we need an even number of bridges from each landmass (even number of edges out of each vertex) — and that is not the case with this configuration. I love that we can give a mathematical argument, a proof, of the impossibility. It’s much more informative and satisfying than just trying all possible routes!

It turns o5B Adding_turquoise_beadsut that this problem is relevant for my beading too. I’ll illustrate this using the most familiar of the polyhedra, the cube. To add the turquoise beads, I have to work along each edge of the cube.

It’s wasteful of thread (and fiddly) to go along an edge twice, so ideally I’d just go along each edge exactly once. Is this possible?

We can represent the three-dimensional cube by a two-dimensional graph.5B Cube

And now Euler’s argument tells us that we have no hope of passing along each edge exactly once, because there’s an odd number of edges at every vertex! This is a little irritating for the beading, but at least I knew in advance and so didn’t waste time trying to do the impossible!

Dr Vicky Neale
Former Fellow

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References:

http://www.beadinfinitum.com/ — website from which I got the beading instructions

Out of the lab: from science to business

_DSC6732CareerI’m the co-founder and CEO of a start-up, GeneAdviser, which is making it easier for doctors to order life-saving tests for genetic diseases and cancers.

I came to Cambridge for a PhD in genomics and computational biology. I chose to study genetics because it is the fundamental language describing how life is built, so it is exciting scientifically as well as being a very powerful tool. I find all life fascinating, and it’s a pleasure to be in a rapidly developing field where there is so much to discover.

But I’ve always had have an entrepreneurial streak! My first business was a fire-twirling and circus performance group at university. We would perform at college balls and other events, which was great fun.DSC_7950_jelena_fire_happy

I’ve also been starting and working for various volunteer groups for as long as I can remember. I find it exciting to promote different causes and help organisations grow. I’m one of the directors of TReND in Africa, a charity aiming to improve scientific capacity across Africa, by training and supporting African researchers working in biomedical sciences. We ship over equipment, help set up genetics labs, and support researchers in setting their own research agendas, and pursuing their dreams.

The field of genetic testing shows so much potential. It’s vital that we share knowledge and work together to turn these scientific advances into results for patients worldwide.

So, how did I get out of the lab and into business?

Jelena in action_002During my PhD and my postdoc work, I met lots of people working in rare genetic diseases and learned a lot about the challenges faced by patients and carers of people with rare genetic disorders.

I was struck by one fact in particular: it takes on average six years for patients to get a diagnosis for a rare disease. People get sent from one doctor to another in search of answers. With the cost of genetic testing falling all the time, there’s no reason why patients should face such a frustrating journey.

I saw an opportunity to help, and to combine my interest in business with my background in science.

Co-founded with my colleague Robert Stojnic, we’re creating a website that lets doctors across the world find and order genetic tests. In time, we hope that more doctors will feel confident to use genetic testing in their practice, so that patients can get diagnostic answers sooner.

Building a community to make a difference

With another colleague and fellow entrepreneur, Tim Guilliams, I co-founded the Cambridge Rare Disease Network to bring together people who research, support and advocate for people with rare genetic conditions.

We’ve just held our first conference, and we were thrilled that Prof Stephen Hawking sent us a message of support. What started as just a few like-minded individuals with a shared interest in rare disease is quickly growing into a group of people who can really make a difference.

It’s been great to have this community of support. Within the Network, we have other young business people and researchers, as well as more experienced mentors and leaders in rare disease advocacy.

My secrets of success

I love the process and the kind of work that keeps me awake at night. Rather than the concept of success, I try and focus on creating something valuable, that solves a real need, and doing that to the best of my ability.

The start-up scene in Cambridge is really supportive. Anyone with a good idea can run with it and with passion, the right resources and a bit of luck, it feels like anything is possible.

Dr Jelena Aleksic
Founder & CEO at GeneAdviser
Founding Director at Cambridge Rare Disease Network
Director of Bioinformatics and UK Coordinator at TReND in Africa
University of Cambridge alumna

Science at Cambridge: The Laboratory of the Night Skies

4D Helen Piatkowski (1)
The 10-inch telescope and me in our garden.

University
My name is Helen and I’m about to start studying Natural Sciences (Physical) at Murray Edwards College. Never thinking I stood a chance of getting a place, I honestly can’t believe I’m only a few weeks away from going to Cambridge.

Studying natural sciences will give me a broad understanding before specialising in the third year; my current plan is to choose astrophysics and hopefully continue into research. A range of modules in the first year allows me to pick topics that interest me and perhaps focus on those most beneficial for studying astrophysics.

Murray Edwards runs an offer holder overnight stay where I was able to sleep in student accommodation, attend lectures, go to a formal hall and meet other students as well as my Director of Studies. This was a wonderful opportunity to soak in the atmosphere of the college and get to know the other offer holders. It was a comfort to find that I was not the only person who was worried they might struggle with their course!

Discovering that Dame Jocelyn Bell Burnell, who first detected radio pulsars, attended Murray Edwards is a huge motivation for me. It will be incredible to study and live at the same college as such brilliant woman as her.

M81 and M82 photographed with our 10-inch telescope
M81 and M82 photographed with our 10-inch telescope

Nothing is more fascinating than science and I am thrilled to have the opportunity to study it throughout my degree. Science stretches your imagination to the limit. Let’s take an example that is right there in front of all of us to see. Go out on a clear night and you will see stars. One of the stars you might know is the Pole Star, or Polaris, which is a whopping 3,784,211,360,000,000 kilometres away. What a ridiculous distance! Due to the immense scale of space the lightyear is used as a unit of distance measurement and is equivalent to 9,460,528,400,000 kilometres. 

Andromeda galaxy circled in red.
Andromeda galaxy circled in red.

If you’re lucky enough to have little light pollution on a dark night you will be able to see a faint band of glowing misty-like light stretching across the night sky.  This is part of the Milky Way (our own galaxy) which is a crazy 150,000 lightyears in diameter and contains around 100,000,000,000 stars.  Then, if you happen to own some binoculars and look at the right place in the constellation of Andromeda you can see a small faint smudge of light which is the Andromeda Galaxy – labelled in the image.  This galaxy is a staggering 2.5 million lightyears away and is one of our closest neighbours within our “Local Group” of galaxies.

Crab Nebula (M1) photographed with our 10-inch telescope
Crab Nebula (M1) photographed with our 10-inch telescope

One step further, with a small telescope you can even see objects that are a billion lightyears away. You need to have a strong imagination to even have a chance of comprehending these enormous distances. The excitement for me is that this generates so many questions, plus the laboratory is out there in front of us all and easily accessible.

Providing a beautiful backdrop for my studies, Cambridge’s great facilities and support from my college and peers will, I’m sure, bring out my best. Small group tutorials will really help me get to grips with course content. Becoming much more independent and stretching myself further is a part of college life to which I’m really looking forward.

Helen Piatkowski
Undergraduate student

Setting up the 8-inch telescope for a recent outreach event with the Guildford Astronomical Society
Setting up the 8-inch telescope for a recent outreach event with the Guildford Astronomical Society

School Winner: Nimble Fingers and an Inferior Equilibrium

Winning Entry Sutton Coldfield4C Joanna (Sutton Coldfield) photo

SchoolThis item touches on Economics which is a little outside the normal focus for this blog but we thought the analytical curiosity would interest scientists and economists alike.  Young women are underrepresented (at A-level) in Economics as well as Maths and Physics.

As Apple roll out their newest collection of iPads, MacBooks, and iPhones, onto the shelves in the coming weeks, it seems like an apt time to discuss the economics behind the humble QWERTY keyboard.

One would think that such a ubiquitous piece of technology should maximise efficiency-perhaps by reducing the distance between the most frequently used keys, thus increasing the speed of typing, and leaving more time to pursue other tasks. Alas, it is not so!

Indeed, the crux of the problem lies in QWERTY’s very existence. Developed by Christopher Scholes in the 1800s, QWERTY was introduced to increase the distance between the most frequently used keys on typewriters, in order to reduce jamming; it did in fact increase efficiency at the time. However, the Dvorak keyboard is by far the most efficient for today’s modern society, (where we don’t have to contend with nuisance typewriters), reducing the distance travelled by our nimble fingers by up to 50%, and so resulting in a 5-10% decrease in the time spent typing. With statistics such as these we can’t help but wonder why the QWERTY keyboard hasn’t been ousted from the market.

We turn to Game Theory to look at the problem. Take the following graph, comparing the percentage of typists using QWERTY with the probability that the next typist will use QWERTY.

When more than 72% of the population use QWERTY, QWERTY is superior, and so the probability that the next typist will learn QWERTY increases. This becomes self-perpetuating, because as more people turn to QWERTY there is a greater incentive for others to adopt the layout also. To understand this, imagine that you are a new typist. You have to decide which keyboard layout to learn to type with.

We assume that there are increasing returns associated with the adoption of keyboard layouts, due to economies of scale. Due to a larger percentage of the population using QWERTY, there is a greater demand for QWERTY layout keyboards and other such compatible products, (e.g. laptops), which in turn increases the supply of such products, increasing variety and also decreasing price. Also, with a larger percentage of the population using QWERTY, it seems rational to choose to learn it because your human capital is more easily transferable as you change jobs, as most businesses, (and schools), use QWERTY. You will not have to undergo retraining in order to learn a new layout. This is further reinforced by the fact that as people expect QWERTY to remain dominant, they choose QWERTY. This outweighs the potential benefit of increased efficiency in using the Dvorak layout. Therefore, your payoffs will be higher if you choose the QWERTY layout and you, rationally, reason to do so.

This in turn means that you increase, (extremely slightly), the percentage of the population choosing QWERTY, creating an even greater incentive for others to also learn to type with QWERTY. It’s a bandwagon effect. The choice becomes self-reinforcing. As a result, we reach an equilibrium where just under 100% of society uses the QWERTY layout.

However, this is an inferior equilibrium. Whilst the individual static payoffs favour QWERTY, the long-run payoffs for society as a whole favour the Dvorak layout, which would increase the productive potential of the economy by reducing the time spent typing. It is just another co-ordination problem. Individually, we have little incentive to choose the Dvorak, but if we all agree to co-ordinate our efforts, we are all better off. Such a co-ordinated effort to choose Dvorak is needed, but this is highly complicated to organise. Thus, we are stuck in the inferior equilibrium.

Despite Apple providing the option to set the keyboard to the Dvorak format, via the settings, I won’t be choosing it any time soon. As Keynes notably said, (although, out of context I hasten to add), “In the long run, we’re all dead!” Although the Dvorak layout would be better for society once the change has been made, for the present, whilst (almost) everyone continues using QWERTY, I will too, guiltily contributing to the inferior equilibrium.

For more on Game Theory, I recommend reading Dixit and Nalebuff’s ‘Thinking Strategically’, where they discuss Game Theory in relation to tennis, trains, and thermonuclear war.

Joanna Samardzija
Sutton Coldfield Grammar School for Girls

Hello, my name is Joanna Samardzija. I am currently studying Mathematics, Further Mathematics, History, Biology and Chemistry A Levels at Sutton Coldfield Grammar School for Girls and I hope to study Economics at University next year.

I find it fascinating how Economics can explain the world around us, even the humble keyboard. This blog post provided me with the opportunity to explore Game Theory in relation to everyday life and to further my understanding of the part Game Theory has played, and continues to play, in the evolution of our society.

Science issue: Your experiences could last for generations

4B Olivia Walker DNA photo
Olivia with the Watson and Crick DNA model at the Medical Research Council Laboratory of Molecular Biology.

NewsHow similar are identical twins? This question has occupied scientists for over a century.1 At first, you may not notice any differences. Their hair colour, facial features and height are often indistinguishable. This is explained by their origin: identical twins are derived from one fertilised egg, which ultimately forms two cells carrying the same sequence of genetic information.2 This is stored in the form of deoxyribonucleic acid (DNA) and acts as a code to produce cellular components essential for cell survival. However, despite their mutual origin, differences have been reported between identical twins, such as in their disease susceptibility.3 A rapidly expanding field called epigenetics is providing insight into the molecular basis of some of these differences.4,5 Scientists have identified chemical modifications on DNA and on the proteins that DNA is wrapped around, which do not affect the genetic sequence but alter its interpretation.

Imagine the genetic code as a music score.6 Two violinists are given the same score and are asked to play the notes they see in sequence. A bystander notices that the resulting melodies sound very different. One violinist plays the notes softly, whereas the other accentuates each note heavily, creating a sense of suspense. In the absence of any guidance on how loudly the music should be played, two contrasting interpretations result. Like musical annotations, epigenetic modifications are thought to act as signals, controlling how the DNA sequence is interpreted by the cell’s machinery. If the levels of these signals vary between identical twins, their DNA sequences may be interpreted differently, modulating the levels of specific cellular components. Epigenetic differences could therefore help explain the distinctions observed between these twins.5 But why would identical twins be epigenetically different? External stimuli are thought to affect the levels of epigenetic modifications and therefore contrasting lifestyles, such as diet, exercise or smoking habits, could all play a role in shaping an epigenetic profile that is unique to one twin.

Epigenetics has wider implications on society. A very recent study indicates that these modifications can be inherited, with reports that trauma-induced epigenetic changes in Holocaust survivors are detectable in their children.7 This raises important questions about the impact our experiences and lifestyle choices have on future generations. Great care is taken during pregnancy to ensure that the developing embryo is not exposed to potentially harmful substances such as alcohol. However, perhaps we are responsible for the health of our children well before they are conceived? The last generation solved the structure of DNA,8 leading to major advances in our understanding of human biology, inheritance and disease. Now it is our chance to make a contribution to this field by delving deeper into the intricacies of our DNA modifications and their potential impact on future generations.

Olivia Walker
Alumna

References

  1. Galton, F. History or Twins. Inquiries into Human Faculty and its Development, Macmillan, 1883.
  2. Singh, V. Textbook of Clinical Embryology; Elsevier, 2012.
  3. Poulsen, P.; Esteller, M.; Vaag, A.; Fraga, M. F. Paediatric Research 2007, 61, 38R.
  4. Flintoft, L. Rev. Genet. 2005, 6, 667.
  5. Fraga, M. F.; Ballestar, E.; Paz, M. F.; Ropero, S.; Setien, F.; et al. PNAS 2005, 102, 10604.
  6. Jablonka, E.; Lamb, M. J. Evolution in four dimensions: genetic, epigenetic, behavioral, and symbolic variation in the history of life; MIT Press, 2005.
  7. Yehuda, R.; Daskalakis, N. P.; Bierer, L. M.; Bader, H. N.; Klengel, T.; et al. Psychiatry (in press).
  8. Watson, J. D.; Crick, F. H. C. Cold Spring Harbor Symp. Quant. Biol. 1953, 18, 123.