Category: news

What is the blast radius of an atomic bomb?

Picture the scene: you’re a scientist working for the US military in the early 1940’s and you’ve just been tasked with calculating the blast radius of this incredibly powerful new weapon called an ‘atomic bomb’. Apparently, the plan is to use it to attack the enemies of the United States, but you want to make sure that when it goes off any friendly soldiers are a safe distance away. How do you work out the size of the fireball?

One solution might be to do a series of experiments. Set off several bombs of different sizes, weights, strengths and measure the size of the blast to see how each property affects the distance the fireball travels. This is exactly what the US military did (see images below for examples of the data collected).

Picture1 Picture2

These experiments led the scientists to conclude that were three major variables that have an effect on the radius of the explosion. Number 1 – time. The longer the time after the explosion, the further the fireball will have travelled. Number 2 – energy. Perhaps as expected, increasing the energy of the explosion leads to an increased fireball radius. The third and final variable was a little less obvious – air density. For a higher air density the resultant fireball is smaller. If you think of density as how ‘thick’ the air feels, then a higher air density will slow down the fireball faster and therefore cause it to stop at a shorter distance.

Now, the exact relationship between these three variables, time t, energy E, density p, and the radius r of the fireball, was a closely guarded military secret. To be able to accurately predict how a 5% increase in the energy of a bomb will affect the radius of the explosion you need a lot of data. Which ultimately means carrying out a lot of experiments. That is, unless you happen to be a British mathematician named G. I. Taylor…

Taylor worked in the field of fluid mechanics – the study of the motion of liquids, gases and some solids such as ice, which behave like a fluid. On hearing of the destructive and dangerous experiments being conducted in the US, Taylor set out to solve the problem instead using maths. His ingenious approach was to use the method of scaling analysis. For the three variables identified as having an important effect on the blast radius, we have the following units:

Time = [T],       Energy = [M L2 T-2],      Density = [M L-3],

where T represents time in seconds, M represents mass in kilograms and L represents distance in metres. The quantity that we want to work out – the radius of the explosion – also has units of length L in metres. Taylor’s idea was to simply multiply the units of the three variables together in such a way that he obtained an answer with units of length L. Since there is only one way to do this using the three given variables, the answer must tell you exactly how the fireball radius depends on these parameters! It may sound like magic, but let’s give it a go and see how we get on.

To eliminate M, we must divide energy by density (this is the only way to do this):

eqn1.png

Now to eliminate T we must multiply by time squared (again this is the only option without changing the two variables we have already used):

eqn2.png

And finally, taking the whole equation to the power of 1/5 we get an answer with units equal to length L:

eqn3

This gives the final result that can be used to calculate the radius r of the fireball created by an exploding atomic bomb:

eqn4

And that’s it! At the time this equation was deemed top secret by the US military and the fact that Taylor was able to work it out by simply considering the units caused great embarrassment for our friends across the pond.

I love this story because it demonstrates the immense power of the technique of scaling analysis in mathematical modelling and in science in general. Units can often be seen as an afterthought or as a secondary part of a problem but as we’ve seen here they actually contain a lot of very important information that can be used to deduce the solution to an equation without the need to conduct any experiments or perform any in-depth calculations. This is a particularly important skill in higher level study of maths and science at university, as for many problems the equations will be too difficult for you to solve explicitly and you have to rely on techniques such as this to be able to gain some insight into the solution.

If you’re yet to be convinced just how amazing scaling analysis is, check out an article here explaining the use of scaling analysis in my PhD thesis on river outflows into the ocean.

And if that doesn’t do it, then I wish you the best of luck with those atomic bomb experiments…

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Arithmophobia – the fear of maths

New research shows that most parents can’t help their kids with maths homework because they have a fear of numbers. Here’s me being asked about the problem (and setting the presenters a farm animal themed maths puzzle) along with Martin Upton of the Open University on BBC Radio Scotland…

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2018 Abel Laureate Robert Langlands

The Norwegian Academy of Science and Letters kindly provided me with a scholarship to attend the Abel Prize week in Oslo earlier this year where I interviewed the 2018 Abel Laureate Robert Langlands.

In the first of a series of videos documenting my experience, Robert describes how he came to do Mathematics at university…

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Spring into action and get ahead of the competition

Wherever we look in the world, we see competition between different groups or beings. Whether it’s two animals trying to earn the right to a watering hole, people trying to assert their social influence, or simply two sports teams playing against each other, this sort of interaction appears in many different situations. As humans, we have a natural desire to rank things that are in direct competition: which is better? Who would win if they faced each other? How does their rivalry compare to others?

We want to know the answers to these questions because it makes us enjoy the competition more, and we feel that we learn more about it. Imagine being able to correctly predict who would win every football match for the rest of the season, you’d probably feel pretty pleased with yourself… But, apart from the inevitable bragging rights, being able to rank competing entities and predict outcomes is an extremely useful skill in many different areas of research, including sociology, economics and ecology.

Of course, you need a bit of maths if you’re going to rank things reliably; you can’t just trust a hunch! There are many different methods that have been used before for rankings, but a group of scientists at the Santa Fe Institute in the USA have come up with a new way of doing it using springs!

So, the ranking system is… a trampoline?! Not exactly. This ingenious method, called SpringRank, treats each interaction as a physical spring, so the model is a whole system of connected springs. Think of a football league: between each pair of teams there is a spring in each direction, and the force of each spring is determined by how many times they have beaten each other in the past. For example, Manchester United have played Liverpool 200 times, winning 80 matches and losing 65. In our spring system, this means that the spring connecting the two teams is biased towards Manchester United – it requires more force to move closer to Liverpool than it does to move towards Manchester United. With this setup, it turns out that the best ranking of the teams is found when you make the total energy in all of the springs as low as possible.

But why use springs? The bonus is that we’ve been studying springs for hundreds of years and so we know the physics behind how they work, which makes it easy to do the calculations. We can use the positions of the springs to work out the rankings of millions of different teams in just seconds! Not only is the maths simple, but it’s also very effective, especially compared to other methods currently used for ranking. In tests run by the researchers, SpringRank performed much better at ranking competitors, as well as predicting the outcomes of future clashes, than existing methods. The data set covered topics as varied as animal behaviour, faculty hiring and social support networks, demonstrating just how versatile the method can be.

This research is a wonderful example of how different areas of science can be combined to create a tool that can actually be put to use in the real world. When learning the subjects separately at school, it’s hard to imagine that you could take centuries-old ideas from physics, turn them into mathematical models, and stick them into a computer program! But here we are, able to work out who is likely to become friends (and enemies), which animals will make it through the heatwave, and whether it’s worth bragging about your favourite team before the game has even happened. So next time you’re challenged to guess the league winner, reach for SpringRank and jump ahead of the competition!

Kai Laddiman

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This robot is a ‘Cheetah’

Robots are developing at an incredible rate, with their ability to perform real-world tasks improving almost by the minute. Such rapid development doesn’t come without downsides, and there are many people who believe that artificial intelligence (AI) could become too powerful, leading to the possibility of robots taking our jobs, or perhaps even taking over the world! Whilst these fears might not be completely unjustified, let’s instead focus on the positives for the time being and marvel at the astonishing accomplishments being made in the field of robotics.

The Cheetah robot, developed by scientists at MIT, is roughly the same shape and size as a small dog, and has been designed to be able to walk across difficult terrains efficiently and effectively. Such a trait is particularly useful when we need to explore dangerous and hazardous environments that may be unsuitable for humans, such as the Fukushima nuclear power plant that collapsed in Japan in 2011. Like all robots, it uses algorithms to help it to navigate, stabilise itself, and ensure that its movements are natural. The latest version, the Cheetah 3, was unveiled in early July, and I think it’s fair to say that it wouldn’t look too far out of place in the animal kingdom!

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[Image courtesy of Sangbae Kim, MIT]

Perhaps the most impressive feature of the Cheetah 3 is that the strangely adorable hunk of metal performs the majority of its navigation without any visual input, meaning that it is effectively blind. The researchers at MIT believe that this is a more robust way to design the robot, since visual data can be noisy and unreliable, whereas an input such as touch is always available. Let’s imagine that you are in a pitch-black room; how would you find your way around? Your eyes are pretty much useless, but you can use your sense of touch to feel around the environment, making sure that you don’t bump into walls or obstacles. It’s also important to step carefully, so that you don’t misjudge where the floor is, or tread too strongly and break through something. The Cheetah 3 takes all of this into account as it gracefully glides across even the roughest terrain.

One of the key ideas that was addressed in the new model is contact detection. This means that the robot is able to work out when to commit to putting pressure on a step, or whether it should swing its leg instead, based on the surface that it is stepping onto. This has a massive impact on its ability to balance when it is walking on rough terrain, or one that is full of different obstacles; it also makes each step quicker and more natural. Going back to our dark room, you are likely to step quite tentatively if you can’t see where you are going as this will allow you to react to whatever surface you come into contact with, and adjust your motion as required. With the latest update, the clever ‘canine’ can make these adjustments by itself in a natural manner.

The Cheetah 3 also contains a new and improved prediction model. This can calculate how much pressure will need to be applied to each leg when it experiences a force, by estimating what will happen in half a second’s time. Returning once again to our pitch-black room, imagine how great it would be if you were able to predict what you’re about to step on and adjust your path accordingly – no more treading on sharp objects or stubbing your toe! The scientists tested the power of the new model by kicking the robot when it was walking on a treadmill. Using its prediction algorithm, the Cheetah 3 was able to quickly calculate the forces it needed to exert in order to correctly balance itself again and keep moving. Whilst I can confirm that no animals were harmed in the making of this robot, whether or not the robot itself felt harm is perhaps a question for another day…

The new and improved Cheetah 3 is certainly one of the more remarkable recent accomplishments in the field of robotics. Its natural movements and quick corrections mean that it excellently mimics animal navigation, and it is easy to see how such a robot would be extremely useful for exploring dangerous terrains. Such incredible progress in the study of robotics is as impressive and exciting as it is scary. While it is extraordinary that we are able to replicate animal movements so closely, it has rightly made many people slightly worried; will robots eventually be able to completely replace us? We can only cross our fingers that these critters have no plans for world domination just yet…

Kai Laddiman

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Play Nice!

Whoever said having fun is more important than winning was not a game theorist. Game theorists are mathematicians who study games, and how to win them. But they aren’t just interested in Snakes and Ladders – game theory also involves studying ‘games’ like nuclear standoffs, trade wars and even the competition of species as they evolve.

New research from the Institute of Science and Technology in Austria (http://dx.doi.org/10.1038/s41586-018-0277-x) might help us to use game theory for environmental good. Their findings look at perhaps the single most important problem we face in looking after the environment – ‘the tragedy of the commons’.

The tragedy of the commons plays out all around us, and relates to situations where everybody stands to benefit from damaging a useful shared resource. Everybody in the office exploits the ‘commons’ of the biscuit tin by taking a biscuit, but nobody can be bothered to go out and buy a new packet to keep the tin full. Eventually, the tin is empty, and everyone has to endure life without biscuits whilst someone looks for more. Such pointless suffering could have been avoided if only someone had acted sooner!

In a more serious setting, the tragedy of the commons can lead to catastrophic results. Take deforestation – the shrinking of the world’s forests as we use trees faster than they can grow back. It is in the individual interests of each logging company to spend all their time chopping down trees (which makes them money) and to waste none of it replanting them (which doesn’t). At least, in the short term. But over time, this clearly won’t work – the ‘commons’ of the world’s forests will be so damaged that everyone will lose out. Bad news for all you atmosphere fans out there.

The new research uses game theory to study the tragedy of the commons, to try and understand what we can do to prevent it. To stick with the logging example, the researchers treat logging companies as players competing in a series of very simple games, over and over, learning each other’s tactics. Each game is just a matter of choosing one of two options: Chop down trees without bothering to replant them, or take the time to replant them as well. Each time the choice is made, the company gets a reward depending on what they picked; they will get a bigger reward if they don’t use any of their time replanting. It looks like companies that are perfectly happy to drop-kick Dr Seuss’ orange defender-of-the-trees, the Lorax, are going to do better than their greener rivals.

At least, initially. The key to the new research is that in it, the games that have already been played affect the rewards up for grabs in the next game. If you keep choosing not to replant trees, then you may do better than your opponents in each game, but you’ll gradually make the rewards smaller and smaller as you start to run out of trees to cut down. So, you can’t just think about the profits to be won in today’s game – you have to think about what you’ll be playing for in tomorrow’s game too.

The researchers found that this makes a big difference to how companies will play. If previous games made no difference to the current game, then companies which don’t replant trees would do better than their replanting rivals. But, given that failing to replant the trees you cut down means worse prizes in the future, the companies which do replant end up doing a lot better than those that don’t bother. In other words, it pays to play nice.

The one catch to this is that the prizes have to get significantly worse when you choose not to replant. So, in practical terms, these findings suggest ways to make the ‘game’ of logging less environmentally devastating – by changing the rules. For instance, governments could pass laws which force any companies failing to replant trees to pay an increased tax on any future trees they cut down (or maybe pay for the Lorax’s extensive pension plan). This makes logging more like the game the researchers studied, where past choices quickly and significantly affect future rewards. So based on the researchers’ findings, such a law would make sure that doing the right thing and replanting trees is the better choice.

Yes, game theory is about winning. But by figuring out which rules reward the sort of people who go out and buy more biscuits for the tin, we can make sure the ‘winning tactic’ for the world’s most dangerous games is to play nice.

Joe Double

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