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).

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):

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):

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

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

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…

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!

[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…