Equations Stripped: Logarithms

Stripping back the most important equations in maths so that everyone can understand…

Logarithms turn multiplications (hard) into additions (much easier) which enabled scientists in the 1600’s to calculate the trajectories of comets and the orbits of the planets around the sun. Nowadays, they are mainly used in Information Theory and Thermodynamics, but still have an important role to play mathematically in helping us to understand trends in experimental data.

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How many ping pong balls would it take to lift the Titanic from the ocean floor?

The answer to the latest question sent in and voted for by YOU.

Lifting the Titanic with ping pong balls was a real suggestion put forward in the 1970’s that needless to say did not happen. Let’s pretend it is possible and work out how many we would need using Archimedes Principle…

 

To vote for the next question that you want answered remember to ‘like’ my Facebook page here.

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BBC News – Maryam Mirzakhani’s Legacy

Live interview on BBC News about the legacy of Iranian Mathematician Maryam Mirzakhani who tragically passed away today (July 15th 2017). She was the first female winner of the Field’s Medal – the mathematical equivalent of the Nobel Prize.

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Naked Maths Trailer

Naked Maths is finally here!

Here’s the trailer for the new video series I’m making with the Naked Scientists taking a look at the maths that’s all around us.

 

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What do Aircraft and Fish have in Common?

What do fish and aircraft have in common? Well, water and air are both fluids. And when fish move their tails and bodies from side to side, they push against the surrounding water and leave behind a mini whirlpool or vortex, which contains information about the drag forces experienced by the fish as it moved along. If you can wind back the events that produced the vortex you can work out how it formed in the first place and therefore how much drag the fish felt. This is what Florian Huhn, from the German Aerospace Centre, has managed to do. And because aeroplanes produce very similar vortices in the air, the same technique can be used to develop improved aircraft designs, as he explains…

Florian – We were looking especially at the swirls, at the vortices that the fish typically create. The water slides really close to the skin of the fish, then the water gets some rotation with it and the result of this rotation put into the water when the fish passes are the vortices. Once we have found these vortices behind the fish, what we do is we use the velocity data from the simulations to move this piece of fluid backward in time.

Tom – By tracking the vortex backwards in time Florian and his time are able to see where the fluid making up the vortex originally came from. Interestingly, they found that water from both sides of the fish flows along its body and merges together at the tail where the vortex is then formed. This not only gives us an insight into how fish swim but can also be applied to many other problems.

Florian – At the tip of the wing – take a typical airplane – and we have a huge vortex but its bad for the pilot because if you land at the airport of course there were other planes before you and they all left their wing-tip vortices in the air somewhere. And you don’t want to hit those with your plane because that really shakes the plane.

Tom – Are they what cause the delay between other planes landing?

Florian – I know that there are other causes away form the runway and all these things, but I know its one limiting factor.

Tom – Understanding how these vortices form, that would give us an idea about how to make them smaller or how to make them go away more quickly and therefore leading to potentially more efficient airports.

Florian – Yeah that would be a good thing if that was possible.

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

Let the Floodgates Open

If you’ve been following so far, we know why it’s useful to know where river water goes when it enters the ocean, why we can build a model of the situation in the lab, what the most important variables in the problem are and what the experimental setup looks like. I suppose you’re probably itching to know what actually happens when we open the floodgates and release the freshwater from the model river into the spinning saltwater tank that is our ocean. In which case, let me point you in the direction of the video below.

It probably doesn’t make much sense without some added explanation so here goes… This is a false colour image of an experiment viewed from above. The freshwater from the river is dyed red with food colouring which means that we can convert the colour intensity into a depth measurement. The more intense the red food colouring, i.e. the more of it there is, the deeper the current must be. The scale starts with black to represent no current (as is the case for the saltwater ocean), then increases with the current depth through red, yellow, green and finally blue for the deepest parts of the current.

If you look at the very beginning of the video you will see that as the river water is released from the source it travels forwards and then is immediately forced to turn to the right. This is due to the Coriolis force arising from the rotating tank (see article 3). As it turns back on itself it eventually collides with the tank wall where it then propagates as an anticlockwise boundary current. The anticlockwise direction is set by the direction of the rotation of the tank (also anticlockwise). The boundary current continues to travel around the edge of the tank, eventually filling the whole perimeter and returning back to the source by the end of the experiment.

As well as the propagating current, a second persistent feature can also be seen in the video: the outflow vortex. This is the large whirlpool-like feature that forms next to the source of freshwater. As the initial current turns to the right and back on itself to collide with the tank wall, the flow divides. One part moves anticlockwise and forms the boundary current that moves around the tank edge, whilst the other part continues in a clockwise direction and re-joins the initial jet of freshwater from the source. The result of this is to form a whirlpool next to the source which grows in size as the experiment progresses. Both features are labelled in the image below so that you can recognise them in the video.

Screen Shot 2017-11-15 at 10.45.52.png

Now that we’ve identified the two main features of the flow – the boundary current and the outflow vortex – the next step is to try to understand them in more detail. This is exactly what the first two sections of my thesis are about, beginning with the boundary current. For example, we might want to know how fast the current moves around the tank, how its depth changes as it does so and whether or not it has a constant width. For the outflow vortex, we are interested in similar properties, such as how deep the vortex is at its centre, what shape it forms at the surface and how it grows in size during an experiment. By looking at the experimental video you can begin to get a grasp on some of these questions, but to really understand them in detail you need two key ingredients: measurements and a mathematical theory.

In my thesis, I begin by discussing the mathematical models used to describe the flow and then compare this theory with the data collected from the experiments, with the hope that they will agree. The theory can only be correct if the measurements support it – which is pretty much my thesis in a nutshell. Do the predictions from the mathematical equations agree with the observations in the experiments? If we’re going to compare the two, we’d better start by forming some equations, which brings me nicely onto the next topic…

 

You can read the rest of the articles explaining my PhD thesis here.

Exploring Saturn’s Newest Ring

Saturn is one of the most well-known planets in the solar system, perhaps owing to its distinctive set of rings. The largest of these rings, the H-ring, was only discovered as recently as 2009 and cannot be seen from Earth. Now, using images taken by NASA’s WISE spacecraft, scientists at the University of Maryland have given us the first insights into the structure and formation of Saturn’s outermost ring. I spoke to lead researcher, Doug Hamilton, to try to remove the shroud around Saturn’s most mysterious ring…

Doug – We’ve been able to look at the structure of the ring in these images and what we’ve discovered is that the particles are dominated by fairly small dust grains. So, 10 microns in size, a hundredth of a millimetre, very very small.

Tom – The rather catchily named H-ring was discovered in 2009 by the Spitzer space telescope, but what led to its formation?

Doug – This one seems to have come form the satellite Phoebe. So, Phoebe is a distant object of Saturn, its got a tilted orbit around Saturn, which means that it goes up and down and the height that it goes up and down matches the height of our ring. It tells us that Phoebe is the source of the dust particles.

Tom – So how does the material come off Phoebe to actually form this ring?

Doug – Its an impact onto Phoebe. There are collisions form other objects out at Phoebe’s distance, so think of a comet tail and all of the debris that come off of it, all of those bits are hitting the satellites and kicking off debris that form part of the ring. And then where Phoebe is located there are another 50 small satellites and occasionally they collide producing more debris.

Tom – The main rings of Saturn are visible from Earth – you can even see them with your own telescope on a clear night, so why can’t we see this new ring?

Doug – Since the dust particles are black they are really hard to see with visible light. You’ve got black dust particles set against the blackness, the darkness of space. So what we do is we realise that those black dust grains will absorb sunlight quite efficiently and they heat up and they re-emit heat, they re-emit red light. We use telescopes that are sensitive to that light.

Tom – You visualise this using infrared light, what does it look like on these images?

Doug – This is pretty neat aswell. It’s fairly planar, so it’s flat. You want to think of a coin, sort of a coin edge on. It’s a flat, yet extended structure. And we see it just because of how the dynamics of the particles work, we always see this thing edge on from the Earth. So you have your coin and you flip it on its side and you’re looking at it like that. So what we see in the sky is a rectangle.

Tom – What makes this ring particularly interesting?

Doug – It’s turned our whole idea of what rings are on its head, because we always think that rings are formed close to their parent planets and then you get moons and satellites further away. And this has just flipped it around – so we’re as far away from Saturn as you can get and still be in orbit around the planet and there’s satellites out there, but there’s also a ring. And so this teaches us – it makes us think differently about rings and hopefully we’ll open our minds a bitand learn a bit more abut how they form.

Tom – The analysis of these new images from the WISE spacecraft has given us greater insight into the structure of Saturn’s latest ring, but how does this new ring compare to the others?

Doug – We have the massive rings that you see on the telescope images, those have been known for a long time. Those are composed of big house-sized chunks of ice. And then we have Saturn’s E-ring. So that’s a ring that’s bigger than the main rings of Saturn and it’s extremely tenuous and it’s made up only of small dust grains. This large ring that we have at great distances is in-between. So we have big particles like you find close to Saturn, we have the small particles like you would find in Saturn’s E-ring and everything in-between.

Tom – Are there any other applications, any other knowledge you’ve gained from studying this ring?

Doug – We like things that are the biggest, the brightest, the furthest and so on… And this is the largest ring in the Solar System. So that’s kind of nice just on a gee-whizz level. On a deeper level what it teaches us is not to be too bound by our expectations. So nobody expected a ring to be this large and so as we go forth and look at exoplanets, and we’re looking in the data for signals, we’ve been surprised over and over with exoplanets. And back here in the Solar System we’ve been surprised by this ring and what it teaches us is to be open-minded. The universe has a lot of things it can do – it hasn’t shown us all of its secrets.

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

 

Optogenetics: the algae that started it all

It may seem like science fiction, but with optogenetics scientists can control the behaviour of animals by simply shining a light into their brains. And believe it or not this technology began… in algae! These single-celled plants are powered by the sun and contain built-in light detectors to control their behaviour. This discovery, and the isolation of the light sensitive protein that is responsible, led to the birth of the science we now call optogenetics. I went to see Cambridge University’s Otti Croze and Kyriacos Leptos to try to catch some of these incredible life-forms…

  • The algae Chlamydomonas Reinhardtii are invisible to the naked eye at around one hundreth of a millimetre or one tenth of the width of a human hair
  • Chlamydomonas contain a light-sensitive protein called channelrhodopsin which triggers the algae to swim using tiny arms called flagella
  • They are phototactic which means that they move towards light which they need to photosynthesise and survive
  • By introducing the light-sensitive protein into nerve cells in the brain scientists can use it as an on/off switch to control the cells by shining light onto them

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

 

Simulating Operations

Doctors need to practice for years to get good at performing often very tricky procedures, but rather than make mistakes on real patients, modern technology means it’s now possible to rehearse complicated operations using simulators first. And it can be very realistic, and very stressful, as I found out when Gareth Wills from Vascular Perspectives had me threading a tube into a pretend coronary artery, one of the blood vessels that can become blocked and cause a heart attack…

  • The wire is 35 thousandths of an inch in diameter and is inserted into the radial artery int the wrist
  • It’s then threaded up through the arm and across the top of the chest until descends down the aorta and joins up with the heart
  • A tube is then threaded along the wire to allow fluorescent dye to be injected onto the blood vessels of the heart
  • This can be seen under X-ray and any problems or blockages can be identified
  • The procedure is generally performed on patients complaining of chest pain at rest or after exertion
  • Using the simulator allows doctors to practice in a low-risk environment

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

Funbers 16, 17 and 18

The fun facts about numbers that you didn’t realise you’ve always wanted to know…

16 – SIXTEEN

Not only is sweet sixteen the title of one of the best/worst (delete as appropriate) television shows ever to exist (which is somehow still going – the trailer for the new season is below), it is regarded around the world as the first real step into adulthood. In the UK you can leave school and find a job, you can play the national lottery and you can also legally have some fun under the sheets…

Sixteen is also big in the world of sleep. Most humans will spend an average of 16 hours awake per day. I say humans, because in the animal kingdom it varies considerably. Your pet dog for example is awake for around 10 hours per day and your cat only 9 (classic cats). However, the laziest animal in the world is the Brown Bat, sleeping on average 20 hours per day. I’d like to think the other 4 are spent biting the necks of sleeping humans and practicing Transylvanian accents.

One of the most interesting facts about the number 16 is that it is used for the RGB colour coding system in web design. Have you ever played around in Photoshop (or the more budget Paint) and tried to select a custom colour? You’re faced with three numbers for Red, Blue and Green, that range from 0 to 255. In web design these numbers are converted into base 16, where 0-9 appear as numbers and then 10-15 are assigned the letters a-f. Consider the colour purple for example, its RGB value is (128, 0, 128). Converting that into base 16 you have (8 x 16) + (0 x 1) = 128, (0 x 16) + (0 x 1) = 0, (8 x 16) + (0 x 1) = 128 and so the final value is 800080. The two numbers/letters represent how many lots of 16 = 16¹ and how many lots of 1 = 160 you need to make the colour value between 0 and 255. There’s a few more examples below for good measure.

Screen Shot 2017-10-23 at 21.51.01

17 – SEVENTEEN

Pick a number between 1 and 20… how many of you said 17? Seventeen is called a ‘psychologically random number’ because it comes up more often than it should when people are picking random numbers. The even numbers have a nice pattern, as do numbers ending in 0 or 5, which just leaves those ending in 1, 3, 7 or 9. A lot of these numbers also tend to be prime numbers (only divisible by 1 and itself) which makes them even more appealing. 17 fits both rules which is why it’s so popular.

Now remember when I said thirteen was the unluckiest number in the world? (If you don’t you should go back and read the previous funbers article – it’s great) Well, seventeen wants a word. In Italy, the number 17 is feared because in Roman numerals it reads XVII, which can be rearranged to spell Vixi meaning ‘I am dead’. It may sound a little far-fetched, but Renault was so worried by the superstition that it changed the name of its R17 model to R117 for the Italian market. Looking at the R17 below I’d say shoddy engineering is more likely to be the cause of death rather than the number 17… how small are those wheels?!

1976_Renault_17_Gordini_(21398064092).jpg

18 – EIGHTEEN

Eighteen is another big number for symbolising adulthood in many cultures, perhaps none more so than in democratic countries where in most cases it is the age at which you are eligible to vote. This hasn’t always been the case, however, as during the Vietnam War the voting age in the US was 21, which meant that you could be conscripted into the army to defend the democratic freedom of the people of Vietnam, but yet not be able to vote yourself back at home. Fortunately this did lead to the age being lowered to 18 in 1971.

In the UK, most people celebrate their eighteenth by having an alcoholic drink or two… or more accurately getting drunk/smashed/wasted/bungalowed/any other word used to describe being intoxicated. The reason being that eighteen is of course the legal age of drinking in the UK and is the most popular choice around the world, with a few notable exceptions… To drink in Egypt or the US you must be 21 and it’s completely illegal in some countries such as Libya, Saudi Arabia and Iran. Other countries are a little more ‘lax’ with their laws, – I’m looking at you Haiti – as they have no restriction on the age at which you can consume alcohol, although apparently in Haiti it is often restricted to those of at least ‘school age’, which is the ripe old age of 6…

 

You can find all of the funbers articles here.

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