Carnival of Mathematics 167

You know you’ve made it as a maths communicator when you have the honour of hosting the Carnival of Mathematics (if you have no idea who I am or what I do then check out this interview for St Hugh’s College Oxford). But, before we get to the Carnival proper, as the creator of ‘Funbers’ I can’t help but kick things off with some fun facts about the number 167:

  • 167 is the only prime number that cannot be expressed as the sum of 7 or fewer cube numbers
  • 167 is the number of tennis titles won by Martina Navratilova – an all-time record for men or women
  • 167P/CINEOS is the name of a periodic comet in our solar system
  • The M167 Vulcan is a towed short-range air defence gun
  • 167 is the London bus route from Ilford to Loughton

Now that we all have a new-found appreciation for the number 167, I present to you the 167th Carnival of Mathematics…

Screen Shot 2019-03-06 at 00.15.32

Reddit’s infamous theydidthemath page tackles ‘fake news’ on Instagram with a quite brilliant response to a post claiming that avoiding eating 1 beef burger will save enough water for you to shower for 3.5 years. Whilst the claim is hugely exaggerated we should still probably stop eating beef…

Next up, Singapore Maths Plus take a light-hearted look at the definition of ‘Singapore Math’ on Urban Dictionary – which is apparently the world’s number one online dictionary (sounds like more ‘fake news’ to me).

Math off the grid jumps in ahead of hosting next month’s Carnival to discuss the book ‘Geometry Revisited’ with a re-examination of the sine function as a tool for proving many fundamental geometric results. Scott Farrar also has the sine bug as he encourages us not to reject imprecise sine waves, but instead to consider the circle that they would form (warning contains a fantastic GIF).

soviet_license_plate

John D Cook introduces what is now my new favourite game with his explanation of the ‘Soviet Licence Plate Game’. Have a go at the one to the right – can you make the four numbers 6 0 6 9 into a correct mathematical statement by only adding mathematical symbols such as +, -, *, /, ! etc. ? Send your answers to me @tomrocksmaths on Social Media or using the contact form on my website.

If by this point, you’ve had enough of numbers (which apparently happens to some people?!), then here’s a lovely discussion of ‘numberless word problems’ from Teaching to the beat of a different drummer. If that doesn’t take your fancy, how about some group theory combined with poetry via this ridiculous video of Spike Milligan on The Aperiodical

If like me you’re still not really sure what you’ve just watched, then let’s get back to more familiar surroundings with some intense factorial manipulation courtesy of bit-player. What happens when you divide instead of multiply in n factorial? The result is truly mind-blowing.

koncnaslika-krog

Finding our way back to applications in the real world, have you ever wondered how the photo effect called ‘Tiny Planets’ works? Well, you’re in luck because Cor Mathematics has done the hard work for us and created some awesome mini-worlds in the process!

Sticking with the real world, Nautilus talks to Computer Scientist Craig Kaplan who discusses how the imperfections of the real world help him to overcome the limitations of mathematics when creating seemingly impossible shapes. They truly are a sight to behold.

With our feet now firmly planted in reality, let’s take a well-known mathematical curiosity – say the Birthday Problem – and apply it to the 23-man squad of the England men’s football team from the 2018 World Cup. Most of you probably know where this one is going, but it’s still fascinating to see it play out with such a nice example from Tom Rocks Maths intern Kai Laddiman.

The fun doesn’t stop there as we head over to Interactive Mathematics to play with space-filling curves, though Mathematical Enchantments take a more pensive approach as they mourn the death of the tenth Heegner Number.

Focusing on mathematicians, Katie Steckles talks all things Emmy Noether over at the Heidelberg Laureate Forum Blog, whilst I had the pleasure of interviewing recent Fields Medal winner Alessio Figalli about what it feels like to win the biggest prize of all…

And for the grand finale, here are some particularly February-themed posts…

The next Carnival of Mathematics will feature mathematical marvels posted online during the month of March, which of course means ‘Pi Day’ and all the madness that follows. Good luck to the next host ‘Math off the grid’ sorting through what will no doubt be an uncountably large number of fantastic submissions!

Funbers 21

Funbers has reached the age of adulthood as the series turns 21 today! Twenty-one is a popular number in gambling, sports and politics, as well as being the number of shots fired in a ceremonial gun salute. To find out why you’ll have to listen to the latest episode below…

You can listen to all of the Funbers episodes from BBC Radio Cambridgeshire and BBC Radio Oxford here.

The Right Ingredients

If you’ve ever had a sprinkler in your garden and like me used to run through it in the summer as a kid, you’ll have noticed the different pattern of the water for when the sprinkler is on compared to when it’s off (or maybe I was just a weird kid). When not rotating, the water shoots out in a straight line in all directions. If you stand at the centre and look directly North, the water will shoot out in a straight line and land directly North of where it started. Now if you turn the sprinkler on so that it starts rotating the patterns become curved. As the sprinkler turns it will still release water directly North, but because of the rotation a sideways force will push it to the left as it’s released (for a sprinkler going anticlockwise). This gives a curved path and the water will actually land a fair bit to the left (or East) of direct North.

The same thing that happens with your sprinkler is also happening with the Earth. This sideways force is quite weak so it only really affects large distances, which means Usain Bolt running the 100 m is safe from its effects, but Mo Farah running the 10,000 m – he will feel it. And the force, called the Coriolis force, is stronger the further you go North or South from the equator. That means it’s very important in controlling how river water moves once it enters into the ocean. As I hinted at in the previous article, it is one of the most important parameters affecting the motion of the water and as such we have to make sure that it is represented correctly in the lab experiments.

For my experiments we did this by mounting a giant fish tank (literally the best way to describe it) on top of a turntable whose speed could be controlled by a computer. The Earth rotates at a very high speed of 1000 mph, but the actual Coriolis force arising from the rotation is relatively small at f = 0.0001 s-1 at mid-latitudes (i.e. around the UK). As with building a scale model of a river in the lab, we can’t just scale down the Coriolis force. We know that it’s important because without it the water moves in a completely different direction and so we have to make sure it’s represented in the experiments, but knowing what value to use requires a little more thought.

This is where we come to the real heart of applied maths – scaling analysis – and it works as follows… Suppose we have 3 parameters that we know are important in our problem, i.e. without any of them the outcome changes dramatically (think of a sprinkler with no rotation for example). Taking one of my favourite problems of the spread of heat through a rod, we might expect that the material the rod is made from and the length of the rod will both affect the time it takes for the heat to spread out. In fact, the particular property of the material that we want here is the thermal diffusivity – a fancy way of saying how quickly the material heats up. Now, if we look at the units of these three quantities, we have time T in seconds (s), length L of rod in metres (m) and thermal diffusivity α in metres squared per second (m2s-1). Fiddling around with these quantities we can create a dimensionless parameter (one without units) given by

dimensionless

and this then tells you that the time taken relates to the length and the thermal diffusivity in the following way

time

At this point it may seem quite random and not really of any use, but when you actually write out the full problem mathematically and solve the resulting system of equations, your solution is given by

heat solution

The only part we need to look at is the exponential term (e) as this is the only part of the solution that has a dependence on time (t). You’ll notice that the exponential power is the same as our dimensionless quantity above (plus some numbers n and π). Solving the equations is great if you need a full solution, but if you only need certain information, like which factors affect the time taken, then we actually already had that from our scaling analysis T ∼ L2/α. To speed up the spread of the heat and decrease the time (T) we just increase the thermal diffusivity (α) or we decrease the length of the material (L). The solution tells us this, but so too did our scaling analysis (with much less effort).

Hopefully you’ve spotted that the trick is to first identify the parameters that are most important to a given problem, and then to combine them in such a way that all of the units disappear – making them dimensionless. For a river our parameters might be rotation f in s-1, the volume flux Q (amount of water flowing out of the river) in m3s-1 and the density difference between the freshwater of the river and the saltwater of the sea, which we represent as the reduced gravity g’ with units ms-2. Combining the units, we find that the dimensionless parameter is given by

I.png

For real rivers we can compute the value of I, and when designing the lab experiments we need to make sure that the value of I for the experiments matches up with the values for the real rivers – that’s the key to capturing the most important real-world behaviour. It also has the advantage of allowing us to vary the parameters by quite a large amount depending on what is easiest to achieve in the lab. For example, a volume flux of 2,200,000 cm3s-1 of the River Rhine cannot be used in the lab (the maximum I used was about 100 cm3s-1), but what we can do is increase or decrease the other parameters to keep the value of I about the same. Here we can increase f and increase g’ which are must easier to do in the lab, we just need a fast turntable and some seriously salty water, which brings me nicely onto our next topic: my experimental setup.

 

All of the articles explaining my PhD thesis can be found here.

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.

 

The Game of Life

The simple mathematical game that inspired the pattern on the facade at Cambridge North station – live interview with BBC Radio Cambridgeshire.

 

Examples (image credits: Wikipedia)

Stable ‘still-life’ patterns that remain fixed for every turn.

Block                 Beehive             Boat                Loaf                    Tub

Game_of_life_block_with_border       98px-Game_of_life_beehive   82px-Game_of_life_boat     98px-Game_of_life_loaf     Game_of_life_flower

‘Oscillators’ that cycle through a number of designs repeating every few turns.

Beacon (period 2)     Blinker (period 2)     Toad (period 2)

Game_of_life_beacon                 Game_of_life_blinker                Game_of_life_toad

Pulsar (period 3)          Pentadecathlon (period 15)

Game_of_life_pulsar                I-Column

Spaceships that travel across the board forever.

Glider             Lightweight Spaceship

Game_of_life_animated_glider        Game_of_life_animated_LWSS

You can play the game of life for yourself online here.

Funbers 4.6692… 5 and 6

4.6692… – FEIGENBAUM’S CONSTANT 

A new addition to the list of important mathematical numbers, Feigenbaum’s constant was only discovered in 1978 through the study of chaotic systems. Chaos in the mathematical sense is pretty much what you might expect – it describes something that is completely unpredictable. My favourite way to think of it is using magnets. If you have two magnets each with a North and South pole (those funky blue and red ones from school) and hold opposite ends near to each other, then they are attracted and will quickly move together. This is a nice stable system – we know what will happen every time. And if you slightly adjust how far apart you put the magnets, or at what angle you hold them near to each other, it won’t make a difference; they will still join up. Now, what happens if you try and hold two of the same ends near to each other? Well, for starters they’re now going to repel. But, what’s important here is that they move around unpredictably. Try it. If you force them towards each other they push back and can end up moving sideways, up, down, pretty much in any direction. And if you try to repeat the same movement you can’t. That’s chaos.

So, what does Feigenbaum’s constant have to do with all of this? The answer lies in fractals — a repeating pattern that continues to look the same despite zooming in further and further (see image below). The rate at which the image zooms in is Feigenbaum’s constant — cool, huh?

Mandelbrot_zoom

Feigenbaum’s constant is important because it describes the rate at which the simplest mathematical systems (called one-dimensional maps) descend into chaos. The really awesome thing is that you can create all kinds of different systems (following a few basic rules) and they will all go into chaos at the same rate given by this exact constant.

5 – FIVE

Apart from being a hit(?) boyband from the 90’s, five is also a number. It has the fun quirk of being the fifth number in the Fibonacci sequence: 1, 1, 2, 3, 5… where the next number is the sum of the two before it. It’s also the number of human senses (or at least the main ones) and the number of rings in the Olympic symbol. My favourite use of five, however, has to be the 5 Platonic Solids. These are the most, regular, symmetrical and beautiful shapes in all of maths – some people would say that they are so beautiful that they should be permanently inked onto your body… who are these crazy people? Hint: it’s me.

To be a Platonic Solid you need to be 3D, with each face the same shape, and at each corner the same number of faces must join together. Take the simplest example: the cube. We have 6 faces that are all squares and at each corner 3 squares join together. The smallest Platonic Solid is the Tetrahedron or triangle-based pyramid which has 4 faces that are all triangles. After the square comes the Octahedron which has 8 faces that are all triangles – basically two square-based pyramids stuck together so that the square face is inside the solid. The last two are the Dodecahedron with 12 pentagon-shaped sides and the Icosahedron which has 20 triangles all stuck together. These are the only 5 shapes that satisfy the very simple set of rules and they appear everywhere in nature, from the shape of viruses to the structure of molecules. In short, they’re grrrrreat (credit to Tony the Tiger).

6 – SIX

The number of impossible things that the Queen from Alice in Wonderland believes before breakfast and also the number of legs on every insect on earth. Insects are in fact the largest group of species we have on the planet, and even outnumber all of the other species combined… We also have 6 Quarks, which as well as being some of the fundamental particles that make up our universe, have the fantastic names of up, down, bottom, top, charm and strange. Sounds like a fun weekend…

Mathematically, six is what we call a perfect number: all of its factors (the numbers that divide it) add up to give six 1 + 2 + 3 = 6. It also happens to be the only number in existence where not only do all of its factors add up to give the number, they also all multiply together to give the number too: 1 x 2 x 3 = 6. Perfect numbers (the ones that equal the sum of their factors) are a little more common, the next three are: 28, 496, 8128. There are more of course, but they get a little tricky to find. Be my guest and see if you can work them out… sounds like another fun weekend to me.

You can find all of the funbers articles here and all of the episodes from the series with BBC Radio Cambridgeshire and BBC Radio Oxford here.

The Prisoner’s Dilemma

I demonstrate the classical game theory problem of the prisoner’s dilemma live on BBC Radio, with a short introduction to the subject from Sergey Gavrilets. You can listen to the full interview via the Naked Scientists here.

A lot of our decisions, although we may not realise it, rely on maths. Game Theory is the study of the decision-making process and it can be applied to almost any subject including economics, political science, psychology, and even biology. The last one is of particular interest to Sergey Gavrilets, a mathematician at the University of Tennessee, who uses game theory to model early human behaviour…

  • Game theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers
  • An example would be if your phone was ringing in the next room whilst you are watching a movie with your family – the group benefits from answering the phone, but no-one wants to make the sacrifice themselves to do so
  • The prisoner’s dilemma looks at whether or not you should snitch on your friend and have a chance of going free, or stay quiet and hope your friend doesn’t snitch
  • Ultimately the best solution is for you both to stay silent, but it relies on trusting your friend for it to work!

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