Today, we are going to be talking about electrons. The little charged guys that wizz around the atom and lead to the properties of ever material we will ever see:

Now, what you see up there is the most simplistic image of an atom you will see in high-school. The nucleus in the middle, a dense cluster of particles, with electrons in circular shells around it. I assume when you were studying it you never gave thought into why the electrons were arranged like this, yet another thing you were told happen but not the reason why. But today is your lucky day! You see, this is where our quantum physics comes into play.

The idea we need to look at is called the Pauli Exclusion Principle. The main idea with this is that no two electrons can ever be in the same quantum state. So what does that mean? Well a quantum state relies on 4 quantum numbers or just simple descriptions of what the particle is doing at a certain time. I feel that if I were to go into these, it would be both confusing and unnecessary, but if you are interested, at the end of the post there will be a paragraph on it.

Anyway! What does this mean for us? Well, here is something pretty cool. This is observable in the atom (not the cool part yet), you see, the electrons have to fit into individual “shells”, and “orbitals” that are subsets of these shells and you can’t have more than two electrons in each orbital. This makes sense, however, and here comes the cool part, it is also relevant on a universal scale. Because all electrons seem to be connected in this way, that means that any time something happens to an electron somewhere it must mean that an incredibly small change overcomes all the other electrons in the universe! The light coming out of this webpage and hitting atoms in your eyes? It is making all the other electrons in the universe rearrange themselves so that it works! Every time you turn on any electron device, move a muscle or even just think, you are rearranging the universe!

Unfortunately, these changes are so small and unnoticeable that we can’t use it for communication but it is an incredibly interesting and useful concept in other ways. For example, semi-conductors, the materials that create the complicated components you will find in laptops and phones and other electronic devices, rely entirely on this effect to work, with electrons effectively stacking to form interesting and useful effects.

So, just to illustrate again that there are parts of the most basic quantum physics that can be interpreted in so many ways, there is a very interesting and strange ways in which this theory can be interpreted. A particularly famous one explains it with a very simple idea. There is only one electron. One electron in the whole of spacetime. However, that electron can travel in space and in time, and because it can travel in time, this means that it appears more than once in any time frame. Every single electron in the universe…is the same electron! It would explain why they are all so similar at least!

Anyway, I hope you enjoyed this post!

For those of you interested in quantum states, there are 4 quantum numbers and these describe how the electron is in an atom. The first says which shell it is in. It is a simple number, if you look to the diagram at the top, there are two shells. Then, the next two describe the orbital of the electron. Orbitals are very interesting, they look like this, and arise due to the wavelike nature of electrons. In fact, you can see distinct similarities between the way that vibrations can appear in a membrane. The final one is something called spin. You don’t need to know very much about this but you can have positive and negative spin. This is how you can have two electrons in each orbital, they have opposite spin.

Firstly here is a link to the original post analysing the cost on centives.net, an excellent blog on “quirky economics” from how much of a cut Domino’s take on your toppings to the game theory involved in The Hunger Games. I recommend that you take a bit of time to read this before getting started on our article.

The major premise of their calculation is that we can assume the same steel density in the Death Star would be about the same as that on HMS Illustrious. However, this doesn’t take into account the fact that the Death Star is largely empty inside, so really, the vital statistic is in fact surface area, as this is where the majority of the mass of the steel will be. In addition, by scaling up directly with a volume ratio, the post doesn’t consider that that will involve massively thickening the walls of the Death Star to a completely unnecessary level.

Using the geometry of HMS Illustrious, the best estimate I can get for the surface area of the steel hull is about 60000 m². The weight of the steel is given as 22,000 tonnes, which gives us a “mass per unit surface area” for the ship hull of 377 kg/m².

The online Star Wars holy grail that is Wookiepedia gives the Death Star a diameter of 140km. The Death Star is (roughly) spherical, so we can find its surface area with the formula A=4πr², which gives the Death Star a surface area of 6.15 × 10¹⁰ m². Multiplying this by our “mass per unit surface area” constant, we can find the total mass of the Death Star, which is only 2.32 × 10¹³ kg, or about 23 trillion tonnes. This sounds like a vast mass, but when you compare that to centives’ original estimate of about 1 quadrillion tonnes it is actually pretty trivial in comparison. In fact, the difference is so great that our estimate requires nearly 500,000 times less steel!

This is a complete game changer. With the original calculations, even if the entire world’s resources were put into a giant Death Star piggy bank, it would take 13,000 years just to get the funds to buy enough steel, and then it would take more than 833,000 years to produce the steel! However, with the revised calculations, a death star is beginning to flirt with possibility.

The GWP (Gross World Product) as of 2012 is just over $69 trillion (all values in USD), and using centives’ steel prices, the steel would now only cost about $18.3 trillion. In other words, with the whole world putting money towards it, the funds could be raised in just over three months! However, seeing as the petition was put to the White House, we find it unlikely that the rest of the world is likely to want to fund the USA to build a death star, so they may have to go it alone. Fortunately for them, they are the world’s largest economy, and could raise the funds purely out of their own GDP in just 15 months.

And then we get to steel production. Without international cooperation, this could be problematic. At current rates, the US are producing 86 billion tonnes of steel per year. Whilst better than 833 millenia, it would still take them 270 years to produce enough steel. Should production actually commence, one would hope they’d make steel more of a priority, but if the rest of the world did cooperate and not refuse to sell steel to them on the grounds that they were building a weapon capable of destroying the planet, then it would only take 15 years for enough steel to be produced. This would mean that in order for production to be completed though, the Democrats would either have to win the next three elections (at least), or some cross-party consensus would have to be reached to finish what they started.

So within 15 years, the US could have their very own Death Star! But once this happens, they must choose their first target, and there is no doubt as to where that should be. No, not Swindon, but Mars: that evil red planet which is (maybe) teeming with intelligent life that seeks nothing but to destroy us. It’s the only logical option.

But at this point, all they have is a giant steel sphere. That wouldn’t do much to blow up Mars.

Fundamentally, in order to “blow up” a planet, you need to get the entire mass of the planet to the escape velocity so that it can escape its own gravity. On Mars, this escape velocity is 5000 m/s. We can use the following equation for kinetic energy (assuming a 100% efficiency) to find out how much energy would need to be inputted to blow Mars up.

Where E is energy in Joules, m is mass in kg, and v is velocity in m/s.

We already know that v is 5000, and the mass of Mars is 6.4 × 10²³ kg, so plugging those numbers in, we get a required energy of 8 × 10¹⁵ Petajoules, or 8 thousand billion billion billion Joules. That’s 40 trillion times more powerful than the most powerful nuclear bomb ever, the Tsar Bomba, but surprisingly, this is still potentially possible – this would require less than a third of the obtainable Uranium-238 reserves in the world in a fusion reaction!

So, Mr Obama, on the day of your inauguration, we do not accept your claim that building a Death Star would be impossible! We urge you to adopt this ambitious and exciting policy at once, and take your opportunity to be remembered as a truly great president!

So lets start with what Schrödinger’s cat really is. Erwin Schrödinger was a fantastic physicist in the 20th century. His influence spread across numerous fields, but he is best known for his work in quantum mechanics, even forming a whole new part, involving waves and which spawned the second most famous physical idea with his name associated with it, Schrödinger’s wave equation. But that isn’t what we are talking about today. Quantum physics is a very strange subject, we have talked about it before so I’m not going to go into much detail, but because of its complexity, there are many ways to interpret it. At Schrödinger’s time, the most popular was something called the Copenhagen interpretation.

Schrödinger found a part of this interpretation very difficult to accept, concerning an idea called superposition. It said that a particle could be in two states at the same time until it is measured. Schrödinger’s cat was Schrödinger’s attempt of showing how this was impossible, through applying it on a macro scale. The experiment involved putting a cat in a box with a geiger counter and a radioactive substance. After an hour, there is a 50/50 chance that the radioactive substance would have emitted a particle.  When the geiger counter detects that this has happened, it will break a vial of poison, thus killing the cat. Schrödinger said that, after an hour has passed, using this quantum interpretation, the radioactive substance has both emitted a particle, and not (so far so accurate). As the chain of events would follow, the cat is then both alive and dead.

The main problem that Schrödinger was trying to show here that the idea that something so obviously either one way or the other, could be in this state until we observed it was entirely possible with this interpretation. However, nothing in quantum says that isn’t possible. Superpositions have been proven to exist, the question is of when this state collapses into one outcome. This problem is one that has taxed physicists for decades and is one of the great unanswered questions in physics. Minute Physics did a great video on this:

We hope you’ve enjoyed this post! If you did then please check out our last two posts:

How to win at Monopoly: how Game Theory shapes your family games – Monopoly is a very mathematical game. How can statistics tell you how to win?

Santa Claus is coming to town and it is going to cost him £40m this year! - Though christmas is past, I’m sure you would like to see how we calculated the Santa’s schedule and why he needs 540 million litres of oats!

The strategy is clearly to get a monopoly on one specific colour group (or more if you can!) rather than get several individual properties, but not all colour groups were created equal! As you go round the board they get more expensive, the cheapest being Old Kent Road at £60, to the most expensive, Mayfair, at £400.

To best decide what colour groups to go for, we need to put together some tables about the total cost of the colour group, and the income that would be received from landing on each of the properties.

In the tables, I have considered the income from each property being landed on once. Although there are only two properties in the Brown and Dark Blue groups, this is accounted for in the fact that the costs for houses and the properties themselves are lower. I then took the income that the number of properties or houses would get and divided them by the costs of the properties and improvements. Generally, the value gets larger (and better) the greater the number of houses are placed: the game is trying to provide an incentive. However, what happens to this ratio as we move round the board?

 

In the later graphs, a “wavy” pattern becomes clear. This arises because two adjacent colour groups have the same prices to add a house or hotel, for example, the Brown and Light blues will both cost £50 to put a house on. The Pinks and Oranges both cost £100, the Red and Yellows £15

According to these graphs, by far the best investment in the game would be building hotels on the Light Blues: at a total cost of just £1070, each of them being landed on once would return you £1700.0 and the Green and Dark blues £200. However, each successive property group has higher rent, meaning that the more expensive of the two property groups that share a house development price will give better value as more of them have been built, so this leaves the better half of the colour groups as being Light blue, Orange, Yellow and Dark blue.

There are two sets of properties that we haven’t yet considered: the Stations and the Utilities. You cannot build on either of these properties, so the options remain quite limited, but owning 3 or 4 railway stations is an excellent investment – the main advantage is that the profitability is extremely high for such a small input of capital, only £200 per station and no building is required.

The Utilities are on average a mediocre investment, although your income from them will depend much more on luck as the rent is directly proportional to the roll of the dice that brings the player to that square.

In this post, we are going to be talking about how Santa will travel. I’m sure I’m not the only one that wondered about the logistics of one man, with a vehicle powered by flying reindeer, visiting every celebrating child in world so this year, I decided to actually look into it! We will be looking at the distance he travels, how much energy he needs and how much that could cost.

The first thing I considered was the distance he has to travel. Whilst researching the post this week, I came across this website. It is trying to calculate the speed that santa has to go, something I will do later. They take the approach of calculating the number of households he has to visit and assuming, though acknowledging it is not as simple as this, that they are equally distributed across the world. I think we are going to take a different route than this, the result just didn’t seem far enough. Instead, I think that most efficient method is similar to what you would find on the back of your fridge:

This pattern of rows means that he can cover as much of the earth as possible in the shortest distance. It also means he can take advantage of time zones. Starting in the east, he could travel for a full 31 hours under the cover of one night. I decided that 160 rows would be a fair number of rows. This is the equivalent of going around the earth itself 160 times from north to south to north again. The maths is a bit harsh, so if you want to hear about that go down to the end of the post where I will be going through specifics. The distance he travels then, is about 5 million kilometers. This is about a thirtieth of the distance from the earth to the sun and thirteen times the distance from here to the moon. Suffice to say that he would have to moving pretty fast!

As I said, if he is tactical (and seeing as he has done it for at least 200 years, you can assume he is), he would take advantage of the fact that night moves across the world, giving him 31 hours to cover the 5 million kilometers. That means he has to travel at about 160,000 ^km⁄_h or 45,000 ^m⁄_s. That is 132 times the speed of sound and 46 times the speed of the fastest vehicle we, as a species, have ever created, the Lockheed SR-71 Blackbird plane.

This speed, though immense, is not actually the biggest problem for santa at his point. That comes when you start to think about air resistance. Again, the physics here involves some nasty equations, but there are a few things to take into account. The amount of drag increases exponentially as the velocity of the object increases, but the higher you go in the earth’s atmosphere, the density of air decreases. Overall though, at a speed as high as this, the drag force will be huge! Again, look at the bottom for the actual calculations, but it comes to about 207 million newtons, the equivalent of the weight of the amount of water in 8 olympic swimming pools pushing on every square meter of the vehicle.

The amount of power required to over power this is the force times by the velocity and seeing as this velocity is so high, this means a huge amount of power is required. Namely 9.3 million MegaWatts.You would need to open up 414 of the largest power stations in the world (the Three Gorges Dam) to create this amount of power. From this, by dividing by the time he is doing this for, we get the amount of energy he needs, 300,000 MWh (MegaWattHours) or  1,081,356,687 MJ (MegaJoules).

When you burn 1000 litres of petrol (gasoline), this releases 10 MWh of energy. This means that if santa was using petrol and the heat from it to power his flight, he would need 30 million litres of it! The price of this varies where he buys it but if he was to buy it in the UK, it would cost  almost £36,000,000 ($58,000,000) though buying in the USA would be a bit cheaper at only £14,000,000 ($23,000,000).

But Santa doesn’t use petrol. His vehicle is animal powered. So, how much would that cost? Well lets assume the diet that santa feeds his reindeer is pretty bland, they just eat oats. Oats are 66% carbohydrates and 2% fat. We also know that within every kilo of carbohydrates, there is 17MJ of contained energy, and within every kilo of fat there is 37MJ. We also know that respiration, the way that animals get energy , is not totally efficient, 60% of the energy is released as heat.  Using all this and the energy we know we need, we can calculate the mass of oats we need…and it is a lot! Santa would have to stockpile over the course of the year, as he couldn’t buy it all at once, 226,000 metric tons of oats! That is 540,000,000 litres of raw oats! This could cost him around £40 million ($63 million).

So I ask you all, lets spread the word! Not only does santa deliver presents, not only is he a symbol of christmas, but he is a millionaire that spends huge amounts of money on oats, he has a supersonic sleigh that is 46 times faster than any man-made vehicle ever and his reindeer can eat 540 million litres of oats over the course of one night! Anyway, we wish you a merry christmas and a happy holiday from us here and hope that you get a visit from the fat, red, supersonic man from the north pole.

We hope you’ve enjoyed this post!

If you did like the post then please check out our last two posts:

Nuclear Power: What is it and why is it important. (Fusion + Fission) – We take a look at the mechanics behind the power source of 18% of the world’s energy.

Matter and Antimatter: Why hasn’t the whole universe exploded? - When they come together, matter and antimatter explode, so why hasn’t that happened to the whole universe?

Appendix

When calculating the distance santa travelled, I thought of 160 separate rings equal distance apart around earth. I assumed the earth to a perfect sphere, though I know it isn’t, as this makes the calculation easier and does not corrupt the results very much.

The equation of each radius is simple. We use Pythagoras:

Where r_o is radius of the earth and n is the number ring it is, where the equator is 0, the north pole is 80 and the south pole is -80.

The circumference of all of these rings is therefore the sum of all of them from ring -80 to ring 80 times by 2 pi:

 Inputting the radius of the earth, 6371 Km, we get our distance.

Here is how we worked out the drag:

Key

Those are the main calculations that I did, however, if you want to look at others such as details on the oats and look at our sources, download the excel spreadsheet I used for this:

Calculations

Well in fission, we take one heavy thing and split it in two. Fusion is the opposite. We take two very light particles and push them together to make a new one.

The two particles we push together are called deuterium and tritium. They are different isotopes of hydrogen, in other words, they both have one proton, making them hydrogen, but they have different numbers of neutrons. Deuterium has one, and tritium has two. Pushing them together is difficult. The glue of neutrons only works when the particles are touching, but the protons repel each other larger distances. But once they get very close, the neutrons grip on and energy is released.

Once the atoms have gripped onto each other, an entirely new substance is made, Helium. Helium has two protons and two neutrons, so we have an extra neutron which is released:

;

So how do we get the atoms to fuse? In the same way that we do a great many other things in physics, with giant lasers! The deuterium and tritium are packed together into a tiny pellet about 4mm in diameter. On the outside you have a simple containing layer and then you have different forms of the DT (deuterium and tritium) mixture, either ice or gas. Now comes the lasers.

The outside layer of the target is heated to very high temperatures using the lasers(1). Eventually the temperature becomes so high that the outside shoots off. Due to Newton’s third law, every action has an equal and opposite reaction, the rest of the target is pushed inwards with an incredible amount of force(2). It compresses to about a 16th of it’s original size, to a diameter of 1/4mm(3). This compression leads to a rapid heat increase and the reaction starts, at this point the density at the core is 20 times that of lead and is 100,000,000˚C(4). Though the initial power input is very high, the energy that we receive from the reaction is much more.

This method is much nicer than fission. It is more efficient, does not create any radioactive byproducts and if the reactions starts to get out of hand, can be shut off immediately with a flick of a switch.

So why don’t we use it? Because it isn’t as easy as fission. The supply of uranium you have in a fission core lasts a year. On the other hand, the reaction in a fusion core takes place within a few minutes and then a new target needs to be put in and the reaction initiated again. Seeing as each of these reactions don’t release very much energy on their own, we need to be doing thousands a day and currently our facilities just aren’t ready for that.

Nuclear power is an incredible thing. The fact that we have managed to harness energy we did not know existed until within the last century or two amazes me. However, right now it is not capable of being our main new source of energy. Though we can control fission quite well, events like Chernobyl mean that the public feel threatened by it, and though fusion is almost the perfect energy source, we cannot yet harness it well enough. So watch this space, as our technology improves, nuclear power will become safer and easier. The futuristic image of nuclear energy powering the world is just around the corner.

We hope you’ve enjoyed this post! Check out the first half of this post series on nuclear power.

If you did like the post then please check out our last two posts:

Matter and Antimatter: Why hasn’t the whole universe exploded? – When they come together, matter and antimatter explode, so why hasn’t that happened to the whole universe?

What is Graham’s Number? – Sometimes we need really big numbers in maths. We aren’t talking a million, a billion or even a trillion. No, this is the largest number ever used in a mathematical problem, and saying it was big would be an understatement.

The atom is a strange place. Gravity, which is integral for everything we do in life, is barely involved, and forces such as the strong force, something we never see in real life, take over. The atom is made of a nucleus and electrons orbiting about it. In the diagram, the small blue particles are electrons, don’t worry about those, they aren’t very important right now. However, in the middle, the cluster of red and green circles is the nucleus and that is very important to us. In any atom there are a certain amount of the red particles (called protons). They define what the substance is. For example, every copper atom, the regular metal you see in wires all over your house, has 29 protons, and iron, a metal with completely different properties has 26. The fact that a change so apparently small can lead to such a different substance is a topic for another day, but it shows you how sensitive the conditions in the nucleus can be.

The red particles, do not like to stay together. They repel each other with a very large amount of force. So how do they stay together in copper and iron and every other substance on earth? Well that is where the green particles are important. They are called neutrons. The neutrons are very similar to protons. In fact, they are pretty much exactly the same. However, the neutrons act like the glue in the nucleus. They do not repel each other or the protons and instead are the reason they stick together (due to the strong force).
Each dot on this graph is a naturally occurring atom on earth. On the x-axis we see the number of protons and on the y-axis we see the number of neutrons. As we can see, all the dots fall about the blue line. All the red dots, the “beta plus emitters”, do not have enough neutrons in their nuclei. The protons are not stuck together well enough and they are radioactive because of this. As you can see, the line is a curve and this shows that if you add another proton to an atom you need to have more than one extra neutron to hold it together and the more protons you have to start with the more you would have to add. This is the main reason that heavy elements, like uranium are radioactive when lighter ones like oxygen are not.

Because all these particles are held together, there is a lot of energy within these connections. So, how can you release it? Well here we come to the matter of fission. Take a very heavy nucleus (one with a lot of protons and neutrons, some are better than others), such as a type of uranium and with a bit of coaxing you can get it to split in half, releasing some of the energy previously trapped in the nucleus. By shooting a neutron into the nucleus, there is just enough energy to split the nucleus in two smaller, lighter elements. This splitting also releases three neutrons and energy. Some of you may have heard of chain reactions in nuclear power plants and this is where the extra three neutrons come into this.

If they fly out and one hits another atom of uranium, that atom will split releasing energy and neutrons. Then one of those neutrons could hit another uranium atom and it will split and so on. Once the reaction starts, it is very difficult to stop it. The idea of not being able to control a reaction that creates large amounts of heat, on its own is a scary idea, and the fact that the elements the uranium splits into are usually highly radioactive means that nuclear power is considered dangerous and frightening. Indeed, disasters like when a nuclear power plant had a meltdown in Chernobyl, are much more common than we would hope. However, the efficiency of this process cannot be over emphasized and if it can be made more safe, then many believe it could power the world.

We hope you’ve enjoyed this post! Return on Sunday for the second post on this subject, on nuclear fusion.

If you did like the post then please check out our last two posts:

Matter and Antimatter: Why hasn’t the whole universe exploded? – When they come together, matter and antimatter explode, so why hasn’t that happened to the whole universe?

What is Graham’s Number? – Sometimes we need really big numbers in maths. We aren’t talking a million, a billion or even a trillion. No, this is the largest number ever used in a mathematical problem, and saying it was big would be an understatement…

Firstly, lets define some things. What is matter? This, surprising, is pretty tricky to define. Matter is the name we give to parts of space with properties such as mass, in other words particles. However, not all particles are matter, and here we can describe antimatter.

Imagine putting a normal particle in front of a mirror. In the mirror you see an exact replica of the particle, but reversed. Right is left, left is right. This is what antimatter is. The antimatter version of a particle is its mirror image, but not only does it look reversed, all its properties are reversed. What is even more interesting is when you take two of these particles and let them hit into each other. You get a very interesting reaction. They destroy each other, evaporating into pure energy ( E=mc² and all that)!

So lets give you an example. Think of an electron. An electron is a piece of normal matter, it is very light, has an electromagnetic charge of -1 (if you don’t know what electromagnetic charge is just imagine -1 as one side of a magnet and +1 as the other), and other than that is a pretty unremarkable particle. And then we take its antiparticle, called the anti-electron but more commonly referred to as a positron. The mass of the positron is the same as the electron, you can’t get negative mass (at least to the best of our current knowledge) after all. However, the charge is not. The charge of a positron is +1. Because the electron’s charge is negative and the positron’s charge is positive, they are attracted to each other (like two magnets). They eventually collide and both disappear, shooting out a large amount of energy.

This video takes a bit of a different approach but makes that same overall points, check it out.

Now we have a conundrum. I have told you, and we know, that antimatter and matter are pretty much exactly the same thing. So, from the big bang, they should have appeared in equal measure. And almost the instant after it all appears, all the matter would find corresponding antimatter and it would all explode! The universe would be left as a very large space filled only with left over energy in the form of photons flying about the universe, we would never be able to exist. So something must have been different, for some reason, more matter was created than antimatter.

The reason I decided to do this post is because we have been able to observe something called CP violation in B mesons very recently in the wonderful particle smasher we call the LHC. Alright, a few more definitions needed here too. B mesons are groupings of a bottom quark (a simple particle) and an up antiquark, don’t worry about the specifics. Bottom quarks are very, very heavy particles. This means they don’t like to stick around for long, in fact, they transform very quickly into other particles. This becomes more important later.

Our second definition is of CP violation. CP stands for Charge-Parity. It refers to how our universe has symmetry within it. Parity is simply where something is in space and we have talked about Charge before. According to our best ideas of the universe, there is CP symmetry. If you take an interaction, for example, our electron and positron collision, and you switch the directions the particles come from (up-down, left-right, forward backwards), this is Parity, and then you make all the particles their antiparticles, this is Charge, the whole reaction should be exactly the same. CP symmetry also includes another aspect, time. You also have to reverse time.

So how was there CP violation here? Well what the mesons turned into should have been antimatter 50% of the time, and matter the other 50% but that isn’t what happened. Instead they found a 1% difference between the two. This had been observed before a different type of meson before. So what does this mean? Well it gives us proof that the universe is perhaps not as symmetric as we think, it will lead on from there to new theories and new developments to describe why that is and predict more CP violations that we can try to observe. Though it starts to show us how the imbalance in matter and antimatter came about, it cannot explain all of it, so our research must continue. This also means something else. Because the interaction violates the time part of CP symmetry, it suggests that time can only go in one direction. Why is this interesting? Well the so-called arrow of time is a great mystery of physics, we have already posted on the classical ideas about this so go check that out!

We hope you’ve enjoyed this post! If you did then please check out our last two posts:

What is Graham’s Number? – Sometimes we need really big numbers in maths. We aren’t talking a million, a billion or even a trillion. No, this is the largest number ever used in a mathematical problem, and saying it was big would be an understatement…

Your Questions:  If your car could travel at the speed of light, would your headlights work? – There is more to this question than you would assume and the answer may surprise you!

The first question you are probably asking is “Who’s Graham?”. This is Ronald Graham, an American mathematician born in 1935 from California. He began looking into Ramsey Theory, a field in Combinatorics, which generally poses problems in the following style.

Ramsey Theory often considers some sort of set. A problem would then ask how many members need to exist in the set for a certain fact about the properties of the set is true. In other words, Ramsey Theory is about when order will exist out of chaos due to volume, and connections between objects.

Like many fields in combinatorics, this has the potentials to have very large numbers as part of their solutions. And one problem that Graham was trying to solve certainly would not disappoint.

The simplest was to state this problem is to consider every possible committee from some number of people and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find the smallest that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees.

This is still quite a mouthful. But it is concerning sets within sets, and the relationships between particular objects within the sets.

Graham didn’t solve the problem. What he did do though, was find a lower and upper bound, and thus knowing that N* is between those two numbers. The lower bound is innocent enough: 6. The upper bound is a monstrously big number that we now know as Graham’s number, that I will begin the rest of the post talking about.

This number is bigger than the famous ‘googol,’ 10¹⁰⁰, and even bigger than a googolplex, 10^googol. In fact, this number is too big for our universe to contain when written out using any traditional notation. Even if each digit occupied just one Planck volume, the smallest amount of space that can exist in the universe, Graham’s number would exhaust all of space. How can it possibly be described then?

The only sensible way to write Graham’s number is using Knuth’s up-arrow notation.

You can think of multiplication of being

b × a = a + a + a + a… until there are b copies of a.

In the same way, exponentiation becomes the following

a^b=a×a×a×a×a… until there are b copies of a.

Knuth then notes a^b as a↑b

a↑↑b is defined as a↑a↑a↑a↑a until there are b copies of a, or in conventional notation, a “power tower” of a’s with b stacks.

a↑↑↑b is defined as a↑↑a↑↑a↑↑a until there are b copies of a, and similarly, a↑↑↑↑b is defined as a↑↑↑a↑↑↑a↑↑↑a until there are b copies of a.

g₁ is defined as 3↑↑↑↑3. To start imagining this, let’s consider 3↑3. That is just 3³, which is 27. What about 3↑↑3? That is (3³)³, which is already about 7.6 trillion. 3↑↑↑3, therefore, would be 3↑↑3↑↑3↑↑3… until there are 7.6 trillion threes! This is almost unimaginably big already.

3↑↑↑↑3 is 3↑↑↑3↑↑↑3↑↑↑3… with 3↑↑↑3′s. To aid understanding, I’m going to begin to label some of the really large numbers, so we’re going to call 3↑↑↑3 (3↑↑3↑↑3↑↑3… with 7.6 trillion threes) h₁ (for huge). So g₁ is 3↑↑↑3↑↑↑3↑↑↑3… with h₁, that unimaginably big number, of 3′s. Let’s also give g₁ the name h₂, for our second huge number.

But that isn’t Graham’s number. Not by a longshot. Let’s look at how we form g₂.

g₂ is 3↑↑↑↑↑…↑↑↑↑↑3, where the number of arrows is g₁, the stupidly large number formed by our stupidly large number h₁. g₃ is, you guessed it, 3↑↑↑↑↑…↑↑↑↑↑3 but with g₂ 3′s! For me this is the point where the numbers are so huge that the all seem to mesh together in my mind. But Graham wasn’t done yet.

Graham’s number is g₆₄.

This number is ridiculously vast. When talking to someone about this, they asked me to describe how big it was, and I couldn’t do it. I could think of no possible analogy, simply because our universe is not big enough for such analogies. They asked me how many digits it had, and I told then that no one could describe the number of digits it had, the number you’d need to use to describe it would be too big for comprehension!

The human brain is actually rather poor when it comes to dealing with very large numbers. Part of this is evolutionarily: there is no reason that the caveman who can visualise Graham’s number will be more likely to survive or reproduce. When a sense of numeracy was most useful to our ancestors was when evaluating the numbers if potential adversaries. To do this, it is much more important to consider ratios than absolute values. 15 men face a reasonable chance in a fight against 18 men, but if it is 1 v 4 then your best shot is to run. It is for this reason that differences begin to feel smaller the larger the numbers you are talking about; past a certain point it doesn’t matter. Our minds are not made to cope with it! But perhaps it is precisely this that fuels our imagination about them.

If I wanted, I could define an even bigger number, let’s call it Caplan’s number, and define it as Graham’s number squared, but that’s not particularly useful. It’s possible to think of a bigger number than any finite number. What makes this number special is not just that it is very large, but that it is the largest number we have actually found a use for.

I hope you’ve enjoyed reading about this gargantuan number. It lies on the edge of human comprehension, and possibly beyond it, but I think that it is a testament to the power of the human mind that we have managed to work with such a vast number.

What is your first reaction when you read this question? I’m sure a large amount of you who are only familiar with classical mechanics would have thought that they would work, but the light would just travel at the same speed as the car that is moving.  Those of you that are familiar with what we do here would realise for it to warrant a post, it just couldn’t be that simple!  So what is happening?  Well, for this, we have to bring in a new type of physics, relativity.

The Theory of Relativity was a revolutionary theory created by Albert Einstein at the beginning of the 20th century. Einstein’s genius was half in his ability to form theoretical experiments that the physics of the time could not explain and half in his ability to come up with solutions with entirely new methods. Around Einstein’s time, a shocking discovery was made. It seemed that, after a few pieces of experimental evidence, light moved at the same speed relative to you no matter what speed you moved at. Think of it as if someone is running along side your car whilst you drive a long. To someone standing still, that person appears to be moving fast, however, if you were to look at them, in fact they look not to be moving at all. In relativity, that person would appear to be moving fast for someone standing still and for you! It is a very strange phenomenon and it led to a number of discovery’s by Einstein, some of which we will be going in to later.

Anyway, just reading that paragraph, what can we assume would happen? Well, from the point of view of me standing on the side of the road, both your car and the light rays are moving at the speed of light. This means that the light rays can’t move away from the car and therefore I wouldn’t see the headlights. However, what about from your point of view? Relative to you sitting in it, the car isn’t moving, but we learnt that the light is going at the speed of light relative to you. So, you would see the headlights stretch away from you and the car! See, relativity leads to some strange results, and actually, we haven’t even  used all of the effects of relativity yet, so it is going to get stranger still.

You see, Einstein realized something strange about relativity, you can’t tell which object is actually moving, no matter where you are observing from and this leads to some very strange effects. For you in the car, the light would indeed stretch away, however, everything else around you would stretch in very weird ways. This video with Carl Sagan explains all these effects very well (despite the slightly dark ending):

From the point of view of me, watching you, you and the car would shrink in the direction of travel. So what does this mean overall? Well…I haven’t been completely honest with you guys here. You see, the experiment is broken.

It isn’t possible for a car to travel at the speed of light. Not that it is difficult, it is simply impossible. So what happens in an impossible situation? We don’t really know. We can look at all the supposed effects but really, it just gets very strange overall. For example, for you in the car, time shouldn’t move, it would just stop. So you couldn’t perceive the light leaving the car, because you wouldn’t be able to perceive anything at all! Anyway, at the end of the day, if you are in a car moving at the speed light, in the time it takes for you to turn on the headlights, you probably would have hit whatever it was you were trying to avoid!

We have posted before on Relativity: The Irregularity of Time and Gravity

We hope you’ve enjoyed this post! If you did then please check out our last two posts:

Spacetime, Extra Dimensions and a TARDIS: Is it possible for an box to be bigger on the inside?  - In the TV show, Doctor Who, the Doctor’s famous Tardis is bigger on the inside than out, appearing from the outside to be a simple police box. This is science fiction of course, but could any real objects have this effect under some circumstances.

Let There Be Light - In 1861, James Clerk Maxwell released a paper in which he created four equations. These equations described the relationship between electricity, magnetism and light. So what were they, and what does each one mean?