Mathematics: What is the most beautiful theorem proof, and why?

I am sure that there are some out there that would say that this does not constitute a proof, but I have always found this to be a beautiful justification for the following identity. 

∑n=1∞12n=1${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{2^n}=1}$

Proof:

How is gravity the weakest fundamental force?

Gravity is pathetically weak. If you drop an iron nail on the floor, you can pick it up with a small bar magnet from a child's toy. In doing so a titanic battle takes place. On the one side we have the entire mass of the Earth (which is quite big compared with the nail or the magnet or you). All of its mass is sucking on the nail trying to stop you from picking it up. On the other side you have the flimsy magnet that probably weighs not much more than the nail.

Which wins? The magnet. The electromagnetic force overcomes the gravitational force generated by the entire mass of the Earth.

I still wouldn't jump out of a window, though.

Complexity: What is the most complicated thing ever?

Do you think a Rubik’s cube is the most complicated mechanical puzzle ever invented? You have no idea how complicated mechanical puzzles can be.

A Standard 3x3x3 Rubik’s cube can have 43,252,003,274,489,856,000 combinations, but only 1 solution.

To put this into perspective, if we have one Rubik’s cube for each possible combination, then all the cubes will cover the whole surface of 275 earth-sized planets. and among all of them, only 1 cube will be in perfectly solved state.

But for some people, this was not complicated enough. they created 4x4x4 cubes, and also 5x5x5 cubes.

A 4x4x4 cube has 7401196841564901869874093974498574336000000000 combinations. If you have that many teaspoons of sugar, it will fill a sphere the size of the solar system 3.5 times over.

a 5x5x5 Rubik’s cube has 282870942277741856536180333107150328293127731985672134721536000000000000000 combinations, this is close to number of atoms in the known universe. and yet, people can solve it within few minutes.

So, some people thought it was not complicated enough, they created this monster:

Say hello to 11x11x11 cube.

Is it world’s most complicated cube? Nope. The largest cube ever made is 17x17x17. Having a total combination of 66.9 * 10^1053. This number is so huge it’s digits can not even be written in here.

Yet, someone solved this thing in 7.5 hours. (reference link below).

Are Rubik’s cubes are the most complicated puzzle ever? Nope! Lets meet Minxes.

Minxes are like Rubik’s cube, but with more than 6 sides. Having more sides also makes them more complicated.

This is a 3x3x3 Megaminx. It has 12 faces and total 50 moving parts, whereas the Rubik’s cube has only 6 faces and 20 moving parts.

A 3x3x3 megaminx has 100669616553523347122516032313645505168688116411019768627200000000000 combinations, i.e. 1000 billion billion (yes there are two billions) more times than a 4x4x4 rubics cube.

The world record for fastest megaminx solving is 37.58 seconds.

Still not complex enough? meet the Gigaminx, with a 5x5x5 structure, the Teraminx with 7x7x7 structure and the Examinx, with 9x9x9 structure.

And this is how it looks when scrambled.

The combinations possible with a Examinx is not worth calculating…

But wait.

Someone has made a Yotaminx, a minx with 15x15x15 structure.

Not sure if anyone was able to solve it or not.

And for people with nerve of steel, here is a Tuttaminx (Thanks Shrey Gadiya for the reference) with 32 faces and 150 movable parts.

If you scramble this, it looks like this:

After seeing all these, if you think a puzzle can not be more complicated than a Tuttaminx, you are wrong!

Multiple Minxes can be fused together to form a combined minx. Here is a triple fused Petaminx.

These puzzles have no practical use for mortals. Although in Hell, they may ask you to solve one.

Update: as mentioned by Digvijay Pr , there are Shape shifting puzzles with non-linear movements as well, like the Ghost cube.

This is how it looks when scrambled, Goodbye brain.

Due to each piece being of a different shape, this puzzle is notoriously difficult to solve. Unlike a Rubik’s cube the movements of the pieces are restricted due to their shape, so it’s very difficult to mathematically calculate the number of possible moves with this one.

Update 2: If your brain is still not fried, let’s meet Sudoku cube, as suggested by Gourab Chowdhury.

The Sudoku Cube or is a variation on a Rubik's Cube in which the faces have numbers one to nine on the sides instead of colours. The aim is to solve Sudoku puzzles on one or more of the sides.

In the Sudoku Cube, the player must place the numbers one to nine on each side with no repetition.

This cube is one of the most difficult of all the cubes, because as well as having to know the basic cubes concepts you also have to know basic sudoku concepts, and unlike normal sudoku any turn of the cube can wipe out work you just did.

What makes this more complicated than a normal Rubik’s cube is unlike a normal colored Rubik’s cube, there is no way to identify which piece originally belonged to which side !!!

Look how confusing a Sudoku cube can make you when it is scrambled. Not only you have no idea to which side the numbers actually belong, but they are also tilted, rotated and can become completely upside down.

So, this must be the most complicated puzzle right? because it has the combined evil nature of both Rubik’s cube and Sudoku…

I also thought so, until I saw this guy… which has 81 numbers each side compared to just 9 of a regular Sudoku cube.

R.I.P brain :-(

Update 3: Some friends have asked how to calculate the total number of combinations possible with a cube. Here is the mathematical formula as taken from Welcome to speedcubing.com! [see reference 3]. In the below formula, n is the number of pieces each side. for a 3x3x3 cube, n = 3

Most amazing fact about black holes.

Accretion disks.

It’s easy to think of black holes as the ultimate cosmic vacuum, that just sucks everything in cleanly and refuses to let anything back out. Well, if you were a cosmic custodian, a black hole would be the last thing you’d want to clean up your spilled stars and planets.

Black holes don’t just drag its neighbors inwards and then gobble them up in one gulp. Quite the contrary. Large objects, like stars or planets, that fall in directly won’t go out like that. Instead, what happens is an astronomically magnified form of spaghettification, where one side of the star is significantly more attracted to the black hole than the other side. The tremendous difference in the gravitational pull felt by different sides of the star is enough that the entire thing is literally torn to pieces while it’s being killed.

It’s more like a cosmic meat grinder than a cosmic vacuum.

Oh, but black holes aren’t finished yet. That’s not enough for them. They need to announce to the rest of the universe their power and superiority. As they violently rip stars large enough to make our Sun look like a peanut into pieces, they wear the removed gas and debris around their bodies like a trophy of victory. Take a look at the swirling disk of matter in the picture above.

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At this stage, the name “black hole” becomes misleading, because the black hole is anything but black to our eyes. The murderous celestial object gains a new name; you’ve probably heard of it. It’s now known as a quasar, or quasi-stellar object.

After successfully demolishing a star, the quasar will acquire an accretion disk, a swirling mass of gas and debris that used to be the body of a plump, happy star. This terror will then pull the shredded corpse of its victim around its event horizon at absolutely unfathomable speeds. Friction caused by the accretion disk material smashing together generates heat. A lot of heat. Heat of a magnitude that is even more difficult to fully comprehend. And with heat, comes light.

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Take a look at this photo.

On the right is a star glowing brightly a few hundred light years away.

On the left? It’s a quasar…9 billion light years away.

Think Eta Carinae is scary? A supernova? A hypernova? None of them have got an ounce of crap on what a quasar can do - if it’s feeling nice. These luminous beasts emit so much light that they can literally mask entire galaxies, containing millions upon millions of stars, with their brightness.

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…And we’ve got one in the center of our galaxy. :)

What will earth be like in one billion years?

Well, much will have changed - obviously.

• After about 250 million years - continental drift will probably cause all of our present continents to fuse into a single giant super-continent as the pacific ocean gradually shrinks. Exactly how this will play out is uncertain - but North America will almost certainly hit Eurasia.
• After around 500 million years - that gigantic continent will have been ripped apart again - it’s unclear what new continents might form from it.
• At intervals of around 500 million years, large gamma-ray bursts happen close enough to the Earth to cause mass-extinction events if they happen to point in our direction. We can expect this to happen a couple of times over the coming billion years…but it’s a matter of chance whether we get cooked.
• At around the same time, the sun will start to become gradually brighter - the consequences of this are huge - but at some point it’ll get hot enough to disrupt the carbon cycle and to start pouring water vapor into the air. That can cause a run-away global warming effect so that within about 600 million years, plant life will likely be impossible - and the knock-on effects of that would probably end all life on Earth by about 800 million years due to all of the oxygen in the atmosphere having been converted into CO2.
• And at 1 billion years, the sun will be about 10% brighter than it is now - the average temperature will be up to 48 deg C. This will cause all of our oceans to evaporate - except perhaps at the poles.

Over this time, the sun will have orbited the Milky Way galaxy about four times.

Interestingly, a billion years is considered to be the likely life of the two “Golden Records” that were placed aboard the Voyager 1 & 2 space probes…and these may be the last remaining traces of humanity in the universe.

How does NASA detect something that is light years away?

Well…quite easily, really.

Go to the countryside. Wait until night. Go outside. Look up.

Voila!

You have detected things which are light years away!

The closest star to us is Proxima Centauri at 4.24 light years away, but is invisible to the naked eye - blotted out by its far brighter cousins in the binary system of Alpha Centauri A and B.

These stars are 4.37 light years away.

This star system is the third brightest object in the night sky - the brightest (Sirius) is a measly 8ly away - but the second brightest (Canopus) is 310 light years away!

Depending on how you classify “visible", the furthest object that can be seen unaided is one of the stars in Casseiopeia - between 4,000 and 15,000 light years away.

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So there you go, using the good old Mk I Eyeball, you can detect objects which are thousands of light years away!

Obviously, however, you can only see the huge stuff which is chucking out loads of light - which makes it really easy to see them!

Distance isn’t the big achievement, really.

The clever stuff that NASA does is spot tiny, dark things across that distance!

Exoplanet hunting is done through a variety of methods, such as:

Direct Imaging

Which you do through a whopping great telescope, or lots of little telescopes all wired up into an “interferometric array”, which (if you have a supercomputer) allows you to treat them as a single whopping great telescope.

This accounts for only a tiny number of detected exoplanet - but produces cool pictures!

Transit Detection

Rather than looking at pretty pictures, you simply take a graph of the light intensity emitted by a star.

Normally the power output is pretty much a constant, but when a planet passes in front of the star, the light emitted drops - like in the image.

By timing the gaps between each drop, you can work out how many planets there are, how far they are from the planet and so on.

This is by far the most common way of detecting exoplanets.

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There are a few other ways of doing this - such as using Doppler shifts to work out how much the star “wobbles” due to the gravitational influence of orbiting bodies, and so on.

In the context of the furore about the recently discovered earth-like planets in the TRAPPIST system, it's also worth mentioning that 40 light years really isn't that far away!

It's practically in the same village as us - the neighbour a few doors down!

Can there possibly be a connection between black holes and time travel?

Black holes are interesting, in every possible way, including the way that gets us to time travel.

They are “born” when stars “die”, stars which are at least 3 times more massive than our Sun. Now, every object in the Universe which is made of matter, from the smallest grain of dust, to the most ginormous galaxy, has got mass. You can imagine “mass” as the property of matter to modify, stretch if you want, the space around it.

Visualise balls, some of them smaller, some bigger, that you can put on a very elastic surface, and watch them sinking on this surface. The more massive they are, the more they will deform that surface. Gravity is a direct consequence of that curving of the surface.

But that surface isn’t just space, it's space-time. The more an object is capable of deforming that mesh, the more it will deform time. Ordinary stars can be pretty massive, and have enough of an effect on space-time that if you could get close to them with a spaceship and then go back home to your family, you’ll find that the watch that you’re wearing and the one hanging on the wall in the kitchen are not synchronised anymore, by seconds or minutes, depending on the star and how close to it you dared to go. And for how long of course.

If you have an extremely precise watch, you can measure this effect even just going to the moon and back. You’ll be measuring tiny fractions of a second in difference, but that’s all you need to do to travel through time. Go somewhere where the gravity is stronger and you’ll effectively travel to the future. Or go away from Earth’s gravity well and you’ll see your friends on Earth age very slightly slower than you. That’s exactly what happens to the astronauts in orbit in the ISS.

Now let’s go extreme and back to black holes. Black holes are not more massive than the stars they originate from, actually a bit less (because the stars explode violently and lose the outer shell when they “die”). But, and this makes all the difference in the Universe, they are immensely more dense than the original star. Basically infinitely dense. Most of the original mass has been compressed into an infinitesimally small point. How can we visualise black holes using the same image of the balls on the elastic mesh? Imagine that instead of putting a ball as big as a soccer ball of the weight of 1 kg on an elastic sheet, we exert the same force of 1 kg using a long needle, or, like in the next picture, a tiny metallic ball made of an ultra dense metal.

The force is the same but it acts much more strongly on the area immediately around the point where the ball is touching the mesh.

What happens if you take your spaceship and go where the mesh is deformed violently? You’ll experience a much more extreme time-dilation. For every second you spend there, many seconds or minutes might pass on Earth, and the extent of the time dilation can increase or decrease a lot just with small movements towards the bottom of the pit or away from it. You might zap there with your super fast ship, orbit for maybe 30 minutes, just the time to prepare a cup of tea and sip it calmly, zap back home and realise that just in the time you were having tea, your little brothers have done a whole semester at school (this, only if you can avoid going beyond the point of no return, called event horizon. After you pass that, there’s nothing in the Universe that can make you see your family again).

In all of these cases you’ll be travelling to the future. There is no known mechanism in the Universe which isn't wildly speculative that could make you go back to the past. No way to recover the time you have been away from your family.

As you can see I haven't mentioned wormholes yet. Because they are rather unnecessary at this point, since black holes are all you need to travel to the future. And for other reasons: wormholes are only theoretical, again wildly speculative. They are sort of a stretch of the laws of physics that we know, but nobody really believes that they can exist, and if they exist nobody thinks that they could be actually useful. And, they wouldn't even occur naturally, hanging around black holes. We would have to produce them ourselves, using preposterous amounts of energy.

I say wormholes are only useful as movie tropes. I’d be happy to be wrong.