Wonders Of Mathematics - Magic Squares

So one of the books that I really enjoyed recently is the famous Dan Brown's "The Lost Symbol". If you read the book (and you should) you might have noticed that it has many interesting maths references. One of which is hidden in the famous Albrecht Dürer's 1514 engraving Melencolia. It's a very special magic square.

But what are these magic squares? Magic squares are very interesting square arrangements of numbers. The numbers are arranged in such a way, that the sum of all columns, rows and diagonals is equal to the same number.

Magic squares come in different sizes that are called orders, for instance 3x3, 4x4 and so on (it's just a number of rows and columns). As for the sum that is constant for columns, rows and diagonals it's called the magic constant.

 

 

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The most simple magic square is of order 3x3 (it's easy to see that 2x2 squares are not possible). But how do you construct such a magic square, without relying to guess work? There are various rules, which can help you construct the square faster, however some guess work is still essential.

So when stumbled across these magic squares, I decide it would be fun to found a 3x3 magic square without any help. It's quite fun, well at least much more fun than sudoku, so I encourage you to try it.

Thanks for reading. Next time we're gonna look at all possible 3x3 magic squares and at the mysterious Melancolia engraving.

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More Open Yale Courses

Hey I have some great news. The guys from the Yale university added more free courses that are open to the public at their website.

For those of you who don't know what open Yale Courses are - well it's a collection of free university lecture series in a variety of subjects starting with psychology ending with philosophy and physics. In addition to these lectures, the exams, tests and other learning materials are also open to the public.

Here's a preview video of all the new added courses:

And here's the link to the Yale open courses website - LINK

Thanks for reading and good luck learning!

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The Great Debate of the Nature of Reality

Edinburgh this week is a very interesting place to be at, due to the famous international science festival. This science festival in short is two weeks of pure fun for science fans. It has everything, starting with talks by famous scientists and writers, events for kids, screenings of Brian Cox's newest film, talk by Richard Dawkins and so much more.

If you have the chance to visit Edinburgh during the festival I highly recommend it. You can find all the needed information at the following link.

But why am talking about this festival? Well, simply I had a chance to visit some great talks. One of them was by Manjit Kumar, a scientist and a philosopher, who is the author of the new book "Quantum - Einstein, Bohr and the Great Debate About the Nature of Reality".

It was a great talk, which involved a lot of historical facts and amazing pictures. Needless to say it involved a lot explanations about the discoveries of quantum mechanics. The most interesting part was about the famous Einstein and Bohr debates regarding the interpretation of quantum mechanics.

So without spoiling the fun I can say that it was a great talk and it's an even greater book, so don't hesitate and buy it as it's really cheap. I especially recommend it for all of you science history lovers.

PS:  the author of the book has a very nice blog, which can be found here.

Also you can buy the book here:

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Review of "The Road to Reality" by Roger Penrose

So recently I managed to buy some great books about physics at a low price and one of them was another great book by Sir Roger Penrose "The Road to Reality".

My first impression when I saw the book at the store was that it was huge, and I mean huge, it has almost 1100 pages. So if you're thinking of buying this book be ready for long weeks of pure maths and science fun. But is it really that fun?

The book starts with some remarks from the author about the notation, which is useful, and some remarks about overall math usage in the book, which is also handy. The prologue is really amazing, as it drags you in, by telling a short story how Pythagoras, who had a hunger for knowledge, joined the brotherhood of 571 wisemen, and began his journey to secrets of mathematics and science. At this point you feel like reading an interesting novel that shows the fun side of science as well as dragging you in with a mysterious narrative. But what is the rest of the book about?

Well, it's about maths and physics of course. It has basically everything covered. And I mean everything! Starting with the roots of physics and mathematics, ending with string theory, quantum mechanics, general and special theories of relativity and even some speculative modern science theories. The book is divided in 34 chapters in total. Also it has some great diagrams and drawings, which really help you understand the physics in some parts of the book. The final part of the book contains some thoughts by the author about the nature of reality itself.

Naturally, every layman would like to know how much maths is used in the book. And the answer is - not that much. But I have to tell you, it's get's quite confusing after a couple of first chapters. And as you progress through the book, maths equations, diagrams and some ideas become incredibly confusing and hard. So I wouldn't recommend this book for those who hate maths or get a headache from even a simple equation. Also the physics is also quite hard, so I recommend this one only for experts.

Also it's worth noticing that almost all concepts of physics are explained from a perspective of mathematical theories and ideas, after all Roger Penrose is famous for mathematical physics, so be ready for more maths than physics.

So the final score:

Content: 9.7/10 ( contains almost everything you need to know about physics )
Beginner "Friendliness":  2/10
Narrative: 6/10 ( start's with a bang, but eventually ends up as another book written in a "dry science" style)
Illustrations: 9/10 (contains some amazing geometrical figures and diagrams)

Overall score 6.7/10

Final verdict: a great book for experts or hardcore science and maths fans, however, a little two confusing for the laymen.

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BBC - Everything And Nothing

Here we have another great documentary from the BBC team. This time is literaly about everything and nothing. It's great as all the other BBC documentaries. Enjoy!

Thanks for reading! Be sure to comment and subscribe ;]


You might be interested in:

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Basics of Quantum Mechanics Lesson 4

So last time we calculated the wave function of a free moving particle and got the following answer:

SE for a free moving particle:

Solution for this equation:  

So what do these equations tell us about the free moving particle? To find the answer to this question we need to find the probability density function. If you recall, the probability density function is basically just the square of an absolute value of the wave function. So by calculating the probability density function we would get:


|Psi |^2 = Psi x complex conjugate of Psi =
= A(cos(kx − wt) + i sin(kx − wt))A(cos(kx − wt) − i sin(kx − wt))
= A^2(cos2(kx − wt) + sin2(kx − wt))
= A^2


Now if you look at the result more carefully, you should realise that there's some strange stuff going on. The probability density function is constant, that means that the probability of finding a the particle in any point of space is equal. In more simple words, you have the same probability of finding the free moving particle in your room and somewhere in the other side of the universe. This is due to the fact that we have not taken into account the uncertainty of the momentum of a particle.



So you might be asking  - why are we studying free moving particles, while in reality most of the matter is situated in confined atoms and molecules. And you're definitely right, so let's look at another system, which might help us understand the behaviour of electrons, which are "trapped" in an orbit of an atom.


The easiest way of understanding simple atoms or similar systems is using a thing called infinite potential well. That is a system in which a particle is trapped between infinite potential "walls", which can be for instance be various electric of magnetic fields or anything similar, which confines the particle in a given space. After all, in a sense electrons orbiting a nucleus are also in a similar potential well - if they get too close to the nucleus, they are pushed back, if the get too far they are attracted back. 


Infinite potential well:

So how does particle behave in such a well? Well let's firstly imagine what would happen in a classical situation, let's say for a tennis ball trapped between two walls. It's clear that (neglecting gravity, which would eventually bring the ball down) any position between the walls has the same probability of finding the ball there.

However, in quantum mechanics, things are different. We already know that particles tend to behave as probability waves, so it's clear that the probability of finding the particle in any point of space will not be equal( a good analogy of a wave in a sort of potential well is a rope fixed at two places and oscillating).

So how can we find out at which points of space we are most likely to find the particle? By solving the Schrodinger equation of course.

And since this is a basic course we're not gonna bother ourselves with the process of solving the SE. We're just gonna skip right to the solution which is:

Wave function for a particle confined in a infinite potential well: Psi(x) = Asin kx

Now to calculate the probability density function, we need to apply the initial conditions of the system - as we know the particle can't be in points of space, which are beyond the boundaries.

In an infinite potential well the boundary conditions imply:
 Psi (0) = Psi  (L) = 0.
Resulting in the limitation that k must have one of the discrete set of values
kn = (pi . n)/L , where
n = 1, 2, 3 . . .


Now when we know k, we can find allowed energies.
Using  E= (p^2)/m and De Broglies wavelength p= h/ lambda = hk/ 2pi gives the following:

En = (h^2 . n^2)/(8mL^2)
 n = 1, 2, 3 . . .


The allowed energies for the trapped particle are quantised according to the value of
n which is known as a quantum number.


Note that we can calculate the quantum number using simple calculations, which we'll look at later. It's important to notice that for macroscopic objects n is very larger, which gives rise to the so called correspondence principle - the idea that if n is very large (which is definately the case for macroscopic objects), quantum mechanics prediction become similar to classical physics prediction.

And we can see this by finding the probability function of the infinite potential well, and graphing it with different values of n:

 

As we can see as n grows, the wavelenght of the wave in the graph becomes shorter. In n was very very big, it would look as every point in the confined space has the same probability (like in the classical physics case).


So that's all for now, thanks for reading!


LINK to lesson 3
LINK to lesson 2
LINK to lesson 1

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