Cool Maths Things

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The Golden Icosahedron

The Platonic Icosahedron has twenty faces which are equilateral triangles. In the Golden Icosahedron, twelve of those triangles (the yellow ones) have been replaced by acute, isosceles triangles with a leg:base ratio which is the Golden Ratio.

[coolmathsthings]

Pythagoras' theorem: A PROOF

Remember that theorem about right-angled triangles from school?

The one that you had to do endless examples of for homework to cement it into your brain?

Well, you know what your teachers never showed you? The cool bit. The (or rather a) PROOF. The easy, straight forward proof.

Take this square

Now those four triangles are exactly the same (or rather congruent) size wise. You may skip to next paragraph if convinced: They have two sides equal length - a and b - and right angle between those two sides (as it’s a square). According the funky side angle side rule (proof you may look up if you wish) and the fact they are equal, they are the same.

This of course means the third side is the same - called c. and of course lastly we know that diamond in the middle is also a square too (hurrah!)

Now, some equating:

So let’s look at the area of this whole square. We have two expressions for it.

1) using the formula for area of triangle 1/2 x base x height = 1/2 ab, and area of a square = c^2, multiplying your triangle area by four we have and simplifying…

Area of big square = 4(1/2 ab) + c^2 = 2ab + c^2

2) using the fact that one side of our big square is a + b

Area of big square = (a + b)^2 = a^2 + 2ab + b^2

Now for the triumphant part – equate and cancel!

2ab + c^2 = a^2 + 2ab + c^2 

Now take away 2ab from both sides et voila

Or another even more geometric way is taking a look at this first figure

You can clearly see the formula!

Now you can impress all your friends and inspire the world with the cool side of maths 

[scienceisbeauty]

The 17 Equations That Changed The Course Of History (Business Insider)

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[coolmathsthings]

I always struggle to know what numbers are and are not divisible by 3 - is 41 divisible by 3 or is it 42? Is 57 prime? Well now we can be sure with just some primary school addition - even for the big numbers!

[coolmathsthings]

What do you expect to happen when you release a double pendulum (one pendulum connected to another) from the horizontal? Pretty interesting. There are plenty more videos on YouTube about this. This is to do with chaos theory. If you want to learn more, here is a nice introduction http://youtu.be/EF5Wvi_Iiy4