BBC -- Beautiful Equations

 

Artist Matt Collings chats openly with scientists as he takes a tour of some of the most important equations in physics. Equations are explained to be an unambiguous generalized shorthand for expressing relationships about physical quantities. "A little increase in mass means a gigantic increase in energy." How much, exactly? Well, the details are actually worked out in a way that anyone can follow.

This documentary is well worth watching as it gives an approachable and concrete explanation of what all the symbols on the page actually mean and how truly remarkable it is that such universal truths, which literally apply to all objects in the universe, can be expressed in such a compact manner. 

The concept of "Mathematical Beauty" is explored and (spoiler alert) it basically means that a single, compact equation can very simply and very generally express the relationship between quantities (mass, energy, time, velocity, force, etc.).

BBC Page for "Beautiful Equations" http://www.bbc.co.uk/programmes/b00wltbm

Matt Collings's website: http://www.emmabiggsandmatthewcollings.net/recent

The five equations presented: 

1) The relationship between mass and energy: E = mc^2 
(E = energy, m = mass of an object, c = the speed of light)

2) The force of gravity between two bodies: F = (G * m1*m2)/(r^2)
(G = the gravitational constant, m1 and m2 = "the first object having mass m1, the second object has mass 2" and r = the distance between the two objects)

3) Time dilation: t' = t*sqrt[ 1 - (v/c)^2 ]
 t = time in your frame of reference, as you move at velocity v.
t' = time according to someone watching you from another frame of reference at rest to the first. c = the speed of light. 

4) Dirac Equation: (i*gamma*del - m)*phi = 0 
Difficult to summarize without getting into complicated mathematics. Essentially it predicted that all matter has anti-matter components and it described how the most elementary of particles behaved. More here.

 "What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating." Paul Dirac 1969 full speech here

5) Hawking Equation:  S = A/4
S = entropy (disorder) , A = the area of the black hole

It is worth mentioning another blog post which goes nicely with this theme.

"Math doesn't use fancy words and weird notations because mathematicians are trying to confuse non-mathematicians. It’s not done out of spite, or out of some desire to exclude non-mathematicians from the club. It’s about precision."

http://www.goodmath.org/blog/2014/03/24/yet-another-cantor-crank-size-vs-cardinality/

As a particularly nice example, I offer up the equation for the Fourier Transformation.



At first, it can seem intimidating and overwhelming -- so many symbols, fractions within superscripts, what does it all mean? Well, each symbol has a very specific and unambiguous meaning, quite a lot of punch is packed into this array of symbols!


The image above was originally seen here and explained for the lay reader in this blog.