# Beautiful Equations

Matthew Collings is an artist and art critic, that's what he knows and understands, but he's about to enter an alien world. To him equations have always been incomprehensible hieroglyphs. What do they describe? Are they just a mathematical game? In this documentary he'll learn about some of the most important equations in science, which are actually masterpieces that explain the universe we live in.

With art Matthew thinks beauty is very important and he's always trying to define it and work out what it is. Now he wants to apply that knowledge to mathematics and maybe understand why scientists talk of beautiful equations. He's glad the most respected living scientist, Stephen Hawking, thinks he's onto something. So, he's come to the University of Oxford to find out more about the most famous equation of all, the one that everyone's heard of.

This equation provokes a whole load of thoughts in Matthew's mind but the main ones are that it's got something to do with the atomic bomb and of course it's by Einstein. But there's cultural knowledge and then there's math, and he doesn't know anything at all about how **E=mc ^{2}** works. When Einstein first published the equation in 1905 it started a scientific revolution.

Scientists live and breathe abstract numbers, but Matthew is an art guy who left school when he was thirteen. However he can see that **E=mc ^{2}**, like all equations, is about balancing two sides. That's what the equal sign is all about. So this equation allows us to calculate how much energy is contained in any given mass. It's a surprise to Matthew that it applies to everything... toothpaste, book, a nail, or uranium for that matter.

This equation is universal. Since **c ^{2}** is such a big number, a tiny lump of matter contains an enormous amount of energy. But what this equation doesn't tell you is how to unlock that energy. The most dramatic proof that the equation was true came forty years after Einstein first worked it out when the atomic bomb was dropped on Hiroshima. Matthew is impressed that

**E=mc**was created before it was shown to be true. The equation was a prophecy. The five symbols explained the link between the energy and all matter across the cosmos.

^{2}
Although they were all good, I thought the Dirac segment was the best. The music for this is also very expertly done, logically matching the content step for step (I kept having to click back a bit to see what words I missed hearing). Not groundbreaking, we've probably all seen docs like it before, but very enjoyable.

Having only studied math and physic for half of my life, I have formed a similar conclusion about beauty in equations like Dirac's. Namely, that it is something one experiences, kinda like art. It is different for everyone and it is something one experiences after a long time of trial and error. You will know it when you get there but not before. People can tell you their definition of what is beautiful, like compactness, simplicity, utility, symmetry. But, trust me, it feels more like falling in love. Sure, all those qualities are there, but words fail to describe it. And once you have experienced it with one equation, it becomes easier to see beauty in all sorts of equations. In my case, my first love is the compressibility equation. She is not famous, not very slim, not so practical, not so symmetric, but she is beautiful in all her imperfections.

To capture beauty, whether it is in an equation, or in a work of art, only has meaning in the eye of the beholder. For an artist such as myself, I was never fond of cubism, because it is not elegant, (pleasing to the eye) and is a poor metaphor for the works of Einstein, Newton, Dirac, and Hawking!

hawking is not seen as the most respected living scientist by his peers...he hardly makes it into the top 20......that compliment may be true for us 'mathmatical-civilians' but that is a different equation.

Yeah, he sure isn't. When I found this out a few years back, I was surprised. I'm sure that ALL of those scientists would say that this assessment is solely due to the work he has done, but I'd be willing to bet you that at least SOME degree of it is founded in plain resentment over his wealth and celebrity, and even the suspicion that among the ignorant masses the dramatic story of his overcoming his condition has to be the prime reason he was able to attain those. Its been said before, that working in those rarefied heights amounts to a kind of "priesthood," and the more monkish you are about what you do, nose kept to the grindstone of your illuminations, the better your reputation is likely to be among your peers. As a lifelong lover of Asimov's non-fiction, in particular, I'm personally very grateful to those who have the expertise trying to make their work accessible in some way to those who don't, dramatic backstories notwithstanding.

i want to know, if time changes - goes slower - relative to your movement speed, and when you get to the speed of light it stops moving, then how can light, (which moves at the speed of light) take time to get somewhere? is it because it is going a VERY small amount slower than the actual unaffected speed of light due to the pull of gravitational objects and other variables?

That's a great question, actually, one I'm surprised isn't asked more often. As far as I can understand, photons are massless, so the rules don't apply to them. Time doesn't mean anything to them, so to speak. They don't experience it, since the speed of light is the only state they know..

thanks, and yes, i too wonder what exactly is light, is it massless? or is it a particle? but if time means nothing to them then how do Light Years even exist? if so then what about black holes sucking it in? it shouldn't be able to affect something timeless, this is quite interesting though, sorry its a little difficult to wrap my head around this stuff at 17 ^^ how ever i love to try!

ah, i was very confused with what time dilation means and when to look it up, (i'm really not the guy for nitty gritty details but i tried anyway) i think i understand, so basically time dilation is something like: a rocket would appear to be moving faster if i saw it on earth than if you saw it on Jupiter?

if so, then what you mean is, it only seems like light takes time to get were it is going because of OUR perception of it relative to were it is suppose to be getting?

It is infamously counter-intuitive stuff, for sure. Type in 'What is time to a photon' in Google. The very first link that comes up should take you to askamathematician/physicist dot com, under the heading 'If a photon doesn't experience time, then how can it travel?' Go there! The explanation is short, but it may help answer your question better than I could, and some of the comments below are also pretty interesting.

You need to stick IMO after that. It's fallacy that nothing can travel faster than light. Einstein did not accept Quantum theory, ergo his equations are incomplete at the sub-atomic level.

Off the top of my head here are three things that travel FTL. Admittedly, the third one has not been discovered yet. But gravity existed long before Newton .....

1. Gravity waves (...or Gravitons ...whatever you want to call them) travel FTL. Obviously.

2. Information. (Through non-locality / Bell's Theorem as proved by Alain Aspect 1981.

3. Tachyons

Light is photonic in nature - it's just photons - if it travels distance then it must take time. Time does not stop. We haven't discovered Tachyons yet either, they are supposed to be FTL also. Einstein didn't give us a full picture of the Quantum universe, so don't hold his laws to be perfect, they are not. We have no gravitational laws for the Quantum level.

1. Where is the rocket?

2. Is the second observer standing on Jupiter, or are they observing the rocket which is located on Jupiter?

The second question makes no sense. Light takes time "to get somewhere" because T = Distance / Speed (in this case, c)

Photonic theory is incomplete. It is by the definition of light as energy which makes it ridiculous to say that photons have no mass. Energy is mass. Light is energy. Therefore light (a photon) has mass. They say the rest mass is zero, but light is never 'at rest'.

Hawking has given the scientific world Hawking Radiation and that's it. His thing is to make grandstand appearances at conventions, come out with a radical unprovable statement, and then whirr off into the wild blue yonder again.

You would never have heard of Stephen Hawking if he didn't have ALS. Fact.

Hawking Radiation is not exactly an insignificant achievement, nor is it his only one, just the most well-known, and one well beyond most of his colleagues, who, if he hadn't have more or less stumbled (no pun intended) into fame and fortune through an unhappy quirk (also no pun intended) of fate, would have had a lot more respect for him, I have no doubt. Scientists are not moral giants, and are as prone to such commonplace things as jealousy as the rest of us, maybe even more so, as their need for achievement tends to be higher.

Light is never at rest relative to an observer

other thanthe photon. " time dilation (no time for photons) goes hand in hand with infinite length contraction (there’s no distance to the destination)."At times, i feel, there's more beauty in equating than in an equation. But perhaps the real beauty lies in actually understanding it. And if that's not there, like the Dirac equation that went way over my head, then i can revert to the easthetics only. Or my imagination maybe.

But exactly the same goes for the arts.

More beauty in painting than a painting, even more in comprehending it. And if not, an aesthetics for the senses only, I imagine :)

I'm not sure, but there might be equations out there that aren't pleasing to the eyes either. Of course, it's a matter of taste, but i can quite appreciate some cubist work, like 'lady in an armchair' presented here in this documentary.

Well, first of all, the the rocket is jut flying through space...simply that both of us can see it from our planets. you are standing on Jupiter looking at the rocket fly by and i am stand on earth doing the same.

Now i have two conflicting opinions right now so i don't know who to agree with, not sure who is right. i do think light takes time to move, but still not sure. if light has a specific speed then it must move right? unless its 136,00 mps in relativity...

As an artist, I can appreciate all styles - aka creativity; however, abstract work,( while creative) is an affront to one's senses, the opposite of elegance/beauty. Complex mathematical equations are an affront to everyone's mind, even the mathematician.(this part, we agree) It is taking something complex, and rendering it simplistic, especially to the layman, is what makes it elegant. Perhaps the best example is James Maxwell. To simplify all electromagnetic phenomenon into 4 simple equations, is creativity, and elegance at it's finest. For me, (my opinion) the finest artists were the likes of Da Vinci, Michelangelo, and Rembrandt. Picasso may have been creative, but he was anything but elegant.

P.S: As I stated in my previous comment, beauty is in the eye of the beholder.(subjective) I accept that many people can derive enjoyment from 'cubism', it just isn't my 'cup of tea'. Take care, and best wishes NX2.

At the risk of sounding contradictory, I will try to answer your questions.

1) A photon is massless elementary particle/wave (0 rest energy) that mediates the observation of the electromagnetic spectrum. A gravitron mediates the force of gravity, a gluon mediates the strong nuclear force, and a boson the weak nuclear force. (4 forces govern the universe - that we are aware of)

2) Photons exhibits wave/particle duality. In essence, when you are observing, or measuring it, it is a particle of certainty; when you are not, it is a wave of probability. (is everywhere at once)

3) In a vacuum, at the speed of light, time stands still. (perfect symmetry = no mass, no position) Once you drop below this velocity, you enter back into relativity; somewhere you are, and somewhere you are not. (broken symmetry = mass and position)

4) Light speed is very quick compared to anything you encounter in daily life, but it is nothing compared to the size of the universe, so a 'light year' (app. 6 trillion miles) allow us to gauge insanely large distances, and reminds us how long it takes to traverse them, at insane speeds (look up at the big dipper on your 75 birthday, and you will see it as it was on the day you were born - over generalization, but you get the idea)

5) Technically, black holes do not suck in light. They warp the 'fabric' of the space/time continuum so dramatically, the path that light itself follows forbids the massless particle's escape, as well as objects of mass (exception: black holes emit radiation via entropy and virtual particles - they are not truly black)

P.S: Don't beat yourself up; scientists themselves don't truly understand it, and probably never will. Take care, and best wishes Matuvo.

If you are interested, check out a documentary on Hawking called, (you guessed it) a Brief history of time. Like Einstein, by Hawking's own admission, he was a terrible student, who preferred to get drunk, and never really took his schooling seriously until he was diagnosed. With that said, colleagues of his all admit that as a mathematician, he was in a class of his own. In reality, there is far more 'brilliant' people than we will ever be aware of; sadly, many of them never get the recognition they deserve. All that aside, it is amazing that he has lived as long as he has with his condition. Perhaps that in itself is a testament to the mental strength that is Stephen Hawking.

If I am reading you correctly, you are trying to determine why a photon needs to move, if it is everywhere at once.

From our relative perspective , a photon takes many thousands of years just to be able to leave the sun. It is absorbed, and re-emitted many times over before it clears the surface, simply because everything is so tightly packed. When it finally does, it will continue to travel until it collides with something else; in this case the earth. (8 minutes later) Thus, individual photons move, and require 'time' to do so, but only from our perspective, not theirs. The fact of the matter is there is more than one photon in the universe, and when you consider the nature of their duality, (certainty vs probability) only then can you realize 'your question' denotes your need for 'one or the other', when in fact just posing the question itself, is where the problem begins. In other words, we do not have a way of bringing our perspective of observation into the equation, without disturbing the result. (Heisenberg uncertainty principal)

Time dilation is best understood as the 'difference' or 'changes' in clocks from one observer to another, relative to their own velocities, or gravitational fields/acceleration. (equivalence principal ) As Einstein reminded others, in extreme velocities, and gravitational fields, the measuring sticks for distance, and time become very distorted. This is precisely why the confusion occurs regarding a photons duality, the size, and age of the universe/multiverse, etc. Perhaps the hardest part for people to wrap their mind around is that 2 separate observations of the same occurrence can appear completely different, yet both are correct, and valid, relative to their own unique situation.

P.S: The speed of light is 186,000 miles per a second. I hope I am helping, not hurting. If it means anything to you, I started with a book called Einstein's universe by Nigel Calder at age 19. (I am now 50) It is a great read, and only after going through it 20-30 times, could I begin to even grasp the contradiction that is relativity. Quantum mechanics is far worse, and not even the physicists have clarity regarding it. (can you think in 11 dimensions?) Take care Matuvo,

Yeah, I did see that one, it's great. And considering the severity of the depression he fell into after finding out his prognosis, the fact that he was able to eventually shake that off apparently permanently is also quite a testament to his mental strength.

Actually, gravity waves are supposed to travel at precisely the speed of light. This is why Einstein was attempting to unify electromagnetism, and gravity, because their appeared to be some connection between them. This is where the similarity ends, since electromagnetism is 1000 billion billion billion billion times more powerful than gravitation. (ten 39th)

I did not like this documentary. I wanted to like it, but I did not. I should have liked it because I know that mathematical equations - and the concepts that they describe - really are beautiful. I did not like it because the narrator did not have the necessary background to explain why mathematics is beautiful. Not really his fault, I suppose; it's just that he hasn't spent the past 20-40 years studying the subject. Let's be honest, this is not an easy topic. It is impossible for anybody who has not spent at least 20 years as a mathematician or theoretical physicist to fully appreciate the elegance and beauty of this subject.

I understand that the intent of this documentary was to introduce mathematical physics to lay-people. But the attempt does its intended audience a terrible disservice. Mathematical physics is an extraordinarily difficult subject that cannot be condensed into a 1-hour long film. As a result, this documentary will produce one of two effects: 1) it will confuse and misinform an audience of non-scientist, or 2) it will frustrate and insult a audience of scientists and mathematicians.

4/10, Narrator is flat out the wrong person to cast in this role. Besides his ignorance the show seems made on impulse rather than planning. Certainly a B or C rank documentary.

I really enjoyed this documentary because at age 46 I have finally decided to overcome my fear of science and have been on a similar journey to the narrator. The explanations of the equations were helpful to me and reinforced what I have been learning in the last few weeks. I think there is a lot of beauty to be found in science and I can understand why an equation can be perceived as being elegant - each new discovery we make is like a carefully sculptured part that fits exactly into the gap that was there in our knowledge allowing us to move forwards with our understanding and search for more answers to more questions.