Your Favorite Deep, Elegant, or Beautiful Explanation

By Sean Carroll | January 15, 2012 9:40 am

The annual Edge Question Center has now gone live. This year’s question: “What is your favorite deep, elegant, or beautiful explanation?” Find the answers here.

I was invited to contribute, but wasn’t feeling very imaginative, so I moved quickly and picked one of the most obvious elegant explanations of all time: Einstein’s explanation for the universality of gravitation in terms of the curvature of spacetime. Steve Giddings and Roger Highfield had the same idea, although Steve rightly points out that Einstein won’t really end up having the final word on spacetime. Lenny Susskind picks Boltzmann’s explanation of why entropy increases as his favorite explanation, and mentions the puzzle of why entropy was lower in the past as his favorite unsolved problem — couldn’t have said it better myself. For those of you how prefer a little provocation, Martin Rees picks the anthropic principle.

But as usual, the most interesting responses to me are those from far outside physics. What’s your favorite?

Full text of my entry below the fold.

Einstein Explains Why Gravity Is Universal

The ancient Greeks believed that heavier objects fall faster than lighter ones. They had good reason to do so; a heavy stone falls quickly, while a light piece of paper flutters gently to the ground. But a thought experiment by Galileo pointed out a flaw. Imagine taking the piece of paper and tying it to the stone. Together, the new system is heavier than either of its components, and should fall faster. But in reality, the piece of paper slows down the descent of the stone.

Galileo argued that the rate at which objects fall would actually be a universal quantity, independent of their mass or their composition, if it weren’t for the interference of air resistance. Apollo 15 astronaut Dave Scott once illustrated this point by dropping a feather and a hammer while standing in vacuum on the surface of the Moon; as Galileo predicted, they fell at the same rate.

Subsequently, many scientists wondered why this should be the case. In contrast to gravity, particles in an electric field can respond very differently; positive charges are pushed one way, negative charges the other, and neutral particles not at all. But gravity is universal; everything responds to it in the same way.

Thinking about this problem led Albert Einstein to what he called “the happiest thought of my life.” Imagine an astronaut in a spaceship with no windows, and no other way to peer at the outside world. If the ship were far away from any stars or planets, everything inside would be in free fall, there would be no gravitational field to push them around. But put the ship in orbit around a massive object, where gravity is considerable. Everything inside will still be in free fall: because all objects are affected by gravity in the same way, no one object is pushed toward or away from any other one. Sticking just to what is observed inside the spaceship, there’s no way we could detect the existence of gravity.

Einstein, in his genius, realized the profound implication of this situation: if gravity affects everything equally, it’s not right to think of gravity as a “force” at all. Rather, gravity is a feature of spacetime itself, through which all objects move. In particular, gravity is the curvature of spacetime. The space and time through which we move are not fixed and absolute, as Newton would have had it; they bend and stretch due to the influence of matter and energy. In response, objects are pushed in different directions by spacetime’s curvature, a phenomenon we call “gravity.” Using a combination of intimidating mathematics and unparalleled physical intuition, Einstein was able to explain a puzzle that had been unsolved since Galileo’s time.

  • Plato


    Some of my favorites are listed at link.

  • Andy J

    Zero-knowledge proofs. This is stolen from Arora & Barak’s fantastic “Computational Complexity – A Modern Approach”, and follows in the vein of the more detailed “How to Explain Zero-Knowledge Protocols To Your Children” (found here:


    “As an intuitive example for the power of combining randomization and interaction,
    consider the following scenario: Marla has one redsock and one yellow sock, but her
    friend Arthur, who is color-blind, does not believe her that the socks have different
    colors. How can she convince him that this is really the case?

    Here is a way to do so. Marla gives both socks to Arthur, tells him which sock is
    yellow and which one is red, and Arthur holds the red sock in his right hand and the
    yellow sock in his left hand. Then Marla turns her back to Arthur and he tosses a coin.
    If the coin comes up “heads” then Arthur keeps the socks as they are; otherwise, he
    switches them between his left and right hands. He then asks Marla to guess whether he
    switched the socks or not. Of course Marla can easily do so by seeing whether the red
    sock is still in Arthur’s right handor not.

    But if the socks were identical then she would not have been able to guess the answer with probability better than 1/2. Thus if Marla manages to answer correctly in all of, say, 100 repetitions of this game, then – though colourblind – Arthur can indeed be convinced that the socks have different colors”

  • Greg Egan

    Sticking just to what is observed inside the spaceship, there’s no way we could detect the existence of gravity.

    Sticking just to what is observed inside the spaceship over a short time compared to its orbital period, there’s no way we could detect the existence of gravity.

    Tidal forces might be extremely weak, and in practice could be damped by air resistance, but in the interior of an airless spaceship full of free-falling objects they will lead to very visible effects on the time scale of the orbit. An object that’s initially displaced one metre from the centre of the ship will typically travel several metres over one orbit, in some combination of simple harmonic motion and exponential motion away from the centre. With careful experiments, the astronauts could determine the ship’s orbital period without looking outside.

    Sorry to nitpick, since obviously both you and Einstein knew this perfectly well, but this thought experiment gets stated so often without any caveat about observation time that some people end up believing it’s true on arbitrary time scales.

  • Chris

    For me it would be the covariant formulation of classical electromagnetism. You take Maxwell’s 4 equations, put them in one antisymmetric matrix which is simply related to the four potential. Now you have an understanding of the EM field in any reference frame.

  • Gizelle Janine

    Why did the 80′s exist?

    Just to prove everything else blows. I think that’s pretty beautiful…

    in some way…


  • Jon Teets

    RE: Andy J’s comment – A similar kind of test using the Stroop effect ( is used to verify grapheme-color synesthesia. People who see letters in color which the rest of us see in B&W read more slowly when the letters are presented in colors different from the mapping their brain uses.

  • Craig McGillivary
  • Jim Cross

    What I find unusual is the interpretations of the contributors of what it means to be a “deep, elegant, and beautiful” explanation.

    I think some of them really miss the mark.

    Lisa Randall – The Higgs Mechanism. Did I miss the announcement that this has been proven?

    Max Tegmark – “What caused our Big Bang? My favorite deep explanation is that our baby universe grew like a baby human—literally.” Interesting to know we understand what caused the Big Bang and how it grew but (assuming our understanding is even correct) where is the explanation?

    Jared Diamond – The Origins of Biological Electricity. Huh? Very interesting but deep and elegant?

    Gregory Benford – “I find most beautiful not a particular equation or explanation, but the astounding fact that we have beauty and precision in science at all. That exactness comes from using mathematics to measure, check and even predict events. The deepest question is, why does this splendor work?” This seems like a deep, elegant question but not an explanation.

    Carl Zimmer – “A Hot Young Earth: Unquestionably Beautiful and Stunningly Wrong”. Carl’s choice is a deep, elegant explanation that is wrong? Enough said.

    Vilayanur Ramachandran – “What’s my favorite elegant idea? The elucidation of DNA’s structure is surely the most obvious, but it bears repeating. I’ll argue that the same strategy used to crack the genetic code might prove successful in cracking the “neural code” of consciousness and self. It’s a long shot, but worth considering.” I admit there’s some merit to the DNA structure idea but when he goes to suggest an interesting hypothesis (that I guess he believes in) is a deep, elegant explanation he might be stretching it. I would say we should at least stick to things that are pretty well accepted today not things that might be accepted tomorrow.

    To me, Leonard Susskind comes closest to the intent of the question.

    “Personally my favorites are explanations that that get a lot for a little. In physics that means a simple equation or a very general principle. I have to admit though, that no equation or principle appeals to me more than Darwinian evolution..” As a physicist, he goes on to cite “Boltzmann’s explanation of second law of thermodynamics: the law that says that entropy never decreases” as his favorite.

  • Chandrashekhar

    that was one awesome thought experiment.. never heard of it before till now (even though I’m knda sure that it must be so very popular amongst physicists and students alike).. thanks for putting it up here..

  • Joel Rice

    I would go with W. R. Hamilton’s explanation of plain old rotation in 3d space with Quaternion algebra. It ignited a huge revolution in physics and enabled Maxwell to formulate Electromagnetism – thinking in complex quaternions. Nature appears to love this algebra, since it keeps showing up – in Pauli’s account of Spin, and again in Isospin, and Weak Isospin. Not to mention that it inspired Graves to discover Octonion algebra – which does not seem to make much sense to physicists, probably for thinking it ought to be a tool one might use, but I hazard a guess that if quaternions make sense of space then octonions make sense of particles.

  • Bill

    While the double helix of DNA gets a few nods, I like Matt Ridley’s short essay on the change of perception brought by this visual model. The idea of complementary base pairing is so fundamental to understanding inheritance, inter- and intra-cellular information transfer, and nucleic acid dynamics that one almost forgets its importance because of its ubiquity. It’s the F = ma of biology.

    As I perused the essays, I was happy to see that Nigel Goldenfeld chose to write about one of my other favorite explanations: the nature of the genetic code. He highlights an early theoretical paper (interestingly categorized under Physics) on the code,

    Codes without commas
    Crick FH, Griffith JS, Orgel LE.
    Proc Natl Acad Sci U S A. 1957 May 15;43(5):416-21.

    which partly addressed Gamow’s fascinating, though flawed, hypothesis. Equally important, at least to me, were the experimental tests of these theoretical ideas. The paper describing the test results is among my favorites. Its logic is simple and the experiments are low tech (petri dishes and toothpicks), yet the paper reads like a tightly woven proof of a mathematical theorem — elegant. Moreover, Crick et al. use language that should be familiar to many. The only arcane word that pops up is “cistron”: they use this term instead of “gene” because it has a precise experimental meaning (derived from cis-/trans-complementation experiments), whereas the term “gene” had (and still has) a flexible and context dependent meaning.

    General nature of the genetic code for proteins

    Crick FH, Barnett L, Brenner S, Watts-Tobin RJ.
    Nature. 1961 Dec 30;192:1227-32

  • Kevin

    I’m always surprised that the Principle of Least Action, or even just the action itself, isn’t brought up in these sorts of things. If you wanted a famous source, Dr. F was a big fan, and for good reason. The idea of energy is already pretty deep and elegant, a concept which is only as intuitive as our constant study and use of it has made it since its early physical definitions and essentially the first step on physics’ path towards the beautiful symmetries it trades in today. But somehow, it gets even better: the idea that the universe prefers energy of motion and the energy of possible motion to be as close to each other as possible over the lifetime of any system is as elegant as it gets. And deep – it’s a principle applicable to both quantum and classical systems, which is a rarity.
    Just wanted to give a little plug to my personal favorite “deep and elegant explanation”.

  • Arun

    My favorite deep and elegant explanations are the ones I understand. While it is easy to take someone else’s word that some result has been established or that something is well-understood, it is hard to take someone else’s assessment that it is deep, elegant, etc.

  • noen

    Does mathematics count? It’s not really an explanation, more like a technique but I would nominate the diagonal argument made famous by Godel and it’s various incarnations. Some of those do not work but lead to the liar’s paradox. The fact that the diagonal argument can become paradoxical I think does say something… important? Deep? Well…. I like it anyway.

  • Igor Khavkine

    A follow up to Greg Egan’s comment. If Greg won’t plug his book, then I will. :-)

    This very phenomenon (gravitational tidal forces observable over long periods of time) was used to great dramatic effect in his recent novel Incandescence.

  • Francois Demers

    Hackneyed, overdone but still: E=MC squared. Explains the relationship of matter to energy. Elegant in its economy of means for the vast domain it explains. Deep in its extraodinary impact on modern physics.

    For pure elgance, Euler’s postulate.

  • Eunoia
  • Sam Gralla

    The equivalence principle is a healthy #2, but of course #1 is Newton’s realization that apples fall and planets orbit for the same reason. The concept of the same force causing both “linear” and “circular” motion is amazingly unintuitive (as well as deep, elegant, and beautiful).

  • Jeff

    My favorite was always the proof, by the Pythagorean theorem, of the time dilation formula of special relativity, based on the assumption that the speed of light is invariant. This is based on using a clock that “ticks” each time a beam of light bounces between two mirrors, where the two mirrors are each parallel to the direction of motion. For a moving clock, the beam of light makes a zigzag pattern and the Pythagorean theorem can be used to determine the length of each diagonal leg of the zigzag — viola, the time dilation formula!

  • Dave

    Noether’s theorem. I’ve always found how it explains the conservation (or even existence) of the notion called energy, momentum, etc. in terms of symmetries particularly beautiful.

  • rww

    Feynman posed a similar question in his Einstein lectures (Six Easy Pieces, ch 1):

    “If in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words?”

    “That all things are made of atoms” was his reply.

  • Justin Loe

    Nonlocality. Clearly nonlocality is counterintuitive to everyday experience and there have been many speculative books for an amateur audience on this topic. Alain Aspect seems to have confirmed nonlocality as an explanation for quantum phenomena but it is fundamentally unclear what this means. In a sense then, this is an explanation that opens up a whole series of additional puzzles that contradict the usual commonsense intuitions with which we perceive everyday phenomena.

  • Peter

    Me too for Noether’s theorem.

  • chas

    It belongs in the future ; it will be in the proof of Goldbach’s Conjecture : ‘Every even
    number is the sum of two prime numbers ‘ .

  • Jimbo

    As beautiful as GR is, don’t we suspect that a Deeper underlying source for its differential geometric formulation is in fact thermodynamics ? Einstein himself mused that,
    “Thermodynamics is the only physical theory of universal content which will never be overthrown”.
    Hawking, Ted Jacobson, T. Padmanabhan, & E.Verlinde have shown such connections underlie the esoteric math of gravity/differential geometry, & empower it with the physical reality of heat. Thus the synergy of the two manifests from cosmology to coffee cup.

  • Debra Leddon

    My favorite is the one liner mentioend by Carl Sagan years ago; “we are all ‘star-stuff’”. It’s a very simple but powerful statement of how a collapsing star creates the heavy atoms needed for life and for the rocky planets on which life exists (that we know of). There’s more to say on this but it is best said by Sagan himself in an episode of ‘Cosmos’.

  • Jimbo

    Debra: I think its much more beautifully Sung, by Joni Mitchell in her `Woodstock’, beating Sagan by ~10yrs: “We’re StarDust…We are billion yr-old Carbon, & we got to get ourselves back to the garden”.

  • sjn

    I second #12 and #20: Least Action and Noether’s theorem.

  • Gizelle Janine

    I really loved that I saw the scientific method on here.

    Underappreciated and taken for granted, me thinks.


  • Ahmed

    Rolf Landaur’s explanation of why (and how) information is physical. Nothing is more intriguing to me than the eerie relation between something we used to think of as a product of our own reflective recognition and categorization of patterns, and the very real and unmistakable flow of energy.

  • Elliot Tarabour

    My vote is for Shannon’s information theory.

  • Ben

    I am not shure I have one.

    But I can pronounce a couple of questions.

    Can – theoretically- the strings or gravitones be the same as the dark energy dARK matter, if they were the first matter that scattered out from the BB- event and could they work as a “forcefield” pulling bringing the clumsier “ordinary mass” to or from its end station position?

  • David Brown

    I have two candidates for deep, elegant, or beautiful explanation. In probability theory, one explanation of data is the “Law of Large Numbers”.
    In the philosophy of measurement, one explanation of measurement is the “Law of Stray Numbers”. This law has been elegantly expressed by Shawn Achor as, “We know that’s a measurement error because it’s messing up my data.”

  • BobC


    I’ve been working my way through Edge’s Deep/Elegant/Beautiful list, and about 20% of the entries have made me pause to learn more. There is lots of beauty and elegance out there, and my main ‘take-away’ so far is how lucky we are to be able to perceive and delight in it.

    Also, ‘deep’ has new meaning for me. It used to be synonymous with ‘difficult’ or ‘obscure’. Now I see Deep as meaning ‘revealing’. A truly equal peer with Beauty and Elegance when defining the degree of satisfaction we feel concerning our attempts to explain our world.

    I’m about half-way through now. Yes, it’s been weeks. But it seems things keep getting more interesting the further I go. A wonderful journey with many surprises along the way. My favorite surprise so far must be the results of combining statistics and graph theory with gastronomy. (No names or links: Look for it!) That simply tickled my brain in all the right places, and makes me want to be more adventurous in the kitchen.


  • JoeTurpin

    The first definition given in Euclid’s Elements of Geometry: “A point is that which has no part”


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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