Chapter One
Beauty and the Beast
What is beauty? What is it that makes certain works of art,pieces of music, landscapes, or the face of a person soappealing to us that they give us an enormous sense ofexcitement and pleasure? This question, with which many philosophers,writers, psychologists, artists, and biologists have struggledat least since the time of Plato, and which led (among other things)to the birth of the notion of aesthetics in the eighteenth century,is still largely unanswered. To some extent, all of the classical approachesto beauty can be summarized by the following (clearlyoversimplified) statement: Beauty symbolizes a degree of perfectionwith respect to some ideal. It is strange, though, that somethingwhich has such an abstract definition can cause such intensereactions. For example, some accounts claim that the Russianwriter Dostoyevsky sometimes fainted in the presence of a particularlybeautiful woman.
In spite of some changes in taste over the centuries (and someobvious differences among different cultures), the perception ofwhat is beautiful is very deeply rooted in us. It suffices to look at afew paintings like Botticelli's Primavera or Leighton's Flaming June,or at a majestic mountain landscape, to realize this.
The term beautiful has evolved from being identified with"good" and "real" or "truthful" in ancient Greece to a quality thatis confined merely to its effects on our senses, in the second halfof the eighteenth century. It is interesting to note, though, that theapproach of the philosophical book of Proverbs in the Bible hasbeen rather dismissive and closer to the more modern definitionin its assertion: "Charm is deceitful, and beauty is vain."
But, even if only affecting our senses, the effects of beautyshould not be underestimated. The ancient Greeks certainly didnot underestimate them. Greek mythology contains the famousstory of the goddess Eris, who, insulted because she was not invitedto the wedding party of King Peleus and the sea nymph Thetis, decidedto take revenge by throwing into the banquet hall a goldenapple that carried this inscription: "For the Fairest." After a longdebate among the goddesses, the choice was narrowed down tothree contestants for the title: the goddesses Hera, Athena, andAphrodite. The matter was brought for a decision to Zeus who(very wisely) passed the task on to Paris, the son of the King of Troy.As it turned out, Paris's job was reduced to an evaluation of thebribes offered to him by each of the goddesses.
Hera whispered to him that she would make him the Lord of Europeand Asia; Athena promised him victory over the Greeks; andAphrodite made him an offer he could not refuse—she promisedhim that the most beautiful woman in the entire world would be his.Paris gave the apple to Aphrodite, an action that can only be describedas a mistake of historic proportions. The most beautifulwoman on earth was Menelaus's wife, Helen, whose face "couldlaunch a thousand ships." The end of the story is tragic. After Pariskidnapped Helen and brought her to Troy, a fierce and bitter warbroke out, which led eventually to the total destruction of Troy.
Helen's beauty is described as being so intense, and its effectsso devastating, that when Menelaus decides to execute Helen, aTrojan hero's mother forces him to swear that he will perform theexecution without looking into Helen's eyes, because "through theeyes of men she controls them and destroys them in the same waythat she destroys cities."
Some speculate that Helen's beauty was of the cold, unapproachabletype, and that its overwhelming effects stemmed fromthe fact that Helen appeared as unattainable as the understandingof the concept of beauty itself.
Sometimes one can understand a certain concept or qualitybetter by examining something that epitomizes the opposite. Thisnotion is partly responsible for such pairings as heaven and hell,Dr. Jekyll and Mr. Hyde, Stan Laurel and Oliver Hardy, and, indeed,as the title of this chapter implies, Beauty and the Beast. Sofar I have only discussed beauty, but where is the "beast"?
The beast, in this case, is—physics! To many of my personalfriends and to a large number of students of humanities whom Ihave met over the years, there is nothing more remote from the notionof beauty and more antithetical, from the point of view of thesensation that it induces in them, than physics. In fact, the disgustand fear that physics stimulates in some people is rivaled only bytheir feelings toward cockroaches. In an article in the Sunday Times(quoted in a BBC lecture by Richard Dawkins), columnist A. A. Gillcompared observations in the sky to movie and theater stars by saying,"There are stars and there are stars, darling. Some are dull,repetitive squiggles on paper, and some are fabulous, witty, thought-provoking ..."Believe it or not, those "dull, repetitive squiggles"represented the discovery of pulsars, objects so dense that one cubicinch of their matter weighs a billion tons, and that take a fractionof a second to rotate, instead of the earth's twenty-four hours!
I hope that this book will convince even skeptics that "beautyin physics and cosmology" is not an oxymoron. I remember a certainscene in the movie Good Morning, Vietnam in which a soldier isasked to which unit he belongs. His answer, "military intelligence,"provokes an immediate reaction from the general: "There is nosuch thing!" In relation to science, the English poet Keats virtuallyaccused Newton of ruining the beauty of the rainbow by his theoreticalexplanations of how it is formed, using the laws of optics. InKeats's words:
Philosophy will clip an Angel's wings Conquer all mysteries by rule and line, Empty the haunted air, and gnomed mine— Uneave a rainbow ...
Incidentally, some readers may find the latter story surprising,given that Keats is often quoted as having said: "Beauty is truth,truth beauty." In fact, Keats said no such thing. It is what he saidthe Grecian Urn depicts, in his criticism of works of art that deliberatelyeliminate existing difficulties of life.
Keats's complaint merely reflects the general feeling that magicians'tricks often lose their charm once we know how they areperformed. However, in physics, very often the explanation is evenmore beautiful than the question, and even more frequently, thesolution to one puzzle helps uncover an even deeper and more intriguingmystery. I therefore hope to be able to demonstrate thatreactions like Keats's merely represent a misunderstanding that isbased on false myths.
What Is Beautiful?
Definitions are always difficult, especially when we are dealing withsomething that is (largely) subjective. In this sense, even the definitionin the Oxford English Dictionary—"Impressing with charm theintellectual or moral sense, through inherent fitness or grace"—whichsurely does not involve all the intricacies originating fromphilosophical interpretations, is not particularly useful.
I will attempt to make my task easier by answering at this pointa much simpler question. Since my goal is to discuss beauty inphysics and cosmology, I will address the question: When does aphysicist feel that a physical theory is beautiful?
Any endeavor aimed at answering this question is bound to resultin the use of an entire vocabulary of concepts, mostly borrowedfrom the arts. The list of such concepts may include symmetry, coherence,unity, harmony, and so on.
Probably not all physicists agree on which subset of conceptsfrom this list should be used. However, I will argue later that at leastthree requirements are absolutely essential and must be fulfilled:
1. Symmetry 2. Simplicity 3. The Copernican principle
A fourth element, elegance, is also considered by some to bean important ingredient of a beautiful theory, but, as I will explainlater, I do not consider it essential.
I realize that at this stage the meanings of all of these conceptsare vague at best (if not totally obscure), but I will now explain insome detail what I mean by each one of them.
1. Symmetry: When Things That Might Have Changed Do Not
Everyone is familiar at least with symmetries of pictures, objects, orshapes. For example, our face and body have an almost exact bilateralsymmetry. What this means is that if we reflect each half ofour face, we obtain something that is nearly identical to the original(strangely enough, this is true even for the one-eyed giant Cyclopswhom Ulysses encountered in his travels).
Some shapes are symmetric with respect to rotation. For example,a circle drawn on a page remains the same if we rotate thepage on the desk.
Other arrangements are symmetric under certain displacementsor translations. For example, if we stood in front of some ofthe row houses in Baltimore facing one unit, and someone were todisplace the entire row by one unit, we would not notice any difference.Similarly, if we look at one Campbell soup can in a paintingcontaining rows of identical cans by the pop artist Andy Warhol,and the painting is shifted slightly sideways or upward, we see anidentical picture.
Notice that in all of these examples the object or shape did notchange when we performed the symmetry operation, reflection, rotation,or translation.
The association of symmetry with beauty does not require elaborateexplanations. Anyone who ever looked through a kaleidoscopehas experienced the sensation of beauty that symmetry inspires.In fact, the word kaleidoscope itself comes from the Greekwords kalos, which means "beautiful," and eidos, which means"form" (skopeein means "to look").
It is important to emphasize, though, that when the concept ofsymmetry is introduced into physics, it is not the symmetry ofshapes that we are interested in but rather the symmetry of the physicallaws. As we shall soon see, in this case, too, the symmetry is associatedwith things that do not change. In order to explain thisconcept better, let me first describe briefly the nature of these entitiesthat we call laws of physics.
The laws of physics, sometimes referred to as the laws of nature,represent attempts to give a mathematical formulation to thebehavior that we observe all natural phenomena to obey. For example,in classical physics, Newton's universal law of gravitationstates that every particle of matter in the universe attracts everyother particle through a force called gravity. It further gives a quantitativemeasure of how this attraction is larger the more massivethe particles (doubling the mass of one particle doubles the force),and how it decreases when the distance between the particles is increased(doubling the distance weakens the force by a factor offour). To give another example of a law of physics, one of JamesClerk Maxwell's equations, the laws that describe all the electricand magnetic phenomena, states that there are no magneticmonopoles (single magnetic poles). Namely, there cannot be amagnet that has only one, say, north pole. Indeed, we know thateven if we take a bar magnet and chop it up into smaller andsmaller pieces, each piece will have north and south poles.
Now, what is the meaning of symmetries of the laws of physics?These symmetries are certain fundamental properties of the laws,which are somewhat similar to the symmetries described for shapesor objects. For example, all the laws of physics do not change fromplace to place. A simple but remarkable manifestation of this propertyis the fact that if we perform an experiment, or study any physicalphenomenon, in Russia, in Alabama, or on the moon, we obtainthe same results. This also applies to different parts of theuniverse; when we observe a star that is located trillions of milesaway from us, it still appears to obey the same laws of physics thatwe find here on Earth. This means that we can apply the same lawsthat we have deduced from laboratory experiments to the understandingof the universe as a whole. This universal transportabilityof the physical laws is encapsulated in the statement that the lawsof physics are symmetric under translations. This property is not to beconfused, for example, with the fact that the strength of the forceof gravity is not the same on the earth and on the moon. Gravityon the moon is different (weaker) because both the mass and thesize of the moon are different from those of the earth. However,given that we know the mass and radius of the moon, we would useexactly the same formula to calculate the force of gravity there, as wedo on Earth.
The laws of physics also do not depend on the direction inspace. For example, they would not change if the earth started torotate in the opposite direction. Were this not the case, then experimentsmight yield different results in the Southern Hemispherethan they do in the Northern Hemisphere. Furthermore,we might obtain different results if we perform an experimentlying down, rather than standing up, or we might find that lighttravels faster to the north than it does to the east. Note again thatI do not refer to the fact that, for example, different stars happento be seen in the night's sky from Australia than from Alaska (norto the fact that different rock music groups may be popular in thetwo places), but to the fact that the laws that describe all the naturalphenomena do not have a preferred direction. Thus, the lawswould not change if someone took our entire universe and rotatedit somehow. This property is expressed by the statement that thelaws of physics are symmetric under rotation.
I would like to further clarify the difference between a symmetryof a shape and of a law. The symmetry of the laws of physics underrotation does not mean, for example, as it was believed in ancientGreece, that the orbits of planets must be circular. A circle, as a shape,is indeed symmetric under rotation. But this has nothing to do withthe symmetry of the law—in this case the law of gravity, which governsthe motion of the planets around the sun. In fact, since the timeof Johannes Kepler, a German astronomer who worked in Praguein the seventeenth century, astronomers have known that the orbitsof the planets are not circular but elliptical. The symmetry of thelaw means that the orbit can have any orientation in space (Figure 3).
Another symmetry that the basic laws of physics exhibit concernsthe direction of time. Curiously, the laws would not change if timewere to flow backward. This is true for both mechanical and electromagneticphenomena at both the macroscopic and the subatomiclevels. For example, there is nothing in the basic laws to indicatethat the phenomenon of a plate falling from a shelf andshattering to pieces on the floor should be allowed, while that ofthe scattered pieces flying up from the floor and assembling to anintact plate on the shelf should be forbidden.
Interestingly enough, as far as we know, the laws of physics alsodo not change with the passing of time. Astronomy proves very usefulin demonstrating this property. We can observe, for example,galaxies at distances of millions, and even billions, of light-years.One light-year is the distance light can travel in one year, which isabout six trillion miles. What this means is that the light thatreaches us now from a galaxy that is over 100 million light-yearsaway left that galaxy more than 100 million years ago. Therefore, whatwe see today is really the way the galaxy was 100 million years ago.Astronomy thus truly allows us to look back into the past. The mainpoint to note is that by analyzing the observed light, we can establishthe fact that the same physical laws that govern the emissionof light by atoms today also applied in the distant past. In fact, wecan now state with a high degree of confidence that the laws ofphysics have not changed at least since the time the universe wasonly about one second old (see chapter 3).
As we have seen, when objects or shapes possess a certain symmetry,this is related to something that does not change, an invariant.For example, because of its left-right symmetry, a mirrorimage of the Notre Dame Cathedral in Paris looks identical to thecathedral itself. Symmetries are therefore related to the indiscernibilityof differences. Since we are discussing symmetries of thelaws that govern the behavior of all natural phenomena, the propertyof having things that do not change is in this case translatedto universal entities we call conservation laws. A conservation lawsimply reflects the fact that there exist physical quantities in theuniverse that remain constant in time. Namely, if we were to measurethe value of such a quantity today, one year from now, or a millionyears from now, we would obtain exactly the same value. Thisis to be contrasted, for example, with the stock market, wheremoney is definitely not a conserved quantity—that is, on a givenday everybody may lose, with no one gaining.
The two symmetries of the laws of nature I have already mentioned,the symmetry under translations and the symmetry underrotations, indeed result in conservation laws. For example, the linearmomentum of a body is equal to the product of its mass and itsspeed, and its direction is the direction of the motion. Thus, thelinear momentum of a body defines in some sense the quantity ofmotion this body possesses; it is larger the larger the mass and thespeed. A stampeding buffalo has a larger linear momentum thanthat of a man running at the same speed but a smaller one than thatof a rocket moving much faster. The symmetry under translationsis manifested in the fact that linear momentum is conserved.Namely, momentum can neither disappear nor be created; itcan just be transferred from one body to another. In everyday life,we see directly the consequences of conservation of linearmomentum—for example, in the trajectories of colliding cars, ofcolliding billiard balls, and in the motion of the puck in ice hockey.The speed and direction of all of these motions are determined insuch a way that the total momentum of the system is conserved.The motion of rockets is also a consequence of the conservationof linear momentum. When the rocket is resting on the launchingpad, its momentum is zero (because its speed is zero). This meansthat as long as external forces do not interfere, the momentummust remain zero. When the rocket starts to eject gases downwardat a high speed, the rocket itself acquires an upward speed, tocounterbalance the momentum of the gases.
The angular momentum of a rotating body is a measure of theamount of rotation it possesses. For example, if two identical spheresare rotating around their axes, the angular momentum is larger forthe one that rotates faster. If two spheres of the same mass are rotatingat the same rate, the angular momentum is larger for the onewith the larger radius. The symmetry of the laws of nature under rotationsis manifested in the fact that angular momentum is also aconserved quality.
Ice skaters make frequent use of the conservation of angularmomentum. In one of the popular routines, skaters start spinningslowly with their arms stretched sideways, and then they bring theirarms to the sides of their body, thus increasing dramatically therate of their spin. I still have a picture in my mind of a young ScottHamilton, with his red hair spread almost horizontally, as he spinsincredibly fast. This behavior results from conservation of angularmomentum—reducing the distance of the arms from the rotationaxis results in an increase in the speed of rotation. Conservationof angular momentum is also responsible (among other things)for the fact that moving bicycles and spinning Hannukah dreidelsdo not fall, for the stability of the axis of gyroscopes (which areused to determine directions accurately), and for the stability ofthe orbits of the planets around the sun.
Another symmetry that was mentioned above, the fact that thelaws of nature do not change with the passing of time, is responsiblefor the existence and conservation of the quantity we call energy.We all have a certain intuitive understanding of what energy means;after all, we pay energy bills to gas and electric companies, andmany of us still remember the energy crisis in 1979, when gasolinewas expensive and hard to find. In some sense, energy reflects theability to do work. Very broadly speaking, energy can be associatedwith motion (in which case it is called kinetic energy), can be storedin some form (e.g., chemical, electrical, gravitational, nuclear; inwhich case it is called potential energy) , or can be carried by light (radiativeenergy). Again, conservation means that energy is neithercreated nor destroyed. It can merely be transferred from place toplace or be transformed from one form to another. For example,when we drop a spoon, gravitational potential energy is transformedinto kinetic energy of motion, and the latter is transformedinto heat and acoustic energy as the spoon hits the floor.
Having briefly explained the concept of symmetry, I will nowturn to the second requirement for beauty, that of simplicity.
2. Simplicity: Less Is More Beautiful
Simplicity is to be understood in the sense of reductionism. Namely,the goal of physics is to replace many questions by very few, basicquestions; or a description of nature that involves many laws ofphysics by a complete theory that has only a few fundamental laws.Physicists have been driven for centuries by a feeling that underneaththe enormous wealth of phenomena that we observe, there exists anunderlying relatively simple picture. The great seventeenth-centuryFrench philosopher and scientist René Descartes once said: "Methodis necessary for discovering the truths of nature. By method, I meanrules so clear and simple that anyone who uses them carefully willnever mistake the false for the true, and will waste no mental effort."We have already seen an example of the application of this type ofthinking in chapter 1, in the search for one mechanism to explain allthe shapes of the nebulae.
We can identify in this drive for reductionism some of the sameelements that perhaps formed the basis for the notion (in theJudeo-Christian cultures at least) that monotheism represents amore advanced (more beautiful?) form of faith than polytheism.The order for one God is expressed very clearly in the first twocommandments: "I am the Lord your God ..." and "You shall notmake for yourself an idol ..." I remember that as a skeptical child,I used to be somewhat puzzled by the statement made by a teacherthat the move to monotheism represented progress. After all, Ithought, if it is all a matter of faith anyhow, then what differencedoes it make if you believe in one God or in many gods, eachof whom is responsible for a different phenomenon in nature?Today, I can identify in that statement the same requirement forreductionism, for simplicity.
Given two theories that explain a given phenomenon equallywell, the physicist will always prefer the simpler one, for this aesthetic(and not just practical) reason. I want to emphasize that thisdrive toward reductionism does not mean that the physicist fails torecognize that there is beauty in the richness and complexity ofphenomena. After all, physicists realize, too, as did the poet WilliamCowper in the eighteenth century, that "variety's the very spice oflife." The emergence of complexity in our universe, with life beingperhaps at the pinnacle of this complexity, is what makes it so beautiful.However, in evaluating the beauty of a physical theory, the physicistregards as an essential element of beauty the fact that all of thiscomplexity stems from a limited number of physical laws.
I would like to further clarify this idea with a simple example.Imagine that we draw a square, and then on each side of the squarewe draw another square with a side equal in length to one-third theside of the original square, and we repeat this process many times(Figure 4). Now, almost everyone will agree that the final patternis quite beautiful to the eye (because of its symmetry). However,the physicist will recognize an additional element of beauty in thefact that underlying this relatively complex pattern, there is a verysimple law (algorithm) for its generation.
The great German philosopher Immanuel Kant had similarideas (in the eighteenth century) concerning the ideal of humanconsciousness. He defined this ideal as the attempt to establish ourunderstanding of the universe on a small number of principles, fromwhich an infinity of phenomena emerge. He went on to identify abeautiful object as one that has a multitude of constituents, all ofwhich at the same time obey a clear, transparent structure that providesthe big picture.
Quite frequently, the mutual influences between the arts and thesciences are exaggerated. As a physicist who also happens to be anart fanatic, I can testify that the direct, immediate, conscious influenceis minimal. Nevertheless, it is true that in some epochs, peoplefrom different disciplines sometimes think along similar lines. Forexample, a part of the title of this section, "Less Is More," was a popularaphorism with the twentieth-century architect Ludwig Mies vander Rohe. The feeling that one has to search for the most fundamentalcharacteristics of things, which has guided physicists duringall ages and in particular in this century, found its way also into someof the art movements of this century. Specifically, the roots of minimalart and conceptual art are clearly in this type of feeling.
An excellent example for this revolution in art is provided bythe transition from the very realistic description of erotic attractionand the act of love, as in the sculptures Desire by the Frenchartist Aristide Maillol or The Kiss of Auguste Rodin, to the very minimalisticdescription, as in the sculpture The Kiss by the French(Romanian-born) sculptor Constantin Brancusi. The entire trainof thought (from realism to minimalism) is exposed in a series oftree paintings by the Dutch painter Piet Mondrian, in which onecan literally follow the transformation of a very realistic renditionof a tree into a very abstract, minimalistic painting that involves aseries of lines symbolizing the leaves.
Another painting by Mondrian that encapsulates the idea ofreductionism, of grasping the most basic characteristics, is BroadwayBoogie Woogie (currently in the Museum of Modern Art in NewYork City), in which the essence of the title is captured in a collectionof squares and rectangles in bright, almost luminous, neon-likered, yellow, and blue colors.
Interestingly, while this reductionistic approach was adoptedonly by certain movements in Western art, by contrast, in Japaneseart, as in physics, reductionism has been regarded as an elementof beauty for centuries. It suffices to look at a lyric landscape paintingby Toyo Sesshu from the fifteenth century, or to read a poemby Shikibu from the eleventh century
Come quickly—as soon as these blossoms open they fall. This world exists as a sheen of dew on flowers
to realize that the Japanese culture has brought simplicity and reductionismto aesthetic peaks. In fact, ever since the eighth century,the most popular poem structure in Japan has been the shortpoem (tanka), which has only five lines and thirty-one syllables(arranged in 5, 7, 5, 7, 7). In the seventeenth century, an evenshorter structure appeared (haiku), with three lines and seventeensyllables (in a 5, 7, 5 pattern).
I will now explain the third element that I regard as essentialfor a physical theory of the universe to be beautiful—the Copernicanprinciple.
3. The Copernican Principle: We Are Nothing Special
Many recognize Nicolaus Copernicus as the Polish astronomer wholived in the sixteenth century and reasserted the theory (based on asuggestion by Aristarchus some eighteen hundred years earlier) thatthe earth revolves around the sun. However, Copernicus was in fact(although not intentionally) responsible for a much more profoundrevolution in human thinking. The early models of the universe followedreligiously the ideas of Aristotle, and were all geocentric.Namely, they all assumed that the earth was at the center of the universe.The most detailed, and most successful, model along theselines (in terms of explaining the observed paths of the sun, themoon, and the known planets) was due to the Greek astronomerPtolemy, who lived in the second century A.D. This model survived,amazingly enough, for nearly thirteen centuries. One can only assumethat it was the withering of intellectual curiosity during theDark Ages, combined with the dominance of the Catholic Church,which regarded Aristotle's teachings as entirely consistent with itsown doctrines, that granted the Ptolemaic model its longevity. In fact,following the integration of Aristotle's teachings into Christian theology,which is credited to Saint Thomas Aquinas (in the thirteenthcentury), Aristotle achieved an almost reverential status. Copernicuswas the first to point out clearly that we do not occupy a privileged placein the universe. He discovered that we are nothing special. This hasevolved to become known as the Copernican principle. In retrospect itis very easy to understand why there should be a Copernican principlein relation to the existence of "intelligent" creatures. After all, ofall the places in the universe where intelligent creatures couldemerge, assuming that there are many such places, very few, by definition,are "special." Therefore it is infinitely more likely for us to findourselves in a nonspecial rather than in a special place. Put differently,the Copernican principle is a principle of mediocrity.
Since the time of Copernicus, the Copernican principle hasbeen substantiated even further. Not only has the earth been dethronedfrom its central position in the universe, in fact, at the beginningof this century, the astronomer Harlow Shapley demonstratedthat our entire solar system is not even at the center of ourown Milky Way galaxy. Indeed, it is about two-thirds of the way outfrom the center, completing a revolution around the center inabout 200 million years. As we shall see in the next chapters, thisvulgarization of the earth's location continued even much further.
The Copernican principle can be expanded and generalized toinclude theories of the universe in general. In other words, everytime that a certain theory would require humans to occupy a veryspecial place or time for it to work, we could say that it does not obeythe generalized Copernican principle. To give a specific example,if a theory were suggested in which the origin and evolution of humanswas entirely different from that of all the other species, sucha theory would not have obeyed the generalized Copernican principle.Darwin's theory of the origin of species by means of naturalselection is thus a perfect example of a theory that does obey theCopernican principle (I will always mean the generalized principlefrom here on) and is therefore, from this point of view, beautiful.
I would also like to note that some theories are considered "ugly"because they violate something that can be regarded as intermediatebetween simplicity and an even more general interpretation of theCopernican principle. I include here all the theories that are extremelycontrived, or that necessitate some very special circumstancesor fine-tuning for their validity (even if they do not involve an explicitrole for humans). The reluctance to associate beauty with such theoriesis a bit like the disbelief that we would surely feel if someone toldus that he can flip a coin and make it land on its side. We will encounterexamples for such fine-tuning in chapters 5 and 6.
I will now explain the concept of elegance, which, as I said, Ipersonally do not regard as a necessary ingredient for beauty in aphysical theory, but which can certainly enhance the beauty of certaintheories.
4. Elegance: Expect the Unexpected
In mathematics and physics, and indeed in almost any discipline,it sometimes happens that a very simple, unexpected new idea resolvesan otherwise relatively difficult problem. Such brilliantshortcuts lead to what are considered to be very elegant solutions.Interestingly, in chess, prizes for beauty are given for precisely thistype of exceptional quality. It is amusing to note that one of themost elegant games of the brilliant American chess player PaulMorphy was played in Paris in 1858, in a box at the Paris Opéra,while Rossini's Barber of Seville was being performed on stage!
It is important to understand that elegance has nothing to dowith reductionism (what I called simplicity). For example, thePtolemaic model for the motion of the planets offered in fact an elegantsolution to a difficult observational problem, in that it founda clever way to explain remarkably well the observed motions of theplanets. The model, however, was not simple at all. It required eachplanet to move around a small circle, called epicycle, the center ofwhich moved around the earth on a large circle (called deferent).To explain all the observations, the Ptolemaic model required nofewer than eighty circles! It was only following Kepler's discoverythat the planetary orbits around the sun are elliptical that a simplemodel for the solar system emerged.
An example of elegance can be found in the following, well-knownmathematical puzzle. Suppose we are asked: can one coverthe board shown in Figure 5 by dominoes (each one having the areaof exactly two squares), so that only one corner is left uncovered?The answer is very simple: no. Since the board has an even number(sixty-four exactly) of squares, and since each domino covers twosquares, we can only cover an even number of squares and thereforewe cannot leave one square open. Suppose, however, that weare now asked: can we cover the board in such a way that we leavetwo diagonally opposite corners uncovered? Clearly, in this case wewill be covering an even number of squares, and so it is less trivialto determine immediately if this can be done or not (try thinkingabout this a little). It turns out, however, that with the help of an extremelysimple trick we can answer the question right away. Theidea is to think of the board as if half of the squares are paintedblack, as in a chessboard. Now, since each domino piece covers oneblack square and one white square, it is clear that it is impossible toleave uncovered two diagonally opposite squares, since they havethe same color! So, the extremely elegant idea of turning the boardmentally into a chessboard helped us to solve immediately a problemthat appeared otherwise to be much more complex. Isn't thiselegant? However, I want to emphasize again that elegance, whichI consider superfluous in a beautiful theory, should not be confusedwith reductionism, which I regard as absolutely essential.
She Walks in Beauty
The beauty (or absence thereof) of physical theories is the maintheme of this book. Beauty, like love, or hatred, is also almost impossibleto define properly. In fact, probably any definition is likelyto raise some objections. I therefore feel that it is worthwhile, evenat the risk of some repetitiveness, to attempt to reconstruct brieflythe thought process that led me to my requirements. I should firstnote that because of the fundamental role I think beauty does playin physics, one cannot be satisfied merely with the attitude "I'llknow it when I see it," which some physicists have adopted towardbeauty in physical theories. I claimed to have identified three elementsthat are absolutely necessary for a physical theory to be beautiful.These are symmetry, simplicity, and a generalized Copernicanprinciple. It is perfectly legitimate to question this identificationand ask: (1) what is it that makes these elements essential to beauty?and (2) are there other elements that may be equally important?
The answers to these questions are not trivial, partly because ofthe fact that the recognition of beauty in a physical theory is to alarge extent dependent on the scientist's intuition and sometimeseven on taste (not unlike the dependence on the artist's aestheticsensibility and taste in relation to a work of art). Nevertheless, I willattempt to outline now a certain logical process that can at leastgive us some guidance toward an answer. Instead of starting withthe elements and trying to justify them, let us try to work our waybackward, starting from a beautiful theory and retracing our stepsin the direction of its basic ingredients.
First, it is important to realize that since the ultimate goal ofphysical theories is to describe the universe and all phenomenawithin it in as perfect a way as possible, a theory cannot produce a realsensation of beauty unless it can be regarded as a major step towardperfection. We therefore need to identify which combination ofproperties can be regarded as constituting perfection. Since theuniverse involves an immense number of phenomena, and maygenerally appear quite chaotic, it is clear that what is required is theintroduction of some regularity, organization, balance, and correspondenceinto the description of nature. These properties can allowmore encompassing perceptions, which eventually enable scientiststo identify common characteristics of different phenomena.Next, since we are interested in beauty, we want to identify classicalaesthetic constituents or concepts that can contribute. Adoptingideas from the arts, the list of such concepts may include symmetry,simplicity, order, coherence, unity, elegance, and harmony. Thequestion now is which of these truly plays a central role in science.In order to answer this last question I will attempt to rank-orderthese concepts in terms of their contribution to scientific thinking.
Symmetry definitely occupies the top position in this hierarchy,since it literally forms the foundation on which physical lawsare built, as I explained in the last section (and as we shall see furtherin chapter 3).
Simplicity comes next, since it allows scientists to choose fromamong all the different possible hypotheses and ideas the mosteconomical ones. Albert Einstein himself wrote once: "Our experienceuntil now justifies our belief that nature is the realization ofthe simplest mathematical ideas that are reasonable."
I did not list order, coherence, harmony, and unity as separateelements that are essential for beauty since in physics these are notindependent concepts. For example, what is meant by order is thatsimilar physical circumstances should produce similar consequences.However, we have already seen that symmetry and simplicityachieve precisely this goal. Furthermore, in the next sectionwe will encounter an excellent example of the unifying power ofsymmetry and simplicity. Werner Heisenberg, one of the foundingfigures of quantum mechanics, the theory of the subatomic world,stated once as a criterion: "Beauty is the proper conformity of theparts to one another and to the whole." As we shall soon see, thisis what symmetry and simplicity (reductionism) are all about.
Finally, as I discussed in the last section, elegance can certainlycontribute to the feeling of attractiveness that is associated with acertain solution to a scientific problem. However, I do not regard itas an essential element of beauty in a physical theory. In this assertionI humbly disagree somewhat with the sixteenth-centuryphilosopher and statesman Francis Bacon, who was described bythe poet Alexander Pope as "the brightest, wisest, meanest ofmankind." Bacon wrote that "there is no excellent beauty that hasnot some strangeness in the proportion." Since "strangeness in theproportion" is to be understood at least partly as an element of surprise,Bacon's "definition" is more closely associated with what Icalled elegance than with beauty. However, Bacon's criterion refersalso to the unification of otherwise seemingly independent concepts,and as such to symmetry and reductionism. My point of viewon elegance is best expressed by a famous quote from the nuclearphysicist Leo Szilard: "Elegance is for tailors."
I hope that the above discussion clarifies the first two elementsin my definition of beauty in physics. The third element, however,requires some further explanation. I have included in my essentialingredients the generalized Copernican principle, which is not traditionallylinked with aesthetics. This is really a property that is peculiarto the sciences, and to theories of the universe in particular.In general, scientists absolutely detest theories that require specialcircumstances, contrived modeling, or fine-tuning. In this sense, asparticle physicist Steven Weinberg puts it in his book Dreams of aFinal Theory, a beautiful theory must be seen as essentially inevitable.A violation of the generalized Copernican principle, in the formof statements like "the universe must be so-and-so because we humansare so-and-so," or in the form of fine-tuning, is certainly a slapin the face of all encompassing inevitability, and is therefore ugly.I will return to this question in the discussion of intelligent life andthe anthropic principle in chapter 9. At this point I will concludeby noting that placing humans center stage may be regarded as adesirable property from theological, psychological, or even theatricalperspectives, but it is not a property most scientists wouldlike to associate with a beautiful theory.
I do not wish to leave the reader with the impression that intheir quest for beauty in the theory physicists lose sight of thebeauty of the universe itself. This is certainly not the case. WhenEinstein once said that the only incomprehensible thing about theuniverse is that it is comprehensible, he referred precisely to thesetwo "beauties." The reality of Marc Chagall's Lovers with Half-Moon(currently at the Stedelijk Museum, in Amsterdam) is much morethan the chemical composition of its paint. Humans are on onehand able to access the beauty of the universe and on the otherable to comprehend the beauty of its workings.
I have always admired the theater. I regard the art of presentationof ideas succinctly through dialogues and monologues as being aclose cousin to the "art" of scientific presentations. Consequently,I have decided that in a few places in this book, I will abandon themore expository style in favor of a more theatrical style. The aimof these short "scenes" is to provide an introduction to new concepts,on the basis of the ideas that have already been developed.
Galileo
In a dark room, lit only by a few candles, five men dressed in red sit behinda long table. The grave expressions on their faces make them look almostidentical.
Man at the center of the table (Grand Inquisitor): Bring him in! [The steps of a guard echo from the stone walls and the ceiling. The guard enters, pushing in front of him a bearded old man, who clearly has trouble walking.]
Grand Inquisitor: We would like to ask you again a few questions about your crazy and, may I add, dangerous ideas.
Old Man: All of my ideas merely represent the progress of scientific knowledge.
Grand Inquisitor: We will be the judges of that. The proposition that the sun is in the center and immovable from its place is absurd, philosophically false, and heretical. Do you agree that the earth stands still at the center of the universe?
Old man [in a weak voice]: Certainly not. The earth rotates around its axis and at the same time revolves around the sun. The entire solar system is not even at the center of our own galaxy.
Inquisitor at far left end of the table [with surprise]: Galaxy? What galaxy? What does the Milky Way have to do with it?
Old Man [straightening himself up a bit]: While the faintly luminous band seen across the sky at night is referred to as the Milky Way, a galaxy is really a collection of about one hundred billion stars like our sun.
Grand Inquisitor: Are you out of your mind? Where are all these suns?
Old Man: As I said, our own solar system belongs to such a galaxy. Most
of these stars are too faint to be observed with the naked eye.
Grand Inquisitor: Why do I not see that the earth revolves around the sun as you say? I see everything here standing still, while the sun is moving around the earth.
Old man [somewhat scornfully]: That is because you can only see relative motions. Everything on the earth is moving with the earth, so you do not see it moving with respect to you.
Second Inquisitor from right: This is the greatest nonsense I have ever heard, even if I were to ignore the desecration in your words. Soon you will be telling us that the sun is not revolving around the earth but around something else.
Old man: Indeed, the sun is rotating around its own axis, and it is revolving around the center of our galaxy.
Grand Inquisitor [raising his voice in great anger]: Do your ears hear what your mouth utters? Everything is rotating! [With great scorn.] What else do you think is rotating?
Old man [now clearly hesitant]: Well, the electrons in the atom, for example, revolve around the nucleus.
[The inquisitors are visibly baffled by this statement and start whispering amongthemselves. Finally, the Grand Inquisitor resumes the interrogation.]
Grand Inquisitor: It is becoming more and more obvious to us that you have lost your faculties. While we do not have the faintest idea of what you are talking about, I point out to you that the word atom in Greek implies that it is indivisible and therefore [raising his voice] there cannot be anything in it!
[The old man remains silent.]
Grand Inquisitor: Well?
Old man [feebly]: Matter as we know it is made of atoms, this is true. But the atoms themselves have a very dense and compact nucleus at their centers. In this nucleus there are particles called protons and neutrons. Other tiny particles, called electrons, revolve around this nucleus.
Grand Inquisitor [clasping his hands and looking upward in despair, then, after looking at his colleagues, turning to the old man with a resentful face]: At least I hope that your protons and electrons are not rotating around their axes?
[All the inquisitors laugh loudly.]
Old man [with some determination]: Well, the electrons and the protons have a quantum mechanical property called "spin," which in some respects can be thought of as if they are rotating around their own axes.
Grand Inquisitor [shouts in rage]: Shut up! I've had enough of this. The earth does not revolve, it is at the center of the universe, and all of these so called electrons and protons do not even exist. [Turning now to the guard.] Put him in a damp cell in solitary confinement, where he will have time to think until his own head will start spinning!
[The guard starts to drag the old man out of the room. The old man, his eyes open wide with fear, can hardly keep up with the guard's huge steps. As he is being dragged past the large wooden door he murmurs to himself]: Eppur si muove. (And yet it moves.)
Needless to say that in Galileo Galilei's time (1564-1642) galaxies,atoms, nuclei, quantum mechanics, and electrons had not yetbeen discovered. But had they been, it would not surprise me ifGalileo would have used about them similar phrases to the ones Ihave put in his mouth.
Spinning
Electrons are the smallest (in mass) known particles that are electricallycharged. Galaxies are huge collections of tens of billionsstars. Yet precisely the same physical laws govern the behaviors ofboth. This is the true meaning of simplicity and symmetry—ofbeauty in physics. How do we know this? The following exampledescribes these principles in action.
Electrons and protons, the basic building blocks of atoms, havea property called spin. Strictly speaking, this property can be describedonly by quantum mechanics—the theory that governs thesubatomic world. However, for some purposes spin can be thoughtof as a rotation of the electron or the proton around its own axis.Therefore, as for all rotating bodies, the properties of this spin aredescribed by the conservation of angular momentum. Because ofthe fact that electrons and protons also have an electric charge, thisspin makes them behave like small bar magnets, since a magneticfield is generated when electric charges are moving. Now, a simpleexperiment can be performed with two small magnets, such as twocompass needles. Imagine that you suspend the needles on thinthreads attached to their middle points. If you place the two needlesalong a straight line, with two equal (say, north) poles facingone another, you will observe that one of the needles will spontaneouslyflip, so that two opposite (one north and one south) poleswill face one another. Energetically speaking, when a system undergoessuch a spontaneous transition, this means that the second configurationis at a lower energy state than the first, since physicalsystems tend to be in their lowest possible energy state (as I am sureeverybody recognizes from their own tendencies).
In an analogous manner, when the hydrogen atom, which iscomposed of one proton around which one electron is orbiting, isin its lowest orbital energy level, the spins of the electron and protoncan be parallel, which is equivalent in some sense to the electronand proton rotating in the same sense, or antiparallel (namely, theelectron and proton rotating in opposite senses; quantum mechanicsonly allows two possible states for the spins of the electron andthe proton). Consequently, the hydrogen atom can spontaneouslyundergo a spin-flip transition, from the parallel to the antiparallelstate. Since the latter is at a lower energy state, the difference in theenergy is emitted in the form of radio waves. Waves in general arecharacterized by a property called wavelength. For example, when wethrow a pebble into a pool, we can observe a series of concentricwaves, and the distance between two crests is the wavelength.
The wavelength of the radio waves emitted by the spin-flip transitionis 21 centimeters. So, how is all of this connected to galaxies?As it turns out, this 21-centimeter radio wave plays a crucialrole in our exploration of galaxies. For example, in order to determinethe global structure of our own galaxy, the Milky Way, andthe motions within it, observations must be made to distances ofthe order of 30,000 light-years. The problem is, however, that theinterstellar medium, diffuse gas and dust in which clouds are dispersed,prohibits optical observations to such large distances becauseit is opaque to visible light. The idea is therefore to identifya wavelength of electromagnetic radiation to which the interstellarmatter is relatively transparent, so that the source of the radiationcan be seen to large distances. In addition, the wavelength hasto be such that either stars or cold gas clouds, which are the mainand most abundant participants in the motions in the galaxy,would be strong emitters in that wavelength, to allow its detection.
The 21-centimeter radiation proves perfect for this purpose.Not only does it allow observations to penetrate to the farthest cornersof the Galaxy, it is also emitted by neutral hydrogen atoms,which constitute the main component of interstellar matter, in theform of cold gas clouds.
There is one more effect that needs explanation in relation toobservations aimed at determining the speeds of gas clouds, andthis is known as the Doppler effect, after the Austrian physicistChristian Doppler, who identified it in 1842. We are familiar withthis effect in everyday life, in relation to sound waves. When asource of sound (e.g., a car or a train) is approaching us, the soundwaves bunch up and we receive waves at a higher frequency(higher pitch). The opposite happens when the source of soundis receding from us. The waves are spread out and we hear a lowerpitch. The effect is particularly noticeable when the source of thesound passes us by rapidly, since the sound changes from a high toa low pitch (recall, for example, the "eeeeeeoooooo" sounds ofcars on the Indy 500 race track). A similar phenomenon occurswith light or any electromagnetic radiation. Therefore, when asource that emits the 21-centimeter radiation is receding from us,the radiation will be detected at a longer wavelength (lower frequency),while when it is approaching us, it will be observed at ashorter wavelength (higher frequency). From the detected shift inthe wavelength, the speed of the source can be determined, sincethe shift is larger the higher the speed.
The stars and the gas clouds in the Galaxy participate in a globalrotation around the galactic center. Radio observations of the 21-centimeterradiation allowed astronomers to determine the structureand the rotation pattern of the entire galaxy interior to the sun'sorbit—namely, between the solar system and the center of the Galaxy.
A detailed mapping of our galaxy reveals that it is a disklike,spiral galaxy. Namely, it has the shape of a flattened pancake, witha spiral pattern observed on the face of the disk. The disk structureis probably a consequence of the formation process of the Galaxy,and it is related at least in part to the amount of angular momentum(amount of rotation) possessed by the gas cloud from whichthe Galaxy formed. The exact process by which galaxies form is stillnot fully understood; however, the following general remarks canbe made. When a rapidly rotating cloud of gas collapses to form agalaxy (due to the force of gravity), it will tend to form a flatteneddisk. This is a consequence of the centrifugal force, which tends topush material away from the rotation axis. The centrifugal force isvery familiar to anyone who has tried to negotiate a sharp turn witha car moving at a high speed—a strong force is felt, pushing thecar off the road. The centrifugal force is stronger when the"amount of rotation" of the cloud is higher. Consequently, for highangular momentum clouds, the collapse forms a flat, disklikestructure.
So, what have we have discovered here? This same entity, calledangular momentum, the conservation of which resulted from thesymmetry of the laws of physics under rotation, explained to ussomething about the electron, which has dimensions of about [10.sup.-13](0.0000000000001) centimeter, and about the Galaxy, which hasdimensions of about [10.sup.23] (100,000,000,000,000,000,000,000) centimeters.What's more, we used one (the spin-flip of the electron) todiscover the other (the structure of the Galaxy). Now, I am asking you,isn't this absolutely BEAUTIFUL?
Continues...
Excerpted from The Accelerating Universeby Mario Livio Copyright © 2001 by Mario Livio. Excerpted by permission.
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