Cosmos 3: Harmony of the Worlds

by Shane L. Larson

People often ask me why I became a physicist. I often respond with stories about my great mentors (some of whom I’ve written about, like Doc  or David Griffiths). But more often than not I simply tell them the truth: physics was the easiest science I could major in!  This declaration is usually met with gasps of horror, looks of incredulity, or laughter.  In response to these varied expressions of disbelief, I quip, “Have you ever taken organic chemistry?”  Quick, look to your nearest organic chemist — they are nodding knowingly. There is a LOT to remember in a science like chemistry or biology; I have much respect for my colleagues in those fields (and sympathy; their brains must be unbearably full).  All fun and games aside, there is some truth to my statement — physics is exceedingly fundamental, and that makes it easy in ways other science cannot be.

There is an internet comic circulating from one of my favorite web comics, Abstruse Goose.  In the upper frame it shows an idyllic sketch of a park with a cute fuzzy little bunny enjoying a meal of a carrot near a babbling brook on a warm spring day.  In the lower frame, it shows the same picture, overlaid with a great many mathematical equations that represent the processes going on in the picture.  The caption reads, “This is how scientists see the world.”

Many who look at this image may shake their heads at the perceived gobbledygook, perhaps they suppress a shudder as their psyche ripples with a memory of a long-ago science class for which they hold few fond memories.  But scientists like myself, we look at this image and we nod sagely.  Yep, that’s what makes everything go.  But I don’t think it is how scientists see the world.  I think it is how we explain the world.  The Cosmos is a great machine that obeys a set of rules that govern how it works — what it does each moment, how all the parts interact in harmony, what happens if something changes, and how the past influences the future.  Scientists have a name for the rules that govern the great machine of the Universe — they are called The Laws of Nature.

Physics is a science that seeks to understand the most basic, the most fundamental Laws of Nature.  Why do apples fall to the ground?  What is light? Why does my Dr. Pepper get warm and why does my coffee cool off?  What is the smallest thing that other things are made of?  Why does the Earth spin, and why do distant galaxies seem to be running away from us?  These are all deceptively simple but intensely probing questions about the nature of the Cosmos.  Surprisingly, physics has discovered that there are answers to all of these questions!  From a set of only perhaps 10-20 fundamental equations that summarize the basic Laws of Nature, all of the basic behaviours of the Cosmos can be deduced; all of these questions can be answered.

Now anyone who has ever picked up a physics textbook is probably scratching their heads right now.  There are a LOT more than 10-20 equations in most textbooks.  But the point is there are only a few fundamental rules, what physicists call “First Principles,” from which all the others are derived.  It is a bit like alphabets.  In modern English, there are only 26 letters, but from those 26 letters we can create, literally, a billion words!  And we are making more all the time (I’m pretty sure Abraham Lincoln did not know the word “email”). Physics is the same way; we take a few basic ideas, we mix them together in a variety of ways that brings out depth, richness, and sophistication that is awe-inspiring in its scope, and breathtaking in its beauty.

A typical page out of an introductory physics textbook (this one was my first textbook, Halliday & Resnick).

One of the most important truths of science is that the Laws of Nature are not the story; they are the words of the story, not the tale itself.  Knowledge and wisdom come from reading the book of Nature, and realizing that knowledge and wisdom are not immutable — they evolve with each reading of the book.  Let me tell you a story about one chapter of Nature’s book that we’ve managed to puzzle our way through.

The beauty of the Laws of Nature is they are everywhere, and they are the same everywhere.  The beauty of our brains is that we have managed to learn what some of those Laws are, and have discovered how to use them and apply them far afield from how we discovered them.  Consider the following interesting factoid: statistics suggest that roughly three quarters of the population needs corrected vision; that means a great many of you reading this blog right now are looking through glasses or contacts; perhaps some of you are still using old fashioned chatelaine glasses or monocles (historical investigations have suggested that the first glasses appeared in Pisa, Italy around the end of the 13th Century).  The physics of how glasses help you see, indeed the physics of how your eye collects and focuses all light, is called optics.

Imagine looking at an object.  What you see is light traveling from each point on the object to you.  In most cases it is reflected light, but in some cases (like with a flame, a light bulb, or the Sun) the light is generated by the object itself.  In either case, what you see is a consequence of light flying from the object to you — a little packet of light from every part of the object.  To represent this idea, physicists draw little arrows from the object to you, representing the light that you are seeing.

Ray tracing light from an object (skydiver) to your eye.

Suppose, in the interest of exhibiting the Laws of Nature, I go skydiving while you watch.  You, and all your friends, can see me while you stand on the ground.  I can show how the light gets to you by drawing little arrows that represent the light flying (in physics-speak we say “propagating”) from the skydiver to your eyeball.

In reality, every piece of the skydiver is sending light in every direction. That is why you can see the skydiver no matter where you are standing.  When your eye (or your camera) looks toward him, it gathers all the little pieces of light that are arriving at that point in space, and it makes an image.

(L) Every little bit of an object sends light in all directions. (R) The result is that you can see the entire skydiver, no matter where you are standing.

There are subtleties to be sure, but this is the basic law of Nature that forms the foundation of optics: light propagates in a straight line (a ray) from the point where it was emitted.  Whether it is a skydiver, the words you are reading in this blog, or the light from a distant star or planet, the light got to you flying on a straight line.

But what about your glasses or contact lenses?  That is what started this conversation.  When light passes through materials, like water, or glass, or plastic, the direction it travels is changed — the ray bends.  You can see an excellent example of this at home by filling up a glass with water.  Here I’ve filled up a wine glass, and if I look through it, everything looks upside down!  That’s because the water in the glass bends the light.  The direction it bends it depends on where the light hits the wineglass, and the result is to make an upside down picture when you look through it.  Try it!  As you get older, the lens in the front of your eye changes its overall composition, generally hardening so the muscles in your eye cannot flex the lens to send all the rays of light to the proper place on the back of your eyeball.  When this happens, we can help your eye out by bending the light before it even shoots inside your head.  All the machines and diopters and numbers used by your eye-doctor are about how much to bend the light before it gets in your eye.

Tracing rays through a glass of water is more complicated, but the rules are well understood. The result here is to make an upside down image of the Christmas tree!

So now we have a new addition to our law of Nature: light bends when it flies through an object of different material. We didn’t know this part when we started.  We just drew some rays to explain how both you and I could see the skydiver from different places.  We had to add to our rule about light when we started thinking about light flying through curved glasses of water.  This is the nature of how we understand the Cosmos — we write down the laws of Nature that explain everything we have seen.  But when we discover something new, we either rewrite the law of Nature to be more inclusive, or we write down a new law of Nature!  Science is a constantly evolving process.  It represents, to the best of our ability, our understanding of how the great machine of the Cosmos works.

One of the most important pieces of the history of discovery in optics was its application to astronomy. The obvious application was in the invention of the telescope, but let’s pass on that for a moment and think about how light travels to us from the most distant places in the Cosmos.  In 1979, a team of scientists using the 2.1 meter telescope at Kitt Peak discovered a pair of quasars now known to astronomers as QSO 0957+561 A/B, in the constellation of Ursa Major.  They are colloquially known as the Double Quasar. At the time, the discovery was remarkable because the two quasars were very close together on the sky.  If you hold a dime at arms length, then both quasars would fit behind Roosevelt’s eye!  Most things in the sky change with time, so astronomers like to periodically go back and observe objects over and over again.  This kind of persistent habit revealed something spectacular about the Double Quasar: if quasar A suddenly got brighter, then quasar B would follow suit 417 days later.  It was like clockwork — anything quasar A did, quasar B mimicked 417 days later.  How could this be? Quasars are energetic galaxies, powered by black holes, billions of light years away. They are giant, insentient objects in the Cosmos, not annoying siblings engaged in a game of copycat!  What was going on?

As it turns out, the answer had been discovered by Albert Einstein more than 60 years before. If gravity is strong enough, it can bend the path that light travels on, just like a lens or a glass of water.  The Double Quasar is really just two images of the same quasar.  The gravity of a galaxy between Earth and the quasar is bending the light, like a lens!  Like a bad pair of glasses though, the gravitational lens (a giant elliptical galaxy, in this case) is not perfect, and it doesn’t bend all the light toward Earth so it arrives at the same time.  The result is we see two images of the quasar, delayed by 417 days.

The light from a distant object (a galaxy or quasar) is bent by the gravity of massive objects (galaxies in this image) on the way to Earth. We see the two orange rays, but they travel different distances, so the light arrives at different times!

The same laws of physics — light, rays, optics — but with gravity playing the role of the lens. In this case, the discovery was made long after a physicist had predicted that Nature would permit such behaviour.  This too is the way science works — we have ideas about how Nature might work, about how the different pieces of Nature might work together in harmony. Some of those ideas might be right, and some of those ideas might be wrong. All we can do is look for a sign, look for a clue to tell us whether we are on the right track or not!

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This post is part of an ongoing series, celebrating the forthcoming science series, Cosmos: A Spacetime Odyssey by revisiting the themes of Carl Sagan’s classic series, Cosmos: A Personal Voyage.  The introductory post of the series, with links to all other posts may be found here:  http://wp.me/p19G0g-dE

The Teacher, the Law, and the Freshman

by Shane L. Larson

My mother has a theory — whatever your great passion is in the first grade, that is in all likelihood the best indicator of what you should do with your life. It’s what will make you the happiest.   For most of my years up through about fourth grade I wanted to be a scientist.  That changed to astronaut after the launch of the space shuttle Columbia in 1981.

The astronaut thing stuck for a long time, until I started college.  Growing up during the space shuttle era, I knew that to be an astronaut did not mean you had to be a pilot, because you could be a mission specialist — someone who did science on orbit.  At some point in the years leading up to college I had decided the thing that would maximize my chances of flying as a mission specialist was to be an engineer.  So the fall I went to college, my education did not start in science at all — it started in mechanical engineering.

Being an engineer was okay; we learned computer programming (in FORTRAN, unfortunately), we got to build balsa wood bridges and then crush them under a two ton load tester, and we got to peek under the hood of all kinds of devices and see how the world works when we’re not looking. I have a strong independent streak, and generally regard my destiny as my own to control. This caused a great deal of strife as a  young engineering student because I didn’t feel compelled to follow the list of courses that had been outlined for me. Unbeknownst to me, a battle was brewing over this fact.  I became suddenly aware of this near the end of my first winter quarter when the dean of engineering called me into his office.  My memory of the conversation is something like this:

• DEAN: You are off track. You’re not taking the courses you are supposed to.
• ME: So?
• DEAN: You’re taking a random physics course. You can’t do that.
• ME: Why? I want to take astronomy; I think it will be useful.
• DEAN: The recommended courses are what is going to be useful.
• ME: Well I’m going to take astronomy. I’ll get to those courses eventually.

In the end, I got to stay in the astronomy class, because spring quarter wasn’t too far away.  The dean did, however, make me sign a contract that said I would adhere to the recommended engineering schedule during all future quarters.

The late Dr. David Griffiths (Oregon State University, Department of Physics).

The astronomy course in question was Introductory Astronomy: Stars and Galaxies, taught by the late Dr. David Griffiths of Oregon State University (not to be confused with the other Dr. David Griffiths, at Reed College, of textbook fame).  Dr. Griffiths was one of the formative figures in my scientific youth, and is responsible for setting me on the path I am on today.  Three days after starting his astronomy course, I changed my major to physics (*).  While I gained some deep satisfaction from delivering my change of major paperwork to the College of Engineering, and even more satisfaction from tearing up the silly contract I had signed the quarter before, you must be wondering what it was that sparked such a drastic alteration in my destiny?  It was Dr. Griffiths.  And Johannes Kepler.

On the first day of astronomy class, I sat squirming in my seat, breathless with anticipation of learning about quasars, dark matter, and black holes. Dr. Griffiths strode into class, his salt-and-pepper curly black hair slightly wild and unkempt in a way that is typical for many scientists.  On that first day, he launched into a story about Johannes Kepler and the laws of planetary motion, which apply to all orbits not just to planets. This was a story I was familiar with from many long hours spent watching Cosmos (Ep. 3, “The Harmony of the Worlds”).  This wasn’t what I was exactly waiting for, but it was astronomy and so I enjoyed myself.

The story focused on Kepler’s Third Law, which tells us that the length of time a planet takes to complete an orbit is related to how big that orbit is.  This is called Kepler’s “Harmonic Law” and is usually stated as: “The square of a planet’s period (the time it takes to complete one orbit) is proportional to the cube (third power) of its semi-major axis (the radius of the orbit if it is circular).”  Mathematically, it is written as

P² ~ a³

Kepler had deduced this result by studying careful observations of the positions of the planets made by his contemporary, Tycho Brahe.  This made perfect sense to me.  It was the way I had always been taught that science worked: you make observations, then deduce the Laws of Nature from those observations.

At this point in the story, Dr. Griffiths did something that changed my destiny forever.  He turned to us, and looking through glasses that had slipped to the end of his nose, said, “We could have figured that out even if we didn’t have any observations.”

I sat up in my chair a little straighter at this point. What?

Dr. Griffiths continued: “We can mathematically fit any data we want to — that’s what Kepler did. But whatever the fit is, it had better come from the fundamental Laws of Nature. We can derive Kepler’s Third Law from the basic rules of physics.”  Which he did.  With a deft hand, in white chalk, he completely blew my mind in just a few short lines.  That derivation is replicated below (in my own handwriting — this moment in my formative history is far too important to ever be reduced to mere typeset equations).  While it would be easy to include this just for the aficionados (and any engineers who think they may want to be physicists…), you should look closely at this.  This is the beauty of the Laws of Nature.  What makes the planets go, and why they move the way they do, was once one of the greatest mysteries of science, yet through sheer force of imagination we humans figured out how to explain it concisely enough that it can be written on a napkin and explained as a motivational exercise for students!

Derivation of Kepler’s Third Law from first principles.

This was a revelation to me.  It was the first time that I truly felt I had a glimpse into what the scientific enterprise was all about.  It was the first time that I had ever encountered the awesome power of science, and the first time I had ever been confronted by the sheer scope of what we could accomplish with our naked intellect.  But most importantly, in the span of those few minutes, as the scritch scritch of Dr. Griffiths chalk outlined the foundations of Kepler’s Third Law, I learned one of the most valuable lessons of my scientific career, which still guides me today: everything is connected, and knowing as much as possible about everything you possibly can will bear unexpected and beautiful fruit in the end.  Before that class was over, I had decided that if this is what physics was all about, then this was a profession for me.  I walked out that day and started figuring out how to change my major.

The Laws of Nature would exist whether you and I were sitting here talking about them or not.  Atoms would continue to bond together and make stuff, apples would continue to fall out of trees and to the ground, and stars would continue to burn and flood the Cosmos with light.  The fact that the Laws of Nature can be figured out is one of the great gifts of the Cosmos to its inhabitants.  We live in a world filled with predictable patterns.  Rocks that I throw up in the air always fall back to the ground. A cup of coffee on my counter always cools to the ambient temperature of the room.  The constellations rise in the east each night at dark, and slowly march across the sky westward over the course of the night.  These tantalizing patterns are clues that Nature provides us about the underlying order of things, breadcrumbs that lead us to science, the express goal of which is to understand the Laws of Nature that make all the wondrous order of the natural world.

Science is a unique creation of our species, the paramount expression of our ability to see the world around us, and imagine why it is the way it is.  The culmination of our creative ruminations are the Laws of Nature, written in the language of mathematics, another invention of our species. As we practice science today, it is a self-correcting framework.  At any given moment, it captures our imperfect understanding of what we have observed about the Cosmos. When we make new discoveries, we reexamine the Laws of Nature as we understand them, and expand our thinking to correctly explain what we have seen.   Johannes Kepler published his Harmonic Law in 1619.  It was perfectly adequate for explaining observations of the planets in the solar system.  But it was not until 1687 that Newton took that understanding and expanded it using the Universal Law of Gravitation.  I’m sure he didn’t imagine that the consequence of that simple expansion of human consciousness would be to create a scientist out of a university freshman 300 years later.

Dr. Griffiths passed away in early 2005, returned to the embrace of the Cosmos from whence we all came.  It is a great sadness to me that I never made it back to Oregon State to talk to him again after I graduated.  Teaching is an artform which requires immense amounts of patience and practice.  I think back on that day in Dr. Griffiths’ class often, deeply cognizant of the fact that that was the moment the changed my career forever.  Today, I teach my own classes, and I hope that I also occasionally stumble through moments of revelation for my students.  I don’t know what those moments might be, or when they might occur.  I just hope they happen.  But I take my cues from Dr. Griffiths: I always teach the derivation of Kepler’s Third Law, just like it was taught to me.  🙂

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(*) This is how I always tell this story, because that is the way I remember it in my head. However, having written it down and looking at it critically, it does seem the exact time that passed between events is not relayed accurately here. However, this is my story, and I’m sticking to it. 😉

Symmetries in the Harmony of Nature

by Shane L. Larson

In 1619, Johannes Kepler published “Harmonices Mundi” (Latin for “The Harmony of the World”), in which he described his discovery of the Third Law of Planetary Motion.  It was the first empirical description of the  “musica universalis,” the ‘music of the spheres’ that medieval philosophers had appealed to in order to describe the motions of celestial bodies.  This naming was not by chance; it reflects the rather profound fact that a simple, consistent mathematics underlies many phenomena in Nature.

The mathematics describing sound and music shares a deep heritage with the mathematics that describes the repetitive motion of the planets in their orbits, which in turn shares a deep connection to objects that exhibit repetitive motion like pendula or spinning wheels.  In science we give a name to these phenomena that will be at once familiar and pleasing to music aficionados: harmonic motion.  The “harmony” of the motion derives from two fundamental observations.  First, the repetition of the motion can be thought of like a wave, a motion that repeats itself in time.  Sound itself is a wave phenomena, an idea that was first put forward by Greek philosopher Chrysippus in 240 B.C. and was solidified by Marin Mersenne (the “Father of Acoustics”) in 1636.  Sound, like ripples on the surface of a pond, is little pieces of air moving back and forth in place.  Second, increasing the speed of repetitive motion exhibits pleasing symmetries.  With sound, this is most familiar in the “octave scale” of music.  Two musical notes, separated by eight well defined steps in pitch (a sound one octave above another is double the frequency of the first sound), resonate together, often in a pleasing way.  Orbits can be the same — if you change the size of an orbit, you change how long it takes a planet to repeat each circuit.  Neptune and Pluto show a “harmony of motions” — for every two times Pluto orbits the Sun, Neptune orbits exactly three times.  Similar harmonies are seen in the cold shining moons of Jupiter — for every single orbit made by grey Ganymede, icy Europa makes two orbits and volcanic Io makes four orbits.

The mathematical connections between sound and music and harmonic motion are astonishing, but common.  As physicists often quip, “Everything in Nature is a harmonic oscillator.”  The metaphorical idea that the Cosmos is alive with musical harmonies is appealing, owing to the deeply pleasing nature of music. Humans ascribe great meaning to music, witnessed by the fact that we often label our most successful achievements with musical adjectives — a monumental work is a “symphony,” and peace among the myriad cultures of the world is “harmony.”  Why is music so special?  By and large, humans find symmetry and patterns pleasing in many facets of our sensory interface with the Cosmos.  Our brains are good at discovering and parsing patterns; they always have been.  Our recognition of systematic behaviour in Nature naturally leads to one of the most profound ideas of the collective human consciousness: the Universe is knowable.

There are rules that describe the goings-ons of everything: if you increase the pitch of a note by an octave, it will resonate with the first note.  If you have a planet farther from the Sun, it takes longer to come back to its beginning point.  If you throw a stone up into the air, it comes back down to the ground.  We call these rules the “Laws of Nature.”  We write these rules down in a common language, called mathematics.  We can use these rules to figure out how the world is going to behave, how things are going to change, and what the consequences of certain actions might be.  The Laws of Nature embody all that we know about the regular patterns and symmetries of the world.  Of course, Nature does not have to play fair nor make these rules easy to discover or understand.  But if they can be discovered, if they can be understood, then we can know the world around us.

Our brains are so good at deducing symmetry, that an entire field of science has emerged whose specific goal is to destroy and obfuscate symmetry.  We call this science “cryptography.”  The basic premise of of cryptography is to systematically hide easily recognizable symmetry (like the words on this page) using a set of rules that can easily be undone (if they are known).  One of the simplest encryption algorithms is the “substitution cipher,” which employs a letter by letter substitution to hide a message.  To create a simple substitution cipher, write out the whole alphabet twice, once forward and once in reverse:

ABCDEFGHIJKLMNOPQRSTUVWXYZ   ZYXWVUTSRQPONMLKJIHGFEDCBA

To encrypt a message, you take a letter from the top row and substitute the corresponding letter from the bottom row.  The message “music is the sound of Nature laughing” would encrypt as “nfhrx rh gsv hlfmw lu Mzgfiv ozftsrmt.”  An encrypted message such as this appears as utter gibberish, but can easily be understood if you know the rules (let’s call these rules the “Laws of Cryptography”) that created the message.  If you don’t know the rules, then you must look for patterns and hidden symmetries that reveal the true nature of the message.

There is an easy way to break a substitution cipher like this, with its own musical name: frequency analysis.  Written language is the repetition of a few letters, organized in patterns called “words.”  Some letters in English appear more often than others.  In any given body of English text, the letter E is about 12% of the letters, T is about 9.1% of the letters, A is about 8.1% of the letters, and so on.  With enough text, one can look at the whole encrypted message and evaluate the number of times a given symbol appears, surmising that the encrypted letter appearing the most often might be an E, and the next a T, and so on.  One rewrites the message using educated guesses about the identity of letters to decode enough parts of the pattern that your brain can deduce what the true text really says.  Even in our attempts to hide and destroy the symmetry of Nature, patterns remain that can be used to discover that which is hidden to cursory inspection.

Like Nature, a cryptographer does not have to play fair.  Exceptionally complicated mathematical algorithms are utilized in text encryption, immunizing messages to simple symmetry investigations like frequency analysis.  If the algorithms are known, messages can easily be understood.  If the algorithms are not known, it can be all but impossible to decipher messages, even using today’s modern computers.  Interestingly, one important approach to cryptography is to use prime numbers, a subject for which the father of acoustics, Marin Mersenne, is well known.  The mathematical algorithms used to encrypt messages are often seeded by a large prime number which is kept secret.  The resulting encryption can be alarmingly random in appearance.  But with the correct prime number in hand, the mathematical algorithms can be accurately unfolded, reversing the encryption in a precise mirroring of the original Law of Encryption that was used in the first place.  Mersenne probably would have found this symmetry pleasing, a harmonious merging of math, language, art and subterfuge that shows Nature’s unfailing knowability.