Tag Archives: symmetry

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:


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.