IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Music of the Spheres

Daniel Litt

Stanford University

November 2, 2013

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

What Does the Sky Look Like?

5600 stars visible in a dark sky (in a modern city, more like200 - 500).

Distinguished features:

The sun and the moon,Certain bright lights (planets) move in apparently erraticpatterns, within a narrow strip: Mercury, Venus, Mars, Jupiter,and SaturnThere are some differences in the night sky depending on oneslocation and the time of year.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

What Does the Sky Look Like?

5600 stars visible in a dark sky (in a modern city, more like200 - 500).

Distinguished features:

The sun and the moon,Certain bright lights (planets) move in apparently erraticpatterns, within a narrow strip: Mercury, Venus, Mars, Jupiter,and SaturnThere are some differences in the night sky depending on oneslocation and the time of year.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

What Does the Sky Look Like?

5600 stars visible in a dark sky (in a modern city, more like200 - 500).

Distinguished features:

The sun and the moon,

Certain bright lights (planets) move in apparently erraticpatterns, within a narrow strip: Mercury, Venus, Mars, Jupiter,and SaturnThere are some differences in the night sky depending on oneslocation and the time of year.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

What Does the Sky Look Like?

5600 stars visible in a dark sky (in a modern city, more like200 - 500).

Distinguished features:

The sun and the moon,Certain bright lights (planets) move in apparently erraticpatterns, within a narrow strip: Mercury, Venus, Mars, Jupiter,and Saturn

There are some differences in the night sky depending on oneslocation and the time of year.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

What Does the Sky Look Like?

5600 stars visible in a dark sky (in a modern city, more like200 - 500).

Distinguished features:

The sun and the moon,Certain bright lights (planets) move in apparently erraticpatterns, within a narrow strip: Mercury, Venus, Mars, Jupiter,and SaturnThere are some differences in the night sky depending on oneslocation and the time of year.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

What Does the Sky Look Like?

Retrograde Motion of Mars

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Pythagoreans

Pythagoreans Celebrate Sunrise, Bronnikov

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Pythagoreans

The Pythagorean Theorem

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The First Great Discovery

The Pythagoreans discovered: we can hear rational numbers.

p

q, p and q whole numbers.

Suppose one has two strings made from the same material oflength x and y . Then plucking both sounds harmonious only ifxy =

pq is a rational number with p, q small (say, less than 7 or so).

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The First Great Discovery

The Pythagoreans discovered: we can hear rational numbers.

p

q, p and q whole numbers.

Suppose one has two strings made from the same material oflength x and y . Then plucking both sounds harmonious only ifxy =

pq is a rational number with p, q small (say, less than 7 or so).

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The First Great Discovery

The Pythagoreans discovered: we can hear rational numbers.

p

q, p and q whole numbers.

Suppose one has two strings made from the same material oflength x and y . Then plucking both sounds harmonious only ifxy =

pq is a rational number with p, q small (say, less than 7 or so).

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The First Great Discovery

Harmonics (the harmonic series)

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The First Great Discovery

An Inharmonious Note

A Fifth

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Back to Cosmology

The Pythagorean Scale

According to Pythagoras, the planets were arranged according themost harmonious scale. This was the music of the spheresand theMaster, Pythagoras, could hear it.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Back to Cosmology

The Pythagorean Scale

According to Pythagoras, the planets were arranged according themost harmonious scale. This was the music of the spheresand theMaster, Pythagoras, could hear it.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Tragedy

All was rational and harmonious, until:

2

It is told that those who first brought out the irrational fromconcealment into the open priced in shipwreck, to a man. For theunutterable and the formless must needs be concealed. And thosewho uncovered and touched this image of life were instantlydestroyed and shall remain forever exposed to the play of theeternal waves. Proclos

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Tragedy

All was rational and harmonious, until:

2

It is told that those who first brought out the irrational fromconcealment into the open priced in shipwreck, to a man. For theunutterable and the formless must needs be concealed. And thosewho uncovered and touched this image of life were instantlydestroyed and shall remain forever exposed to the play of theeternal waves. Proclos

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Tragedy

All was rational and harmonious, until:

2

It is told that those who first brought out the irrational fromconcealment into the open priced in shipwreck, to a man. For theunutterable and the formless must needs be concealed. And thosewho uncovered and touched this image of life were instantlydestroyed and shall remain forever exposed to the play of theeternal waves. Proclos

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Other Greek Theories

Other Greek Theories

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

A Problem

But the Greeks knew: the opposite side of the world wasalways shrouded in darkness.

Possible explanation (Philolaus): a Central Fire about whichthe Earth orbited, and a counter-Earth, to block out its light.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

A Problem

But the Greeks knew: the opposite side of the world wasalways shrouded in darkness.

Possible explanation (Philolaus): a Central Fire about whichthe Earth orbited, and a counter-Earth, to block out its light.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Cosmas and the Medievalists

Cosmass Theory

Explains why rivers run faster towards the South-East.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Cosmas and the Medievalists

Cosmass Theory

Explains why rivers run faster towards the South-East.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Plato, Aristotle, Ptolemy, and Copernicus

Plato

Plato decreed that all heavenly bodies move in perfect circles

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Plato, Aristotle, Ptolemy, and Copernicus

Plato

Plato decreed that all heavenly bodies move in perfect circles

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Aristotelian Conception

The Crystalline Spheres

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Some Problems

The Orbits of Mercury, Earth, and Mars

Not to mention: the planets dont orbit around the Earth!

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Some Problems

The Orbits of Mercury, Earth, and Mars

Not to mention: the planets dont orbit around the Earth!

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Ptolemy

Ptolemy

Idea: More circles!

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Ptolemy

Ptolemy

Idea: More circles!

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Epicycles

Epicycles

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Not Bad!

Retrograde Motion of Mars

Epicyclic Motion

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Not Bad!

Retrograde Motion of Mars

Epicyclic MotionDaniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Not Bad!

Ptolemy needs 39 epicycles.

But reasonably accurate 1000 years after publication, in 1504!

Unclear that Ptolemys goal was to accurately describejustcompute.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Not Bad!

Ptolemy needs 39 epicycles.

But reasonably accurate 1000 years after publication, in 1504!

Unclear that Ptolemys goal was to accurately describejustcompute.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Copernicus

Copernicus

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Copernican System

48 epicycles, though they were smaller than Ptolemys

Copernicus claims Ptolemy had 80 epicycles!

Serious problemthe lack of stellar parallax

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Copernican System

48 epicycles, though they were smaller than Ptolemys

Copernicus claims Ptolemy had 80 epicycles!

Serious problemthe lack of stellar parallax

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Copernican System

48 epicycles, though they were smaller than Ptolemys

Copernicus claims Ptolemy had 80 epicycles!

Serious problemthe lack of stellar parallax

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Epicycles Get a Bad Rap

An actual orbit:

x(t) = a cos( t), y(t) = b sin( t)

Circle if a = b

Fact (Fourier): Any periodic path can be written as a sum ofcircles!

x(t) =

ai cos(i t), y(t) =

ai sin(i t)

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Epicycles Get a Bad Rap

An actual orbit:

x(t) = a cos( t), y(t) = b sin( t)

Circle if a = b

Fact (Fourier): Any periodic path can be written as a sum ofcircles!

x(t) =

ai cos(i t), y(t) =

ai sin(i t)

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Epicycles Get a Bad Rap

An actual orbit:

x(t) = a cos( t), y(t) = b sin( t)

Circle if a = b

Fact (Fourier): Any periodic path can be written as a sum ofcircles!

x(t) =

ai cos(i t), y(t) =

ai sin(i t)

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Epicycles Get a Bad Rap

Fourier Analysis

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Kepler

Kepler

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Keplers First Question

Why are there only six planets? (Mercury, Venus, Earth, Mars,Jupiter, Saturn)

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Answer

Keplers First Model of the Universe

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Platonic Solids

The Platonic Solids

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Platonic Solids

V E + F = 2

aF = 2E = bV

where a is the number of edges per face and b is the number ofedges per vertexThus

1

a+

1

b=

1

2+

1

E>

1

2

Only possibilities are

(a, b) = (3, 3), (3, 4), (4, 3), (5, 3), (3, 5).

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Platonic Solids

V E + F = 2

aF = 2E = bV

where a is the number of edges per face and b is the number ofedges per vertex

Thus1

a+

1

b=

1

2+

1

E>

1

2

Only possibilities are

(a, b) = (3, 3), (3, 4), (4, 3), (5, 3), (3, 5).

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Platonic Solids

V E + F = 2

aF = 2E = bV

where a is the number of edges per face and b is the number ofedges per vertexThus

1

a+

1

b=

1

2+

1

E>

1

2

Only possibilities are

(a, b) = (3, 3), (3, 4), (4, 3), (5, 3), (3, 5).

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Platonic Solids

V E + F = 2

aF = 2E = bV

where a is the number of edges per face and b is the number ofedges per vertexThus

1

a+

1

b=

1

2+

1

E>

1

2

Only possibilities are

(a, b) = (3, 3), (3, 4), (4, 3), (5, 3), (3, 5).

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Enter Tycho Brahe

Tycho Brahe

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Keplers Laws

Essentially by carefully studying the orbit of mars, Keplerpostulated:

The orbits of the planets are ellipses, with the sun at one focalpoint

The line joining a planet to the sun sweeps out equal areas inequal time

The square of the orbital period is proportional to the cube ofthe semi-major axis of the orbit.

Essentially extracted by Newton from a long and mystical tome ofKeplers.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Keplers Laws

Essentially by carefully studying the orbit of mars, Keplerpostulated:

The orbits of the planets are ellipses, with the sun at one focalpoint

The line joining a planet to the sun sweeps out equal areas inequal time

The square of the orbital period is proportional to the cube ofthe semi-major axis of the orbit.

Essentially extracted by Newton from a long and mystical tome ofKeplers.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Keplers Laws

Essentially by carefully studying the orbit of mars, Keplerpostulated:

The orbits of the planets are ellipses, with the sun at one focalpoint

The line joining a planet to the sun sweeps out equal areas inequal time

The square of the orbital period is proportional to the cube ofthe semi-major axis of the orbit.

Essentially extracted by Newton from a long and mystical tome ofKeplers.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Keplers Laws

Essentially by carefully studying the orbit of mars, Keplerpostulated:

The orbits of the planets are ellipses, with the sun at one focalpoint

The line joining a planet to the sun sweeps out equal areas inequal time

The square of the orbital period is proportional to the cube ofthe semi-major axis of the orbit.

Essentially extracted by Newton from a long and mystical tome ofKeplers.

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Keplers Actual Theory

A Baroque Construction

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

Keplers Actual Theory

Music of the Spheres

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Modern Day

But why should we care about all this stumbling about in the dark?Weve escaped our mystical obsessions in the modern day, right?

Right?

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Modern Day

But why should we care about all this stumbling about in the dark?Weve escaped our mystical obsessions in the modern day, right?

Right?

Daniel Litt The Music of the Spheres

IntroductionThe Pythagoreans

Plato, Aristotle, Ptolemy, and CopernicusKepler

The Modern Day

The Modern Day

The central idea of string theory is quite straightforward. If youexamine any piece of matter ever more finely, at first youll findmolecules, atoms, sub-atomic particles. Probe the smallerparticles, youll find something else, a tiny vibrating filament ofenergy, a little tiny vibrating string. Brian Greene

Daniel Litt The Music of the Spheres