Meeting 13: 17 Dec 2002
Reading for Everyone:
 Adams, The Knot Book
 Mumford et al, Indra's Pearls
 Hayles, Complex Dynamics in Literature and Science
Projects for Everyone:
 Why are knots knotted?
 What surfaces can you make from triangles coming together
7 to a vertex?
 Work on your final project, and prepare its documentation to hand in.
Individual topics and projects:
 Final projects: Josi, Jason, Ben
New and pending questions and topics.
 Why are knots knotted?
 Quasifuchsian groups
 How does it all fit together?
 Thursday, 9 Jan, 2 pm: reunion

Meeting 12: 10 Dec 2002
Reading for Everyone:
 Thurston, ThreeDimensional Geometry and Topology
 Mumford et al, Indra's Pearls
 Adams, The Knot Book
Projects for Everyone:
 Try to visualize the 3torus, the 3sphere and some other
3manifolds.
 Why are knots knotted?
 Work on your final project.
Individual topics and projects:
 Dynamical systems: Brendan
 Final projects: Brendan, Walker, Dina, Will
New and pending questions and topics.
 What happens if you glue together a mass of triangles, 7 to a vertex?
 The Borromean rings and the cube
 3manifolds, knots and Not Knot
 Quasifuchsian groups
 Hayles, Complex Dynamics in Literature and Science
Schedule of final presentations:
 17 Dec: Josi, Jason, Ben
 Thursday, 9 Jan: reunion

Meeting 11: 3 Dec 2002
Reading for Everyone:
 Rademacher and Toeplitz, Pedal triangle
 Mumford et al, Indra's Pearls
 Thurston, ThreeDimensional Geometry and Topology
Projects for Everyone:
 Prepare to continue discussion of RLR, Memento and Borges.
 Work on your final project!
Individual topics and projects:
 Dynamical systems: Brendan
 The pedal triangle: Brendan
 Final projects: Susannah, Andrew, Brendan
New and pending questions and topics.
 The hexponential
 3dimensional manifolds
 Knots and Not Knot
Schedule of final presentations:
 10 Dec: Walker, Dina, Ben
 17 Dec: Josi, Jason, Will

Meeting 10: 26 Nov 2002
Reading for Everyone:
 Borges, The garden of forking paths
 Series, Continued fractions
 Rademacher and Toeplitz, Pedal triangle
 Mumford et al, Indra's Pearls
Projects for Everyone:
 Compute some continued fractions (e.g. sqrt[n], e, pi, ...)
 Prepare to discuss RLR, Memento and Borges.
 Work on your final project!
Individual topics and projects:
 3D life and movies  Brendan
 Dynamical systems: Brendan, Susannah and Will
 The pedal triangle: Brendan
New and pending questions and topics.
 Is life a garden of forking paths?
 Points of confusion: what is the significance of the
Feigenbaum constant? What does T(f(x)) = g(T(x)) have to
do with chaos?
 What does `sensitive dependence on initial conditions' mean?
 Expansion factors for 2x mod 1, x^22, ax + x^2
 Are billiards and irrational rotations chaotic?
 In the square, not all itineraries are possible
 The hyperbolic plane; surfaces of higher genus
 In the curved torus, all itineraries are possible
 Tilings, continued fractions.
 The hexponential
Schedule of final presentations:
 3 Dec: Susannah, Andrew, Brendan
 10 Dec: Walker, Dina, Ben
 17 Dec: Josi, Jason, Will

Meeting 9: 19 Nov 2002
Projects for Everyone:
 Read Tabachnikov on Billiards.
 Study the dynamics of rotation: f(x) = x + a mod 1.
Show when a is irrational, all orbits are dense.
 Invent and investigate your own dynamical system.
 See "Memento" and "Run Lola Run".
Individual topics and projects:
 3D life and movies  Brendan
 Escher and the Droste effect  Walker
New and pending questions and topics.
 Peano animation
 What does `sensitive dependence on initial conditions' mean?
 x + a mod 1: are orbits dense? Closest returns?
 Billiards.
 Continued fractions; find your own!
 Is life necessarily balanced on the edge of chaos?
 Is life a garden of forking paths, as in Run Lola Run and Borges?
 Points of confusion: what is the significance of the
Feigenbaum constant? What does T(f(x)) = g(T(x)) have to
do with chaos?
Upcoming reading:
 Borges, The garden of forking paths
 Series, Continued fractions
 Rademacher and Toeplitz, Pedal triangle
Organization:
 Schedule final presentations, Dec. 31017.

Meeting 8: 12 Nov 2002
Projects for Everyone:
 Write up your midterm report.
 Invent and investigate your own dynamical system.
Questions to answer:
 What is (or are) the attractor(s)?
 What are the periodic points?
 Which are attracting? Which repelling?
 What is the doubling time?
 Study the dynamics of rotation: f(x) = x + a mod 1.
 How does the bifurcation picture for f(x) = ax(1x) relate to
the Mandelbrot set?
 See "Memento" and "Run Lola Run" by 19 Nov 2002.
Individual topics and projects:
 Complexity of English  Josi
 3D life and movies  Brendan
 Escher and the Droste effect  Walker
 What is a Julia set?  Dina
 What is the Mandelbrot set?  Susannah
 More on the halting problem  Ben
New and pending questions and topics.
 Peano animation
 What does `sensitive dependence on initial conditions' mean?
 x + a mod 1: are orbits dense? Closest returns?
 What is a Julia set?
 What is the Mandelbrot set?
 Can a computer be programmed to discover every mathematical theorem?
 Is the world even the solar system "random and unpredictable"?
 Is life necessarily balanced on the edge of chaos?
 Billiards.
 Continued fractions.
Upcoming reading:
 Tabachnikov, Billiards
 Hardy and Wright, Continued Fractions

Meeting 7: 5 Nov 2002
Projects for Everyone:
 Invent and investigate your own dynamical system.
Questions to answer:
 What is (or are) the attractor(s)?
 What are the periodic points?
 Which are attracting? Which repelling?
 What is the doubling time?
 Study the dynamics of rotation: f(x) = x + a mod 1.
 Read Milnor on Julia sets. Browse Marmi on chaos in the solar system.
 How does the bifurcation picture for f(x) = ax(1x) relate to
the Mandelbrot set?
 See "Memento" and "Run Lola Run" by 19 Nov 2002.
Individual topics and projects:
 Complexity of English  Josi
 3D life and movies  Brendan
 Escher and the Droste effect  Walker
 Chaotic Newton's methods  Will
 Chaos of 4x(1x)  Jason
 What is a Julia set?  Dina
 What is the Mandelbrot set?  Susannah
 More on the halting problem  Ben
New and pending questions and topics.
 Peano animation
 The halting problem
 Can a computer be programmed to discover every mathematical theorem?
 Universality
 What is a Julia set?
 What is the Mandelbrot set?
 Behavior of x+a mod 1; closest returns
 Pappus's theorem
 Memento and Lola Rennt
 Is the world even the solar system "random and unpredictable"?
 Is life necessarily balanced on the edge of chaos?

Meeting 6: 29 Oct 2002
Readings: May; Feigenbaum; browse Marmi; read "What is Life?"
Topics/Projects for Everyone:
 Graph the `attractor' of f(x) = ax(1x) for a in [0,4].
(The attractor can be drawn as the set of points f^n(1/2)
for 100 < n < 1000, for example.)
 Investigate what happens in the windows of calm.
 Investigate the rate of period doubling.
 What happens for other functions like f(x) = a sin(x)?
Individual topics and projects:
 Fractal ruler  Josi
 Curves of dimension 1.9 or so  Walker, Will
 3D fractals and movies  Brendan
 Life and computers  Brendan
 Iteration of Newton's methods for x^2+1  Will
 Iteration of 4x(1x)  Jason
 What is the complexity of the English language?
 Generation of babble.
New questions and topics.
 Peano animation
 Impossible questions  the halting problem
 Chaos: 2x mod 1, Newton's method for x^2+1, 4 x (1x), z^2
 Attractor of a x (1x), cascade of period doublings
 Expansion, contraction and borderlines
 Behavior of x+a mod 1; closest returns
 RATS
New project for Everyone:
Invent your own dynamical system and investigate its behavior.
Your dynamical system might be an automaton, a sequence like RATS,
or an iteration like f(x) = x^2+c.
Questions to answer:
 What is (or are) the attractor(s)?
 What are the periodic points?
 Which are attracting? Which repelling?
 What is the doubling time?

Meeting 5: 22 Oct 2002
Readings: May; Feigenbaum; browse Marmi.
A good book on matrices: Linear Algebra with Applications, O. Bretscher.
Topics/Projects for Everyone:
 Graph the `attractor' of f(x) = ax(1x) for a in [0,4].
(The attractor can be drawn as the set of points f^n(1/2)
for 100 < n < 1000, for example.)
 Invent your own dynamical system and investigate its behavior.
Individual topics and projects:
 Fractal ruler  Josi
 Curves of dimension 1.9 or so  Walker, Will
 3D fractals  Brendan
 Dimension of Brownian graph  Andrew, Walker
 How are random numbers generated (Knuth)?  Will
 Cellular automata and randomness (Wolfram)  Susannah, Dina, Jason
 How to map [0,1] onto a cube  Ben
 What is the complexity of the English language?
 Iteration of Newton's methods for x^2+1  Will
 Iteration of 4x(1x)  Jason
Question to keep in mind on dynamics.
 What is (or are) the attractor(s)?
 What are the periodic points?
 Which are attracting? Which repelling?
 What is the doubling time?
New questions and topics.
 Impossible questions  the halting problem
 Invent your own random number generator.
 Life and computers
 Behavior of x^2 and 2x mod 1
 Attractor of a x (1x), cascade of period doublings
 Universality
 Chaos: 2x mod 1, Newton's method for x^2+1, 4 x (1x), z^2
 What is a Julia set?
 What is the Mandelbrot set?
 Behavior of x+a mod 1; closest returns

Meeting 4: 15 Oct 2002
Readings: Feller 6797;
Knuth: How to generate random numbers (137);
What is a random sequence (142177).
Another good probability book: Jim Pitman, Probability
Topics/Projects for Everyone:

What are the chances you never see a run of 5 0's in a random sequence
of length N?
 Figure out the relation between positive paths and
returning paths (Feller, Prob. 7, p.96).
Other topics and projects:
 What is the dimension of an actual computer image, or an
actual coastline?  Andrew, Susannah
 Fractal ruler  Josi
 Curves of dimension 1.9 or so  Walker, Will
 3D fractals  Brendan
 Random walks in 2 and 3 dimensions  Josi, Andrew, Dina
 Dimension of Brownian graph  Andrew, Walker
 Bias  Susannah
Further topics:
 Does a random walker in the plane visit every point?
 How complex is the Enligsh language?
 What is a random number?
 Dynamics of 2x mod 1
 The map a x (1x), e.g. a=2, 3, 4; stability of cycles
 How does Newton's method behave for p(x) = x^{ 2 } + 1?
 Bifurcations and x^2 + c
 What is the Mandelbrot set? a Julia set?

Meeting 3: 8 Oct 2002
Readings: Feller 6797;
Knuth: How to generate random numbers (137);
What is a random sequence (142177).
What are the chances you never see a run of 5 0's in a random sequence
of length N?
Invent your own random number generator.
Draw a picture of a long random walk on the line, in the
plane or in space.
 `The dilema of probability theory'  Brendan
 Random walks  Will
 Does a random walker always return home?  Jason
 Is it hard to write down a random sequence?  Dina, Susannah, Walker
Questions about random walks to keep in mind:
 Does randomness = ignorance?
 Why is the distance from home after n steps
expected to be Sqrt[n]?
 What is the probability the walker is home at the 2nth step?
 What is the probability the walker is home for the
first time at the 2nth step?
 Why is it common for a random walker
to spend a great deal of time to one side of home?
 What is the dimension of the graph of Brownian motion
on a line?
Further topics:
 Random walks and Pascal's triangle
 Long onesided excursions
 Harmonic functions via random walks.
 Recurrence in dimensions one, two and three
 Positive paths and returning paths (Prob. 7, p.96)
 Hausdorff dimension of graph of Brownian motion on a line.
 Computergenerated random numbers (Knuth)
 Generating random numbers via automata (Wolfram)
 What is a random number?
 Dimension, information, entropy and language
Other fractal projects: Compute or estimate the dimension of an actual
coastline or boundary of a geographic entity.
Construct a fractal ruler. Draw some 3D fractals.
Draw a curve of dimension > 1.9 and a surface of dimension > 2.9.
Why doesn't the Koch curve meet itself?

Meeting 2: 1 Oct 2002
Readings: pp.5173 of the Mandelbrot selection;
Littlewood, `The Dilema of Probability Theory';
Doyle and Snell, 37, 119121. Browse Feller 6797.
Invent your own fractal  a curve, a surface or another
type of object. Compute its dimension. Render it
graphically.
Be ready to discuss the following topics.
(Leader follows topic.)
 Cantor function  Dumas
 The Gosper snowball (and selfsimilar tilings?)  Josi
 Windows in the Sierpinski triangle  Ben
Find an image showing the selfsimilar nature of an internal
biological structure, such as the lung, the intestine or
the capillary system.

Meeting 1: 24 Sept 2002
Read pp.2550 of the Mandelbrot selection.
This begins with `5. How long is the coast of Britain?',
and continues into `6. Snowflakes and other Koch curves.'
(Readings are from coursepacks
249 and 250, available in the Science Center basement and on
reserve in the Birkhoff library.)
Pick an example related to the seminar topics:
a fractal, a dynamical system, a geometric object, a random process 
mathematical or from nature  to briefly present to the class.
(You can present your example in a variety of ways, e.g.
on the blackboard, with a magazine clipping, with
a natural object, etc.)
Pick an image from among those on the web page
math.harvard.edu/~ctm/gallery that you would like to
discuss  for example, one that is familiar to you already, or
one you would like to know more about.
Print out your picture and bring it to the seminar.
