| # | Date | Topics | Homework |
|---|---|---|---|
| 1 | F 9/8 | Introduction | Reading: Prologue and The butterfly effect |
| 2 | M 9/11 | The k-th stage of the Koch curve has length (4/3)^k. The Koch curve itself is infinitely long. | Questions to think about: (1) How can an infinitely long curve fit into a finite amount of space? (2) If you start walking on the Koch curve, can you reach the other end? |
| 3 | W 9/13 | The Sierpinski carpet. The k-th stage has area (8/9)^k. The carpet itself has area 0. Also the Menger sponge. | - |
| 4 | F 9/15 | Self-similar sets. The numbers N of smaller copies making up the whole and the scale factor r | - |
| 5 | M 9/18 | Fractal (self-similarity) dimension | HW 1 (due M 9/25) |
| 6 | W 9/20 | Introduction to iteration. Terminology: initial value, orbit, fixed point, cycle | - |
| 7 | F 9/22 | Find fixed points algebraically by solving f(x) = x for x. | - |
| 8 | M 9/25 | Repelling and attracting fixed points. Four ways to visualize iteration (list of numbers, number line with arrows, time series, histogram). Chaos. | - |
| 9 | W 9/27 | More on chaos. Some properties of chaos: (1) repelling periodic points are everywhere, (2) it has sensitive dependence on initial conditions (the butterfly effect), and (3) it is transitive (almost all points travel everywhere) | Homework 2 (due 10/3). |
| - | - | The small pendulum demonstrates chaos. | - |
| 10 | F 9/29 | Iteration with a spreadsheet. (handout) Measuring boarders and coastlines. | Reading: Gleick: "Nature's ups and downs" |
| 11 | M 10/2 | Measuring the length of the coastline of Great Britain, also measuring the length of a straight line, a curved line, and the Koch curve. | Homework 4 (due 10/13) |
| 12 | W 10/4 | The ruler dimension | Homework 3 (due 10/11) |
| - | - | - | Homework 5 (practice with logs and exponents; due 10/16) |
| 14 | F 10/6 | [No class] | - |
| 15 | W 10/11 | - | HW 6 (Find the ruler dimension of the Koch curve) |
| 16 | F 10/13 | [no class] | - |
| 17 | M 10/16 | Introduction to Boston University's iteration applet | - |
| 18 | W 10/18 | Video "Chaos, fractals and dynamics" | - |
| 19 | F 10/20 | Box dimension | - |
| 20 | M 10/23 | HW 7 done in class | Test 1 handed out. Topics: up through HW 5. |
| - | - | - | - |
| 21 | W 10/25 | Random Koch curves and Sierpinski triangles, fake landscapes | HW 9: Draw a random Koch curve. (Do with someone else, one tossing the coin, the other drawing) (Due 11/1) |
| 22 | F 10/27 | HW 8 done in class (due before beginning of class 11/1) | - |
| 23 | M 10/30 | Introduction to the complex numbers: addition, subtraction, multiplication, division | - |
| 24 | W 11/1 | How HW 7 and HW 8 together lead to the orbit diagram | - |
| 25 | F 11/3 | More on complex numbers: graphing, representing in polar coordinate form. Multiplication of complex numbers is adding the angles and multiplying the distances. | Homework 10 (due 11/8) |
| 26 | M 11/6 | Iterating z^2 + c using complex numbers. Terminology: prisoners, escapees. | - |
| 27 | W 11/8 | Reading: Gleick: "Universality" | Homework 11 ("Lab 3"): Windows in the orbit diagram of the family cx(1-x). |
| 28 | F 11/10 | (con't) | - |
| 29 | M 11/13 | - | - |
| 30 | W 11/15 | Period doubling. The orbit diagram is a fractal. | HW 13 (the first one numbered 13--examples of self-similar fractals with increasing fractal dimension). The curves become "rougher" |
| - | - | The map x -> rx(1-x) is essentially the same as x-> x^2 + c. The former is related to polulation growth in a limited environment; the latter is used for the Mandelbrot set and its Julia sets. | - |
| - | - | Test II handed out (on HW 6-10) | pdf version of test, without pictures |
| 31 | F 11/17 | For f(z) = z^2 the motion on the unit circle is chaotic | Test II due |
| 32 | M 11/20 | - | - |
| 33 | M 11/27 | Introduction to Julia sets | Reading: Gleick "Images of chaos" |
| 34 | W 11/29 | (con't) | - |
| 35 | F 12/1 | Homework 13 (the second one numbered 13) "A tour of Julia sets" | Finish the second HW 13 by 12/8 |
| 36 | M 12/4 | [go over test] | - |
| 37 | W 12/6 | The Mandelbrot set | Reading: Gleick "A geometry of nature", "Images of chaos" |
| 38 | F 12/8 | - | Second version of test II |
| - | - | - | HW 14: Exploring the Mandelbrot set |
| 39 | M 12/11 | - | - |
| 40 | W 12/13 | The sound of iteration (demo) | All homework (except numbers 13, 14) is due at 5pm today. |
| - | - | Can chaos authenticate Pollock's paintings? http://www.phys.unsw.edu.au/PHYSICS_!/FRACTAL_EXPRESSIONISM/fractal_taylor.html ; http://www.sciencedaily.com/releases/2006/12/061204123447.htm | - |
| - | - | The Lorenz attractor: http://www.sat.t.u-tokyo.ac.jp/~hideyuki/java/Attract.html | - |
| - | - | Fractals and the 12c. pulpit of Ravello Catherdral: http://www.christianhubert.com/hypertext/fractals.html | - |
| - | - | Fractals in nature: http://www.ba.infn.it/~zito/project/nfractals.html | - |
| - | - | The dinosaurs live on! | - |