3.9. Lab 8: Physical Modelling

Due Date: 11:59pm March 3, 2024 Labs should be submitted as a single Jupyter notebook to CourseLink Dropbox This lab counts for 4% of the final grade

To do in advance of this lab:

• Make sure you have run the code in Allen Downey’s Chapter 7 notebook.

3.9.1. Exercises

Complete the exercises corresponding to Exercise 7.1 and 7.2 in Chapter 7 of Think Complexity (version 2):

1. In Section 7.6, Downey showed that the Rule 18 CA produces a fractal. Can you find other 1-D CAs that produce fractals?

   Note: the Cell1D object in Cell1D.py does not wrap around from the left edge to the right, which creates artifacts at the boundaries for some rules. You might want to use Wrap1D, which is a child class of Cell1D that wraps around. It is also defined in Cell1D.py.

2. In 1990 Bak, Chen and Tang proposed a cellular automaton that is an abstract model of a forest fire. Each cell is in one of three states: empty, occupied by a tree, or on fire.

   The rules of the CA are:

   1. An empty cell becomes occupied with probability p.

   2. A cell with a tree burns if any of its neighbors is on fire.

   3. A cell with a tree spontaneously burns, with probability f, even if none of its neighbors is on fire.

   4. A cell with a burning tree becomes an empty cell in the next time step.

   Write a program that implements this model. You might want to inherit from Cell2D. Typical values for the parameters are p=0.01 and f=0.001, but you might want to experiment with other values.

   Starting from a random initial condition, run the CA until it reaches a steady state where the number of trees no longer increases or decreases consistently.

   In steady state, is the geometry of the forest fractal? What is its fractal dimension?

3.9.2. Submission

Please submit a single Jupyter notebook which completes the above exercises, filling in Allen Downey’s Chapter 7 notebook. There are already placeholders in the notebook for your work.