WittSem 100L: Patterns in Nature

Assignment 8 for Thursday, October 25, 2007

To hand in (on a separate sheet of paper):

1. Exploring percolation clusters (forest fires, epidemics, behavior of alloys and other mixtures of materials, Jell-O)

Go to http://polymer.bu.edu/java/java/blaze/blaze.html

Read the information on this page. When you get to the model for growing a forest, you don't need to work in pairs, and instead of creating the forest by hand on a square grid and rolling dice (directions 1-11, and the paragraph that talks about placing a tree if the die comes up 1, 2, 3, or 4), simply read the directions, and then run the Blaze Applet with a probability that trees grow in each location of 0.5. (Note: the page talks about being able to control the dryness; I don't see how to do that in the Blaze Applet, so just ignore that part of it.)

Now do the following:

a) Try the Blaze Applet for a number of different probabilities for trees growing in each location, from near 0 to near 1. Don't put water on the fires; simply let them burn till they burn out. Keep a record of the number of unburned trees for each value of probability that you use. Make a graph (preferably using Excel) of number of unburned trees (y axis) vs. probability (x axis). What conclusions can you draw from your data about how to get the highest score of unburned trees? (Include your data and graph as part of your answer, but be sure to also explain in words what you've concluded. Make sure to label axes of the graph.)

b) In the Peterson reading, he discusses a related problem: a plague invading an orchard and spreading from tree to tree. He mentions a value for a critical concentration of trees. How does his value compare to the results you obtained from the Blaze Applet? Explain.

c) If you now are allowed to put water on the trees, do your conclusions in a) and b) change, and if so, how? Use the Blaze Applet to gather data to answer this. Again, as part of your answer include data and any graphs you make.

2. There is a wonderful website on snowflakes (snowcrystals.com) by a Cal Tech professor of physics, Kenneth Libbrecht. On this website, read about Snowflake Branching: http://www.its.caltech.edu/~atomic/snowcrystals/dendrites/dendrite.htm

also follow the link on this page to Crystal Faceting and read that page as well. Answer the following questions in your own words.

a) What does diffusion have to do with the growth of snowflakes?

b) What's a dendrite?

c) What affects the number of branches in a dendrite?

d) What kinds of structures are produced by faceting? What symmetries do ice and snow crystals that have grown by faceting have?