In my Fall course of Math Models, I have three groups working on projects to finish up the semester. One of the groups have an assignment to explore a model of the spread of a forest fire. The assumptions are that the trees are on a rectangular grid, or a lattice. The time is a discrete variable and at each time step the probability that the fire spreads from one point in the lattice to an adjacent point (up, down, left or right) is given by p. For simplicity, the event that the fire spreads to each point is assumed to be independent of any other point.
Part of their project is to implement a numerical simulation of their forest fire. I couldn't let them have all the fun, so below is an example of my version of the simulation in MATLAB. I have to hold off on posting the code until after they have handed in their project.
In the graphical representation of my simulation, green represents an unburnt tree, black is burnt and red is currently on fire. The fire lasts for exactly one time step. I also implemented a 3-D version, where a height of 1 is unburnt, 2 is on fire, and 0 is burnt. I'll confess to having way too much fun with this.
I have used a 200x200 lattice with p = 0.5.
I have the most excellent privilege of teaching a course at Wayland in Mathematical Modeling. The course is designed as a projects course where the majority of the semester is spent working on modeling projects. The typical problem will take groups of 3 - 4 students anywhere from 2 to 4 weeks to solve. They often need to develop and learn new mathematical skills but mostly they will rely upon the mathematics courses they've covered up to this point. This gives them the opportunity to see their mathematics in action; they get to see what an applied mathematician actually does.
I'm always looking around for additional websites that provide real-world problems and not just problems designed for a particular application in a particular course. The problem with many problems that are included with typical textbooks is that they have been shaped and manipulated so that the techniques being covered in the course fit neatly within the problem. In practice, that rarely happens. The problem, not the technique, comes first and a mathematician must develop a reasonable model based on the desired outcome. Then appropriate techniques are used, learned or developed as the case demands.
This post marks the first of many I hope to follow which indexes a number of interesting sites that aid in promoting mathematical modeling and problem solving in the true spirit of serendipitous, constructive learning.
eBourbaki is a mathematical problem-solving company whose mission is to solve the world's mathematical problems using contests to inspire innovation and creativity.
eBourbaki’s mission is to solve the world’s mathematical problems. Our primary role is to host prize competitions focused around pertinent problems on behalf of sponsor organizations. The competition is global and open to everyone through the internet. The only pre-requisite for winning is providing the most innovative practicable solution. Our ultimate and unique agenda is to improve mathematical engagement, education and innovation worldwide.
The site is slick. I like the idea of competing to develop the best solution and I have seen a few different competitions along this line. The first contest was held in 2007. The ultimate goal was to develop a plan to shade downtown Phoenix, AZ during the summer. Below is part of the statement of the problem:
Your task is to devise the most cost-efficient way to distribute trees and structures throughout the downtown area so that the sidewalks and public spaces are shaded for the duration of the working day (8-5).
This year's contest runs from May 5 - 12 and the winning team will receive a cash prize. Here's the teaser (they won't give the full statement of the problem until the contest begins):
London faces serious transportation challenges today. With congestion charges on the rise and increased awareness of the environmental impact of many forms of commuting, cities are turning to bicycle stations to ease traffic, reduce pollution, improve parking, and enhance a green-friendly image. Last summer, Paris joined the ranks, instituting a city-wide network of high-tech low-cost rental bicycle stations.
We ask the question: if London were to embrace this concept, how would it best go about doing so? Where should the bike stations go? How many bikes at each station?
I'm setting a goal of participating in some sort of contest along these lines with a group of our students next year.