They say that the flap of a butterfly's wings can cause a hurricane on the other side of the world. This "butterfly effect" references the fact that weather patters are an example of a chaotic system, where small changes in initial conditions can build up to massive differences in outcomes. Chaos can occur even in systems governed by extremely simple rules: for example, a single pendulum is a simple, well-understood system, but a double pendulum is chaotic. The ability to analyze complexity that arises from simple is very useful when studying fractals, as they are often generated by iterating a short procedure or function, but can contain seemingly infinite complexity. In this semilab, we will study fractals named after Cantor, Koch, Peano, Sierpiński, a dragon, Julia, Mandelbrot, Newton, and the country of England. We will also examine some chaotic systems including the bifurcation function, the double pendulum, rule 30, and the Lorenz attractor. Requirements: Comfort with high school mathematics, approximately on the level of pre-calculus. Recommended (but optional!): Knowing what slopes, rates of change, and derivatives are, as well as how to find the derivative of a polynomial. Familiarity with linear regression (finding lines of best fit). Familiarity with logarithms.
Slides for this semilab can now be found here.