E-320: Teaching Math with a Historical Perspective Oliver Knill, 2011Lecture 10: Fractals1. ObjectiveWe want to compute the dimension of various objects in the plane.2. Definition of DimensionAssume we need n squares of side length r to cover an object X. The dimension is defined asd = − log(n)/ log(r)when r gets zero.3. Dimension of a curve1) Assume a curve is given as the boundary of the unit square. How many squares of lengthr = 1/10 do we need to cover the curve? If we call n this number, what is − log(n)/ log(r)?4. Dimension of a region2) Assume a region is the unit square. How many squares of length r = 1/10 do we need to coverthe square? If we call n this number, what is − log(n)/ log(r)?5. The Sirpinski carpetThe Sirpinski carpet is constructed recursively by dividing a square in 9 equal squares andcutting away the middle one, repeating this procedure with each of the squares etc. At the k’thstep, we need n = 8ksquares of length r = 1/3kto cover the carpet. What is the dimension?6. A castleWhat is the dimension of the following fractal from which we see the first levels of construction?7. The Koch SnowflakeWhat is the dimension of the K och snowflake? How large is n, the number of squares we need tocovert the flake if the square has size 1 /3kassuming that the first triangle has side length 1.Lecture 10: Julia and Mandelbrot set1. ObjectiveWe want to understand the definition of the Julia sets a nd the Mandelbrot set. DefineT (z) = z2+ c ,where c is a fixed parameter. The filled in Julia set Jcis the set of points z for which t he orbitz, T (z), T (T (z))... stays bounded. The Mandelbrot set M is the set of c for which the point 0 isin the filled in Julia set Jc. It is the set of c such that 0, T (0) = c, T (T (0)) = c2+ c, T(T ( T (0))) =(c2+ c)2+ c... stays bounded. The Julia set finally is the boundary of the filled in Julia set.2. Complex Arithmetic1) What is the square root of −9?2) Add 2 + 6i with 6 + 8i.3) Multiply 2 + 6i with 6 + 8i.4) What is the length of the complex number 3 + i4?3. Drawing an orbit5) Assume c = 2. Compute the first 3 steps of the orbit of z = 1 of the quadratic map.4. Drawing the Mandelbrot set6) Verify that 0 is inside the Mandelbrot set. Verify that −1 is inside the Mandelbrot set. Verifythat i is inside the Mandelbrot set.7) Verify that 2 is outside the Mandelbrot set. Verify that 1 is outside the Mandelbrot set.8) Can you verify that 1/4 is t he largest r eal number in the Mandelbrot set and −2 the smallestreal number in the Mandelbrot set?Image credit: