the M-set, you can choose a new starting point and see whether or not it diverges for a given complex number. () = 2 + 0 . For this function the filled Julia set is bounded by a circle of radius one about the origin. Given this, if you have a starting value of that has a magnitude greater than 1, the function will diverge. The relationship between the M-set and the Filled Julia Set is that there are two types of Filled Julia Set depending on whether the complex number was in the M-set or not. If the complex number is within the M-set, the Filled Julia Set is a closed fractal such as [ fig. 10 ]. If is outside of the M-set, the filled Julia set is a disconnected series of points, [ fig. 11 ]. What the filled Julia set looks like for a given value of is incredibly complex, requiring hundreds of thousands of iterations across the entire complex plane. But due to its relationship with the M-set we don’t need to know all of that information to know something about the Filled Julia Set: all you need to know is whether is a member of the M-set. From there we can know what sort of filled Julia set it will be: a closed fractal or a number of disconnected points. The most basic Filled Julia Set is that of the function 0
fig. 10: Filled Julia Set where is an element of the M-Set
fig. 11: Filled Julia Set where is not an element of the M-Set
Fractals are still a very active topic of mathematics with new proofs being discovered much more regularly than in other areas of mathematics: such as the proof that the area of -dimensional Wallis Sieves is equal to the volume of an -dimensional Hyper-ball.
Practical Applications of Fractals
While we have shown that fractals are definitely mathematically magical, are they of any practical use? Well the answer to that lies in the particularly amazing way in which fractals describe the intricacies of nature. Since the time of Euclid, we have focused on the smooth planes of geometry, on spheres and cubes, however you rarely see anything smooth in nature. Fractal geometry provides the perfect way to discuss and understand the formations of nature owing to the iterative process of their development over time. A good example of this is to ask the question: How long is the coastline of the UK? Well to answer this you have to state how long your ruler is.
51
Made with FlippingBook - professional solution for displaying marketing and sales documents online