So one aspect of the Newton fractal is this: when we choose our initial guess for the approximation of a root of a function with Newton’s method, there are initial guesses lying close to each other that converge to similar patterns on different scales. We’ll get to this property below and I’ll show you an example for it. In addition, they often concern only parts of the fractal and not the entire thing. But in the fractal world, the examples are usually more spectacular than that. A square (two dimensional non-fractal) is self-similar as well, because you can cut it up into four identical smaller versions of itself. Self similarity is not restricted to fractals. But we can even understand one of the best-known example properties of fractals easily, namely the property of self similarity. In our case, the fractal dimension is not what we are after, but it is necessary to mention as the basis of the definition of what a fractal is.Įven though, e.g., the concept of a fractal dimension may be hard to grasp, we can all easily appreciate the beauty and complex detail of pictures of various fractals. In this case, we usually talk about the “size” of the object as its “volume”.įor an object of fractal dimension, the dimension isn’t integer, but something else/in between, like 1.538. The 2 in the exponent is the dimension, and the word “size” refers to an “area” in this case.įor a three-dimensional object (like a cube), its size goes up by a factor of \(2^3=8\), if scaled by a factor of two – notice the dimension-exponent 3 here. For example, a two-dimensional object (like a square), if scaled up by a factor of 2, increases its size by a factor of \(2^2=4\). What’s a fractal dimension? Briefly, it is a number that describes the scaling behavior of an object, like the dimensions we are used to. The most important point about Newton fractals being fractals is that fractals scale unintuitively: they have what is called a fractal dimension. If you want to look up all the details, I recommend you read through the Wikipedia article on fractals, but for what we discuss here, you don’t have to. The definition of a fractal in general is much more broad. Definition and meaning of a Newton fractalĪ Newton fractal is a fractal based on using Newton’s method in order to find roots for a particular function, and starting from points laid out on a grid in the complex plane. Most importantly for creating a fractal, this behavior doesn’t change when we zoom in and look at our structure at smaller and even smaller scales. In the examples below we can see that the color changes very interestingly and abruptly in some areas. For the picture of the fractal, different colors are used for different final outcomes of the iteration (finding different roots). In the context of fractals, such a sensitivity to initial conditions is a key property. For the present purpose, however, this behavior is an interesting observation. When we try to get a precise value for a particular root of a function, that can be unwanted. If there are several roots, a small difference in the starting point can change the outcome of Newton’s method from one root to another. The starting point is often called initial guess, when we use Newton’s method to find the root of a function, because there we try to guess the starting point as close to the root as we can, in order to make the method convergence in fewer steps.īut in addition to that, we want the method to find the one particular root we are looking for, sometimes among many. If you’d like to know, why Newton’s method can be used to generate fractals, the main point is this: The outcome of an iteration procedure can depend very strongly on the starting point. There you can find all the necessary details that you need in order to find this article here even more informative. Then, please take a couple of minutes and read my articles Newton’s Method Explained: Details, Pictures, Python Code, Highly Instructive Examples for the Newton Raphson Method, and How to Find the Initial Guess in Newton’s Method. If you are not familiar with Newton’s method yet, watch this video, where I give you some intuition about what to expect in an explanation: You have the same algorithm, starting from some initial guess and then converging (or not) stepwise towards a root of the function under consideration. One of the most beautiful aspects of this is that Newton’s method isn’t more complicated for complex numbers than for real numbers.
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