Newton's Fractals

What is a newton fractal?

Well it is a boundary set in the complex plane, which is characterized by Newton's method applied to a fixed polynomial.

Nick is probably the only person who would understand that. Lets break this down into terms anyone can understand.

Newtons Fractals

Aamir Mansoor

How do we Create Newtons Fractals

So now that we know what the statement I introduced at the beginning of class means we can now come up with a better definition of newton’s fractals. Newton’s fractals are basically visual representations of the iterations experienced when iterating with newton’s method. They basically show the tangents of the lines and the very centre of the fractal, where all the large fractals appear to converge represent the root of the function (where the tangent meets the x-axis). Depending on the function, which will be demonstrated after the presentation, different functions create different roots.

What can we use newton’s method for?

Surprisingly newton’s method does not only have a place in calculus. It was actually created by newton in order to solve Kepler’s Equation about the orbit of a body. However, this has never been completely proven. Also, newton’s method plays a part in financial mathematics. It can be used to approximate the amount of product needed to reach a break-even point. Newton’s method also has its place in calculus. Say you have a complex formula of which you need to solve for the roots. If you were looking at the visual representation of the formula you would observe the approximation of the root (which could get quite precise after several iterations) would be the centre of the fractal.

What is the Significance of Newton’s Fractals?

Besides looking very cool newton’s fractals have a very useful purpose. As we have already learned, newton’s fractals are visual representations of complex roots, approximated through newton’s method. If it were not for this method there would be no way to be able to approximate the roots for such functions and would render many problems in calculus unsolvable. The fractals themselves might not have a very useful purpose, but the math behind it is a key element to calculus and much of advanced mathematics.

Newtons Method

Newton’s method, for those who don’t know, is a simple way to find the roots of a function with complex roots. Think of it as a more advanced version of the quadratic formula we learned in grade 10. It works by using the derivative of a function to improve an approximation given to a root. For example if there was the function f (x) = x^3-x+1 with a derivative of f ’ (x) = 3x^2-1 with use of power rule which basically states that the derivative of a third root equals 3 times the variable squared, the derivative of x is one and 1 is 0.

Now that we know the derivative of the function the last step is to have an approximation to the root of the function, in this case it will be x1=1. Then we plug all these values into the equation given in newton’s method, which is x_(n+1)=x_n-(f(x_n))/(f'(x_n)). This gives us the value of 1.5 for x2. We can subsequently plug x2 into the equation to get x3 and so forth until the wanted amount of iterations are satisfied.

Works Cited

Photo 1: http://media.techtarget.com/WhatIs/images/complex_number.gif

Photo 2: http://www.hdwallpaperspot.com/wp-content/uploads/2013/01/Sir-Isaac-Newton-Pictures.jpg

Video 1: outube.com/watch?v=v-du-KjqniE

Photo 3: http://3.bp.blogspot.com/_3Yi34O67M6o/SPww73SZyBI/AAAAAAAAAbE/WCDgvLWBazg/s400/Fractal+3a.PNG

Chairk, . "Fractals derived from Newton-Raphson iteration." chiark.greenend.org.uk. N.A, 02 Apr 2007. Web. 9 Jun 2013. <http://www.chiark.greenend.org.uk/~sgtatham/newton/>.

Hass, Joel. "The Power Rule." math.ucdavis.edu. University of California, Davis, 26 Mar 1999. Web. 9 Jun 2013. <https://www.math.ucdavis.edu/~hass/Calculus/HTAC/excerpts/node19.html>.

arianabedi, . "Real Work Application of "Newthon-Raphson" method.." 04 Apr 2012. American Network, Online Posting to Physics Forums. Web. 9 Jun. 2013. <http://www.physicsforums.com/showthread.php?t=593265>.

noolouit, . Newton's Method: Example. 2009. Video. Youtube, N.A. Web. 9 Jun 2013. <outube.com/watch?v=0H7L1m4_qvs>

Complex Planes

Basically a complex plane is not much different as a Cartesian plane, a regular grid we use everyday. However complex planes are special in that they can represent complex numbers as well as rational numbers. In this plane rational numbers are represented as on a Cartesian plane along the y-axis, but quite to the contrary irrational numbers are represented along the x-axis. This allows for a visual representation of complex numbers and is an essential tool with dealing with complex roots as in this situation.

What Gives them Their Fractal-Like Qualities

Well, what gives these “newton fractals” the fractal-like properties? Well the area in which the iterations converge into a single point becomes the fatou set of the iteration. The complementary set becomes known as the Julia set. For out purposes the fatou and the Julia sets are two different sets of the iterations. Every fatou set has a boundary, which is a Julia set. Thus, every Julia set is the point in which all the fatou sets meet. Within every fatou set there is a Julia set and this continues to iterate forever. This property is what gives newton fractals their special fractal-like properties.