Drawing fractals with GeoGebra

acron shared this question 8 years ago
Answered

With the Julia set,7508387a755aafb557eecefb6c49bea1The file below is my attempt to start a Julia fractal. Also, I need to learn about Mandelbrot sets.


My problem is I do not understand the basic concepts needed to create either set...in order to teach some this to my students, I need a simple lesson...my students in the past have copied some of the other fractals apps...Koch, Sierpinski , and others...I want more...


Thanks,


Tony

https://ggbm.at/561845

Comments (5)

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maybe http://www.kuro5hin.org/sto... or http://en.wikipedia.org/wik... or http://mathworld.wolfram.co...


Don't worry about the complex plane if you don't yet know about imaginary and complex numbers. You may just view it as an ordinary coordinate system.


The mandelbrot set is built like this:


For each point C0 = (a, b) in the coordinate system you calculate a new point C1 = (a^2 - b^2 + a, 2ab + b) (this is the complex C^2 + C).


Then you keep doing this. Note that the last (...+a, ...+b) don't change. Otherwise just keep on iterating. One of three things will happen:


The points escape from the origin quickly - colour the original point C white

The points are quickly atracted to the origin - colour the original point C black

The points keep jumping about round about the distance 1 from the origin. After you get tired calculating you just colour the original point C grey.


If you are a bit more sofisticated you count how many iterations it will take to escape to, say, a distance of 10 from the origin. This number decides the colour according to some scheme.


The mandelbrot set are those points that take infinitely long to decide... :-)


That's basically it. The Julia set I'm not familiar with but it is related to the Mandelbrot set some how

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I have been looking at fractals for several years, and have tried to do some with GeoGebra. I have had trouble creating the Mandelbrot and Julia sets using GG. Several of the GGT apps have a completed fractal as a background clip, but I want to create a Mandelbrot or Julia graphic using GeoGebra...this is an area where I am a beginner in GG.


Thanks for the resources, I will try investigating them. The Cartesian coordinate plane does a nice representation of the complex plane. Theoretically, I know beside the 3D real plane, there is a 3D complex plane; just like creating a 3D as a 2D representation there is no easy way to accomplish the task...without an actual 3D viewing field, a 4D representation cannot be illustrated in 2D; we can speak of it though...


Part of my problem is creating the iterative methods needed to create a fractal in GG. I am hoping for some simple examples at a near beginner level...this was the reason for posting in this forum...if I can finally understand the process, I can teach it to my students...


Thanks for all of the assistance I can get...


Tony

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Maybe this link can be useful: http://www.mrsankey.com/geogebra/mandelbrot.html


And here are some examples (in Spanish): http://www.geometriadinamic...

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A question relating to the reply of themadmathematician.

Introduction:

Assume that I let both (a,b)->(x,iy) and (u,iv) cover the (x,iy,u,iv) planes. Then I can write

x^2-y^2 = u^2-v^2 and 2xy =2uv. The (x,iy) curves are two hyperbolas for each value of (u,iv). I do

not see hyperbolas in fractals. I do see spirals. This leads me to think that I should be using polar coordinates.

r^2 ( cos(t)^2 -sin(t)^2) = Re(w^2), and r^2*2cos(t)sin(t)=Im(w^2) where w = u + iv is also expressed in polar coordinates.

Assume that I let r=t so that spirals are formed. The parameter t can be common to both (x,iy) and (u,iv). This will sync the two planes. A spiral in (x,iy) then corresponds to a spiral in (u,iv), assuming this is what I want to do.

Question:

Why do I need an iteration process? Each spiral in (x,iy) will either extend off the page or not. The first and second derivatives can be calculated as needed. Each point on each spiral will seed another spiral. The spiral from this seed has already been calculated. In fact all possible (x,iy) spirals have been calculated after one pass over the (u,iv) plane. Of course some points may be in several spirals.


Thank you.

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