Comparison+&+contrast

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The Cycle attractor is more complex than the simple attraction or repulsion type Point attractor. Its analogy in consciousness is the thinking function. Like objective thinking the Cycle attractor recognizes both sides and tends to include a third; for example, the synthesis coming out of the thesis and anti-thesis. ====== Taken from: []

Right vs Oblique Pyramid
This tells you where the top (apex) of the pyramid is. If the apex is directly above the center of the base, then it is a Right Pyramid, otherwise it is an Oblique Pyramid.
 * || [[image:http://www.mathsisfun.com/geometry/images/right-pyramid.gif width="181" height="130" caption="Right Pyramid"]] ||
 * Right Pyramid ||  ||   || [[image:http://www.mathsisfun.com/geometry/images/oblique-pyramid.gif width="181" height="132" caption="Oblique Pyramid"]] ||
 * Oblique Pyramid ||  ||
 * ~ Right Pyramid ||~ Oblique Pyramid ||

Regular vs Irregular Pyramid
This tells us about the **shape of the base**. If the base is a regular polygon, then it is a Regular Pyramid, otherwise it is an Irregular Pyramid. Taken from: []
 * || [[image:http://www.mathsisfun.com/geometry/images/square-pyramid.png width="181" height="130" caption="Regular Pyramid"]] ||
 * Regular Pyramid ||  ||   || [[image:http://www.mathsisfun.com/geometry/images/irregular-pyramid.png width="129" height="129" caption="Irregular Pyramid"]] ||
 * Irregular Pyramid ||  ||
 * ~ Regular Pyramid ||~ Irregular Pyramid ||
 * || [[image:http://www.mathsisfun.com/geometry/images/square.png width="102" height="102" caption="Square"]] ||
 * Square ||  ||   || [[image:http://www.mathsisfun.com/geometry/images/irregular-square.gif width="141" height="106" caption="Irregular Ploygon"]] ||
 * Irregular Ploygon ||  ||
 * ~ Base is Regular ||~ Base is Irregular ||

Assignment
Click the following link and you will find a text called "Mandelbrot and Julia Sets". Please read it carefully and locate comparisons and contrasts among the fractals described there. Explain what made you notice that there were comparisons and or contrasts in the text. As soon as you have your assignment ready please write your answers in your wiki. []

For further information please visit: []


 * __Comparisions__**


 * __Both__** Mandelbrot and Julia sets are types of fractals. However, these are more complicated fractals then the other fractals that have been mentioned (such as the Sierpinski's triangle).**__Both these sets__** require the use of **complex numbers**.

To compute the basic Mandelbrot (or Julia) set one uses the equation f(z) -- > z 2 + c, where **__both__** z and c are complex numbers. To describe what occurs it is easier to view the function f(z) to be a machine that squares a complex number and then adds c to it. Now to compute the sets one takes a starting value for z and places it in the "machine". The number is squared and c is added to it and a new number (most likely) comes out of our machine. Now, one places that new number in the machine and the process occurs again. This process is called iteration and it is how the **__Mandelbrot and Julia sets__** are computed.


 * __For the Mandelbrot and Julia sets__** it can be proved (through a very complex proof) that if the distance, on the Cartesian plane (remember we are using complex numbers here), between the origin and a point resulting from the iteration of some initial value is greater than 2 then the behaviour of that initial value is that it will go to infinity. If, however, after numerous iterations (possibly hundreds, thousands or more) the distance between that origin and the point is never greater than two, it is said that this point is bounded.

Then, knowing that, the definition of the Mandelbrot set is : the set of all the complex numbers, c, such that the iteration of **f(z) -- > z 2 + c** is bounded (starting with z =0 + 0//i//). More simply put, the Mandelbrot set is the graph of all the complex numbers c, that do not go to infinity when iterated in **f(z) -- > z 2 + c**, with a starting value of z =0 + 0//i.//
 * __A Julia set is almost the same thing.__** It is defined to be : the set of all the complex numbers, z, such that the iteration of **f(z) -- > z 2 + c** is bounded for a particular value of c. Again, more simply put it is the graph of all the complex numbers z, that do not go to infinity when iterated in **f(z) -- > z 2 + c**, where c is constant.


 * __Contrasts__**

Thus, **__the basic difference between the Mandelbrot set and Julia set__** is that in any Mandelbrot set, you are plotting various values of c on a Cartesian plane, whereas for a Julia set, you are plotting various starting values of z, and c is kept constant.


 * Underlined words are what helped me to find the comparisons and contrasts in the text.**