(X, Y)

Iterations: 0

Zoom Level: 1

The graph above plots whether or not a complex number `X + Y`

is in the Mandelbrot set. This graph is a beautiful fractal pattern that you can zoom into and explore.**i**

The Mandelbrot Set is a set of `complex numbers`

that behave in a specific way. To determine if a complex number is in the Mandelbrot set, you must take the complex number `C`

and place it into this `iterating function`

X_{n}= X^{2}_{n-1}+ C

where the first value of `X`

is _{n-1}`0`

and where an `iterating function`

is one that you keep doing over and over again. The complex number `C`

is in the Mandelbrot set when the value of `X`

does not approach infinity as you iterate to infinity._{n}

For example, if `C`

is `1`

and you start iterating, you get

X_{1}= X_{0}^{2}+ 1 = 0^{2}+ 1 =1

X_{2}= X_{1}^{2}+ 1 = 1^{2}+ 1 =2

X_{3}= X_{2}^{2}+ 1 = 2^{2}+ 1 =5

X_{4}= X_{3}^{2}+ 1 = 5^{2}+ 1 =26

X_{5}= X_{4}^{2}+ 1 = 26^{2}+ 1 =677

X_{6}= X_{5}^{2}+ 1 = 677^{2}+ 1 =458330

As you can see, continuing this iteration will cause `X`

to approach infinity. Therefore, the number `1`

is **not** in the Mandelbrot set.

For another example, if `C`

is `-1`

and you start iterating, you get

X_{1}= X_{0}^{2}+ 1 = 0^{2}- 1 =-1

X_{2}= X_{1}^{2}- 1 = -1^{2}- 1 =0

X_{3}= X_{2}^{2}- 1 = 0^{2}- 1 =-1

X_{4}= X_{3}^{2}- 1 = -1^{2}- 1 =0

X_{5}= X_{4}^{2}- 1 = 0^{2}- 1 =-1

X_{6}= X_{5}^{2}- 1 = -1^{2}- 1 =0

As you can see, continuing this iteration will cause `X`

to flip-flop between `-1`

and `0`

forever. Therefore, the number `-1`

is in the Mandelbrot set, since it does not approach infinity

A description of each option for this graph is given below.

The `Width`

option is for changing the width of the graph's image.

The `Height`

option is for changing the height of the graph's image.

The `Iterations`

option is for changing how many times to iterate before deciding when a complex number is in the Mandelbrot set. Since we can't wait for eternity to decide that a given complex number is in the Mandelbrot set, we need to break out at some point. The larger this number is, the more accurate the Mandelbrot image will be. Try zooming in 4 levels and then increasing `Iterations`

to 700 and see what happens to the image.

The `Cutt Off`

option is used to quickly decide when a complex number will cause `X`

to approach infinity. If we see _{n} = X^{2}_{n-1} + C`X`

become larger than the _{n}`Cutt Off`

value, then we assume that `X`

will approach infinity_{n} = X^{2}_{n-1} + C

The `Zoom Factor`

option is for changing how far to zoom into on each click or tap.