ov =gew8xpb1ara · mandelbrot set any series of colors used is finite and must be repeated over and...
TRANSCRIPT
![Page 1: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/1.jpg)
Mandelbrot Sets on YouTubeo Mandelbrot Set Zoom
o http://www.youtube.com/watch?v=gEw8xpb1aRA
![Page 2: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/2.jpg)
The Mandelbrot SetHalley Newman
![Page 3: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/3.jpg)
Terminolgyo Mandelbrot Seto Iterated Function
![Page 4: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/4.jpg)
Example—Iterated Functiono Let f(z)=z2+c, c is arbitraryo Let c=2o Our first input will be c.
o f(2)=(2)2+2=6o f(6)=(6)2+2=38o f(38)=(38)2+2=1446o f(1446)=(1446)2+2=2090918
o General Rule = Exponential Growth
![Page 5: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/5.jpg)
Between -2 and 1o Weird Stuff Happenso Example: Let c= 0.1o f(0.1) = (0.1)2+0.1= 0.11o f(0.11) = 0.1121o f(0.1121) = 0.11256641o f(0.11256641) = 0.112671196
o Notice: This will converge, but will never reach 1. This is an exception to the general rule.
![Page 6: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/6.jpg)
o Numbers between -2 and 0 behave differently
o Negative * Negative = Positiveo The square of all negative numbers
is positive.o Adding a Negative = Subtraction
![Page 7: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/7.jpg)
o The positive gain you get between -2 and 0 is diminished by adding back the original negative number
![Page 8: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/8.jpg)
Exampleo Iterate -1 in f(z) = z2 + co f(-1) = 1 + (-1) = 0o f(0) = 0 + (-1) = -1o f(-1) = 1 + (-1) = 0o f(0) = 0 + (-1) = -1o f(-1) = 1 + (-1) = 0
o The outputs alternate between -1 and 0 forever.
![Page 9: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/9.jpg)
Exampleo What about c = -2, iterate this
using f(z) = z2 + co f(-2) = (-2)2 + -2 = 2o f(2) = (2)2 + -2 = 2o f(2) = (2)2 + -2 = 2
o Special Case!!!
![Page 10: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/10.jpg)
o Other values between -2 and 0 behave even more strangely.
o They have chaotic behavioro Hidden Patterno Difficult to guess what the output
sequence will look like
![Page 11: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/11.jpg)
Complex Numberso Real + Imaginary = Complexo Each have their own unique
address on the complex plane.
![Page 12: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/12.jpg)
o Let’s study the same function f(z) = z2 + c
o We understand what happens when we iterate this function with real numbers between -2 and 1
o What happens when we use complex numbers in this same area?
![Page 13: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/13.jpg)
f(z) = z² + co Fix a c on the complex plane.o Plug c into the function
f(z) = z²+ c.o Ex.o Fix c = - 0.5 - 0.5io So f(c) = (-0.5-0.5i)² + (- 0.5 - 0.5i)
![Page 14: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/14.jpg)
Iterate f(z) = z² + (- 0.5 - 0.5i)o f(- 0.5 – 0.5i) = (- 0.5 – 0.5i)² +
(- 0.5 – 0.5i)o = 0.5i + (- 0.5 – 0.5i)o = - 0.5o Ando f(- 0.5) = (- 0.5)² + (- 0.5 – 0.5i)o = - 0.25 + - 0.5i
![Page 15: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/15.jpg)
Iterate f(z) = z² + (- 0.5 - 0.5i)Continuing, we get…-0.6875+-0.25i-0.8984375e-1+-0.15625i-0.51634216+-0.47192383i-0.45610287+-0.1265166e-1i-0.29213024+-0.48845908i-0.6532522+-0.21461266i-0.11932016+-0.21960761i-0.5339902+-0.44759277i
![Page 16: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/16.jpg)
Iterate f(z) = z² + (- 0.5 - 0.5i)And Eventually…-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i-0.40867701+-0.27512526i
![Page 17: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/17.jpg)
Iterate f(z) = z² + (- 0.5 - 0.5i)oSo this function stabilizes
to a single value…
o-0.40867701+-0.27512526i
![Page 18: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/18.jpg)
GraphingoWhenever a function
stabilizes, we will color its associated number (on the complex plane) blue. If, however, the function doesn’t stabilize, we will color its associated point red.
![Page 19: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/19.jpg)
ExampleoThe function of(z) = z² + (- 0.5 - 0.5i)oStabilized too-0.40867701+ -0.27512526ioSo we’ll color - 0.5 - 0.5i blueoThe function f(z) = z² + (0.5 + i)oDoes not stabilize at alloSo we’ll color 0.5 + i red
![Page 20: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/20.jpg)
![Page 21: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/21.jpg)
The Mandlebrot SetoFor every point, we
determine whether the associated function stabilizes or not. If it does, we paint that point blue and if not, we paint the point red.
![Page 22: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/22.jpg)
Warning!!!oSome functions stabilize
to more than one point (they cycle!).
oEx.of(z) = z² + (- 1 + 0i)
stabilizes too-1 and 0
![Page 23: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/23.jpg)
Warning!!!We will still color it
blue!
![Page 24: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/24.jpg)
![Page 25: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/25.jpg)
Filling it in…
![Page 26: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/26.jpg)
Mandelbrot Setso How do we get all the neat
colors in a Mandelbrot set?
![Page 27: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/27.jpg)
Mandelbrot Seto Technically defined
as the set of all numbers c that do not grow exponentially in the iterated function f(z) = z2 + c
o These numbers are typically represented in black
![Page 28: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/28.jpg)
Iterate 1+1i in f(z) = z2 + co f(1+1i) = (1+1i)2 + 1+1io = (1+1i) (1+1i) + 1+1io = 1+1i+1i+-1+1+1io = 2i +1+1io = 1+3i
o This output falls outside our circle so we know that the input 1+1i is not in the set, we color it red.
![Page 29: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/29.jpg)
o If we perform the first iteration of the function for every number within the circle, each number whose first iteration output escapes the circle will also be colored red.
![Page 30: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/30.jpg)
o If we continue this process using green for the second iteration and blue for the third iteration, we get:
![Page 31: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/31.jpg)
o The edge can never be fully realized because it would take and infinite number of iterations to draw the complete boundary
![Page 32: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/32.jpg)
o If we could perform an infinite number of iterations, we could see the boundary of the Mandelbrot Set is infinitely long even though it is contained within a circle of radius two
![Page 33: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/33.jpg)
o Lastly, to display the same information, any sequence of colors can be used (choice of color is arbitrary) unlike the Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations.
![Page 34: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/34.jpg)
Example
![Page 35: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/35.jpg)
Example
![Page 36: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/36.jpg)
Example
![Page 37: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/37.jpg)
Example
![Page 38: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/38.jpg)
![Page 39: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/39.jpg)
![Page 40: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/40.jpg)
Mandelbrot Seto Zoom Applet
o http://math.hws.edu/xJava/MB/
![Page 41: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/41.jpg)
Mandelbrot Sets on YouTubeo Mandelbrot Set Zoom
o http://www.youtube.com/watch?v=gEw8xpb1aRA
![Page 42: ov =gEw8xpb1aRA · Mandelbrot Set any series of colors used is finite and must be repeated over and over with increasing iterations. Example. Example. Example. Example. Mandelbrot](https://reader034.vdocuments.site/reader034/viewer/2022051800/5aca83d17f8b9a40728e45d3/html5/thumbnails/42.jpg)
Questions