MaplePrimes - Questions and Posts tagged with fractal

Lyapunov fractals

acer — Sun, 03 Feb 2013 06:24:33 Z

The following (downsized) images of Lyapunov fractals were each generated in a few seconds, in Maple 16.

I may make an interface for this with embedded components, or submit it in some form on the Application Center. But I thought that I'd share this version here first.

I'm just re-using the techniques in the code behind an earlier Post on Mandelbrot and Julia fractals. But I've only used one simple coloring scheme here, so far. I'll probably try the so-called burning ship escape-time fractal next.

Here below is the contents of the worksheet attached at the end of this Post.

The procedures are defined in the Startup code region of this worksheet.

It should run in Maple 15 and 16, but may not work in earlier versions since it relies on a properly functioning Threads:-Task.

The procedure `Lyapunov` can be called as

Lyapunov(W, xa, xb, ya, yb, xresolution)

Lyapunov(W, xa, xb, ya, yb, xresolution, numterms=N)

where those parameters are,

- W, a Vector or list whose entries should be only 0 or 1

- xa, the leftmost x-point (a float, usually greater than 2.0)

- xb, the rightmost x-point (a float, usually less than or equal to 4.0)

- ya, the lowest y-point (a float, usually greater than 2.0)

- yb, the highest y-point (a float, usually less than or equal to 4.0)

- xresolution, the width in pixels of the returned image (Array)

- numterms=N, (optional) where positive integer N is the number of terms added for the approx. Lyapunov exponent

The speed of calculation depends on whether the Compiler is functional and how many cores are detected. On a 4-core Intel i7 under Windows 7 the first example below had approximately the following performce in 64bit Maple 16.

	Compiled	evalhf
serial (1 core)	20 seconds	240 seconds
parallel (4 cores)	5 seconds	60 seconds

>	with(ImageTools):

>	W:=[0,0,1,0,1]: res1:=CodeTools:-Usage( Lyapunov(W, 2.01, 4.0, 2.01, 4.0, 500) ):

memory used=46.36MiB, alloc change=65.73MiB, cpu time=33.87s, real time=5.17s

>	View(res1);

>	W:=[1,1,1,1,1,1,0,0,0,0,0,0]: res2:=CodeTools:-Usage( Lyapunov(W, 2.5, 3.4, 3.4, 4.0, 500) ):

memory used=30.94MiB, alloc change=0 bytes, cpu time=21.32s, real time=3.54s

>	View(res2);

>	W:=[1,0,1,0,1,1,0,1]: res3:=CodeTools:-Usage( Lyapunov(W, 2.1, 3.7, 3.1, 4.0, 500) ):

memory used=26.18MiB, alloc change=15.09MiB, cpu time=18.44s, real time=2.95s

>	View(res3);

>	W:=[0,1]: res4:=CodeTools:-Usage( Lyapunov(W, 2.01, 4.0, 2.01, 4.0, 500) ):

memory used=46.25MiB, alloc change=15.09MiB, cpu time=33.52s, real time=5.18s

>	View(res4);

Download lyapfractpost.mw

An easy snowflake fractal curve

Carl Love — Tue, 01 Jan 2013 05:36:47 Z

restart;
a:= 2.75: b:= 7.: N:= 2^4:
X:= t-> add(sin(b^k*t)/a^k, k= 0..N):
Y:= t-> add(cos(b^k*t)/a^k, k= 0..N):
plot(
[X, Y, 0..2*Pi], numpoints= 2^12
,scaling= constrained
,axes= none
,color= cyan
,caption= cat("Happy New Year ", StringTools:-FormatTime("%Y"))

faster fractals

acer — Tue, 22 May 2012 11:08:39 Z

Using techniques previously used for generating color images of logistic maps and complex argument, attached is a first draft of a new Mandelbrot set fractal image applet.

A key motive behind this is the need for a faster fractal generator than is currently available on the Application Center as the older Fractal Fun! and Mandelbrot Mania with Maple entries. Those older apps warn against being run with too high a resolution for the final image, as it would take too long. In fact, even at a modest size such as 800x800 the plain black and white images can take up to 40 seconds to generate on a fast Intel i7 machine when running those older applications.

The attached worksheet can produce the basic 800x800 black and white image in approximately 0.5 seconds on the same machine. I used 64bit Maple 15.01 on Windows 7 for the timings. The attached implementration uses the Maple Compiler to attain that speed, but should fall back to Maple's quick evalhf mode in the case that the Compiler is not properly configured or enabled.

The other main difference is that this new version is more interactive: using sliders and other Components. It also inlines the image directly (using a Label), instead of as a (slow and resource intensive) density plot.

Run the Code Edit region, to begin. Make sure your GUI window is shown large enough for you to see the sides of the GUI Table conveniently.

The update image appearing in the worksheet is stored in a file, the name of which is currently set to whatever the following evaluates to in your Maple,

cat(kernelopts('homedir'),"/mandelbrot.jpg"):

You can copy the current image file aside in your OS while experimenting with the applet, if you want to save it at any step. See the start of the Code Edit region, to change this filename setting.

Here's the attachment. Comments are welcome, as I'd like to make corrections before submitting to the Application Center. Some examples of images (reduced in size for inclusion here) created with the applet are below.

Newton-Raphson Fractal Black Areas Revealed

HSybesma — Wed, 04 Apr 2012 01:12:13 Z

NewtonBlackArea.mw

I have been working with Newton-Raphson fractals for some time. Like others it was necessary to deal with the "black areas" many times, so I performed some additional analysis and present some of these results here. This will allow others to stop coloring these areas black and allow visualization of the structure inside these areas. It will also help demonstrate...

Area of a fractal

Red Horse — Thu, 15 Mar 2012 14:47:51 Z

I have the fractcal of a snowflake

Fractal Geometry

tiepdinhvan — Tue, 31 May 2011 06:29:12 Z

I have a pbs: Suppose 3 vector A, B,C with noted [A:B]:=(list)[ A,B,-A,-B] (i.e a paralelogram);

An endomorphism T such that T(.) where (.)=A,B or C are B,C, and (list)[C,-A,-B,-B]
and T([x:y])=[T([x]):T([y])] where x,y = A,B or C.

Problem: draw T^{n}[A:B] , i.e its boundary is a fractal figure when n is big.

The Remarkable Story Behind the Mandelbrot Monk

Samir Khan — Thu, 02 Apr 2009 01:19:00 Z

While visiting a cathedral in Germany, Bob Schipke, a retired Harvard mathematician was astounded to find a glyph in a 13^th century manuscript that looked remarkably like the Mandelbrot set. This led to a remarkable voyage of discovery that was publicised in a