Logo by mauxuam - Contribute your own Logo!

END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

it was a great time but no longer maintainable by c.Kleinhuis contact him for any data retrieval,
thanks and see you perhaps in 10 years again

this forum will stay online for reference
News: Visit the official fractalforums.com Youtube Channel
 
*
Welcome, Guest. Please login or register. August 11, 2022, 10:53:34 PM


Login with username, password and session length


The All New FractalForums is now in Public Beta Testing! Visit FractalForums.org and check it out!


Pages: 1 [2]   Go Down
  Print  
Share this topic on DiggShare this topic on FacebookShare this topic on GoogleShare this topic on RedditShare this topic on StumbleUponShare this topic on Twitter
Author Topic: 1D Mandelbrot  (Read 10237 times)
0 Members and 1 Guest are viewing this topic.
Tglad
Fractal Molossus
**
Posts: 703


WWW
« Reply #15 on: January 08, 2010, 03:15:30 AM »

The OP was looking for the 1d equivalent rather than an arbitrary slice, and since complex numbers just extend real numbers, R^2 + C seems to be the rather obvious 1d analog.
-Mandelbrot is connected, the line segment -2 to 0.25 is connected
-Mandelbrot inside has dimension 2, this line has dimension 1
The Mandelbrot has border points at the connection with the first great circle, and more further to the head in ever increasing numbers. So R^2 + C  has an increasing number of border points (as R goes towards -2), in fact I would conjecture that it has so many border points that the border has fractal dimension 1. So in fact the line is rather a set of lines with border points (but no space) between them. So..
-Mandelbrot border has dimension 2, this line border has dimension 1

It is actually a pretty interesting shape, and I'm in no doubt that this is the 1d equivalent.
The lengths of the connected line segments don't follow a simple pattern, as can be seen by plotting the bifurcations horizontally on the logistic map.
Logged
Timeroot
Fractal Fertilizer
*****
Posts: 362


The pwnge.


WWW
« Reply #16 on: January 08, 2010, 05:52:00 AM »

It's a line. When you say, "border points", this is equivalent to just saying "where the period changes". You'll just get the bifurcation diagram of the logistic map; boring. Just as the Mandelbulb was taken to the 8th power for aesthetic reasons, why not take a slice of the Mandelbrot perpendicular to the real axis? It's way more interesting...
Logged

Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
Tglad
Fractal Molossus
**
Posts: 703


WWW
« Reply #17 on: January 08, 2010, 09:58:39 AM »

Well you can if you like, and it might be more interesting. I'm just commenting on what the direct 1d analog is.
Besides, this line segment is interesting. Imagine a line going diagonally along a chess board like a bishop on black squares. At the exact point it crosses to the next square, is the colour of that point black or white? It can be neither. It is fair to define this diagonal line as single segments of length root 2 attached by border points to each other. Each being the single 0d point where the horizontal borders cross.
So R^2 + C looks like this -------+----+--+------+----+--+-+--+-+  The line segments are connected in the same way that the black squares on the chessboard are connected and the same way that the mandelbrot insides are connected. But the imaginary axis of the mandelbrot is not connected, so is not an equivalent fractal in my opinion.

The R^2 + C also shows the decimals of PI when calculating the number of iterations to escape as shown in this table:
"here's a table for points of the form (.25 + X, 0)

X               # of iterations
1.0               2
0.1                8
0.01              30
0.001               97
0.0001       312
0.00001       991
0.000001       3140
0.0000001       9933
0.00000001       31414
0.000000001   99344
0.0000000001 314157"
This is just like the 2d mandelbrot does.

What seems more interesting than the similarities are the differences. The border of the 2d mandelbrot is completely non-smooth fractal, the _border_ of the 1d mandelbrot isn't connected (so is dust) and, looking to 4 dimensions, the quaternion mandelbrot border is overly smooth.
This is interesting and could be explained by saying that the ^2 operation does not have enough degrees of freedom in 1d, has just the right amount in 2d and it doesn't take up all the degrees of freedom in 4d (hence smoothness). This would explain why the mandelbulb requires two operations (on longitude and latitude) and why a properly fractal 4d mandelbrot would require rotating in 3 directions.

Of course this all kind of undermines my point because it means R^2 + C isn't the true 1d analog... the true equivalent would have a connected border of fractal dimension 1 and the line segments should also be connected. I'm not sure such a shape exists.
Logged
Kali
Fractal Supremo
*****
Posts: 1138


« Reply #18 on: January 24, 2011, 02:44:57 PM »

Hello guys, I was reading this old post, and I invite you to read my recent post:
http://www.fractalforums.com/mandelbrot-and-julia-set/mandelbrot-on-real-numbers-t5375/msg26926/#msg26926
Perhaps you can tell if I'm really on something interesting or just loosing time, because I'm not a fractal expert, just curious about them and I try to research all that I can with my limited knowledge. I'll really apreciate some comment. Thanks.
Logged

|</\|_|
░░░░░░░

http://fractkali.deviantart.com/
Xazo-Tak
Navigator
*****
Posts: 77



« Reply #19 on: September 10, 2011, 01:37:24 AM »

Wouldn't the line have some on/off structure to at least give the appearence of dots and dashes (though obviously some very short/long dots/dashes sometimes).
I think 1d mandelbrot would look like the 2d mandelbrot at the line i0. And i0 so happens to contain an unbroken line. (If you do not know what I mean by i0, y0.)
Logged

I lost the game!
Pages: 1 [2]   Go Down
  Print  
 
Jump to:  

Related Topics
Subject Started by Replies Views Last post
Rhythmic Gymnastic Mandelbrot FractInt Gallery simon.snake 0 650 Last post May 06, 2012, 06:46:51 PM
by simon.snake
A Brief History of the Mandelbrot Set Film Pauldelbrot 0 1899 Last post May 27, 2012, 12:11:56 PM
by Pauldelbrot
Mandelbrot Cakewalk Sampler Mandelbulb3D Gallery tobiles 0 549 Last post October 23, 2012, 10:42:53 PM
by tobiles
Mission Mandelbrot Introduction to Fractals and Related Links Applemanikin 8 1890 Last post December 03, 2012, 04:54:15 PM
by Applemanikin
Anytime, anywhere surround Mandelbrot set Movies Showcase (Rate My Movie) spaciane 4 1213 Last post March 05, 2013, 12:14:27 AM
by spaciane

Powered by MySQL Powered by PHP Powered by SMF 1.1.21 | SMF © 2015, Simple Machines

Valid XHTML 1.0! Valid CSS! Dilber MC Theme by HarzeM
Page created in 0.193 seconds with 24 queries. (Pretty URLs adds 0.007s, 2q)