Math Is Fun Forum

gurthbruins

Member

Registered: 2010-05-09

Posts: 157

Gurth's Fractals

In the years 1989 -2004 I spent many hours computing fractal pictures.

Inspired originally by the exceedingly beautiful pictures of the Mandelbrot Set in The Golden Age of Mathematics.

The following picture represents my final achievement, and the furthest I could get from and beyond the Mandelbrot set.

Comments: This is what I call a Solid Fractal. Meaning:

Whereas the MS and most other fractals can be considered to be functions of points on a plane, a "Solid Fractal" represents a function of the points in 3-dimensional space.

The MS treats the points on the plane as complex numbers, then repeats a Process which produces output numbers from input numbers until the value explodes. The number of repetitions required to do this (R) is the function of the starting point, and is usually portrayed by selecting various colours according to various values of R.

I deviated from using complex numbers, I took points in (x,y) format, and used some Process (differing for each fractal picture) to compute fresh points (x,y), repeating the process until the value of x or y exploded. Here again, the number of repetitions required is then used to compute the colour of the pixel on the screen at each coordinate point (x,y).

For Solid Fractals, I simply added another dimension, taking points in space in (x,y,z) format, processing the points to produce new points until explosion, to arrive at a function for each point.

Actually depicting the solid object on the screen, I used sectional views to enable the viewer to see beneath the surface to some extent.

If anyone is interested, I can furnish the computer code by which this and other of my fractals were produced.

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It's the activity of the intelligence above all that gives charm to existence.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Gurth's Fractals

Good image!

It looks like a garden. And at first it seems random and then you notice patterns in it.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

DaveRobinsonUK

Member

Registered: 2010-04-24

Posts: 123

Re: Gurth's Fractals

I would like to see the code please, looks great

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Can feel it coming together.. Slowly but Surely

gurthbruins

Member

Registered: 2010-05-09

Posts: 157

Re: Gurth's Fractals

Hi, Dave, glad to see someone taking an interest in Fractals.

Here is the essential core of my program, the process by which the points (x,y,z) are processed:

  xv  := xs; zv:=zs;
   For xp := 0 To 799 do
    begin
     allzdone:=false; zv:=zs;
     otd:=td; done:=0;
     while allzdone=false do
      begin
       zv:=zv-dz; x:=xv; y:=yv; z:=zv;
       xq := sqr(x); yq := sqr(y); zq:=sqr(z); r:=sqrt(xq+yq+zq);
       xad:=sqrt(xq+yq);
       yad:=sqrt(yq+zq);
       zad:=z/2;
       
       if zv<-3 then allzdone:=true;
       cou:=0; peint:=false; sq:=2*lim;
       while cou<reps do
        begin
         cou:=cou+1; osq:=sq;
         nx := xq - 2* sqrt(zq+yq)+xad;
         ny := yq - 2* sqrt(xq+zq)+yad;
         nz := zq - 2* sqrt(yq+xq)+zad;
         x:=nx; y:=ny; z:=nz; xq:=x*x; yq:=y*y; zq:=sqr(z); sq:=xq+yq+zq;
         If sq > lim Then cou:=1000;
        end;

This is part of the Delphi4 source code used to draw the picture. You can find the entire code at
http://sites.google.com/site/gurthsfiles/fractal-programs .

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It's the activity of the intelligence above all that gives charm to existence.

DaveRobinsonUK

Member

Registered: 2010-04-24

Posts: 123

Re: Gurth's Fractals

Hi Gurth

At the moment i am doing my graphic stuff using Maya and the like, I want to get into programming my own graphics but I am still getting to grips with the maths behind it all and also need a language that is easier to do graphics on than C++. I see you use Delphi.

btw

Thank you for posting your code.

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Can feel it coming together.. Slowly but Surely

gurthbruins

Member

Registered: 2010-05-09

Posts: 157

Re: Gurth's Fractals

The image (post #1 above) is a bit lonesome I think - after a long time, I developed another 3D fractal formula yesterday - it drew nothing but a black space - at midnight I thought the bug might be in a very obvious place, and yes it was - so I had the whole thing fixed up by about 2 am. Here's the pic - I call it "Bananas" (being a fruitarian!)

PS: I've now translated my drawing program to VB2008EE language. Copies of the program available on request.

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It's the activity of the intelligence above all that gives charm to existence.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Gurth's Fractals

it looks nice!thanks for the pic!

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

gurthbruins

Member

Registered: 2010-05-09

Posts: 157

Re: Gurth's Fractals

Some more fractal eye candy:

Organza

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It's the activity of the intelligence above all that gives charm to existence.

gurthbruins

Member

Registered: 2010-05-09

Posts: 157

Re: Gurth's Fractals

Notice I'm gradually getting the jaggies smaller...

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It's the activity of the intelligence above all that gives charm to existence.