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### Author Topic: Lambda Fractal: Additional Curves Now Seeing Them  (Read 713 times)

#### Boleeman

• Hero Member
• Posts: 639
##### Lambda Fractal: Additional Curves Now Seeing Them
« on: August 07, 2024, 08:41:38 am »
A Lambda Fractal, producing Mandelbrot type fractal curve.

Thanks to tetrastes. Additional Curves Now Seeing Them

The Lambda formula is an alternative version of the equation for Julia fractals. While it is capable of creating the same Julia sets, the corresponding Mandelbrot version looks different.

Paul Bourke at https://paulbourke.net/fractals/lambda/ says "Each value of "a" gives a different fractal image, commonly used value is -(0.85 + 0.6 i)"

Other values:  a = 0.7 + 0.7 i       ,     a = 0.7 + 0.725 i         ,    a = 0.69 + 0.725 i

At  https://www.ultrafractal.com/help/index.html?/help/formulas/standard/lambda.html  they also say to try:

For the standard Lambda Mandelbrot set, this should be set to (0.5, 0), which is what is shown in the attached png screenshot. Other values create distorted shapes that can be interesting, but they are usually not as well-formed as the standard set.
For well-formed sets, the real value should be set to 1 divided by the real value of the exponent.
For example, use (0.25, 0) if Exponent is set to (4, 0).

Exponent:
The default value is (2, 0), resulting in the classic equation.   c * z * (1 - z)

Try (3, 0) and (4, 0) and so on to increase the complexity of the fractal.

I can't seem to see those other fractal curves mentioned at those 2 web sites.

« Last Edit: August 09, 2024, 11:20:24 am by Boleeman »

#### Dzandaa

• Sr. Member
• Posts: 349
• From C# to Lazarus
##### Re: Lambda Fractal: Additional Curves Not Seeing Them
« Reply #1 on: August 07, 2024, 07:13:43 pm »
Hi,

Nice Fractal (again)

B->
Regards,
Dzandaa

#### tetrastes

• Hero Member
• Posts: 517
##### Re: Lambda Fractal: Additional Curves Not Seeing Them
« Reply #2 on: August 08, 2024, 01:51:34 pm »
I can't seem to see those other fractal curves mentioned at those 2 web sites.

You obviously implemented not formula from the 1st site ( zn+1 = azn(1-zn) ).
Attached is your project modified for this formula, though with your default settings (0.5, 0) it draws smth like ass instead of Mandelbrot-like. I used ucomplex unit from DMath package for simplicity.
From the 2nd site I've understood, that this formula is special case of more common one, but I cannot find it.

#### Boleeman

• Hero Member
• Posts: 639
##### Re: Lambda Fractal: Additional Curves Not Seeing Them
« Reply #3 on: August 09, 2024, 09:05:07 am »
tetrastes, thanks for working that out.

That explains why I could not get the other curve types.

Now getting some nice fractal curves.

Dzandaa, thanks for the nice comment.