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.
Both Mandelbrot and Julia versions of the Lambda fractal are available as fractal formulas in Standard.ufm and as fractal formula plug-ins in Standard.ulb.
Because the Mandelbrot version is a map of Julia sets, this allows you to find Julia sets with the Switch feature in a different way than with the usual Mandelbrot set. It is easier to find good spirals and other interesting Julia sets.
The formulas provide the following parameters:
| Start Value (Mandelbrot only) |
For the standard Lambda Mandelbrot set, this should be set to (0.5, 0). 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). |
| Julia Seed (Julia only) |
This parameter specifies the point in the Mandelbrot version that corresponds to the current Julia set. It defines the shape and behavior of the Julia set. Use the Switch feature to select good values. |
| Exponent |
Specifies the 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. Non-integer values for the real part of the exponent will interpolate between these well-formed sets. If the imaginary part is not zero, the fractal will be further distorted. |
| Bailout |
Specifies the magnitude of z that will cause the formula to stop iterating. To obtain well-formed fractals, this should be set to 4 or larger. Larger values tend to smooth the outside areas. Some coloring algorithms require specific bail-out values for good results. |
See Also
Julia sets
Standard formulas